Lecture Notes in Computational Science and Engineering Editors M. Griebel, Bonn D. E. Keyes, Norfolk R. M. Nieminen, Espoo D. Roose, Leuven T. Schlick, New York
17
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Barbara 1. Wohlmuth
Discretization Methods and Iterative Solvers Based on Domain Decomposition With 82 Figures and 25 Tables
Springer
Barbara I. Wohlmuth Institut fur Mathematik Universitat Augsburg Universitatsstrafse 14 86159 Augsburg, Germany e-mail:
[email protected]
Cataloging-in-Publication Data applied for Die Deut sche Bibliothek - CIP-Einheitsaufnahme Wohlmuth, Barbara : Discreti zation methods and iterative solvers based on domain decomposition I Barbara I. Wohlmuth. - Berlin; Heidelberg ; New York; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer , 2001 (Lecture notes in computational science and engineering ; 17) ISBN3-540-41083-X
Mathematics Subject Class ification (1991): 65N30, 65N15,65FlO, 65N55, 65N50 ISSN 1439-7358 ISBN 3-540-41083-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the materia l is concern ed, specifically the rights of translation, reprintin g, reuse of illu str ations, recitat ion, bro adcasting , reproduction on microfilm or in any other way,and storage in data banks. Duplicatio n of this publication or parts thereof is permitted only und er the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obt ained from Springer-Verl ag. Violations are liable for prosecution und er the German Copyrig ht Law. Springer-Verlag Berlin Heidelberg New York a memb er of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2001 The use of general descriptive names, register ed names, trademarks, etc. in this public ation does not imply, even in the absence of a specific statement, that such name s are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: Friedh eIm Steinen-Broo, Estudio Calamar, Spain Cover production: design & production GmbH, Heidelberg Typeset by the author using a Springer TEXmacro package Printed on acid-free pap er SPIN 10725050 46/3142/LK - 5 43 21 0
Preface
DOMAINE: [domen] Ce domaine est encore ferrne aux savants DECOMPOSER : [dek3poze] Decomposer un probleme pour mieux le resoudre
• Micro Robert : Dictionnaire du francais primordial
The num erical approximation of partial differential equations, very often , is a challenging task. Many such problems of practical interest can only be solved by means of modern supercomputers. However , the efficiency of the simulation depends strongly on the use of special numerical algorithms. Domain decomposition methods provide powerful tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applicat ions in science and engineering. Although the first domain decomposition techniques were used successfully more than hundred years ago, these methods are relatively new for the numerical approximation of partial differential equations. The possibilities of high performance computations and the interest in large-scale problems have led to an increased research activity in the field of domain decomposition. However, the meaning of the term "domain decomposition" depends strongly on the context. It can refer to optimal discr etiz ation techniques for the underlying problems, or to efficient iterative solvers for the arising large systems of equations, or to parallelization techniques. In many modern simulation codes, different aspects of domain decomposition techniques come into play, and the overall efficiency depends on a smooth interaction between these different components . The coupling of different discretization schemes , the coupling of different physical models, and many efficient preconditioners for the algebraic systems can be an alyzed within an abstract framework . At first glance these aspects seem to be rather independent. However , all have one central idea in common: The decomposition of the underlying global problem into suitable subproblems of smaller complexity. In general, a complete decoupling of the global problem into many independent subproblems, which are easy to solve, is not possible. Since, the subproblems are very often
VI
Preface
coupled, there has t o be communication between t he different subproblems. Althou gh t he t erm optimal dep ends on t he context, t he pr oper handling of t he inform at ion t ransfer across the int erfaces between t he sub pro blems is of majo r impor t an ce for t he desig n of opt imal methods. In the case of discretization techniques, a priori estimates for t he discreti zati on errors have to be considered. They very mu ch depend on t he appro pr iate couplings across the interfaces which are often realized by mat ching condit ions. The jump across the int erfaces which measur es the non conformity of t he method has to be bounded in a suitable way. In t he case of it er ativ e solvers , t he convergence rat e and t he computational effort for one it er ation ste p measure t he qu ality of a method. To ob t ain scalable it er ation schemes, very ofte n, one has to includ e a suitable global prob lem of small compl exity. In t his work, both discretiz ation t echniques and it erativ e solvers are addr essed . A brief overvi ew of different approaches is given and new t echniques and ideas are proposed . An abst ract framework for dom ain decomposition methods is pr esented and an analysis is carried out for new t echniques of special inter est . Optimal est imates for the methods considered ar e est ablished and numerical results confirm the theoreti cal pr edicti ons. Chapter 1 concerns special discretiz ation methods based on domain decomposit ion t echniques. In particular , t he decomp osition of geomet rical complex structures int o sub domain s of simple shape is of special int erest . Another example is t he decomposition int o subst ructures on which different physical mo dels are relevan t . Then , for each of t hese subp rob lems , an optimal appro ximation scheme involving t he choice of the triangulation as well as t he discretiz ation can be chose n. However t o obt ain optimal discretiz ations for the global problem , t he discret e subpro blems have been glued together appropriately. Here, we focus on mort ar finite element methods. To start, we review t he standard mor t ar setting for t he coupling of Lagrangian conforming finite elements in Sect . 1.1. Both standard mortar formul ations - t he non conforming positiv e definit e probl em and the saddle poin t problem based on t he un constrain ed product space - are given. In Sect . 1.2, we introduce and analyze alternative Lagran ge mul tiplier spaces. We derive ab stract condit ions on t he Lagran ge mul tiplier spaces such t hat t he non conforming discretiz ation schemes obtain ed yield optimal a priori results. Lagrange multiplier spaces based on a du al basis ar e of special int er est . In such a case, a biorthogonality relation between the nodal basis fun cti ons of these spaces an d th e finit e element t race spaces holds . A main advantage of these new Lagran ge multiplier space s is t hat t he locality of the support of t he nod al basis functions of t he constrained space can b e pr eserved. Wi th this observation in mind , we introduce a new equivalent mortar formulation defined on t he un const rain ed pr odu ct space in Sect . 1.3. We show t hat t he non-symmetric formulation can be analyzed as a Dirichlet Neumann coupling. Based on the elimination of t he Lagran ge mul tiplier , we deri ve a symmetric positi ve definit e formul at ion on t he un constrain ed
Pr eface
VII
product sp ace, and t he equivalence to th e positive definite problem on th e constrained space is shown. Two formulations, a variational as well as an algebraic one, are presented and discussed . A st andard nod al basis for th e unconstrain ed product space can be used in the impl ement at ion . The stiffness mat rix associate d with our new variational form can be obtain ed from t he standard one on the un constrain ed space by local operations. Section 1.4 concerns two examples of non-st andard mortar sit uations. Eac h of them reflects an int erest ing feature of t he abstract genera l fram ework , and illustrates the flexibility of t he method. We start with the coupling of two different discreti zation schemes. The matching at the int erface is based on the du al role of Diri chlet and Neum ann boundary condit ions. Two different equivalent formulations are given for the coupling of mixed and standard conforming finit e elements . In our second example, we rewrite the nonconforming Crouzeix- Raviart finit e elements as mortar finit e elements. We consider the extreme case t hat the decomposition of th e dom ain is given by t he fine triangulation and t hat t here fore t he number of subdomain s t ends to infinity as the discreti zation par am et er of t he t ria ngulat ion t ends to zero . Finally in Sect . 1.5, we pr esent several series of num erical results. In part icular, we st udy t he influence of t he choice of the Lagran ge multiplier space on t he discreti zation erro rs . Examples with severa l crosspoints, a corne r singularity, discontinuous coefficients, a rotating geomet ry, and a linear elast icity probl em are considered. A second t est series concerns th e influence of the choice of the non-mortar side. Adaptive and uniform refinement t echniques are applied. In our last te st series, we consider the influence of jumps in th e coefficient on an adapt ive refinement process at the int erface. Chapter 2 concerns it erative solut ion techniques based on dom ain decompos it ion . A bri ef overv iew of general Schwar z meth ods, including multigrid te chniques, is given in Sect. 2.1. Examples for t he standard H I-case illust rat e overlapping , non- overl apping, and hierar chical decomposition techniqu es. The following sect ions contain new results on non-st and ard sit ua t ions; we discuss vector field discretizations as well as mor t ar methods. Secti on 2.2 focuses on an iterative subst ruct ur ing and a hierarchical basis method for Raviart-Thomas finit e elements in 3D. We st art with the definition of th e local spaces and th e relevant bilinear forms and subspaces. The central result of t his sectio n is established in Subsect. 2.2.2; it is a polylogarithmical bound ind epend ent of t he jumps of t he coefficients across the subdomain boundaries of our ite rat ive substructur ing method. The tec hnical tools are discussed in det ail with par ti cular emphas is on t he role of t race t heorems, harmonic extensions, and du al norms applied to finit e element spaces . As in t he 2D case for standard Lagran gian finit e elements , we introduce three different types of subspaces called VII , Vp , and VT . We cannot avoid t he use of a global space to obt ain quasi-optimal bounds. But in contrast to th e standard Lagrangi an finit e elements in 3D , the low dimensional Raviar t - Thomas space associated with t he macro-triangulation formed by th e subregions can
VIII
Preface
be used to obtain quasi-optimal results where the constant does not depend on the jumps of the coefficients across the subdomain boundaries. Sections 2.3-2.5 concern different iterative solvers for mortar finite element formulations. In Sect. 2.3, we combine the idea of dual basis functions for the Lagrange multiplier space with standard multigrid techniques for symmetric positive definite systems. The new mortar formulation, analyzed in Sect. 1.3, is the point of departure for the introduction of our iterative solver. We define and analyze our multigrid method in terms of level dependent bilinear forms, modified transfer operators, and a special class of smoothers which includes a standard Gaufl-Seidel smoother. Convergence rates independent of the number of refinement steps are established for the W-cycle provided that the number of smoothing steps is large enough. The numerical results confirm the theory. Moreover asymptotically constant convergence rates are obtained for the V-cycle with one pre- and one postsmoothing step. Section 2.4 concerns a Dirichlet-Neumann type algorithm for the mortar method. It turns out to be a block Gaufi-Seidel solver for the unsymmetric mortar formulation on the product space. Numerical results illustrate the influence of the choice of the damping parameter. The transfer of the boundary values at the interface is realized in terms of a scaled mass matrix. This matrix is sparse if and only if dual Lagrange multiplier spaces are used. In Sect. 2.5, we study a multigrid method for the saddle point formulation . Two different types of smoothers are discussed; a block diagonal and one reflecting the saddle point structure. In the second case, the exact solution of the modified Schur complement system is replaced by an iteration, resulting in an inner and an outer iteration. This multigrid method is given for the standard mortar formulation as presented in Sect. 1.1. In contrast to the two previous sections, the use of dual Lagrange multiplier spaces does, in general, not reduce the computational costs for one iteration step. Acknowledgments: It is a great pleasure for me to thank my colleagues, friends, and parents for their support. In particular, I wish to thank Prof. Ronald H.W . Hoppe, University of Augsburg, for his support throughout this work, Prof. Yuri A. Kuznetsov, University of Houston, who introduced me to work in this area, and Prof. Dietrich Braess, Ruhr-University Bochum, for his interest in my work since we first met at the annual GAMM meeting in Braunschweig, 1994. This work was supported in part by a Habilitandenstipendium of the Deutsche Forschungsgemeinschaft and in part by the National Science Foundation under Grant NSF-CCR-9732208. In particular , I would like to thank my coworkers, Rolf Krause, PhD Andrea Toselli and Priv.-Doz. Christian Wieners for their support and fruitful discussions . Finally, it is a great pleasure for me to thank Prof. Olof B. Widlund of the Courant Institute, New York University. His criticism, his encouragement, and his preference for simplicity influenced my work. Augsburg, October 2000
Barbara Irmqard Wohlmuth
Contents
Preface .. . . . . . . . . . .... .. . ... ... ... ... ... ... ... .. . .. . . . . . . . . . . .
V
1. Discretization Techniques Based on Domain Decomposition
1.1 Introduction to Mort ar Finite Element Methods. . . . . . . . . . . . . . 1.2 Mortar Methods with Alt ern ative Lagrange Multipli er Spaces . . 1.2.1 An Approximation Property. . . . . . . . . . . . . . . . . . . . . . . .. 1.2.2 The Consiste ncy Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.2.3 Discret e Inf-sup Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Ex amples of Lagrange Mult iplier Spac es 1.2.4.1 The First Ord er Case in 2D . . . . . . . . . . . . . . . . .. 1.2.4.2 The Fir st Ord er Case in 3D . . . . . . . . . . . . . . . . . . 1.2.4.3 The Second Ord er Case in 2D . . . . . . . . . . . . . . .. 1.3 Discretization Techniques Based on the Product Space . . . . . . . . 1.3.1 A Dirichlet-Neum ann Formul ation. . . . . . . . . . . . . . . . . . . 1.3.2 Vari ational Formul ations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.3.3 Algebraic Formul ations 1.4 Examples for Special Mortar Fini te Element Discretizations 1.4.1 The Coupling of Primal and Dual Fini t e Elements. . . . . . 1.4.2 An Equivalent Nonconforming Formul ation. . . . . . . . . . .. 1.4.3 Crou zeix-Raviart Finite Elements. . . . . . . . . . . . . . . . . . . . 1.5 Num erical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.5.1 Influence of the Lagr ange Multiplier Spaces . . . . . . . . . . . . 1.5.2 A Non-op timal Mortar Method . . . . . . . . . . . . . . . . . . . . .. 1.5.3 Influence of t he Choice of t he Mortar Side . . . . . . . . . . . . . 1.5.4 Influence of the Jump of th e Coefficients . . . . . . . . . . . . ..
1 3 11 15 19 24 27 29 33 36 37 40 43 47 50 51 54 58 61 64 71 74 83
2. Iterative Solvers Based on Domain Decomposition. . . . . . . . . 2.1 Abstract Schwarz Theory 2.1.1 Additive Schwarz Methods 2.1.2 Multiplicativ e Schwarz Methods . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Multigrid Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2 Vector Field Discretizations 2.2.1 Raviart-Thomas Finite Elements 2.2.2 An It erative Substructuring Method
85 87 88 95 97 99 101 103
X
Contents
2.2.2.1 An Interpolation Operator onto V H 2.2.2.2 An Extension Operator onto V F 2.2.2.3 Quasi-optimal Bounds 2.2.3 A Hierar chical Basis Method 2.2.3.1 Horizont al Decompo sition 2.2.3.2 Vertical Decomposition 2.2.4 Num erical Results 2.2.4.1 The 2D Case 2.2.4.2 The 3D Case 2.3 A Multi grid Met hod for t he Mort ar Product Space Formul ation 2.3.1 Bilinear Forms 2.3.2 An Approximation Property 2.3.3 Smoo thing and St ability Properties 2.3.4 Impl ement ati on of th e Smoothing Step 2.3.5 Numerical Results in 2D and 3D 2.3.6 Extensions to Linear Elasticit y " 2.3.6.1 Uniform Ellipticity 2.3.6.2 Numerical Result s 2.3.6.3 A Weaker Interface Condition 2.4 A Dirichlet-Neum ann Type Met hod 2.4.1 The Algorithm 2.4.2 Numerical Results 2.5 A Multigrid Method for th e Mortar Saddl e Poin t Formul ation . 2.5.1 An Approximation Property 2.5.2 Smoothing and St abili ty Properties 2.5.2.1 A Block Diagon al Smoother 2.5.2.2 An Ind efinite Smoother 2.5.3 Numerical Results
105 107 113 114 115 118 121 122 123 125 126 129 131 136 137 142 145 149 151 155 155 158 162 164 167 168 171 174
Bibliography
177
List of Figures
187
List of Tables
189
Notations
191
Index
195
1. Discretization Techniques Based on Domain Decomposition
This chapter concerns domain decomposition methods as discretization techniques for partial differential equations. We present different approaches within the framework of mortar methods [BMP93, BMP94]. Originally introduced as a domain decomposition method for the coupling of spectral elements, these techniques are used in a large class of nonconforming situations. Thus, the coupling of different physical models, discretization schemes, or non-matching triangulations along interior interfaces of the domain can be analyzed by mortar methods. These domain decomposition techniques provide a more flexible approach than standard conforming formulations. They are of special interest for time dependent problems, rotating geometries, diffusion coefficients with jumps, problems with local anisotropies, corner singularities, and when different terms dominate in different regions of the simulation domain. Very often heterogeneous problems can be decomposed into homogeneous subproblems for which efficient discretization techniques are available. To obtain a stable and optimal discretization scheme for the global problem, the information transfer and the communication between the sub domains is of crucial importance; see Fig. 1.1.
Fig. 1.1. Decomposition of a global problem into homogeneous subproblems
One major requirement is that the interface between the different regions is handled appropriately. Very often suitable matching conditions at the interfaces are formulated as weak continuity conditions. One possibility is to use a dual set of boundary conditions. Then, the coupling can be realized in terms of a Lagrange multiplier. The analysis of the resulting jump terms across the interfaces plays an essential role for the a priori estimates of the B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition © Springer-Verlag Berlin Heidelberg 2001
2
1. Discreti zat ion Techni ques Based on Dom ain Decompositi on
discretization schemes . In particular, optimal methods can only be obtained if the consistency error is small enough compared with t he best approximatio n error on t he different subdomains. The consistency error measur es the nonconformity of the method and cont rols t he information transfer at t he int erface. In Sect. 1.1, we review the definition of a standard mor t ar finit e element method. For an overview of mortar tec hni ques including spectral elements, we refer to [BD98, Ben99, BM97, BMP93 , BMP 94, LSV94]. Both standard mortar formulati ons - t he nonconfo rm ing positi ve definite pro blem and t he saddle point problem based on t he un const rain ed product space - are given. Bas ic ideas and tec hniques are explained in a standard context. In the following sections, new ideas res ult ing in alte rnative spaces are st udied, and different modificati ons and aspects of t he methods are conside red. We also introduce a new p ositi ve definite mortar formulati on base d on the un constraine d pr oduct space. In Sect . 1.2, a general framework for the const ruction of new Lagran ge mul tiplier spaces is given. Sufficient condit ions on t he Lagran ge multiplier space are introduced to obtain an optimal ord er global method. So far t he Lagrange mul tiplier spaces have bee n defined as finite element trace spaces. We focus on a special type of Lagran ge multi plier spaces based on a du al basis. T his idea is new and simp lifies t he impl ement ati on as well as t he it erat ive solution of t he arising problem . T he nodal basis functi ons of t he Lagran ge mu ltiplier and t he trace spaces satisfy a biorthogonality relation. As a consequence t he constraints at the interfaces can be eliminated locally, and t he local ity of t he supports of t he bas is functi ons of t he constrained space can be guaranteed. We focus on t he int ro du ction of a new equivalent mortar formulati on based on the unconstrained pr odu ct space in Sect. 1.3. T he biorthogo nality relat ion yields a diago na l mass matrix on t he non-mortar side. Then , the Lagran ge mul ti plier can be obtained in a local post pro cessing ste p, involving t he inver se of t hat diagonal matrix. T his observation serves as our starting point for t he int roduction of our new mor t ar formulation on t he un constrained pr oduct space. In a first ste p, we show t hat t he resulting unsym metric form is nothing else t han a Diri chlet-Neum ann coupling between t he different subdomai ns . A symmet rized form gives rise to a positiv e definite variational formulation on t he un constrained product space. T his third mortar formulation can be implement ed mu ch easier and solved more efficient ly than the standard nonconforming and saddle point formul ations. Section 1.4 concerns several nonconforming situations which are analyzed wit hin t he fram ework of mor t ar finite element methods. T he examples illustrate t he lar ge variety and flexibility of mort ar tec hniques . A first example is given in Sub sect. 1.4.1. T he coup ling of primal and du al methods is ana lyzed , using the du ality of essential and natural bound ary condit ions as our point of departure. T wo different formulations are given for t his coup ling, includ-
1.1 Introduction t o Mortar Finit e Elem ent Methods
3
ing a sa ddle point problem where the Lagran ge multiplier space is defined by piecewise constants. The second exa mple is based on Cr ouzeix-Ravi ar t elements . In contrast t o the pr evious example, t he number of subdomains t end s to infinity when the meshsize t ends to zero. Each element is regarded as a subdom ain and the Lagr an ge multiplier space is t he product space of one dim ensi on al spaces associate d with the edges of t he t riangulat ion. F inally in Sect. 1.5, num erical results are given which illustrate the perform an ce of mort ar discreti zation t echniques. We st art with a comparison of t he discretiz ation err ors for t he different choices of Lagran ge multiplier spaces given in Sect . 1.2. In a second part , we focus on t he choice of t he mortar side in case of discontinuous coefficients and highly non conforming triang ulations . Finally, the use of a pos t eriori error est imat ors shows that two completely different sit uat ions arise at the int erfaces depending on the choice of t he mortar side. Throu ghout t his chapte r, we emphas ize the role of stable proj ections for t he best approxi mation prop erty of t he constrained space, the approx imati on prop erty of t he Lagran ge mul tiplier space for t he consiste ncy err or, discret e inf-sup condit ions for t he a pr iori estimates for t he Lagran ge multi plier , mesh depen dent norms for measuring t he nonconformity, and dua l basis fun ctions as Lagrange mul tipliers for t he locality of the support of the nodal basis functions of the constrained space. The followin g elliptic second ord er boundar y valu e problem -div( a'Vu)
+ bu = f
in fl ,
u=O on afl
(1.1)
will serve as our mod el problem. Here, a is a uniformly positive definite matrix , aij E LOO(fl) , 1 i ,j d, f E L 2(fl) , O:S bE LOO(fl), and o C ]Rd , d = 2, 3, is a bounded polygon al dom ain. For simplicity, we assume that the coefficients aij and b are constant on each element of t he t riangulat ions. Alt hough the det ailed analysis is given in 2D, most of our result s also hold in 3D. In the following, we only point out differences between the 2D and 3D case if different t echniques for t he proofs are requ ired or if different qu alit ative results are obtained . In t hose cases, t he analysis of t he 3D case is carr ied out separately. All constant s 0 < c C < 00 throu ghout t his work are generic and might dep end on t he coefficients a and b, the aspec t ratio of the element s and subdomains, and t he order of the discreti zation method but they do not dep end on t he meshsi ze.
:s
:s
:s
1.1 Introduction to Mortar Finite Element Methods In t his secti on , we bri efly review the standa rd mortar method for t he coupling of Lagran gian finit e elements. We recall the non conforming positiv e definit e formul ation as well as the saddle point pr oblem and t he a pr iori est imates .
4
1. Discretization Techniques Based on Dom ain Decomposition
An examinat ion of th e mort ar pr ojection shows t hat t he support of a bas is functi on on t he non-mort ar side is, in genera l, non-local. Let D be decomposed int o K non-overlapping polyhedral sub dom ains Dk such t hat K
D=
U a, .
k =l
We restrict ourselves to t he geometrical conforming sit uation where the int ersectio n bet ween t he boundari es of any two different sub doma ins aDI n aDk, k i= I , is eit her empty, a vert ex, a common edge or face in 3D; see Fig. 1.2. We call it an inte rface only in t he latter case .
/1,'~ V~Yj(~,;
Q6
l~oocl"~P~--"'--,
ro9
~:cL_L-.L-J"-.:
Q y
= 1,.
Fig. 1.2. Geo metrically conform ing (left ) and nonconforming (right) sit uation
Geometrically nonconforming sit uations are technically more difficult to han dle. A possibility t o reduce t hese complications is to require t hat each vert ex of t he decomp osition is also a vertex of each adjacent triangulation; see t he right part of Fig. 1.2. We define for each subdomain a simplicial t riangulation 0. ;h k ' the meshsize of which is bounded by h k . The finit e element space of conforming Pn k elements on Dk associated wit h 0. ;h k ' nk 2:: 1, which satisfy homogeneous Dirichlet boundary condit ions on aDn aDk , is denot ed by Xh k ;n k ' No boundary conditions are imposed on X h k ;n k in t he case t hat aD n aD k = 0. The results can be eas ily generalized to ot her ty pes of t riangulations; see Fig. 1.3.
Fig. 1.3. Decom posit ion and non-m at ching triangulations in 2D and 3D
Each int erface aDI n a[h is associated wit h a (d - I )-dimensional triangulation, inherited from eit her 0. ;h k or Ti;h l ' In general, t hese t riangulations
1.1 Introduction to Mortar Finite Element Methods
5
do not coincide . The interfaces are denoted by I'm, 1::; m ::; M. For each int erface, there exists a couple 1 ::; l < k ::; K such that 1m = 8D l n 8D k . Since we are working with finit e dimensional Lagr an ge multiplier spaces on t he int erfaces, we have to define appropriate triangulations on I'm. The triangulat ion on I'm is called Sm ;h m , and its element s are boundar y edges in 2D and boundary faces in 3D of eit her Ti;hl or ~ ;hk ' The choice is arbit rary but should be fixed. Then , by definition, the Lagran ge multiplier space inh erits its triangulation from t he non-m ortar side. The adjacent side is called t he mortar side . We denote th e subdomain associated with t he non-mort ar side by D n( m) and t he one associated with t he mortar side by Dn( m) ' I
I
II1I1I1I1IIII11111111
h1«h k
Qk Q I
Fig. 1.4. Different non-matching t riangulat ions at an interface in 2D
In general ~ ; hk and Ti ;hl do not match across t he common interface. We remark t hat no condi tions on t he triangulations ar e imposed at the common int erfaces. Figure 1.4 shows two characteristic situations for non-matching trian gul ations at the interface. The situation on t he left typically arises for t ime dependent problems, e.g., in th e case of sliding meshes. In the case of highly discontinuous coefficients a, global triangulations having a jump in t he meshsizes on th e different sub dom ains might yield better results; see t he right of Fig . 1.4. For both situations, th e mor t ar finit e element method yields optimal a priori bounds for th e discreti zation erro r in the energy norm [BMP93, BMP94]. The constant in th e a priori est imate in 2D depend s neither on the ratio of the meshsize of two adjacent subdom ain s nor on th e distortion. In contrast to t he 2D case , the ratio hmor/hnon ente rs, in genera l, into the a priori est imates in 3D; see, e.g ., [BM97, BD98]. Here, h m o r denotes the meshsi ze on the mor tar side and h non t he one on the non-mortar side . Und er some addit ional assumptions on the trian gulation, the factor hmor/hnon can be avoided. We refer t o [KLPVOO, WohOOa] for a mor e pr ecise analysis of the constant s in the a priori est imates and to Subsect . 1.5.3 for some num eric al results illustrating the influence of t he choice of th e non-mortar side. In the rest of t his secti on, we restrict our selves to the 2D case . The discussion of t he 3D case is included in Sect . 1.2. To obtain t he mortar approximat ion Uh , as a solution of a discret e vari ati onal problem , there are so far two main approaches. The first one has been int roduced in [BMP93, BMP94] and gives rise to a positive definit e nonconforming vari ational problem. It is defined on a subspace Vh of the product space , th e element s of which sat isfy weak conti nuity condit ions across t he int erfaces. The const rained finit e element space Vi, is given by
6
1. Discr et izat ion Techniqu es Based on Dom ain Decomposit ion
Vh := { v E L 2 (D ) I Vl nk E X hk;nk' 1 ::; k::; K , I [v ]J.Lda = 0, J.L E Mh mbm) , 1 ::; m::; M} , 1m
where t he test space M hmbm) is given by 2
M hm b m) := { J.L E L b m ) I J.L = w l.,m' W E Xh n(m);nn(m) , J.L le E Pnm-I(e) , if e E S m;hm contains an endpoint of 1m } , (1.2)
where n m := n n(m)' T he indices 1 and k are rese rved for the subdomains wher eas the index m is used for an interface, an d we have to un derst and t he definit ion of n m in t his sense. In 3D, t he definition of the Lagrange mul tiplier space has to be modified in the neighb orh ood of 8 We remark t hat Mh m bm) is a mod ified trace space of codimension two , associated with t he 1D t riangulation on t he non-m ort ar side which is inh erited from In (m);hn(m)' T hus , t he space M hm bm) dep ends on t he choice of the non-mor t ar side . The global product Il~= 1 Mh m bm) is denot ed by M h , and is a subs pace of L 2 (S ), where S := U;;;=I"Ym' In pri ncip le, it is also possible to introduce a new independent 1D t riangu lation on each interface "[m. which is inherit ed neither from In (m);hn(m ) nor from 'Tf. (m);hn(m)' T hen , we have to impose suitable conditions on the meshsize or ad d adeq uate bubble functions to the finit e element space to obtain a discret e inf-su p cond ition. Such stabilization t echniques are discussed in [BMOO, BFMR98] for three-field approaches. The nonconforming formu lation of the mortar method can be given in te rms of t he constrained space Vh : Find Uh E Vh such that
,m,
(1.3) see [BMP 93, BMP94]. Here , the bilinear form a(·,·) is defined as J(
a(v,w) :=
L
J
k=l [ h
a\7v· \7w
J(
+ bv w dx ,
v ,w E
II HI(D
k)
k= 1
We rem ark t hat continuity was impose d at t he verti ces of t he decomposi t ion in t he first pap ers about mortar methods. However , t his condit ion can be remov ed wit ho ut loss of stability. Both t hese set tings guarantee uniform ellipticity of t he bilinear form a( ·,·) on Vh x Vh , as well as a best approximation err or and a consiste ncy error of optimal order; see [BMP 93, BMP94]. Combining Lax-Milgram 's and Strang's Lemm as, it can be shown that a unique solut ion of (1.3) exist s. If the weak solution U of (1.1) is smooth enough and H 2 -regularity holds, then we have the following a priori est imates for the discretization error in the broken HI-norm an d in the L 2 -nor m
1.1 Introduction to Mortar Finite Element Methods
7
(1.4)
see [BMP93 , BMP94, BDW99]. Here, we use a standard Sobolev notation for norms which are not explicitly defined and set IlvilI := ~~=I IlvllLtk ' v E n~=1 HI([h). 11 ·lls;D st ands for the Hs-uoru: on t he open set Den , and 1·ls;D is t he corresponding semi norm. In th e case that D = n, the index n is suppressed . Fur thermore, we have ellipticity of the bilinear form a(' ,') on Y x Y a(v , v) ~
c Ilvlli,
(1.5)
vEY ,
where Y is defined as K
Y := {v E
II HI(n
k)
I
vlen = 0, ![v] da = 0,1 :::;
k=1
m:::; M}
;
(1.6)
1m
see [BM95]. We not e t ha t Vh c Y . Then, the energy norm I I vll1 2 := a(v ,v) , v E Y is equivalent to t he broken HI-norm. In [Gop99, Theorem IV.l], it has been established t hat the ellipticity constant is independ ent of the number of subdomains. A similar estimate is given for th e three field approach in [BMOO] . T he proof of the a priori bounds for t he discretiz ation error (1.4) is based on a best approximation error result and an a priori estimate for th e consistency err or; see [BMP93, BMP94]. We note th at very often the a priori est imate in t he broken HI-norm is given in the weaker form K
Ilu -
uhli l :::; C L h~k lulnk+l ;!h
.
k=1
However , in [WohOOa] it has been shown t hat (1.4) also holds with a const ant ind ependent of the number of subdomains. The const raints at t he interfa ces guar antee th at the consiste ncy err or is at least as good as the sum of the best approximations erro rs on the different subdomains. Replacing Vh in (1.3) by the unconstrained product space
yields a consiste ncy error t hat is not bounded in terms of t he meshsiz e. To prove the best approximation err or of Vh , t he mortar projecti on, lIm : C (')'m) ----t W h m (')'m ), plays an important role. Here, the trace space W h m (')'m ) is given by
W h m (')'m) := {fl' E C(')'m)
I
fl = w l-,.m'
W
E
Xh n (m) ;n n( mJ '
8
1. Discretization Techniques Based on Domain Decomposition
and llm is defined in terms of the Lagrange multiplier space tt;» - v E CoCrm),
J
(v - llmv)p,du
= 0,
P, E
M h= Crm)
(1.8)
1m
We have just recalled the definition of the mortar projection as in the original mortar papers. In Subsect. 1.2.1, we will consider a more natural way in defining it. It can be easily seen that the operator llm is well defined; see [BMP93, BMP94]. For the analysis of the approximation error, it is sufficient to show that the mortar projection is uniformly stable in suitable norms. The HJ-stability of the mortar projection is proved in [B~P93, BMP94]. This operator can be extended in a stable way to a linear and continuous 2Crm) operator from H66 onto WO;hmCrm) := HJCrrr,) n Wh=Crm); we will still denote the extended operator by llm. We find the following inclusions x; n HJ(fl) c Vi, c x; and observe that replacing Vi, in (1.3) by Xh n HJ(fl) or Xh does not provide a good discretization scheme. It is obvious that the quality of the nonconforming approach (1.3) and the properties of Vh depend on the space M h . Let us consider the structure of Vh in more detail. In general, Vi, is not a subspace of HJ (fl) and thus (1.3) is a nonconforming finite element method. Even for a nested sequence of global triangulations, the corresponding finite element spaces are non nested. Secondly in general, no basis of Vi, with local support can be constructed. The constraints are given in terms of a L 2-orthogonality of the jumps, and an element Vh E X h belongs to Vi, if and only if
J
llm([Vh])P,h do
= 0, P,h
E
Mh m Crm), 1 < tri < M
1=
In particular, if Vh E Vi, and [Vh] E CoCrm), 1 ~ tri ~ M, then llm([Vh]) = 0, 1 ~ m ~ M. A nodal basis function on the mortar side has to be extended to the non-mortar side such that the matching conditions are satisfied.
Fig. 1.5. Structure of the support of a nodal basis function in Vh, (standard)
Figure 1.5 shows the typical support of a basis function in Vi, associated with an interface, where the non-mortar side is on the left. The support of such a nodal basis function on the non-mortar side is a strip of length liml and width h m , and the locality of the basis functions is lost.
1.1 Introduction to Mortar Finite Element Methods
9
Figure 1.6 illustrates the trace of a basis function of Vi, on the two different sides of an interface for the two choices of the mortar side. In the left part of Fig. 1.6, the mortar side is associated with the finer triangulation whereas in the right part, it is associated with the coarser mesh. Although the basis functions on the non-mortar side have a global support, their values decrease exponentially.
'.~
1\ '[~}J"'"
.20.6 c 0.8 iJl a 4
~
.~
'1'
.20.6 r c; 0 . 8 . .~ 0.4 t
\ \
~o;J
iot~i\/ _02~ d
c_O.2
§
1
I\'
! I,
~O.6
2°.4
I
~O.2 ~
..
/
a
-I'
~O.6
2°·4
\
:@O.2
\
\
s °l=..:=:=::::'~~'==:J
-
non-mortar side
mortar side
non-mortar side
mortar side
is 1 ~O.8
(I
i o.s
Ij
Fig. 1.6. Nodal basis function on a mortar and non-mortar side, (standard)
These observations motivate a second approach which was introduced in [Ben99] and further studied in [Woh99a]. The matching conditions on the interfaces are not imposed on the global space but realized by means of Lagrange multipliers. The starting point is a constrained minimization problem leading to the following saddle point formulation: Find ('Uh' Ah) E ix; M h ) such that
a(uh' v)
+ b(v, Ah) =
(1, v)o,
v E Xh ,
= 0,
b('Uh,/~)
ME
(1.9)
Mh ,
where the bilinear form b(·,·) is given by the duality pairing on the interfaces M
K
b(V,M):= L([V],P,),m,
v E
II H
(fh ), ME
k=l
m=l
and [v] := via n(m) - vla_n{m) . Here, (H
H
M
1
II (H~(rm))
,
m=l 1 2(rm))' /
denotes the dual space of
1 2(rrn), /
and (', '),m stands for the duality pairing. Since the solution of the positive definite variational problem (1.3) and the first solution component of the saddle point problem (1.9) are equal, we use the same notation. The discrete Lagrange multiplier Ah approximates the flux. A priori estimates for the error A - Ah can be obtained, using the approximation property of M h and a suitable inf-sup condition. Here, Al,m := a ~~, where n is the outer normal on f!n(m)' This issue was first addressed in [Ben99], where a priori estimates in the H~t2 -dual norm have been established. The H~t2 -dual norm is defined by
L M
:=
m=l
IIMI1 2
L M
1 2
( H 0 0 h m ))'
:=
rn=l
sup
V
E
()2 V;M 1m
H~00 ( 1m ) IIVII IIoohm) .~
10
1. Discr etizati on Techniques Based on Dom ain Decomposit ion
where f-L E (H~t2(S)) ' and (H~t2 (S) )' := TI~= l (H~t2bm)) ' . The space H~t\'fm ) can be interp reted as an int erpolation space between £ 2b m) and HJb m). Working with a post eriori est imates , it is ofte n more convenient to deal with mesh dep end ent norm s. Here, we consider a mesh dep end ent £ 2-norm given by M
1If-LI I ~t- ! ;S
:=
L L
m=l eE S "" h",
h ellf-L 116;e'
f-L E £2(S) ,
wher e h e is the diam et er of t he element e; see [AT95]. A pr iori bounds for t his mesh depend ent £ 2-norm ar e der ived in [Woh99a]. The qu ality of the a priori est imat es in the H~t2 -dual and this mesh dep end ent £ 2-norm is t he same . 2 We use II . 11 M' to mean eit her t he HU -dual norm , II . II (H~62 (S) )" or t he mesh depend ent £2-norm, II . IIh-t /2;s . As in th e general saddle-point approach; see, e.g., [BF91], the essent ial point is to establish adequate inf-sup condit ions; such bounds have been est ablished with constants ind ependent of t he meshsize for both these norms; see [Ben99, Woh99 a]. Thus, there exists a constant such that (1.10) We not e t hat , so far , no inf-sup condit ion of t he form (1.10) has been established for the H 1 / 2 -dual norm , and t hat no a pri ori est imates are available in t hat norm. If t he solut ion u is regular enough, we find t he following a priori estimat e for t he Lagran ge multiplier by mean s of (1.10) and t he approximation pr op er ty of M h K
11..\ - ..\hll ~, ::; C L h~nk luli+nk;!h
(1.11)
k=l
T he proof for t he du al norm is given in [Ben99] and t hat for t he mesh dependent £ 2-norm in [Woh99a] . We remark t hat t he bilinear form b(·, '), defined on X h x M h is not uniformly cont inuous. To see this, we consider t he following exa mple. Let Vh E X h be const ant one on one subdomain rho and zero elsewhere, and f-Lh E M h be const ant one on one interface 'Ymo C 8D ko and zero elsewhere . Then , if the triangulation on the non-mortar side of 'Ym is qu asi-uniform with meshsize h m o , we find (1.12)
1.2 Mortar Methods with Alt ernative Lagrang e Mul tiplier Spaces
11
However , b(·, .) is uniforml y cont inuous for both II ·IIM,-n orm s if X h is replaced by a suitable subspace. For th e pr oof of (1.11) , it is imp ort an t that (1.10) also holds if t he supremum is taken over this subspace . Det ails are worked out in Sub sect. 1.2.3.
1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces In Sect. 1.1, the const ra ints at the int erfaces 'Ym are realized by means of a global L 2-projection , with test and t rial spa ces almost the sa me. In t he nonconforming vari ational formulation (1.3) , we have to face the problem th at the basis fun cti ons of Vh cannot be given easily as linear combinat ions of t hose of X h , since th e const ruc t ion of a basis of Vi, involves the solution of mass matrix syst ems for each interfac e. Furtherm ore, th e supports of th e basis functions associated with th e interfaces ar e non-lo cal on the nonmortar sides. Following th e second approach, (1.9) , we can work with th e un constrained product space X h . However , this formul ati on gives rise t o an ind efinit e problem. We note t hat efficient iterative solvers for saddle point pr oblems are often more complex th an those for positive definit e problems. In [SS98], it is shown t hat suit able lower dimensional Lagran ge multiplier spaces also yield optimal discreti zation schemes and t hat, without loss of optima lity, t he ord er of the Lagran ge multiplier space can be redu ced by one compa red wit h t he standard approach given in Sect. 1.1. One characteristic of t he Lagran ge multiplier spaces int rodu ced in [SS98] is t hat t hey are modified trace spaces of lower order conforming finit e element discretizations. Here, we pr opose different , more flexible, spaces for t he Lagran ge multiplier , in par ticular , spaces based on a du al basis. We recall t hat th e Lagran ge multiplier Ah in th e saddle point approach prov ides an approximation of t he flux. A fun ction v E H 1(f h ) has it s trace in H 1/ 2 (8fh ) and th e normal compo nent of its flux is in t he dual space H - 1 / 2(8[h). This observat ion is t he starting poin t for t he const ruction of a new ty pe of discret e Lagrange multiplier spaces. A special exa mple of a du al basis in t he first ord er case, nk = 1, in 2D has already been st udied in [WohOOa]. Here, we pr esent du al basis space s for the qu adratic case in 2D and the first ord er case in 3D. Before we define our new du al bases, we develop a general fram ework for Lagrange multiplier spaces which yield optimal results. We introduce a subspace of L 2 b m) of dimension N m :S dim WO;h = b m), and give appropriate assumpt ions und er which this space can replace the standard Lagr ang e multiplier space defined in (1.2). The spaces W h = bm) and WO ;h= bm ) = W h = bm) n HJ bm ) ar e introdu ced in Sect. 1.1 as finit e element t race spaces on t he non-mortar side . For convenience, we keep the same not ati ons as before, and denot e each element of t he new abstract class of Lagr ange multiplier spaces .by M h = bm) . We assume that th ere exists a bas is {1/Ji I 1 :S i :S N m } of M h Tn bm) satisfying t he following pr operties:
12
1. Discret ization Techniques Based on Domain Decomposit ion
(Sa) Locality of t he support : #(SUpp'¢i) ::; C,
1 ::; i ::; N m
,
#(p) ::; C, p E 'Ym , where # (SUpp'¢i) is t he number of elements in S m ;h m havin g a non-emp ty int ersection with t he simply connected supp ort of '¢i , and #(p) is t he number of functions '¢i such t hat the point p is contained in t he support of '¢i. (Sb) Approximation property of Mhmb m): For each f-t E H nm-I / 2b m) , t here exists a f-t1/J E M hmbm) such t hat
L
eES rn j h
hel/li -
f-t1/J/1 5;e::; Ch;;:m 1f-tI;'m- I/2;'Ym
1n
In 2D, he is t he length of t he edge e and in 3D, he denotes t he diam et er of t he face e. For a given basis, condition (Sa) is easy to verify and nat ur al in t he finite element context. Assumpti on (Sb) requires t hat the constants are contained in t he Lagrange multiplier space . As in t he standard mortar sit uation, a L 2 -pro jection-like operat or plays an essent ial role in establishing t he approximat ion property for th e constrained space . We base t he more general mort ar proj ecti on on a second set of linearl y independe nt functi ons 8i E WO;hmbm ), 1 ::; i ::; u.; having simply connected local suppo rts . The space WO;hmb m) := span {8i I 1::; i::; N m } is a subs pace of t he trace space WO;h mbm) . It is a prop er subspace if N,« < dim WO;hmbm) . Obviously t he space WO ;hmbm ) cannot satisfy an approximation property for H I bm) ' To establish an optima l order upp er bound for t he consiste ncy erro r, we use the following modification
e
where i E W hmb m) form a set of linear ind ependent funct ions havin g simply connected local sup ports. By const ruction, t he t hree spaces M hmbm), WO;hmbm ) and W hmbm) have the same dimension. The following t wo assumpt ions concern t he discret e spaces WO;hmbm ) an d W hmbm) and their relat ion to M hmbm): We assu me t hat for M hmbm ), there exist two sets of bas is functions defining WO;hmbm ) an d W hmbm ) such t hat (Sc) an d (Sd) holds: (Sc) Ap prox imation pr operty of WO;hmbm ) and Wh mbm): For each v E Hob m) and H Sb m), 0 ::; s ::; 1, there exists a Vo E WO;hm bm ) and W hmbm), respectively, such t hat
2: ;;hllv - vo l16;e ::; Clvl;;'Ym ' Ivoll;'Ym::; C lvk'Ym'
eESrn ;hm
e
v E H Ib m) .
1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces
c D1/J :S u .o;'
(Sd) Spectral equivalence:
c D1/J :S u.ir; -
--1
Mr :S C D1/J u,T
(ddij := 6ij IIBiI16;'Ym' zIIB iI16;'Ym' (dd ij := 6ij h
o.
(d1/J)ij :=
,
-
:S C D1/J'
where the elements of the diagonal matrices D 1 , D1/J' are given by
b,
6ijll1/JiI16;'Ym'
(d1/J)ij := 6ijh~ill 1/JiI16;'Ym'
13
i E {I, 2} ,
and D1/J E lRNm «n.; 1 :S i ,j:S N m 1:S i,j:S
,
u.; ,
where hOi and h1/Ji is the diam et er of supp Bi and supp 1/Ji' respectively, and the mass matrix M 1 E lRNm xNm is defined by
(md ij
:=
J 'Ym
1/Ji Bj da,
1:S i,j :S ».;
To obtain D z , Dz , and M z , we replace the basis functions Bi of WO ;h m (')'m), in t he definitions of D 1 , D1 , and M 1 , by the basis functions i of W h m ('Ym) . The mass matrices M i , i E {I, 2}, are sparse due to the locality of the supports . If N m = dim WO ;h m (')'m) , the approximation property (Sc) for WO ;h m (')'m) is automatically satisfied.
e
Remark 1.1. The two approximation properties (Sb) and (Be) are of different nature. (Be) is a low order approximation property which does not depend on the order of the finite eleme nt approximation on the subspaces . In contrast, the order of th e approximation property (Sb} depends on the order of th e finite element approximation on th e non-morta r side. We now use as our new Lagrange multiplier space M h = n~=1 M h m (')'m), wher e Mh m (')'m) is th e abstract discr ete space given by
The alternative non conforming finite element space Vh is defined as before in terms of M h
Vi,:= {v E LZ(fl) I
J [v]Jldo =
'Ym
Vln k E Xhk ;n k ' 1:S k:S K , 0, Jl E M h m (')'m) , 1 :S m :S M}
(1.13)
We rem ark t hat formally (1.13) is exactly the same definition as in (1.2). For simplicit y, we use the sam e notations as before , but we point out that now Vh , Uh, and Ah do not depend only on the order of the discr etization and the triangulation but also on the special choice of M h . We bri efly recall the two different mortar settings. The symmetric positive definite one can be written as: Find Uh E Vh such that
14
1. Discretization Techn iqu es Based on Domain Decompositi on
(1.14) In t he saddle poi nt ap proach, t he unconstrained pr odu ct space X h is independe nt of the choice of M h . The abstract saddle point formulat ion has exactly t he same st ruct ure as in (1.9). The only difference is that now t he special Lagran ge multiplier space (1.2) is replaced by t he genera l one: Find (U h' A h) E (X h , M h ) such t hat (1.15) Within t he genera l saddle point framework, the approximation property of
Vi" is a conseq uence of th e approxima t ion property of X h , t he cont inuity of
the bilinear form b(·, ') , and an inf-sup conditi on; see, e.g., [BF91]. A discret e inf-sup condition is necessar y in ord er t o obtain a prio ri est ima tes for the Lagran ge multiplier. In t he mortar sit uat ion, t he bilinear form b(· ,·) is not uniformly cont inuo us on Xi; x Mh ; see (1.12). Thus one has to be very careful in the a priori analysis. In t he following t hree subsectio ns, we show th at we have t he sa me quality of a pri ori est imates for t he solut ions U h and (U h' A h) as before for the standard case . Sub sect ion 1.2.1 is devot ed to t he analysis of t he best approxim at ion pr op erty of t he constrained space. It is based on t he stability est imate in t he H~{/ -nor m for t he genera lized mort ar projecti on. To establish t his stability, assumpt ions (Sa), (Sc) , and (Sd) are required . In Sub sect . 1.2.2 , we consider th e consiste ncy error of t he abstract nonconforming mort ar formulati on (1.14). In order to obtain a consiste ncy error of at least t he same order as t he best approximation error, assumptions (Sa)(Sd) are necessar y. These are t he two basic tools t o obtain a pri ori est imat es for U - U h . To obtain t he required ord er of t he consiste ncy erro r, t he assumption (Sb) is of crucial imp ortan ce. Roughly speaking, we have t o require t hat t he space M h = hm ) cont ains t he space of polynomials of ord er j, n m - 1. The stability of the sad dle point probl em relies on a suitable inf-sup condition . We establish a discrete inf-sup condit ions in Subsect . 1.2.3. Based on t hese preliminar y considerations, we obt ain a priori estimat es for t he Lagrange multiplier which are of the same ord er as t he a priori est imates for U - U h in t he energy norm . In Sub sect. 1.2.4, we pr esent an d analyze several exa mples for Lagran ge multiplier spaces wit h particular emphas is on dual basis spaces . The advantage of t hose space s satisfying a suitable biorthogonality relation is th at t he nodal basis fun ctions of Vh have a local support . Examples are given in 2D and 3D and for piecewise quadrat ic finit e elements . For higher ord er elements, we refer to [OWOO] .
1.2 Mortar Met hods with Alt ernative Lagrange Mult iplier Spaces
15
1.2.1 An A p proxim ation P r op er t y The essent ial tool in t he pr oof of t he approximation property of t he const rained space Vh is the stability of t he mort ar proj ecti on JIm . In t his subsection , we introduce a mod ified mor t ar pr ojecti on which depends on the choice of t he spaces WO ;h m (,m) and Nh m (,m) . The modi fied mortar projectio n, JIm : L 2 (,m ) --7 W O;h m (,m), is defined by
J
tt; » /-l do =
J
V/-l da ,
/-l E M hm(,m)
(1.16)
1m
1m
We observe that t he dimension of W O;h m (,m) and M h m (,m) is t he same by construction. However, t he mor t ar pr ojection JIm will not be well defined for arbit rary choices of WO ;h m (,m ) and M h m (,m).
R emark 1. 2. I n th e original papers about mo rta r m ethods, th e mo rt ar proj ection fo r the stan dard Lagrange m ultiplier space is given in a different way; see S ect . 1.1. It satisfies an approximation property for H I-functions, and 2 can be exte nded to H 66 -functions in a stable way . H ere, we us e a different f orm which does not tak e the values at the endpoints into account. However, both definitions give rise to operators which are identical when restrict ed to 2 H 66 (,m ). W e note th e follo wing difference: Th e operato r defin ed by (1.16) is in contrast to th e one given by (J.B) n ot H I (,m) -stable but L 2 (,m )-stable. Figure 1.7 illustrates the st abilit y properties of the two m ortar projection s.
\ I VIO-,
-8
j_nw L2 - stability
(1.8) IIn vll o- h (1.16) Ilnvll - 8 o
1
H - stability
Iv1
=0
1
I
_n~
(1.8) Inv11=0 (1.16) Inv1 !;: 0 1
F ig . 1. 7. St abili ty prop erties of (1.8) and (1.16)
In the following, we work with t he mortar project ion JIm defined by (1.16). The stability of t he mortar pr oj ection plays an essent ial role in the analysis of t he best approximation erro r. The following lemma provides uniform stability in t he L 2 _ and H I- norm s.
16
1. Discret izati on Techniques Based on Dom ain Decomp osit ion
Lemma 1.3. Un der the assumpti ons (Sa) , (Sc), and (S d), th e m ort ar projection (1.16) is well defin ed and is £ Z-st able and HJ- stabl e:
IIllmvII O;I~ IllmviI: l~
:::; Cll vll o;I~ ' :::; C l v h ;I~ '
v E
£2b m) ,
v E H J b m) .
P roof. The spectral equivalence (Sd ) shows that M I is non-sin gul ar and t hus t hat llm is well defined. Using the explicit rep resent ation llmv = 2:~1 ai Bi, we find M Ia
wher e r E IE.N~ , with ri := simply connec te d su pport we have N~
IIllmvI1 6;1~
:::; C L
J
=r
v'l/Ji da , 1 :::; i :::; N m . Due to t he locality of t he 1 :::; i :::; s.; and the linear indep enden ce,
OlOi'
a; I I Bi I 16;1~ = C a T o, « = CrT M I- T D IMI-I r
i =1
Finall y, t he ass um ptions (Sa) and (Sd) yield the £ Z-st ability
Reasoning as before, we get the stability of t he general mort ar pr oject ion in a weight ed £ z-n orm
(1.17) Here, we have used t he locality (Sa ) and t he spectral equivalence (Sd) . For t he pro of of t he H J -st ab ility, we use t he approximation property (Sc) of t he space WO ;h ~ (I'm ) an d an invers e estimate for polynomials. By means of t he best approximation ve E WO ;h~ bm ) and (1.17 ) , we find for v E HJ b m) Illm viI: l~
:::;
Illm( v - ve)h ;l~
:::; C (
2:
eES1T1 jh11l.
+ I veiI: l~
; 2 1Illm( v - ve )116;e) e
1
'2
+ IveiI:l~ :::; ClviI:l~ .
0
1.2 Mor tar Methods with Alt ern ative Lagrange Multiplier Spaces
17
H6/
The -stability of the mortar projecti on is guaranteed by t he £ 2_ and HJ -stability and an interpo lation arg ume nt. Following t he lines of [BMP 93, BMP94], it is easy to establish an approximati on pr op er ty for V h in the 2D case provid ed t hat the mor t ar projecti on 2 IIm is -stable, For eac h subdomain fh , we use t he Lagr an ge interpolat ion ope rator h , and define Wh E X h by Wh !n k := h u. We not e t hat Wh , in gene ral, will not be contained in V h . To obtain an element in V h , we have to add appro priate corrections. We obser ve that t he jump [Wh] is in HJ bm) for each interface "[m. in 2D . This is not t rue for 3D , and t he proof of t he approximat ion pr op er ty has to be modified . In the 2D case , we apply t he mor t ar projection t o [Wh] . The resul t llm[Wh] is extended by zero onto ofln (m ) \ "[m. and the exte nsion is still denoted by llm[Wh]. Then , the defini-
H66
t ion of WO;h ~ b m) yields llm[Wh] E H 1/ 2 (o fl n (m ) ) ' Now , llm[Wh] is extended as a discrete harmonic fun cti on into the interior of fln( m )' Finally, we define M
Vh := Wh -
L
H n (m ) (llm[Wh]) ,
(1.18)
m=l
wher e H n (m ) den ot es the discret e harmonic exte nsion operato r sa t isfying
see [Bra66, Wid88]. Here, the extens ion H n (m ) (llm [Wh ]) vani shes outside fln (m ) ' By construct ion , we have
and t hus Vh E V h . Repl acing t he test space M h in t he definition of the nonconforming sp ace, we find t hat t he approximati on prop er ty is pr eserved 2 as lon g as t he modified mort ar proj ection is -st abl e.
H66
Lemma 1.4. Under th e assumption s (Sa) , (S c) and (S d) on Mh ~ bm), 1 :S m :S M , th e nonconforming space Vh satisfies the approximation property in 2D, (1.19 )
if u is regular enough. P roof. We set V h E V h as in (1.18) and find by t he and a coloring argument
H662-st ability of llm
18
1. Discreti zation Techniqu es Based on Dom ain Decomposition
o We rema rk t hat t he constant depend s on t he shape regulari ty of t he t riangulations and t he decomp ositi on but not on t he number of subdomains. In par ti cular , t he ratio of t he length of adjacent edges on t he non-mort ar sides ente rs in the boun ds. But t he ratio between the meshsizes on mort ar and nonmor t ar sides does not enter. For a more det ailed analysis of th e constants in a special case, we refer to [WohOOa].
The 3D ca se . In the 3D case, we have t o face t he fact t hat t he boundar y of 81m is a closed one dimensional cur ve. For th e ana lysis of t he standard mortar situation, we refer to [BM97, BD98 , LSV94]. We cannot expect t hat [wh]1"In> = 0 on 81m in t he case of non-m at ching meshes on 81m, where Wh is defined as before. Since t he mor t ar project ion given by (1.16) is L 2- st abl e bu t not H I-st abl e, no uniform stabilit y results are available for the H~!/ norm . T hus, we can not work with the discret e harmonic extension H n(m) from H I / 2 (8 Dn (m)) onto HI(Dn(m)) as in t he 2D case . We replace H n(m) by iin(m) supporte d only in a small st rip in t he neighb orh ood of "[m. on t he non-mort ar side . It is defined by its valu es at the Lagran gian int erp olation points x p { v(x p ) , X p E "[m , ( H~ n(m)v ) (x ) p := 0, elsewhere . Now, an inverse inequalit y yields, for v E Il ~= 1 M
Wh ",
M
L Iliin(m)Vlli;S]n(",) < e L L
m= 1
m =1 eES",;h",
(')'m ),
~e Il vl1 6;e.
Proceedin g as in t he 2D case and observing t hat JIm is stable in t his weight ed L 2 -nor m finally yield an a pr iori est imate which depends on (1 + hmor/hnon)
where hmor/hnon stands for the maximum ofthe local ratio between meshsizes on t he mort ar and non-m ort ar sides, and S m C Dn(m) is a st rip of width hn(m ) and area 1 1ml on the mor t ar side. We not e that ISml /IDn(m)1 te nd s t o zero if t he meshsize te nds to zero. Alt hough the constant in t he a priori est imate depend s on (1 +hmor/hnon), it might be advantageous to associate t he non-mortar side with t he finer mesh.
1.2 Mortar Methods with Alt ern ative Lagran ge Multiplier Spaces
19
We refer t o Sect . 1.5.3, for num erical results illustrating the influence of t he choice of the mort ar method. In general , using very different meshsizes on t he mortar and non-mort ar sides is only appro priate if the coefficients are st rongly discontinuous. Then , this jump of the coefficient s is also reflect ed in the constants of t he a priori est imate s and t his effect might cancel the factor h m or / h n on . For a more det ailed analysis of the constant in te rms of the coefficients, we refer to [WohOOa]. In par ticular , we find t hat t he ratio a non / a mor ente rs int o t he constant s of the a pri ori estimate in the energy norm. Here, a non and a m or stand for un upper and lower bound for the eigenvalues of the coefficient a on t he non-m ort ar and mor t ar sides, resp ectively. Remark 1.5. W e note th at in th e special situation th at th e triomqulations 'T",( m );hn (", ) and'Tn(m);hii(m) coincide on o"(m; see Fig. 1.8, an d nn(m ) = n n(m)' 1 ::; m ::; M , we can pro ceed as in th e 2D case . Then , applying th e Lagrange in terpolati on yields [wh]I )'", = (In(m)U - I n(m)n)l )'m E H~t2 bm). W e refe r also t o [KLPVOOj for th e special cas e n k 1, 1 ::; k ::; K.
=
Fig. 1.8. Trian gulations on mortar and non-mortar side of "(m in 3D
1.2.2 The Consistency Error T he definition of Vh guarantees t hat t he jump of an element is orthogonal to t he Lagran ge multiplier space M h . However , in general, t he £ 2-norm of t he jump will not vani sh . Thus, we are in a non conforming set t ing and Vh HJ(fl). In t his case, the approximat ion prop erty (1.19) is not sufficient to obtain opt imal a pri ori est imate s for the finit e element solution of (1.14) . According to Stran g's Lemma, the consiste ncy err or
ct
has to be considered ; see, e.g., [Bra97]. The weight ed £ 2-norm of t he jump of an element in Vh measures it s non conformi ty. Using t he definit ion of the mort ar pro jecti on, we find t hat for an eleme nt v E Vh ,
20
1. Discreti zati on Techniques Bas ed on Domain Decomp ositi on
(1.20) where vlnii (m ) is t he t race of v on t he mort ar side and vlnn( m) is t he t race of on t he non-m or t ar side. However , JIm satisfies an approx imation proper ty on ly in HJCtm) bu t not in H ICtm). Since no cont inuity is imp osed on t he space Vh a t t he cross po ints , the jum p [v]l-Y m' v E Vh , will be in general not belon g to HJCtm). In t he first mortar pap ers, t he cont inuity at t he cros spo ints was requ ired and bot h proper ti es, t he approx imation proper ty on H I(im) and (1.20), could be obtained at the sa me time for t he mor t ar proj ecti on. For the pr oof of t he approximation proper ty of the constrained space in 2D , it was impor t an t t o work with an operator guarante eing t hat t he result is zer o on 8i m and can be extended by zero onto 8f2n(m) \ "[m. resul ting in an eleme nt in H I / 2(8f2 n(m))' Her e, we introduce a new pr ojecti on having both pr op er ti es even without cont inuity at t he crosspo ints, bu t which does not guarantee that t he result is zero on 8i m. In the definiti on oft he mortar projection , we repla ce WO ;h mCtm) by WhmCtm) and define Pm : L 2Ctm) ~ W hmCtm ) by V
Jr ; »
u da =
"Ym
J
JL E MhmCtm)
VJL du,
1m
T he pr oj ecti on Pm is well defined under t he ass umption (Sd) . By definiti on , we find , for v E Vh, Pm[V] = 0, 1 ~ m ~ M . We remark t hat Pm cannot replace JIm in t he proof of the approximation pr op er ty. For t he construct ion of an element in Vh sa ti sfying t he approximat ion prop er ty , it is impor t an t t hat H~62 -fun ctions are mapped onto H~62 fun cti ons. This is t he case for JIm but not for Pm. In addit ion to Pm, we now introduce a du al oper ator Qm: L 2Ctm) ~ MhmCtm ) by
"1m
"1m
The followin g lemma shows t hat Pm and Q m sa t isfy cert ain approximat ion pr op er ti es. Lemma 1.6. Under the assumpti ons (Sa) , (Sc), and (Sd), there exist con-
stants such that
L:
e ES m ;h
L:
e ES m ;h
m
m
~
e
Ilv-
Pmvl16·e ~ I
Civil. ,
v E H ~ Ctm) ,
2 ,Tnt
hellv - Qmv l16-e ~ Ch~m l vl 2 m _ I
n
1' 2 ,1'm 2 Vl 1
L: hellv - Qmvl 16;e2 c Ilv - Qm 1
eESm;hm
(H o20 C'Y m))'
V ,V
E Hnm- ~ Ctm ) , E
L 2Ctm) .
1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces
21
Proof. We st art with the proof of t he approximation prop erty of Pmv . It is an easy consequence of the locality of th e supports, (Sa) , th e approximation prop erty, (Sc) , of Wh ", (')'m) , and th e spect ral equivalence, (Sd). As in the proof of Lemma 1.3, (Sa) and (Sd) gua ra ntee t he stability of Pm in the L 2 _ and a weighted L 2 -nor m . Finally, (Sc) yields for s = 1/2
2:
eESm jh m
1 Ilv- Pmv l16;e = e
2:
e E Sm ;h m
1 Ilv- Vo e
:::; Ce E Sm 2:;h 111 v m
e
Pm(v -
Vo 116'e:::; I
vo)116;e
Civil.
2 ' ''''(711
To prove t he second inequality, we first consider the stabili ty property of Qm in a weight ed L 2-nor m . Observing that Jrm Qmv w da = JTTn v Pmwda, v , w E L 2(')'m) and using the stability of Pm in th e weight ed dual L 2-nor m , we get
<
~
L.. eES", ;h",
II 11 2 he V O;e
2: sup
e ES""h",
w EL2(-y", )
~
L IlPm wl16;e
L.. e ES"'; h",
II 11
:::;
1 2 h e W O;e
C
'\'""'
L.J
heIlvllO;e . 2
eE S ",;h",
Now as before, the approximat ion prop erty of Qm follows from the stability an d the best approximat ion prop erty (Sb): Let v1jJ E M h ", (')'m ) such th at (Sb) hold s. Then , we find
2:
e E S m ;h m
hell v -
Qmvl1 6;e=
2: hellv - v1jJ - Qm(v - v1jJ )116;e :::; C 2: hell v- v1jJII 6-e :::;Ch~"'l vI 2 m _ 1 . e E Sm jh ' n ,I'm eESmjh m
2
m
T he proof of th e last inequality is mainl y based on the fact t hat Qm is th e du al operator of Pm. Using the definition of t he du al norm and t he first inequality of Lemma 1.6, we obtain
Ilw 1IH ! (
00 /'"
)
';eIIw( e ES2: "'; h", o
1
PmwIl6;e)
2"
22
1. Discr eti zation Techniques Based on Domain Decomposition
An upper bound for the weight ed L 2 -nor m of th e jump of an element in Vh can be obtained by mean s of t he proj ection Pm . In particular , th e nonconformity of an element can be measured by this norm. The following lemma provides . an upper bound for th e jump in t he weighted L 2 -nor m . Lemma 1.7. Under th e assumptions (Sa) , (S c), and (Sd) , th e weighted L2_ no rm of th e jumps of an eleme nt v E Vh is bounded by M
1
L L
m =l eESm; h m
-, II[v)116;e::; WE inf Ilv - wiIi t H~(Sl )
(1.21)
0
e
Proof. The pr oof uses t he same ideas as in t he case of t he standa rd constrain ed space; see [Woh99a). Using t he orthogonality of th e jump and t he trial space and Lemm a 1.6, we find for v E Vh 1 2 he II[v)IIO ;e=
1 he II[v)
-
2
Pm[v)IIO ;e ::; CI[v)1 h'im .
e E S m ;h m
eESm jh m
Fin ally, using the conti nuity of t he t race operator, we get for each w E HJ (D) ,
~ II[v)116·e < C(IVln n(m)
2:
e E S Tnj h
e
I
-
wl'i.2 ,lm + IVln n(m) _
-
wl'i. ) 2 ' 1m
m
::; C
(11v- wlli;nn(m) + Ilv- wlli;nn(m) ) .
Summing over the interfaces "[ ni gives (1.21) with a const ant independ ent of t he number of subdomains. D The consiste ncy err or of the mortar formulation (1.14) is closely relat ed to t he nonconformity of t he element s in the const ra ined space VhoThe following lemma provides an upp er bound for the consiste ncy err or. The proof is based on Lemm a 1.7 and t he approximation property (Sb) of Mh o Lemma 1.8. Under the assumptions (Sa) -(Sd) and u regular enough, th ere exis ts a cons tant such that
Proof. By mean s of t he ort hogona lity, we find for Vh E Vh and f.Lh E Mh m (1m) th at
Ja~~
1'm
[Vh) dCT =
J(a ~~ -
1m
f.Lh) [Vh) dCT
1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces
23
Using W = a in Lemma 1.7, the approximat ion property (Sb) , a trace t heorem and summing over the interfaces guarantee a consiste ncy err or of optimal ord er. 0 The coercivity of t he bilinear form a(·,·) on Vh x Vh is an easy consequence of (Sb) . From (Sb) , we obtain Poh'm) C Mh m h'm) which yields VI, C Y. Thus, the unique solvability of (1.14) is guaranteed. Now, the approximation proper ty of Vh and the consist ency error guarantee an optimal ord er discretization scheme. We obtain th e sa me quality of the a priori est imates as in the standard case (1.4) . The st ability of the mortar projection yields t he approximatio n property. This argum ent is mainly bas ed on th e spect ral equivalence (Sd) and the approximation property of W h m h'm). The essent ial t ool in the proof of the consiste ncy error is t he approxima t ion property of M h m h'm)' Using Lemmas 1.4 and 1.8, we obtain standard a priori est imate s for th e modified mortar approach (1.14) in the broken HI -norm if the solut ion u is smoot h enough, i.e., K
Ilu - uhlli :::; CL
hink lul;'k+!;fh
(1.22)
k= 1
If we fur thermore assume H 2-regularity, th e discreti zation error u - Uh in th e L 2 -nor m is of ord er h 2 . The proof is based on the Aubin-Nitsche trick. For t he standard mor t ar setting it can be found , e.g., in [BDW99]. Introducing t he dual problems: Find W E HJ(fl) and Wh E VI, such th at
a(w ,v ) = (u - Uh , v )o,
v E HJ(fl) ,
a(wh' v ) = (u - Uh, v)o,
v E Vh ,
we get
Then , t he H 2 -regularity, Lemma 1.7, (Sb) , and observing th at the jump of an element in Vh is orthogonal on M h yield
Ilu - uhl15 :::;
+ + :::; C
f'"!~l., C~, 'ES~, _ ~.II["hlllfi;
a(u - Uh ,U - Uh)I/ 2a(W - Wh, W - Wh)I/2
(1 'EE ,,_ ~.II C~, 'EE,,_
(II
U -
[whlllfi;,
f' f'
~h llfi;. h.llx - ~h llfi;. h.II'\ -
uhlh(llw- wh ill + hllwl12) + Ilw- Whill /lhi~th 11,\ -
P,hllh- ~ ; S)
where XI'Ym := a ~w . Using t he a priori est imate for t he energy norm (1.22) un and the approximation property (Sb) , we obt ain an ord er h 2 a priori est imate for the L 2 -nor m
24
1. Discretization Techniques Based on Domain Decomposition
Ilu -
uhllo :::; Ch
(IIU - uhlh + JL hEMh inf IIA - J.Lhll h _1 ) 2;5 K
:::; Ch ( L: hink k=l
lul;'k+l;.fh )! .
(1.23)
We remark t hat t he global meshsize h ente rs int o the a priori est imat es in t he L 2-norm and not only t hose of th e subdomains . This is in cont ras t with t he Hr-uotu; est imate. 1.2.3 Discrete Inf-sup Conditions
The approximat ion prop erty of M h is in general not sufficient to obt ain a pri ori est imates for th e Lagran ge multiplier. In addition, suitable inf-sup conditions have t o be sat isfied; see, e.g., [BF91]. In this subsect ion, we analyze t he error in the Lagr an ge multiplier for the H~I/ -du al norm and a weighted L 2-norm. For both th ese norms , we obt ain a priori est imates of the same qu ality as for the standa rd mortar approach. In th e following, we show t hat t he inf-sup condit ions, established in [Ben99] and [Woh99a] for th e st andard case , also holds for t he general pairing (X h , M h ) . Before we prov e t he discret e inf-sup condit ions , we consider an exte nsion operator 1-lM= : W O;h= (I' m) --+ X hn(= J;n n<= ) ' It s definition depend s on the choice of II ·11M" In Sect. 1.1, t he norm, II ·11M' , associated with th e Lagr an ge multiplier space was defined in two different ways. For both definitions, we consider a du al norm. We start with the case II . 11M;" = II . II (H~6 2 b= ) ) ' and set
IlvlIM= := Ilvll H 1 b = ) , V E Mm := 020
1
HJo(l'm)
The exte nsion operator 1-lM= is t hen defined as t he discret e harmonic extension, where the elements of WO ;h= (I' m) are exte nded by zero ont o f)fln (m ) \ I'm . In t he case of II ·11M1n' = 11 ·llhm-1/2." m , we set M m := L 2 (1'm) and II ·IIM= := II . 11,1.TJl.'/2". m . The mesh depend ent norm II . 11,lorn'/2.1rm has t he inverse weight compared wit h
II . Il h ;;;'/2;,=
1
2
he 11J.Ll o;e' e E5 =; h=
T he corre sponding operator 1-lM= is defined as t he t rivial extension by zero, i.e. , we set all nod al valu es on f)fln( m ) \ I'm and in fl n (m) to zero . Then , 1-lM= ¢ is non zero only in a st rip of area II'm I and of width h m · For both cases, we obtain t he following stability property
II1-lM=¢ Ih;nn< =)
:::;
CII¢IIM=, ¢ E WO ;h= (I'm )
.
Based on t his st abili ty est imat e, we can easily establish th e following discret e inf-sup condit ion for both choices of II . IIMI.
1.2 Mort ar Methods with Alt ernative Lagrange Multiplier Spaces
25
Lemma 1.9. Under the assumptions (Sa) , (Sc) and (Sd) , there exists a constant in depen dent of h such that (1.24)
Proof Considerin g one int erface "1m at a tim e, we get , by means of the stability of the mortar proj ection in th e M m-norm,
(/-lh , ¢) o ;,~ II ¢IIM~
The maximizing element in WO ;h~ hm) with Mm-norm equal one, is called ¢/-Lh' To obtain an element Vm E X h from ¢/-Lh E WO ;h~ hm), we apply the exte nsion op erator H M~ Vm ln\n
n( m)
:= 0,
Now, we find t ha t for both choices for
II · 11M'
~
0 :::; (/-lh' ¢/-Lh)o; '~ = b(Vm,/-lh) ,
and t ha t
Ilvmlh :::; C1 vmkn n( ~ )
CII¢/-LhIIM~
:::;
= C. Finally, we set
M
V/-Lh :=
L
b(Vm,/-lh)Vm ,
m=l
and observe t ha t V/-Lh = 0 if and only if /-lh = O. Introducing N(k) := {1 :::; m :::; MI n(m) = k} and observing th at th e numb er of element s in N(k) is bounded ind ependently of th e number of subdomains, we find K
Ilv/-Lhili = 112:
2:
k = l mEN( k)
:::; C
K
2: 2:
k=l m EN (k )
M
:::; C
K
b(Vm ,/-lh)Vmlli
2:
m= l
= 2: II 2: k =l
Ilb(vm,/-lh)vmlltn k
II/-lhll ~:,.II ¢/-Lh II~~ = C
:::;
M
2:
m=l
m EN (k)
C
b(Vm,/-lh)Vmllt n k
M
2: b(Vm,/-lh)2
m=l
II/-lhll~:,. = C II/-lhll ~,
Summing over all int erfaces yields
II /-lh 11 2M ' < _
M
C 'Z:: " b(Vm, /-lh)2 = Cb( V/-L h, /-lh ) m= l
<
C b(V/-Lh II ,/-lh) II V/-Lh 1
II/-lhIIM'
26
1. Discretization Techniques Based on Domain Decomposition
By const ruction, we have found for each f.1h E Mh ' f.1 h =f- 0, a non zero E Xh such t hat
V ' Lh
,< C
II f.1h II M
-
b(V/Lh ,f.1h) Ilv/Lh 111
0
We rem ark that by const ruction, the inf-sup condition (1.24) also hold s t rue if t he supremum over X h is replaced by the supremum over a suitable subspace. Mor eover , two stabilit y est imate s hold. The first one is obtained by a coloring argume nt and is already used in t he proof of (1.24) (1.25) T he second follows from the definition of V/Lh and the du ality of t he norms M
M
II [V/L h]111I = I:: (b(Vm , f.1h) )211 [vm ] II1I = I:: (b(Vm , f.1h)) 2 mM=l m=l 'S I:: IIf.1hlllI:" II[vm]lllI = lIf.1hlllI, . m=l m
(1.26)
m
The inf-sup condit ion (1.24) t oget her with t he approximation property, Lemma 1.6, and t he first equation of the saddle point problem give an a pri ori estimate similar t o (1.22) for the Lagran ge multiplier. Lemma 1.10. Un der th e assu mpti ons (Sa) -(Sd) and u regular en ough, th e follo wing a priori estim ate for th e Lagrange multiplier holds K
IIA - AhlllI, 'S C L h%n lul; k+l;!tk . k
(1.27)
k= l
Proof. Followin g [Ben99] and using the first equa t ion of t he saddle point problem , we get
Ch oosing V/Lh - AI. as in t he pr oof of (1.24) , and applying the st ability est imates (1.25) and (1.26), we find
IIf.1h - AhlllI, 'S
C b(V/L h-Ah,f.1h - Ah) = C (a(uh - U,V/Lh - Ah ) + b(V/L h- Ah,f.1h - A))
'S C 'S C
(Ilu - uhlir IIV/Lh -Ah IIr + IIf.1h - AIIM' II[v/Lh-Ah]IIM) (Ilu - uhlll + IIf.1h - AIIM' )11f.1h - AhllM' .
Applying the trian gle inequ ality and choosing f.1hl.,m := QmAI.,m, 1 'S m 'S M , we find by mean s of Lemma 1.6
1.2 Mortar Methods with Alt ernative Lagrang e Multiplier Spaces
II>. -
27
>'hll~, :s c (IIU - uhlh + m~l h;;:m1>'I~m- kfm) :s c 2:: h~n k lul;dl;[./k . K
k= l
Here, we have used th at>. restricted to "1m is a\7u · n and a trace t heorem.
o
There is a structural difference between t he 2D and 3D case only in th e proof of the approximat ion prop erty. The proofs of the consiste ncy error and t he discret e inf-sup conditi on are exactly th e same. Remark 1.11. Th e a priori esti m ate s (1.22), (1.23) and (1.27) can be weakH Sk(n k ) , S k ~ t > 3/2 . ene d fo r the more gen eral case U E Ht(n) n
rrf=l
W e refer to {WohOOa] fo r th e lowest order case.
1.2.4 Examples of Lagrange Multiplier Spaces In the previou s subsect ions, a general fram ework was given for new Lagrange multiplier spaces. Here, we focus on Lagrange multiplier spaces based on a du al basis and present concrete exa mples. The main advantage of such a biorthogonal basis is the locality of t he supports of the nodal basis functions of the const ra ined space Vh. Additionally, th e implementation of the mortar method for such a basis can be carr ied out using th e unconstrained product space; det ails are provided in Sect . 1.3. We consider t he special case where Ni ' 1 :s i :s N m } and {Bi , 1 :s i :s N m } are biorthogonal and sati sfy (Se)
J B(l/Jj da =
~~
c5ij ci
J BJ da,
c :S Ci
:s C
.
/~
Considering now t he support of an element V h E Vh , we find a st ructural difference between t he standard case and the more gener al one satisfying (Sa)-(Se) . We not e t hat WO ;h m ("(m ) C Wh m ("(m), t hus {Bi 11 :s i :s N m} can be easily exte nded to a basis of Wh m ("(m ) with local support; {Bi 11 :s i :s v m }, V ni := dim W h m ("(m )' We recall that N m < t/-« . Furthermore in 2D , we find that N m :s V ni - 2. In the 3D case, we have N m :s V m - N p , where N p is the number of nod es on EJ"fm . Each v E X h , rest ricte d to a non-mortar side "[m , can be written as Vm
vI17n = ""' L...J a ·B·1-
i=l
1,
(1.28)
Let v E Xh rest ricted to "[m be given as in (1.28) . Then , v E Vh if and only if for each non-mortar side "[m.
(1.29)
28
1. Discretization Techniques Based on Domain Decomposition
The proof follows from (1.28) and the biorthogonality relation (Se). Using (1.29) and taking (Sa) into account, it is easy to construct nodal basis functions of Vh that have local support. As in the standard finite element context, nodal basis functions can be defined such that the diameter of the support is bounded by Ch. Here, C depends on the maximum number of edges on the mortar and non-mortar sides that have a non-empty intersection with the supports of'l/)i and e·i , 1 :::; i :::; N m , respectively; see Fig. 1.9. We recall that this is not possible in the standard mortar case; comparing Fig. 1.5 with Fig. 1.9 shows the structural difference. In contrast to the standard case, the value of an element v E Vi, at a point p on the non-mortar side is determined completely by its values in a small neighborhood of p on the mortar side. As a consequence, the constraints at the interfaces can be locally satisfied.
Fig. 1.9. Structure of the support of a nodal basis function in Vh, (dual)
Working with a biorthogonal basis reflects the duality between the trace space of the weak solution and the one of the flux. The basis functions of such a dual basis are, in general, not continuous and cannot be defined as a trace of conforming finite elements. The approximation property, well known for trace spaces, has, in general, to be checked by hand. The following lemma provides a simple tool for verifying the approximation property (Sb) of a Lagrange multiplier space satisfying the biorthogonality relation (Se). Lernrna 1.12. Under the assumptions (Sa), (Sd) , and (Se) , the Lagrange multiplier space M h = (rm) satisfies (Sb) if and only if Pn m - 1(r m) cMh m (rm).
Proof. The proof is based on arguments similar to the Bramble-Hilbert Lemma. Obviously, P n= -1 (rm) c M h= (rm) is a necessary condition for the approximation property (Sb). We define 7/Jv .- ~~"i ai (v )7/Ji for v E H- 1j2(r m) by
I
ai V := ()
"1m
.
c."
veid(J
r e2 da(J ' i
l' N :::; 'l:::; m '
"1=
Then, the spectral equivalence (Sd) and the fact that M 1 is a diagonal matrix yield af(v)II'tPiI16;'Y= :::; Cllvl16; suppe', for v E L 2(rm). Considering an element e E Sm;h= at a time and using the locality of the supports of ei and 'I/)i, we obtain the following stability estimate
1.2 Mortar Methods with Alt ernative Lagrange Multi plier Spaces
II 1Pvll~;e : : ; C
N~
L
i= ] e n suPPVJi #0
29
N~
L
a7(v)ll 1Pill~;l'm::::; C
Il vll ~;suPP8i::::; Cllvll~;De
i=l
e n s u PPWi #0
Here, e C D e is a simpl y connect ed union of at most a fixed number of elements e' E Sm ;h ~ . T he maximum number of elements contained in D e depend s on the number of elements contained in th e supports of Bi and 1Pi but not on the meshsi ze. If P nm-1 ("(m) C Mh m((m), then 1Pv = v for v E Pn~- l ({m) , and we find
Ilv - 1PvIlo;e:: :; Ilv - lInm-1 vllO;e + 111P(v-lInm v)IIO;e : : ; Cllv -lInm- 1 vIlO;De ::::; Ch~:-2I vl nm_ ! ;De - l
1
'
(1.30)
where bo, is t he diam et er of D e, and lIn~-l is a locally defined L 2 -projection onto Pnm- 1 ({m)
JlIn~-l V
De
ui
do
=
J
De
vw da,
wE Pnm- 1 ({m) .
T he local quasi-uniformity of t he triangulation yields ehe' ::::; he ::::; C he',
e' E D e .
Together with th e fact t hat the number of elements in De is bounded by a const ant, we obtain (Sb) by summing over the elements and using (1.30) . D
The rest of t his subs ection is devot ed t o the const ruction of alte rnat ive Lagran ge multiplier spac es, which provide optimal finit e element solut ions . We rest rict our selves to low ord er finit e elements and refer to [OWOO] for t he general order case in 2D. As has been shown earlier, it is sufficient to verify (Sa)-(Sd) . Furthermore if, in addit ion, th e biorthogonalit y relation (Se) is satisfied , a basis of Vh having local support can be const ruc te d. The idea of using du al spaces for t he definition of th e Lagrang e multiplier space can also be carr ied over to 3D. As in the st andard mortar approach [BM97, BD98], the analysis and the definition of t he Lagran ge multiplier space is mor e technical than in 2D. 1.2.4.1 The First Order Case in 2D. We st art with the first ord er case, i.e., nk = 1, 1 ::::; k ::::; K , in 2D. Figure 1.10 shows basis functions of four different types of Lagrange multiplier spaces. All four typ es satisfy the assumptions (Sa)-(Sd) . T he two pictures on t he right represent elements of a du al basis which also satisfy (Se). In each case , we t ake WO ;h ~ ({m) as WO ;hm ({m ) and choose W hm ({m ) equal to th e standard Lagrange multiplier space defined by (1.2) . The basis function s und er considerat ion are the st andard nodal ones. Then , the approxima t ion property (Sc) is sat isfied without any further assumptions.
30
1. Discretization Techniques Based on Domain Decomposition
We consider four different Lagr ange multiplier spac es for the piecewise linear case denoted by Mh, 1 :::; i :::; 4, each defining a constrained finite element space The corresponding mortar finite element solutions of (1.14) and (1.15) are denoted by uh and (uh ,Ah) , respectively. The nodal basis functions of Mh(f'm) ar e denoted by 'ljJf, 1 :::; l :::; N m, where N m is the number of vertices X I in the interior of "[ m - For simplicity, we suppress the ind ex m in the case of the basis functions and the vertices. The enumerat ion of the vertices XI is lexicographically, and the two endpoints of ar e denoted by Xo and XN=+l' Furthermore, the length hi, 1 :::; l :::; N m + 1, is defined by hi := IlxI-l - xLII, and we define the diagonal matrix D by d ii := hi ,
Vi: .
,m
1:::; i:::; s.;
standard Lagrange M~
dual basis (piecewise linear) M~
piecewise constant M~
dual basis (piecewise constant) M~
-1 -------------------------------------------------
Fig. 1.10. Different types of basis functions for Lagrange multiplier spaces
We set M~ equal to t he standard Lagrange multiplier space; see [BMP93 , BMP94], and observe that the nodal basis functions are cont inuous and piecewise linear. The nodal basis functions in the interior of the interface are the st andard hat functions , and they are modified only in the neighborhood of the two endpoint s; see the upp er left of Fig . 1.10. It is then easy t o see that the st andard Lagrange multiplier space (,m) and its set of nodal basis functions satisfy the assumptions (Sa) and (Sb) . Observing that M, is symmetric in this speci al case and that M i , D i , D,p and D,p , D~ and ii; Djl are spectrally equivalent, respectively, we find (Sd) . The second space M~ is based on piecewise constant functions . We define the nodal basis functions 'ljJ; of M~(f'm) by
ML
'ljJ; (X) :=
{
~:
X E [~(X i -l elsewhere,
+ Xi),
~(X i
+ Xi+l )],
2:::; i:::; N m -1 .
1.2 Mort ar Methods with Alt ern ative Lagrang e Multiplier Spaces
31
'l/Jr (X) and 'l/JJvm(x) are equa l one on [x o, ~(Xl + X2 )] and [~(XNm- l + XNm), XNm +l ], resp ectively, and zero elsewhere; see the lower left of Fig . 1.10. Obviously, the locality of the supports (Sa) and the approxima t ion property (Sb) are sat isfied. To verify (Sd) , we consider the mass matrices. We find that (d,p)ii = 0.5(h i + h iH) , 2 ~ i ~ N m - 1, (d,p)l1 = 0.5(2h l + h 2) , (d,p) NmNm = 0.5(h Nm + 2h NmH), (dl) ii = 1/3(h i + h iH ), 1 ~ i ~ N m . The mass mat rix M l in (Sd) is symmetric positive definit e and t ridiagonal
~ hl + ~ h2
3 ~ h2 s(h2 + h3)
~h2
~h3
~ hNm - l ~ (hNm -l + hNm) ~ hNm l8 h Nm ~8 hNm + l2 h Nm +1 The explicit represent ati on of th e matrices shows t he spect ra l equivalences of u ; D i , D,p , Di l and D as well as between D,p and D3. These equivalences gua rantees (Sd) . The next two examples sat isfy th e biorthogonality relat ion (Se) , in addit ion t o (Sa)- (Sd). Two different sets of bior thogonal functions are introduced . In the first case, the funct ions are piecewise linear whereas in th e second case they are piecewise constant. The space MK was originally introduced in [WohOOa] for t he mor t ar set t ing and is spanned by piecewise linear but discontinuous fun ctions. The main advantage of these basis functions is t heir bior thogonality with respect t o the standa rd hat functions. Here, we review the definit ion of t he nod al basis functions and define I~i (21 1x -
'l/Jr (x ):=
{
Xi- III - Ilx - XiII) , h i~ l (211x - xiHII - llx - Xi iI), 0,
x E [Xi-I , Xi] , X E [Xi, XH l] , elsewhere,
for 2 ~ i ~ N m - 1. The two basis functi ons 'l/Jr(x) and 'l/J~m (x) close to t he endpoints of "[m. are equal one on [Xo , Xl ] and [XNm , XNm+ l ], respectiv ely, while they have t he same st ructure as t he oth er basis functions elsewhere; see t he upp er right of Fig. 1.10. Our next example is also a du al basis, but in contrast to MKb m) , its basis funct ions are piecewise constant . For 2 ~ i ~ N m - 1, let
'l/Jt (X) :=
{
~ ' X E [ ~(Xi- l + Xi), ~ (Xi + XiH) ] , - ~ , X E [Xi- I , ~ (Xi -l + Xi)) U ( ~(X i + XiH ), xHd 0, elsewhere,
,
and let 'l/Jt(x ) and 'l/J1vm(x ) be equa l one on [XO , Xl] and [XNm, XNmH], respectively, while t hey have t he same st ructure as t he ot her basis functions elsewhere; see t he lower right of Fig. 1.10. T he condit ions (Sa) , (Sd), and (Se) can be eas ily verified for t he two Lagran ge multiplier spaces ML bm), I E {3, 4}, assoc iated with th e two sets of du al basis functi ons. In both cases, we find t hat
32
1. Discretization Techniques Based on Domain Decomposition
N=
1=
L7/JL
l E {3,4} ,
i=l
ML,
and thus PO(rm) C (rm), l E {3, 4}. By means of Lemma 1.12, we obtain (Sb). Moreover, we find
./7/JiBj do = 6 j'/ ¢j da = i
ITn
ITn
da,
1::;
i.i :
Nm
,
l E {3,4} .
To get a better understanding of the Lagrange multiplier spaces based on dual basis functions, we consider, in the rest of this subsubsection, the mortar projection in more detail. Here, we restrict ourselves to the special case of a uniform triangulation of "[rri with h m := lei and use (im) = span {1;0r 11 ::; i < N m}. Then,
ML
where Po and P1 are the two endpoints of e. Using the definition of the mortar projection (1.16) and summing over all elements, we get
Figure 1.11 shows the trace of a nodal basis function of v,~. The trace on the mortar and non-mortar side is illustrated for the two different choices of the mortar side. In the two pictures on the left, the mortar side is associated with the finer mesh whereas in the two pictures on the right, the mortar side is associated with the coarser mesh.
mortar side
non-mortar side
mortar side
non-mortar side
Fig. 1.11. Nodal basis function on a mortar and non-mortar side, (dual)
In contrast to Fig. 1.6, where the same situation for a basis function in vl is illustrated, the support of the trace is bounded on both sides by a constant times the meshsize. This is the main advantage of using dual basis functions; the locality of the supports of the nodal basis functions of the constrained space is preserved.
1.2 Mortar Methods with Alternativ e Lagr an ge Multiplier Spaces
33
F ig. 1.12. Tr ace of a hexahedral (left) and a simplicial (right) t riangulat ion
1.2.4.2 The First Order Case in 3D. As a second case , we consider first order Lagran gian finit e elements in 3D. Two different sit ua tions are discussed separat ely; see Fig. 1.12. In t he first , the triangulation on the int erface "[m. is a te nsor grid, in t he second, it is a simpli cial t riangulation. For simplicity, we assume that the int erfaces are rect an gles. As in t he 2D case, t he definit ion of th e dual basis functions has to be modified in the neighborhood of the boundary a"lm' We recall that W hmbm ) is the t race space of bilinear or linea r finit e element s on "[m, and th at rPI is the st andard nod al basis fun ction associate d with th e vertex Xl . In a first ste p, we define a basis which is biorthogonal t o the st andard nod al basis of W hmbm ). In a second ste p, we reduce t he number of du al basis functions by t he number of ver tic es on the boundary of "1m with out loosing th e biorthogonalit y and t he approximat ion propertyof t he du al space . We define ¢l elementwise for T E Sm;hm, such that supp e, = supp rPI, by ¢ lIT := (4rPI - 2rPal(l) - 2rPa2(l) + rPO(l) )!T'
T E supp rPI
in t he case of a hexahedral triangulat ion and by
in t he case of a simpli cial t riangulation. The ind exing of a2(1) , and a(l) , is illustrat ed in Fig. 1.13. The indi ces of vertices of Xl on aT are denot ed by a1(l ) and a2(1) and opp osit e one, in th e case of a hexahedral trian gulation , by
a[jlll I
a2(1)
th e nodes, a1(1), the two adjacent the index of t he a(l ).
a1(1)
I
~2(1)
Fig. 1.13. Indices in t he case of a rect an gle and a triangle
T he basis functi ons 'l/JI are associate d with t he interio r vertices Xl of "[mWe now set 'l/JI := ¢ l in th e inte rior of "[m, i.e., if a(suPPrPI) n a"lm = 0. The corresponding vertices are marked by filled circles in Fig. 1.12. For all
34
1. Discretization Techniques Based on Domain Decomposition
other vertices Xl E "[m , we have to modify the definition. The corresponding vertices are marked by empty circles in Fig . 1.12. To do so, we introduce two sets of vertices M and Ml given by Ml := {Xj I Xj E 8"(m n 8(SUPP4>I)} '
M:= {Xj I Xj E 8"(m} \
U
Ml .
X t E 'Ym
The set M can be non-empty only in the case of simplicial triangulations; it is always empty in the case of a tensor grid. In the situation, which is shown in the right part of Fig . 1.12 , M contains two elements associated with the upper left and the lower right corner, marked with filled squares. The modified dual basis functions are given by Xl E "[m.
.
(1.31)
Here, nj is the number of sets M k such that Xj E Mk, and alj = 1 if 8(SUpp4>l) n 8(supp 4>j) contains an edge and zero elsewhere. We refer to [BM97] for an introduction of a standard Lagrange multiplier space, and to [BD98] for some more sophisticated choices of the weights in (1.31). In particular, it is possible to replace Ifnj by a weight involving the areas of the adjacent elements. Obviously (Sa) is satisfied. To verify (Sc) and (Sd), we have to specify the spaces WO ;h m ("(m) and W h m bm) and their basis functions . As in the 2D case, we set WO ;h m ("(m) := WO ;h m ("(m) and choose the standard hat functions as basis functions Bi := 4>i . The space Wh m bm) is obtained from WO ;h m bm) by modifying the standard hat functions in exactly the same way as the Lagrange multiplier basis functions before :
Bt
:= 4>1
+
L
xjE M t
~ . 4>j + J
L
alj4>j,
x; E
"[m. .
xjEM
It is now easy to see that I::Xt E'Ym 'ifJl = I:: XtE 'Ym Bt = 1. Thus (Sc) is satisfied and Pobm) C M h m bm). Furthermore, a straightforward computation shows that the following biorthogonality relation holds
This implies (Se) and (Sd) for i = 1. (Sd), for i = 2, is obtained by observing that M 2 , D 1 and D 2 are spectrally equivalent as well as b, and D2 . Applying Lemma 1.12 yields (Sb) . As in th e 2D case, the biorthogonality relation results in a mass matrix, on the non-mortar side , which has a diagonal block associated with the interior nodes . In addition, the nodal basis functions of
1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces
35
the constrained space have local support. We remark that we do not have supp 't/Jl = supp ¢l in the neighborhood of the boundary of "[niIn the special situation of a tensor product mesh on a rectangular interface, the construction of a dual basis function can be simplified. Piecewise bilinear and piecewise constant dual basis functions can be defined as a product in terms of the dual basis functions 7/Jr and 't/Jt given in the previous subsubsection, respectively. Then, the support of this simplified dual basis function 'I/Jt associated with the vertex .Tl E "[m. is the union of the four adjacent rectangles sharing the vertex Xl. Figure 1.14 illustrates the isolines of such a dual basis function if 8(supp7/Ji) n 8 rm= O. -2
-2
Fig. 1.14. Isolines of piecewise bilinear and piecewise constant dual basis functions
Comparing these simplified piecewise bilinear dual basis functions with the definition (1.31), we find that they are the same if 8(supp't/Ji) n8 rm= O. However, in the neighborhood of the boundary of "[m , we observe a difference. In particular, the tensor product dual basis functions 'IN have a smaller support than 't/Jl' We distinguish between three different types of vertices Xl. The inner ones, i.e., 8(supp ¢z) n 8 rm = 0, are marked with empty squares, the ones close to the corners, i.e., 8(supp ¢z) contains one corner of "[m , by empty circles and all other vertices are marked by filled circles; see Fig. 1.15.
Fig. 1.15. Different types of piecewise bilinear dual basis functions
Figure 1.15 shows the different groups of vertices and one piecewise bilinear dual basis function 7/Jt for each vertex type. The other ones can be obtained by local rotations. For the simplicial case, it is also possible to modify the definition (1.31) in the neighborhood of the boundary of rm; see [KLPVOO]. Then, the nodal basis function ¢l and the dual basis 7/Jl have the same support. The advantage of the simplified dual basis functions is the
36
1. Discretization Techniques Based on Dom ain Decomposition
smaller support. In contrast to supp 'l/Jl' the support of 'I/Jt contains for all indi ces 1 only the element s sharing the node Xl. The assumpt ions (Sa)-(Se) are easy to verify. 1.2.4.3 The Second Order Case in 2D. In our last example, we consider t he case n k = 2 in 2D. Again, we set WO ;h m(')'m) := WO ;h m(')'m), and W hm (')'m) is defined as the standard Lagrange multiplier space (1.2) with nk = 2. Both spaces are associate d with t he set of the corr esponding nod al basis functions. Figure 1-16 illustrat es the numbering of the nod al basis functions ()i , 1 ::; i ::; N m, in WO ;h m(')'m).
Fig. 1.16. Numbering of the nodal basis functions of
The du al basis fun ctions are now given in t erms of define 'l/Ji' for i = 21 + 1,1::; 1 ::; (N m - 3)/2 , by
'l/Ji (X) :=
{t 1 + ~()i)(X),
and for i = 2/,2::; 1 ::; (Nm
'l/Ji (X) := {
-
()i,
1 ::; i ::; N m . We
E SUPP()i ,
elsewhere,
3)/ 2 by
6:- ~()i- l +
The t wo first basis functions, ar e defined differently by
X
WO ;h m (--Ym)
()i -
~ ()i+d (x),
X
E SUPP()i ,
elsewhere .
'l/Jl ' 'l/J2 ' and the two last ones , 'l/JNm -l' 'l/JNm, X
E SUP P ()l ,
elsewhere, X
E supp ()l
,
X E supp ()2 \ supp ()l elsewhere ,
,
with 'l/JNm - l and 'l/JNm given in a similar way. A straightforward computat ion shows th at (Se) is sat isfied. In this case , th e spe ctral equivalence (Sd) follows from t he spectral equivalence of D 1j; and D 1. Fur thermore, we find L:~i 'l/Ji = 1. Observing th at 'l/J2i + O.5 'I/J2i-l + O.5 'I/J2i+l' is t he st andard piecewise linear and continuous hat fun ction associated with t he vertex Xi , 1 ::; i ::; (N m - 1)/2, th e approximation property
1.3 Discretization Techniques Based on th e Product Space
37
(Sb) follows from Lemma 1.12. Figure 1.17 illustrates the du al basis fun ctions
'l/Ji'
1:: : i s s.;
We obs erve t hat a priori est imates for the Lagran ge multiplier of order h" can be obt ain ed by using piecewise polynomials of order (n - 1) for the Lagran ge mul tiplier ; see also [8898] . Since the a priori est imates for the Lagrange mul ti plier depend s on the est imate for U - U h , the a priori est imate for t he flux cannot be improved by choosing higher ord er elements for the Lag range multiplier.
Fig. 1.17. Dual basis functions, (nk = 2)
Remark 1.13. Th e assumptions (Sa)-(Sd) also allow us to work with Lagrange multipli ers which are defined on a coarser triangulation than S m ;h m • For example if S m ;h m is obtain ed by a un iform refine ment step from S m ;2h m , th en M 2 h m bm ) sat isfies (S a)-(Sd) if M h m bm ) does. Finally, we not e t hat the concept of du al Lagran ge multiplier spaces can be gener alized to higher order elements . The const ruc t ion of higher order du al basis funct ions is very t echn ical , and we refer t he inte reste d read er t o [OWOO] . In [OWOO], dual basis fun ctions are const ructed for conforming P n-elements in 2D. The number of edges contained in the support of a du al basis fun ction is at most t hree and ind ep end ent of t he order n . Moreover , t he assumpt ions (8a)-(8e) are satisfied by construc t ion. Thus, optimal higher order mortar methods for du al Lagr an ge multiplier spaces are obtained .
1.3 Discretization Techniques Based on the Product Space In the pr evious subsections, two possibl e approac hes to t he impl ement ation of a mort ar discreti zati on have been discussed . The first gives rise t o a positiv e definit e problem , and is based on t he constrain ed space . The second works with the un const rain ed product space and a Lagran ge mult iplier space and results in a sa ddle point formulation , where t he weak cont inuity is enforce d
38
1. Discreti zati on Techniques Based on Domain Decompositi on
by means of t he Lagran ge multiplier space . In this sub section , we consider discret ization t echniques working only on the un constrained product space . Such a possi bility is also discussed in [BH99, St e98]. An idea introduced by Nitsc he in [Nit70] is rediscovered and generalized to non-m at ching trian gul atio ns . It t urns out t o be a penalty meth od where t he penalty par am et er does not depend on t he meshsize and does not influence t he condition number of t he resulting linear system. We refer to [BH99] for det ails and some num erical results including a posteriori error esti mators. Here, we develop a different idea . We note th at th e num erical solut ion of t he mortar method is, very ofte n, based on its saddle point formulat ion [AK95, AKP95, AMW 96, AMW9 9, BD98, BDH 99b, BDW9 9, EHI+98, EHI+ OO, HIK + 98, Ku z95a, Ku z95b, Ku z98, KW9 5, Lac98, LSV94, WW98 , WW99]. Work ing with t his ind efinit e formulati on has t he advantage t hat t he unconstrain ed product spaces assoc iated with a nest ed sequence of global t riangulat ions are nest ed , while the nonconforming constrained spaces are not . Our approach is based on t his observation, and we introduce an uns ymmetric Diri chlet-Neum ann coup ling and a sym met ric and positi ve definit e var iati onal pr oblem on t he unconstrain ed produ ct space . We t hen show t hat t he solut ion of t hese pro blems coincides wit h t he mortar finit e element solutio n. The idea is not rest rict ed to first order discretizations or to 2D. We assume t hat t he pair (X h, M h ) defines a constrained space Vh such t hat t he nonconforming variational pro blem (1.14) and t he saddle point problem (1.15) are well defined and yield optimal a priori bounds for t he discretization errors , see conditions (Sa)-(Sd) in Sect. 1.2. Then , we have only to ensure t hat t he nodal bas is fun cti ons of M hm(rrm ) an d WO;h m(rrm) form a biorthogonal set
J
'l/JtOk da = Ct 6tk IIOk I16;'Ym'
1 ::; i.e -;
Nm
,
(1.32)
'Ym
where C ::; Ct ::; C, 1 ::; l ::; N m, and n.; = dimMhm(rrm) = dimWo ;h m(rrm ), see condit ion (Se) in Sect . 1.2. We recall that WO;h m(rrm ) is a suitable subspace of t he finit e element t race space on t he non-mort ar side with zero valu e on t he boundar y of "[m - The inde x m is dr opp ed for t he basis functi ons and also for t he constants Ct. Examples of such pairings have been given in t he prev ious subsections for t he first order cases in 2D and in 3D , and for the qua dratic case in 2D. We recall t hat the only exa mples introduced in the previous subsections which do not satisfy t hese condit ions are t he standard Lagran ge multi plier space M~ and t he space M~ . Here, we restrict ourse lves to t he spec ial case N m = dimWo;hm(rrm) and Ot = (Pt, 1 ::; l ::; N m, where cPt are t he standard nod al basis functions . Our int erest in t he int roducti on of a new, t hir d, mort ar formulation is based on the following observation: Saddle point pro blems , like (1.15), are, very often, solved by Uzawa like algorit hms , and inn er and outer it erati ons are
1.3 Discreti zat ion Techniques Based on t he P roduct Space
39
required . Recentl y such a multigrid t echnique has been st udied in [WW99] for t he t raditional mortar approach. In t his case, a modified Schur complement system has to be solved it eratively in each smoothing ste p. This method is a genera lization of th e techniques given in [BD98, BDW99, Br aOl , WW98] where t he modified Schur complement system was solved exa ct ly. Another approac h is analyzed in [Kuz95a, Ku z95b] an d further used in [EHI+98, EHI+ OO, HIK+98, Ku z98]. A good pr econdi tioner for t he exact Schur complement is required , and t he ite rative solution of t he saddle point problem is obtained by a generalized Lanczos method. It is also true t hat standard multigrid t echniques cannot be applied for th e pos iti ve definite formulation (1.14) which is associate d with a nonconforming space. We refer to [AMW96, AMW99 , BDH99b, CDS98 , CW96 , Dr y96, Dr y97, Dry98a, Dr y98b , GPOO] for multilevel and multigrid meth od s and domain decomposition t echniques. Recentl y, multigrid methods for th e non conforming formulation have been developed [BDH99b , GPOO] which involve t he solution of a mass matrix syst em in each smoo t hing st ep . Dirichlet-Neum ann ty pe preconditioners have been proposed for t he positiv e definit e system in [Dry99 , DryOO]; see also [LSV94]. We now introduce a t hird equivalent mor t ar formul ation. It is based on t he un constrain ed product space and gives rise t o a symmetric positiv e definit e formulation on which we can apply standa rd multigrid t echniques. The rest of this secti on concerns th e const ruct ion of a symmetric positive definit e syste m defined on the product space . We present a vari ational formulation on the product space as well as its algebraic form . The st iffness matrix associated with the un constrain ed product space can easily be obtain ed by eliminating the Lagran ge multiplier. The starting poin t is the observation that t he Lagran ge multiplier Ah is given in a postprocessing ste p. But it can be easi ly evaluated in te rms of the right hand side and t he mort ar solut ion Uh only in t he case t hat t he bior thogonality relati on (1.32) holds. Otherwise, t he eliminat ion involves t he inverse of a mass matrix. We introduce an abstract fram ework for a new equivalent mortar setting based on t he un constrain ed product space . The analysis of our syste m on th e product space is carr ied out using the to ols of th e mortar framework. T he new idea is to use a biorthogonal basis for t he Lagrang e multiplier space and t o eliminate t he Lagran ge multiplier in t he saddle point formulat ion. We begin by considering, in Sub sect. 1.3.1, a Dirichl et-Neum ann couplin g for t he special case of two subdoma ins. Subs ections 1.3.2 and 1.3.3 concern t he construction of a symmet ric positive definit e syst em defined on t he pr odu ct space. It s condition number is bounded by a constant times 1/h 2 , as in the usual finit e element conte xt. We pr esent the variat ional formulat ion on t he pr odu ct space in Sub sect . 1.3.2 and its algebraic form in Subs ect . 1.3.3.
40
1. Discreti zati on Techniques Based on Dom ain Decompositi on
1.3.1 A Dirichlet-Neumann Formulation Let us consider t he following special sit ua tio n: The domain D C 1R2 is decomposed int o two nonoverlapp ing subdomai ns , D = D I U D 2 , and meas (8 D n 8Di ) > 0, i E {l , 2} ; see Fig. 1.18 .
Fig. 1.18. Decompositi on into two sub domains
On D I , we consider a Diri chlet bounda ry value problem -div(a'Vu d
= t. in DI , UI = 0, on r l := 8D n 8DI , UI = ut», on T := int(8D I n 8D 2 )
+ bUI
(1.33) ,
where so E H6// (r) · Its variational formulation is: Find UI E H~D ; rl (D I ) such t hat al(uI,vd = (f, VI)O ;{]ll VI E HJ( Dd . Here, H~D ; rl (Dd is t he subset of HI (D I) with a vani shin g t race on r l and a trace equa l to gD on r . The bilinear form s ai(-, '), 1 ::; i ::; 2, are given by
ai(V,w )
:=
J
a'Vv'Vw + bvwdx,
v,w E HI (Di ) .
{] i
On D 2 , we solve a partial differential equation with a Neuma nn boundar y condit ion on r -div (a'Vu 2) + bU2 = i, U2 = 0, afu!:L an = 9 N,
in D 2 , on r2 := 8D n 8D 2
onr ,
,
(1.34)
where n denot es t he outer uni t normal on D 2 • The weak variati onal formulation is given by : Find U2 E H h (D2) such t hat
a2(u2, V2 ) = (f ,V2 )O;{]2 +
J
YNV2 da = : !2(V2 ),
V2 E H h (D2) .
r
Here, Ht i (D i) , 1 ::; i ::; 2, denot es t he subspace of H I(Di) havin g a vani shin g t race on t he oute r boundary rio
1.3 Discr etiz ation Techniques Bas ed on the Product Space
41
Let us assume for the moment that the solution u of (1.1) and the flux on the interface r ar e known. Then, the choice gD := ulr and gN := aBu/Bnlr yields UI = Ul fl l and U2 = Ulfl2' The discrete spaces of conforming PI-Lagrangi an finit e elements on D I and D 2 , associat ed with t he simplicial triangulations Ti and 72 , are denoted by Xl C H}, (Dd and X 2 C Hh (D2 ) , resp ectively. The set of corresponding vertices are denoted by P I and P 2 , and the subs ets of vertices on the int erface by p[ and Pf. Then , (1.33) and (1.34) can be discretiz ed by means of the discret e spaces Xl and X 2 , respectively. We will now use th e same notation for an element in X i , 1 :s; i :s; 2, and its vector repres entation with respect to the nod al basis , {rPP' P E Pd, of Xi . The matrix representation of the Dirichlet problem (1.33) can be given by
(UI) ADUh;1 := (AI/AIr) 0 Id u~ I
=
(iJ) I}
.
(1.35)
Here, A I/ and A Ir are the st iffness matrices associate d with the bilinear form restricted to Xi x Xi and Xi x X}, respect ively. The space X} is spa nned by the nodal basis functions rPP' p E pf, and Xi := Xl n HJ(Dd . The right hand side is associat ed with the linear form I st, I vdx , and I} dep end s on soDirichlet boundary conditions are often realized by a pointwise equa lity at the nodes of the triangulation al (-,.)
iJ
(f})p = gD(p) ,
pE
p[ ,
if the boundary data function gD is cont inuous. Here, we use a different appro ach and sp ecify th e Diri chlet boundary condit ion by a weak int egral condit ion 'l/JpgD da , pE p{ ,
J
where e; := 3rPp- 1. We refer to [SZ90] , for a discussion of nonhomogeneous Dirichl et boundary cond iti ons , and observe that 'l/Jp is t he sam e as a nodal basis function of the du al spac e h'm) in Sect . 1.2, if the vertex p is not next to an endpoint of th e int erface. In cont rast to Dirichl et boundary conditions, Neumann boundary conditions gN ar e natural and ente r dir ectly into the variational formulation. The discret e vari ational formul ation of (1.34) can be written as
ML
(1.36) We use the sa me not ation as before ; and define
jJ.
by
42
1. Discretization Techniques Based on Domain Decomposition
Unfortunately, the solution and the flux restricted to T are, in general, unknown, and the boundary data gD and gN are not available. However, if in the continuous setting we set sr: := 7L2 and s» := aa7Lr!an, (1.33) and (1.34) form a coupled system of boundary value problems. Then, h(V2) can be rewritten, and we find that
Here, v E HJ(fl) is an extension of V2 E H r 2(fl 2 ) . We remark that the jump of the trace of v across r is zero. However, the jump of the flux of v does not have to be zero. These observations form the starting point for our discrete approach. When working with non-matching triangulations at the interface r, it is in general not possible to extend a function Vh;2 E X 2 to Xl X X 2 such that the jump of the trace is zero. However, in the mortar approach, it is standard to replace the strong continuity of the jump by a weaker one. Observing that each element in Xl x X 2 is uniquely defined by its values at the vertices Pl E Pi and P2 E P 2 , a suitable discrete extension EVh;2 of Vh;2 E X 2 is constructed in the following way: P E P2 PE PE
,
v, \ pi , pi .
(1.37)
We note that the extension operator E is not uniformly bounded in the broken Hl-norm.
Fig. 1.19. Support of the extended function
of
The shadowed region in Fig. 1.19 illustrates the structure of the support E Xl X X 2 . Using the definition of'l/;p and E, we find
EVh;2
1.3 Discretization Techniques Based on the Product Space
Xl
We are now ready to formulate our discr et e approach: Find X X 2 such that a l (Uh ;l ' V h ;l )
U h; l (P ) a 2(uh ;2, Vh ;2)
= (I, vh; d o; .l?ll = EUh ;2(p) , = (I ,Evh;2) O -
Vh ;l E
pE al(uh ;1 , E vh ;2) ,
(U h ;l ' Uh ;2)
xl ,
p{ ,
V h;2 E
X2
43
E
(1.38)
,
where EVh ;2 and EUh ;2 are the discret e ext ensions of V h ;2 and U h ;2 given by (1.37) . The solution of (1.38) dep ends on the definition of E on the inte rface bu t not on t he valu es at P, P E PI \ p{ . Thus the t rivial exte nsion E by zero into the int erior of ill could be repl aced by a discrete harmonic exte nsion wit hout influencing the solution of (1.38). The matrix formul ation of (1.38) reflect s dir ectly the Dirichlet -Neumann coupling between t he two subdomains
(1.39)
where M:= (m p 2P 1 ) P2E P f m
,PI EP [
is a sca led mass matrix defined by
2 [supp cPPI I
.- . , - - - - -
P2Pl . -
J
s UPPQ>P I
Before we analyze t he unique solvability of (1.38) and the discretiz ation error in the new setting, we consider our approac h in t he more general sit uat ion with sever al subdomains and crosspoints . We rem ark that we have to modify the approach in the neighborhood of a crosspoint . Using the same notati ons as in Sect. 1.2, we find t hat t he choice of t he Lagran ge multiplier space M h plays an essential role. We show in the next subsect ion that the discret e vari ational problem on the product space can be obtain ed from the sa ddle point problem by locally eliminat ing t he Lagran ge multiplier . R emark 1.14. W e observe that th e stiffness matrix in (1.39) reflec ts the Dirichlet-Neumann coupling between the two su bdom ains . Applying a block GaujJ-Seidel m ethod on this syst em results in a Dirichlet-Neumann type preconditioner. This me thod is well kno wn for conform ing fin it e elem ent m ethods; see [SBG96] and the references th erein , and has been also studied fo r th e stan dard mo rtar approach; see [Dry99, DryDD, LSV94]. For' details, we refer to S ect. 2.4. 1.3.2 Variational Formulations
In t his subsection, we introduce our new abst rac t vari ational formul ation on t he un constrain ed product space. The only necessary condit ion is the
44
1. Discreti zation Techniques Based on Domain Decomp osition
biorthogonality relation (1.32) . An un symmetric formul ation as well as a symmetric one ar e discussed . A car eful analysis shows that th e un symmetric one for t he special pairing (X h , M~) is the Dirichl et-Neum ann coupling described in t he pr evious subsect ion. The symmet ric one is positive definit e, and as we will see in Sect. 2.3, a standa rd multigrid method can be applied for t he resulting linear system. In t he rest of this sect ion, we assume th at the biorthogonality relation (1.32) is satisfied . At first glance the saddle poin t problem (1.15) has t he same st ructure for all Lagr an ge multiplier spaces. However , th ere is an essent ial difference. Given th e solution Uh of (1.14) , t he Lagran ge multiplier Ah l can be obtain ed by solving a mass matrix syste m. This postprocessing ste p involves t he inverse of a mass matrix which is, in general, dense on each int erface "[m - Only in th e case t hat the biorthogonality relation (1.32) hold s will the inverse mas s matrix redu ce to a diagonal one. Then , t he valu e of an element v E Vh at a point on one non-mort ar side is det ermined by it s values in a small neighborhood of that point on the adjacent mortar side. In par ti cular , t he mass matrix associate d with the non-mortar side is replaced by a diagonal one reflecting the biorthogonality. On each non-m ort ar side "[m» we can write Ah as Ah l .,~ = L:~l A m ;i "pm; i. Compared with Sect. 1.2, t he addit ional ind ex m indic at es th e corre sponding non-mort ar side "[m » The coefficients Am;i are given in te rms of Uh and f by ., ~
(1.40)
where cPm;i is t he nod al basis fun ction which has it s support in S?n (m) ' Let us now define linear functionals 91 : X h --+ M h and 92 : X h --+ X h · Both functionals are associated with the int erfaces. We define M N~ """"' """"'
91(V) := ~ Z::
m= l i==l
vI
(p.)
nn ( ~) m;t .t , . . A. .' do "pm;i,
J If/m, wm; 2
v E Xh
,
'1,
,~
where Pm ;i are the nodal points on "(m, i.e., cPm;i (P m ;j) = 6 i j . We fur t hermore use a linear func ti onal 92(-) which vani shes on Vh . It is defined by (1.41) It is easy to see that 92(V) is only supporte d in a sma ll neighb orhood of the non-mort ar side and vanishes on the mortar sides of t he interfaces. Proposition 1.15. Th e kern el of 92( ') is exactly Vh and 92 (9 2 (v)) = 92(V) . Moreover, th e L 2-no rm of 92(V) on S? can be bound ed by the L 2-norm on th e interf aces of th e jump of v
1.3 Discretization Techniques Based on the Product Space
45
M
Ilgz(v)ll& ::; eLL hell[vlll&;e · m=l e ES m ;h m
Proof. The definition of gz(-) and the biorthogonality relation (1.32) yields the first two assertions. A straightforward calculation shows
Z ~ ~( II gz (v )II 0::; mL:1 i~ f
b(v, 'ljJm'i ) ) Zll " liz 'ljJm;i¢r:.;i da 'f'm;i 0
~m
Here, we have used th e assumptions (Sd)-(Se) which guarantee the equivalence of II¢m;illo;'Ym and II'ljJm;illo;'Ym' and of h"'m:i and h m :i' Finally, (Sa) yields that the maxim al number of 'ljJm ;i having an overlapping support is bounded independently of the local meshsiz e. D There are now two possibilities to show that an element v E X h is also in Vi,. The first uses the definition (1.13) of Vh ; it is sufficient to prove that b(v, 'ljJm ;i) = 0 for 1 ::; Tn::; M, 1 ::; i ::; u.; The second possibility is based on th e projection gz(.); v E X h is an element of Vh if and only if gz(v) = O. The following lemm a shows an equivalence between the positive definite vari ational problem (1.14) on t he constrained spac e Vh and a non-symmetric and a symmetric variational problem on the unconstrained product space X h . The idea behind the introduction of t he new variational problem is to use the decomposition v = (v - gz(v)) + gz(v) . Then, v - gz(v) is an element of the constrained space Vi,. Lemma 1.16. Let Uh E Vi, be the unique solution of (1.14) . Then, Uh is the unique solution of the non-symmetric variational problem (1.42)
and also of the positive definite symmetric variational problem a(Uh - gZ(Uh) , v - gz(v))
+ a(gz(uh) ,gZ(V)) =
(/, v - gz(v))o,
v E Xh . (1.43)
Proof. In a first step, we show that the solution Uh of (1.14) satisfi es (1.42) and (1.43) . Using t hat Uh E Vh, we find b(Uh ,gl(V)) = 0 and gZ(Uh) = O. Since gz(-) is a projection, we have gz(v - gz(v)) = 0 and thus v - gz(v) E Vh. Combining these observations , we imm ediately find th at Uh satisfies (1.42) and (1.43) . To obtain t he unique solvability of (1.42) and (1.43) , it is sufficient to show that each solution of (1.42) and (1.43) , respectively, is an element of Vh and thus a solut ion of (1.14) . An element v E X h is in Vh if and only if b(v ,f..l) = 0, /1 E Mh , or equivalent ly if gz(v) = O. Let us assume that Uh
46
1. Discreti zation Techniques Based on Domain Decomp osition
satisfies (1.42) . For each 1 ::; m ::; M , 1 ::; i ::; N m , we have 92(rPm;i) = rPm; i and 91 (rPm;i) = Dm;i1/Jm ;i with Dm;i =I- O. The choice v := rPm;i yields
b(u", 1/Jm;i) = 0 ,
u"
and t herefore is t he unique solut ion of (1.14) . To prove th at E V" if is the solution of (1.43) , we use the second possibility and show th at 92 (U,,) = O. Setting v := 92(U,,) yields
u"
u"
a(92(u") ,92(U,,)) = 0 . Since t he bilin ear form a(·,·) is coercive on 92(X,,), we find that 92(U,,) = O. It is now easy to see t hat the vari ational problem (1.43) is symmetric and positiv e definit e. We define a(v, w) := a(v-92(V) ,W- 92 (w))+a(92 (v), 92(W)), v , w E X " . Then a( ·, ·) is a symm etric bilinear form satisfying a(v , v) ~ O. For each v E X " sati sfying a(v,v) = 0, we find that 92(V) = 0 and a(v ,v) = O. Since t he kernel of 92(-) is V" , an d a(·,· ) is V;,-elliptic, we get v = O. 0 Remark 1.17. Optimal a prio ri estim ates are available fo r th e variation al problem s (1.42) and (1.43) by usi tu; Lemma 1.16.
In th e rest of t his sect ion, we show t he relation between t he new mortar formulation (1.42) for t he special pairing (X" , M~) and t he Dirichlet Neum ann coupling (1.38). It t urns out th at t he non- symmetric product formul ation is almost t he Diri chlet - Neum ann formulation in t he case of two subdomains in 2D. Here, we associate the non-mortar side with t he subdomain VI . The space Xl can be dir ectly decomposed into x! + X[. Each element in X i is extended by zero on the adjacent subdomain , and we will use t he sam e notation for th e extended function. Using t he decomp ositi on for X l and observing that gl (v) = g2(V) = 0 for v E xl and g2(V) = v for v E X [ , (1.42) can be rewrit t en as a coupled system
adu" ,v) = (f ,V)O ;fh , b(U" ,g l(V)) = 0, a(u" , v - g2(V)) + b(U" ,gl(V)) = (f , v - g2(V))O ,
v E xl , v EX[ , v E X2 .
(1.44)
Comp aring (1.38) with (1.44) , we find th at the first equation is exactl y th e sa me. In a second ste p, we consider the second equation in more det ail and obtain for v = rPi
J
(u" ln, - u" ln)1/Jt do = u" ln, (Pi)
r
J
rPi do -
r
Using t he definition of 1/Jr an d t hat int erior verti ces of T
U" In, (Pi)
=
IsuP~ rPi I
Ir rPi do =
J
supp o ,
J
u" ln21/Jt da = O . (1.45)
r
0.5l supp rPi l, we obt ain for the
u" ln21/Jt da ,
1 ::;' i ::; N r ,
1.3 Dis cretiz ation Techniques Based on the Product Sp ace
47
where N r is the number of interior vertices on the non-mortar side of r . Having (1.37) in mind , (1.45) gives rise to almost the same system as t he second equa t ion in (1.38). The only difference is for i = 1 and i = N r : in these cases 'l/Jr :j:. 'l/JPi ' In the genera l mortar approach, t he analysis of t he consiste ncy error requires this modification , but only for th e vertices on t he non-mortar sides sharing one edge with a crosspoint. Fin ally, we have to compare the equa t ions for v E X 2 • St arting with the observation t hat v - g2(V) is almost equa l t o Ev , v E X 2 , and using 91(V) = 0 for v E X 2 , we find equa lity except at t he two endpoints of T of th e coup led syst ems (1.38) and (1.44) . Thus t he non-symmetric variational probl em (1.42) obtained by a local eliminat ion of t he Lagrang e multiplier is nothing else than a Dirichlet Neumann coupling of boundar y valu e problems on th e different subdomains . On t he mortar sides Neumann boundary condit ions and on t he non-mortar sides Dirichlet boundary condit ions are used . 1.3.3 Algebraic Formulations The introduction of t hese t hird mortar formul ations was motivat ed by a need to find new, mor e efficient, iterativ e solvers. The positive definite vari ati onal formul ation (1.14) gives rise t o a non conforming approach where t he const rained spaces are non-nest ed . Working with t he equivalent saddle point approac h yields an ind efinit e syste m for which efficient iterative solvers are ofte n relat ively expensive. Two special iterative solvers will be considered in Sect. 2.3 and Sect . 2.4. One is based on the symmet ric approach (1.43) and the ot her one on th e un symmet ric Diri chlet-Neum ann formul ation (1.42). We point out that t he use of du al basis functions as Lagran ge multiplier space is essent ial for the efficiency of our new iterative solvers. In t his subsection, we consider the algebra ic formul ation of t he vari ational probl ems (1.42) and (1.43) given in Subsect . 1.3.2. We show t hat t he corresponding st iffness matrices can easily be obtained by means of those of t he un constrain ed product space.
BBB B B B
.
o
Set 0
(interior nodes)
Set 1
(mortar side)
Set 3
(non-mortar side)
Set 2
(crosspoints)
Fig. 1.20. Decomposition of the nodes into sets in 2D
We start with a split t ing of the degrees of freedom into different sets ; see Fig. 1.20. All nod es in the int erior of a subdom ain have ind ex zero. T he nodes
48
1. Discretization Techniques Based on Domain Decomposition
on t he interior of th e mor t ar sides have ind ex one . The node s on the int erior of the non-m ortar sides have ind ex zero or three; each nod e corre sponding t o a basis function of WO ;h~ (')'m) has ind ex three, all other nodes have ind ex zero. In t he special case th at WO;h ~ bm) = WO ;h~ bm ), all nodes on the int erior of t he non-mort ar sides have ind ex three, this is t he situation illustrat ed in Fig. 1.20. All nod es on the boundary of the mortar and non-mort ar sides have ind ex two . In 2D, t hese are the vertices of th e triangulations coinciding geomet rically with a crosspoint. There are mor e nod es of this typ e than crosspoints, since each crosspoint p is associated with n p nod es, where n p is the number of subdom ains sharing the crosspoint p. In 3D , th ese are the nodes on the wirebasket of the decomposition, i.e., on U~=l fh m\ see Fig. 1.21. In the right of Fig. 1.21, the sit ua tion of four subdomain s shar ing one edge of the wirebasket is illustrat ed . The dashed lines symbolize the edges of t he t riangulations of the different subdomain s.
an;
..' , :
---_.- -----.......:: =.. i·····
..... :
/1"»tl l
>
--- -- f ~ ~ :::;·..:s:: :-·
------
Fig. 1.21. Wirebasket (left) and detail (right)
T he vari ational problems (1.14), (1.15) , (1.42) , and (1.43) are equivalent, and, in addit ion, the Lagran ge multiplier Ah can be obtain ed from Uh by local postprocessing. However, th e choice of the underlying mortar formu lation will make a big difference for t he it erati ve solut ion. Here, we only remark t hat t he natural setting for man y efficient iterative solvers such as preconditioned conjugate gradient metho ds or multigrid methods is a symmet ric positive definit e problem based on nest ed discret e spaces. In th e case t hat one of these condit ions is violated , standa rd t echniques, very ofte n, have to be modified . The new formulation (1.43) appears t o offer a real advantage since it results in a symmet ric positive definit e problem associate d with nest ed spaces. Formulations (1.14) and (1.43) give rise t o symm etric and positive definit e problems. However in t he case of (1.14) , the discret e const rai ned spaces are non conforming and, in general, non-nest ed . The un constrain ed product spaces are nest ed , and (1.15), (1.42), and (1.43) are associated with th ese spaces. But only (1.43) yields a symmet ric and positiv e definit e system . The saddle point formulat ion (1.15) yields a symmet ric but ind efinite system , and (1.42) gives rise to a non symm etric formul ation. In t his subsection, we discuss some algebra ic aspects of (1.42) and (1.43). In par t icular , it is shown how we find (1.42) and (1.43) from (1.15) by 10-
1.3 Discr etiz ation Techniques Bas ed on the Product Space
49
cal elimination. We consider the corresponding algebraic system, first for the saddle point formulation. Using nodal basis functions and grouping the vee. diIces UT T ) , were h T := (T T) an d tors accor dimg to t hei eir in UR, UN UR Uo, U T 1 ' U2 h = (T UN := U3, (1.15) results in the following indefinite system
G:)
U~) - (0
(1.46)
All indices related to the meshsize ar e suppressed for the stiffn ess matrices and subvectors. In addition, the same notation is used for functions in the discrete spaces and their coefficient vectors with resp ect to the nodal basis fun ctions. A RR, A RN , ANR and ANN ar e the stiffness matrix associated with the bilin ear form a(', ') restricted to the different subsets, and M R and D are associated with b(·, .). We remark that M R is a block mass matrix with many zero blocks. Moreover D is a positive definite, diagonal matrix. This is an essential point, and is, as already pointed out, only true if dual basis functions for the Lagrange multipliers are used . Ex amples of dual basis functions have been discussed in Subs ect. 1.2.4. The non-symmetric formul ation (1.42), based on the product space, is obtain ed by choosing Ah according to (1.40) and setting f-l := 91(V) in (1.15) . This can be expressed in a very simple way in the algebraic setting Ah = WT(f - AUh) , f-l=WTv ,
where W is defined as
W T := (0, D- I )
Then , local elimination of Ah and f-l yields (Id , W)
( AB) ( BT 0
Id A ) Uh = (Id - BW T )f =: f _ WT
A
(1.47)
It can be easily shown that (1.47) yields ((Id - BWT)A + W BT)Uh = l, and that (1.47) is the algebraic form of (1.42) . A different possibility to obtain a reduced system for Uh is to define f-l by f-l := W T A(WB T - Id)v ,
and repl ace Ah = W T (f - AUh) by the equivalent formula
We note that the mortar solut ion satisfies BTUh = O. This choice gives rise to a symmetric positive definite system
50
1. Discretization Techniques Based on Domain Decomposition
Au" := (Id, (B W T - Id) AW ) A
(
AB) (W T A( WIdBT _ Id) ) u"
BT 0
= f
A
(1.48) T he following lemm a establishes a relation betwee n the algebraic formulation (1.48) and the vari at ional pro blem (1.43). Lemma 1.18. The algebraic fo rmulation of the variational problem (1.43) is given by Au" = j . Moreover, th e Lagrange m ult iplier can be obtaine d, once u" has been computed, by A" = W T(j - Au,, ) .
Proof. To start, we rewrit e t he linear funct ional 92 (v) in its algebra ic form
W BTv . T he multiplicat ion with W can be interpreted as a scaled map ping from M" onto X " such t hat t he result ing element of X" is supported in a sma ll st rip of width h on t he non-m ort ar side, whereas t he mult iplicat ion with B T maps each element in Vi, to zero . Using t hese Observations, t he algebra ic form of variational problem (1.43) can be written as (Id - B W T )A (Id - W BT)u" + BW T AWB T = (Id - BWT )f .
U"
A st raightforward comp utation completes t he proof by comparing t his system wit h (1.48) . 0 The matrix A can be easily assem bled from A, M := M RD- 1 , and is sparse. Using t he indices given at t he beginni ng of t his sect ion, it can be written as a 2 x 2 matrix as
A=
(
ARR + 2MA NNM T - M A NR - A RN M T ANNM T
)
(1.49)
1.4 Examples for Special Mortar Finite Element Discretizations In t his subsection , we consider severa l mort ar sit uations. Each of t hem pr ovides an int eresti ng insight into t he abstract general fra mework. The flexibility and t he wide ra nge of app lications is illustrat ed. Duality arg uments play an essential role in our first example. Essenti al an d natural boundary conditions are used to realize t he coup ling of mixed and standa rd finite elements . In our second exa mple, we conside r t he special case t hat each element is one subdomain , and thus t he number of sub domains te nds to infinity as t he meshsize te nds to zero . The set ti ngs are given for simplicial t riangulations in 2D, but can be generalized wit hout any pro blem t o 3D.
1.4 Ex amples for Special Mortar Finite Elem ent Discret izations
51
1.4.1 The Coupling of Primal and Dual Finite Elements
In this subsection, we focus on th e coupling of two different discreti zation schemes . We use a mixed finit e element discreti zation on one sub dom ain and a standard conforming one on the ot her subdomain . The coupling of mixed finite element discreti zations on non-m atching grids has been introduced and analyzed in [WY9S, AY97, Yot97) . In cont rast t o standa rd mortar methods , t he finit e element trace space of th e flux in norm al dir ection at th e int erface defines a Lagr ange multiplier space which does not yield optimal results. In par ti cular, t he consiste ncy error is to o lar ge compared wit h the best approximation error. To obt ain optimal results, t he ord er of the Lagran ge multiplier space has t o be increased by one. In t he lowest order case, the Lagrang e mult iplier space has to contain piecewise linear functions and not only the constants . We observe t hat t he trace space of HI -fun cti ons and th e Lagran ge mult iplier space reflect the du ality between H I / Z and H- I / Z on th e interface. In our approach, we use t he du alit y of t he roles of Diri chlet and Neumann boundary cond it ions in the primal and du al setting. Using t his du ality, we can work on the pr odu ct space without int rodu cing a Lagran ge multiplier. The idea of combining mixed finit e element methods with prima l ones was originally introduced in [WW9S). Recently, efficient iterative solvers were st udied in [LPV99). For simplicity, we restrict our selves t o th e case of two subdomains. Here, il c JRz is decomp osed into two nonoverlapping polyhedral subdoma ins ill and ilz , and we assume that meas(ailz nail ) > O. On ill , we use du al discret ization techniques, whereas st andard primal approaches are used on ilz . The coupling at t he inte rface T := ail l n ailz is realized without a weak cont inuity conditi on between t he spaces and without a Lagran ge multiplier enforcing the orthogonality of t he jump. In st ead we t ake the flux of the solut ion on ill to define a Neuma nn boundary condit ion on T for t he boundary valu e probl em on ilz , and t he solut ion on ilz as a Diri chlet boundar y condition on T for t he bo unda ry valu e pr oblem on ill ; see Fig. 1.22. We show t hat pro ceedin g in this way yields a suit abl e coupling between t he two doma ins. The global discreti zation error can be bounded by the sum of t he local best ap pr oximation errors, and optimal a priori bounds can be established. flux
r>.
mixed FE
primal FE
Dirichlet problem U
Neumann problem trace
Fig. 1.22. Coupling betw een du al an d pri mal finit e eleme nts
52
1. Discretization Techniques Based on Dom ain Decomposition
We introduce simplicial triangulations Th, and Th 2 on the subdomains D 1 and D 2 which do not have to match at the interface. The sets of corresponding edges are called [h , and [h 2 • On D2 , where primal t echniques are used, we work with standa rd conforming P n 2 -element s, Xh2;n2 C HJ;r2 (D 2 ) . Her e, HaIr (D 2 ) is a sub space of H 1(D2 ) satisfying homogeneous Dirichlet , 2 boundary conditions on := 8D n 8D2 · The cont inuous space on D 1 is given by H(div; D 1) x L 2(Dd , where the vector valu ed Hilb ert space H (div ; Dd is defined by H (div ; Dd := {q E (L 2(D1))21 div q E L 2(D)} . For the discretization, we use Ravi art-Thomas finit e element s
n
RTh, ;n, := {q E H(div; D 1) I qlT E (Pn, (T)) 2 + r.; (T)x, T E Th,} , of order n1 2: 0 for the flux; see [RT77], and the space of piecewise polynomials Wh, ;n, := {v E L 2(Dd 1 v lT E Pn,(T), T E Th,} of the same order , for the primal variabl e in D 1. We refer to [BF91] for an overvi ew of mixed finit e element s. We remark that no boundary condit ion has t o be imposed on the int erface T. In cont rast to the standard case, Diri chlet boundary condit ions ar e natural boundary condit ions for the mix ed formul ation, i.e., t hey appear in the definition of the right hand side of the vari ational problem but are not enforced in the const ruct ion of the spaces. For the st andard prim al approach, t he sit uation is exactly reversed . The Neumann boundary conditions ar e the natural ones and t he Diri chlet boundary condit ions are imposed on t he space. This du ality allows t he coupling between Raviart- Thomas and conforming finit e eleme nts without Lagr an ge multiplier on t he interface . Using this Diri chletNeumann coupling, we find t he following discr et e vari ational problem : Find (jh' , Uh2' Uh,) E RTh,;n, x X h2;n2 x Wh, ;n, such that
a1(h"qh) - d(qh ,Uh2) d(jh" Vh)
+ b(qh ,Uh,)
+ a2(Uh 2' Vh)
= 0,
qh E RTh,;n, ,
= (1, Vh)O; il2'
Vh E Xh 2;n2 ,
- C(Uh, , Wh) = -(1, Wh)O; il" Wh E Wh,;n, . (1.50)
b(jh" Wh)
Here, t he bilinear forms ai(-, '), i = 1,2 , b(·, '), c(·, ·) and d(·,·) are defined by a1(P 1, q d: = J a- 1p 1 ' q1 dx, P1 , q1 E H(div;D1) , il, a2(w2, V2) := J (aV'v2 V'W2 + bV2W2) dx , V2, W2 E H 1(D ) ,
b(q1,vd := c(w1, vd
:=
il2
J divqj V1 dx , J bW1 V1 dx ,
il, il,
d(q1 ,V2 ) := (qj n, V2),
2
V1 E L 2(D1), q1 E H(div; Dd , V1, W1 E L 2(D1) , q1 E H(div; Dd , V2 E H 1(D2) ,
where (". ) stands for the du ality pairing of H- 1 / 2(T) and H 1 / 2 (T) . The bilinear forms a1(., .), b(·, ') , and c(', ·) ar e associate d with the du al approach on
1.4 Examples for Special Mort ar Finit e Element Discretizations
53
.01 whereas a2(',') is a standard H6;r2(.o2)-elliptic bilin ear form. The du ality of t he boundar y condit ions on r is realized by mean s of d(·, -). Mor eover , d(·, ·) t ransfers the boundary condit ions between the subdomains. Con sidering (1. 50) in more det ail , we find that it has t he followin g saddle point st ructure ) = ( BAT -B0 ) (/l'h U hl
(h) , h
fLh :=
(jhl) , U h2
where the operato rs B and 0 are associate d with the corresponding bilinear form s. The op erator A is non symmetric and has the form
AI - D ) A:= ( D A 2
'
where the op er ators AI , A 2 and D are associate d with t he corre sponding bilinear form s. The right hand side is given by t he linear form (1, ' ) 0; [./; , i = 1, 2. The kernel of the cont inuous op erat or B'[ : H(div; .od x H6.r (.02 ) ---+ , 2 2 L (.od , whi ch is associated with t he linear form b(·, vd, is KerB'[ := {(ql , V2) E H(div ;.od x H6;r2(.o2 ) I divq, = O} . It is well known; see, e.g., [BF91], t hat we have t he following equivalence : For q l E RThl ;n l ' divq, = 0 if and only if b(ql ' vd = 0, VI E Wh l ;n l . Thus the kernel of the discret e op erator B T : RTh l ;n l X X h 2 ;n 2 ---+ Wh l ;nl is a subs pace of KerB'[We fur t her introduce t he non symmet ric bilinear form a(a, T) := a2(w2,V2 ) + d(P l, V2) + al(Pl , ql ) - d(ql, W2), where a := (ql , V2) and T:= (P l , W2) are elements of the product space H(div; .od x H6;r 2(.o2 ) . The norm II· II on t his product sp ace is inh erited by the pro du ct topology and is defined by IIal1 2 := IIqlI1 6;[./1 + Ildivq lI1 6;[./1 + Il v21I r;[./2· It is now easy to see that t he nonsymmetric bilinear form a( ·, ·) is coercive on KerB'[
Here, we have used t hat a2(' , -)l / 2, rest rict ed to H6;r2(.02), is equivalent to t he H I-norm on .02 and t hat divqj = O. An essent ial tool in establishing a pri ori bounds for t he discretization err or is pr ovid ed by the ab st ract sa ddle point t heory. A suitable inf-sup condition guarantees t hat t he discreti zation error can be bounded in t erms of t he b est approximation err or. Here, t he relevant inf-sup condit ion is nothing else than the standar d one for t he mixed finit e element scheme on .01
Choosin g a as (q l ' 0) , q l E R T h l ;nll this is a standard result for Raviar tThom as finit e element s, and it is sat isfied with a constant ind ep endent of t he
54
1. Discretization Techniques Based on Domain Decomposition
meshsiz e without any further assumpt ions; see, e.g., [BF91]. In addit ion, we obtain unique solvability of the saddle point problem (1.50) by t aking th e continuity oft he bilinear forms, t he inf-sup condition, and th e coercivity of a(·,·) on K erE T , into account; see, e.g., [BF91, WW98] . Following [BF91], an opti ma l a priori bound for the discretiz ation erro r can be established by mean s of t he best approximat ion error. Since we are workin g with un constrained standard finit e element spaces , t he approximation properties are well known ; see [BF91]. With no matching condit ion imposed on th e discret e spaces at th e int erface, the analysis as well as t he impl ement ation of t he method becomes quite simple. A priori est ima tes of the ord er of min(nl + 1, n 2) are also obtained
Ilj - h 111 ~iv; .o1 + lIu -
Uh 1116;.o1
+ Ilu - Uh211r;.o2
::; c (h~(n1 +1) (lul;'1+1 ;.01 + UI;'1+1 ;.01 + [div j 1;'1+1;.oJ + h~n2Iul ;'2+1 ; .o2) (1.51) if the problem has a regular enough solution. Here, j denotes the cont inuous flux defined by j := a\1u . n. Remark 1.19. Th e coupling of m ixed finit e eleme nts on non-matching triangulations has been work ed out in [WY98, A Y97, Yot 97]. A s in th e standard conforming setting , one has to work with Lagrang e multipliers or equivalently a suit able cons traine d space. In the lowest order case, th e Lagrange multiplier space has to be at least piecewis e lin ear, since the optimality of th e me thod is oth erwise lost.
In t he next subsect ion , we show t hat the saddle point problem (1.50) can be rewritten . Introducing a piecewise constant Lagr an ge multiplier on t he interface, we obtain a mort ar finit e element method which couples conforming Lagran gian finit e elements and nonconforming Crouzeix-Raviar t elements. The special characteristic of t his approach is t hat t he piecewise constant Lagran ge multiplier gives rise to a diagon al mass matrix on t he non-mortar sides . In Sub sect . 2.3, we discuss how standa rd multigrid techniques can be applied in such special mor tar sit uat ions . 1.4.2 An Equivalent Nonconforming Formulation It is well known that mixed and nonconforming finit e element methods are
equivalent; see, e.g ., [AB85, BF91]. Introducing Lagr an ge multipliers on the edges of t he t riangulation, th e flux vari able as well as th e primal vari abl e can be evalua t ed locally. The resulting Schur complement syste m is th e same as for t he positive definit e vari ational problem associated with a non st andard non conforming Crouzeix-Raviart discretiz ation; see [BF91]. Furthermore, th e mixed finit e element solut ion can be obtain ed from t he non conforming by local postprocessing. Here, we pr esent a mor t ar coupling of conforming and
1.4 Examples for Special Mortar Finite Element Discreti zati ons
55
nonconforming finite elements which is equivalent to t he present ed primal du al coupling . In t he rest of this sect ion, we restrict our selves to t he lowest order Raviar tThomas space, n l = O. The dimension of t he local flux space is t hree, and the local space for t he discret e primal vari able is one dimension al. An equivalence with Crouzeix- Raviart elements can be obtained if we enrich the nonconforming space by local cubic bubble fun ctions. We t herefore consider t he enr iched Crouzeix-Ra viart space
where C R h , is t he Crouz eix-Raviart space of piecewise linear functions which are cont inuous at t he midp oint s of t he t riangulation '!h I and equa l to zero at t he midpoints of any boundary edge e E Chi naD. B is t he space of piecewise cubic bubble functions which vanish on the boundary of t he elements . Restricting th e vari ati onal problem (1.50), for t he moment , to D1 by setting V h E X h2 ;n 2 equa l zero, we get
'q
a d j"ll q h)
+
b (qh ,uh ,)
bO h " Wh )
-
C( Uh l, W h)
= d(qh , Uh2) ' = -(J, W h) O;D I
qh Wh
,
E RTh , ;o E Wh l ;o
, .
(1.52)
If t he solution U h2 is known, t hen t he Dirichlet boundary value problem (1.52) for the mixed formulation on D 1 can be solved. Compa ring (1.50) and (1.52) , we find t hat both vari ati onal problems have a saddle point st ructure . Introducing a Lagran ge multipli er and eliminati ng t he vari ables j,,1 and v«, in (1.52) yield a symmet ric positiv e definite variationa l problem. The boundar y condition now ente rs into t he definiti on of t he trial space . Let N Cg;h" 9 E L 2 (r ), be t he linear manifold of N Ch , defined by
NCg;h, := {'l/Jh E NCh,
I
J
'l/Jh da =
e
J
q da, e E Chi
n r} .
e
Then , t he solut ion of t he saddl e point problem (1.52) can be obtained , equivalentl y, by solving: Find lJIh l E N CUh2 ;h l such t hat (1.53) where aNc( ¢h,'l/Jh) := L:T EThl I T P a- I (a\l ¢h)\l'I/Jh + bIIo¢h IIo'I/Jh dx. Here, IIo is t he L 2-p ro jection ont o W hl ;O, and P a- I is the weighted L 2-pro jection , with weight a-I, ont o t he local Rav iart- T homas space of lowest orde r which has t hree degrees of freedom per element; see [AB85, BF91]. We remark that t he solut ion space NCUh2 ;h l for the solut ion in D1 depends on t he solut ion in D2 , but that t he right hand side in (1.53) does not depend on t he solution. Here aga in, t he different roles of the boundary condit ions come into play. Using t he equivalence of (1.52) and (1.53) in (1.50), we find a globa l variational problem in D: Fi nd (lJIh ll U h2) E N CUh2 ;h l x X h2 ;n 2 such th at
56
1. Discreti zat ion Techniques Based on Domain Decomposition
The impl ementation of (1.54) is based on a different formul ation. Considering (1.54) , in more det ail, it can be seen t hat it is a minimization problem with constraints, where the const raints depend on th e solut ion. Introducing a Lagran ge multiplier space, we can transform (1.54) into a saddle poin t problem . To do so, we show in a first ste p that t he Dirichl et problem (1.53) can be extended t o a vari ational problem on the whole space NChl . Using that j hl = Pa- l(a'VlJfhl) and v«, = IIolJfhl and applying Gr een 's formul a to t he second equation of t he saddle point pr oblem (1.52), we obt ain
aNC(lJfhll '1/;h) - d(Pa- 1(a'VlJfh l) , '1/;h) = (j , IIO'l/;h)O;f2} ,
'l/Jh E NChl
(1.55)
From t he definition of NCg ;hl' we find that an element X E N C hl is in NCg ;hl if and only if
J
J..l (X - g) da = 0,
J..l E Mh l ,
(1.56)
r
where M hl := {J..l E L 2(T ) I J..ll e E Po(e) , e E [ hI n T} . In particular, M hl is used as a Lagran ge multiplier space. T he dim ension of M hl is equa l to th e number of edges in [ hI n T. Finally, we obt ain a mor t ar coupling between conforming and non conforming finit e element s by means of piecewise constant Lagrange multipliers. Theorem 1.20. Let (lJfhl , Uh2) E NCUh2;hl x X h2;n2 be the solution of (1.54)·
Then, UM := (lJfhl ,Uh2 ) and AM := Pa- I(a'VlJfhl )·nlr is the unique solution of the following saddle point problem: Find (U M' AM) E (N C hl X X h2;n2) X M hl such that a(UM, v) - c!(AM,v ) = j(v) ,
v E N C hl x X h2;n2 ,
c!(J..l,UM)
J..l E M hl .
= 0,
(1.57)
Here, t he bilin ear and linear forms are given by, v := (VI,V2), w := (WI ,W2):
a(w , v ) := a2(w2, V2 ) + aNc( WI,VI), v, W E N Chl x X h2;n2 , d(J..l , v ) :=
I J..l(VI -
r
V2 ) da ,
J..lEM hl ,
j(v) := (j , V2)O;.l?2 + (j, IIovdO;.l?1 Proof. The assertion is an easy consequence of (1.54) by usin g (1.55) and (1.56). 0 Theorem 1.20 states t he equivalence of (1.50) and (1.57) in t he case n l = O. We obtain t he solut ion of (1.50) by a local postprocessing from th e solut ion
1.4 Exam ples for Sp ecial Mortar Finite Element Discreti zation s
57
of (1.57), from the formul as h I = Pa- l (a'Vu MIQl) ' v« , = lIouMIQl and Uh2 = UM!Q2' The a priori bound (1.51) and Theorem 1.20 guar antee t hat t he discretiz ation error of U-UM in th e energy norm is of ord er h . Comp aring (1.57) with (1.15), we find t hat (1.57) repr esents th e saddl e point formul ation of a mortar coupling . We recall t ha t t he saddl e point problem (1.50) realizes th e coupling of du al and primal finite elements methods. Using th e equivalent non conforming approach instead of t he mixed method, we ar rive at a mortar finite element method expressing th e coupling of conforming and nonconform ing spaces. The ana lysis of the resulting discrete problem could be also don e within t he abst rac t mortar framework. The piecewise constant Lagrange multiplier AM = h I .nlr is associate d with the side of the nonconforming discretization , and the mass matrix on t he non-mortar side is diagonal. We find that the mortar side is associate d with the conforming discretization and th e nonmortar side with th e nonconforming discreti zati on. Figur e 1.23 illustrates th e relation between th e different couplings.
c Mixed:5~
= Pa- l (a'V1[thl ) Uhl = IIo1[thl
j hl
IT
La" .., .
<,
~
DBC
Uh2 Uh2
~
DB C
Non conform ing non-mort ar
t\£'\'.
'£,e CO
NBC Conforming mortar NBC
'\ece'll~
_V
_
)\M
Fig. 1.23. Equivalence between primal du al coupling and mort ar coupling
Remark 1.21. For th e implem ent ation, we will eliminate th e cu bic bubble fun ctions in (1.57) locally; see, e.g., [AB85, BF91j. W e th en obta in th e stan dard vari ati on al probl em fo r Crou zeix-Raviart elemen ts, where th e right hand si de f is repla ced by lIo f in th e specia l case of b = 0 and a piecewise const an t diffusion coefficien t a . Th e soluti on lJFhl of th e n onc onf orming probl em is give n by lJFhllr = Uhli r
+
5 12
L h;; lIofl r(A1 A2A3) ' 3
T E 1/,.1 ,
i= l
where Ai , 1 :::; i :::; 3, are th e bary centric coordin ate fun ctions, and he; is th e length of th e edge e: C 8T, 1 :::; i :::; 3. H ere, Uh l stands for th e CrouzeixRaviart part of th e mo rtar- fi nite elemen t soluti on of (1. 57) .
58
1. Discr etization Techniques Based on Domain Decomposition
1.4.3 Crouzeix-Raviart Finite Elements A lar ge class of nonconforming methods can be analyzed within the mortar framework, here we pr esent the well known Crouz eix-Raviart element s in the cont ext of mortar finit e elements; see [CR74]. Reformulating st andard nonc onforming finite element s on matching t riangulat ions as mortar finit e elements can also be carried over to rotated bilinear elements on quadrilateral trian gulations and to 3D. Previou sly, we have decomposed the dom ain f? into a fixed number of subdomains f?k' and we have impli citly assumed th at the number K of subdomains is sma ll compared to th e number of elements N h of the global t riangulat ion Til . In particular , within a multil evel approach the ratio between th e number of subdomains and elements KINh t ends to zero with h -7 O. However , neither th e number of sub domains nor th e ratio KINh ente r into t he constants of th e a priori bounds and the inf-sup condition. To rewrite the Crouzeix-Raviart elements as a mor tar method, we have to consider the extreme case th at the decomposition of f? is given by the finit e element t riangulation and that th e number of subdomains t ends to infinity as the discreti zation par am et er of Tit tends to zero. Here, we assume that Tit is a shape regul ar famil y of globally conforming simplicial triangulations. The set of edges is denoted by Gh . For each int erior edge e = aTi naTo, we have to fix t he orient ation ofthe normal dire ction n e . We then define To such that n , is its outer normal vector, and T; as the adjacent element; see Fig. 1.24. In the case t hat e C of?, we choose th e outer normal of of? as n e . e
Fig. 1.24. Ori entation of t he normal vector n ;
St arting with th e following decompo sition of f? f?-
UT
,
T E7i,
and pro ceeding as in Sect . 1.1, we find that t he int erior edges of the t riangulat ion are the int erfaces. We also include the boundary edges of the t riangulati on into the set of int erfaces. On each subdomain T, we choose t he discret e space of linear funct ions Pl (T) and define th e un constrain ed product space by Xh :=
IT T ET h
r, (T)
(1.58)
1.4 Exa mples for Special Mortar Finite Element Discretiz ations
59
In Sect . 1.1, we have seen that t he bilin ear form a(·,·) a(v ,w) :=
L
!
«o« . \1w + bv w dx ,
TE Ti'T
is uniform ellipt ic on a suitable subspace of TITETh H I (T) X TI TETi, HI (T) . To have ellipticity it is sufficient to guar ant ee that the mean valu e of the jumps van ishes across t he edges . The non conforming space Vh is thus defined as a subspace of Xh satisfying mat chin g conditions at the int erfaces (1.59)
on,
Here, [v] := vITo - vIT; for an inn er edge, while for an edge e c it is defined as the trace. From the definition of Vh, it follows that Vh c Y and t hus t he bilin ear form a(·,·) is uniformly elliptic on Vh x Vh , which gua rantees t he unique solvability of t he symmet ric vari ational problem: Find U h E Vh such that (1.60) The definition (1.59) is equivalent to (1.7) with the special local Lagran ge multiplier space
M(e) := Po(e) , and the global one is given by M h := TI eE£h M( e). We have modifi ed t he definition of Vi, in one resp ect , in comparison t o Sect . 1.1. The boundary of n is now part of t he union of the int erfaces and t hus, t he homogeneous Dirichlet boundary condit ion is sat isfied only in the weak form
!
v do = 0,
eE
on,
v E Vh
.
e
The const raints at t he int erface for v E Vi,
![V]dlJ" = 0,
e E Eh ,
e
force a piecewise linear fun ction to be cont inuous at t he midpoint m e of an interior edge e and t o equa l zero at the midpoint of a boundar y edge e con. Recallin g t he definition of Cr ouzeix-Raviart element s of lowest order CRh := { v E L 2 (n)\ vl T E PI (T) , T E Ti" v(me)ITo = 0 , e E Eh, e vIT; (m e) = vITJme), e E Eh, e C n} ,
c on ,
60
1. Discr etization Techniques Based on Domain Decomposition
we find that Vh = CRh. Thus, the positive definit e nonconforming variat ional prob lem for the Crouzeix-Raviart element s is exa ct ly th e sam e as the non conforming formul ation of t he mor t ar method associated with th e sp ecial decomposition (1.58) and (1.59) . In cont ras t to t he mor e general case of Sect. 1.1, the nodal basis functions of Vh have local support in exact ly two elements. In the rest of t his subsection, we introduce and analyze a simplified a post eriori error est imat or. Th e st ar ting poin t for the const ruc t ion of this err or est ima t or is the corresponding saddle point problem: Find (Uh ' Ah) E X h xMh such that (1.61) where b(w, J.1) := L eE£h Ie[w]J.1 da . Observing that the global Lagr an ge mult iplier space M h is nothing but the product space of one dimensional space s, the unique solvability can be obt ain ed without explicit ly considering an infsup condit ion. Let Uh E Vh be the solution of (1.60). Then , t he Lagr an ge multiplier is given locally by (1.62) wher e vea: and Ve;To E X h ar e local basis functions satisfying IeVe;Ti da = IeVe;To do = <See, supp Ve ;Ti = T i and supp Ve;To = To. It can be easily verified th at t he Lagr ange multiplier is well defined and that (Uh ' Ah) satisfi es (1.61) . Reliable and efficient a pos t eriori erro r est imators are considered in [CJ97, CJ98, DDPV96, Woh99c] for Crouzeix-Raviar t elements. In cont ra st to st andard conforming approaches , local a post eriori err or est imat ors have to include an addit ional t erm measuring th e discontinuity of the finit e element soluti on. Basically, the definitions of the local cont ribut ions are t he sam e and can be characterized by
7J~' := L (~e II[Uh]116;e+ hewell[a::h]116;e) + h}llild eCaT
e
e
bUh116;T , (1.63)
where We , We > 0 are suitable weighting factors which do not depend on the meshsize but, in genera l, on t he coefficients. The L 2-p rojection i ll onto piecewise linear fun ctions can be replaced by t he one ont o piecewise constants. T he difference between them can be bounded by h}lIf - ilofl16 'T which is a higher order t erm compared with the ot her components of t he error est imat or, and it can t herefore be neglect ed . We can now show that a simplified a post eriori erro r est imat or can be obtained. The starting poin t of th e const ruc t ion, is th e explicit representati on of t he Lagran ge multiplier given in (1.62) . This simplified residu al based
1.5 Numerical Results
61
error est imator for Crouzeix-Raviart finit e elements was first introduced in [Woh99c]. By consideri ng t he saddle point problem (1.61) , it can be shown h th at t he t erm ]llo;e is redundant. Appl ying Green's formul a on (1.62) , e the Lagrange multiplier can be obtain ed in te rms of a8uh / 8ne and t he elementwise residual. The explicit form of th e Lagran ge multiplier yields
Il[a::
Since [a ~~: ] is constan t on each edge, we find an upp er bound for the second te rm in (1.63) (1.64) These pr elimin ar y considerations motivat e th e introduction of our simplified erro r esti mat or. The local cont ribut ions fjT are defined by fj} :=
L
eC 8 T
~e II[Uh]116;e+ h}lllId - bUh116;T . e
The following lemma shows a quasi-local equivalence between the two a post eriori err or est imat ors . Lemma 1.22. There exists a constant such that
irr ~ TJT ~ C (fjT +
L
fjTJ
eC 8T fJT e n aT = e
P roof. The proof is an easy consequence of th e definition of t he local contribut ion of t he err or est ima t ors and (1.64). D The evalua t ion of t he local cont ribut ions itr within an adapt ive multilevel scheme is extremely simple. Only one contribut ion per element and one per edge have t o be compute d. In par ticul ar , no linear syste m of equat ions has to be solved. Num erical results for this simplified error est imat or can be found in t he next sect ion.
1.5 Numerical Results In t his secti on , we collect and describ e the results from different series of num erical experiments with several mortar settings in 2D. The methods are tested on different types of examples, and t he num erical results confirm the theory. Throughout t his work , the implementation is based on t he finit e element t oolbox ug, [BBJ+97] . To show the adva ntages of mortar methods, we pr esent some exa mples which illustrate the flexibility and efficiency of t hese dom ain decomp osition
62
1. Discr eti zation Techniques Based on Dom ain Decomposition
te chniques . A rot ating geometry, sub dom ain s with reentrant corners, and a region with an ext remely bad aspect ratio are discussed. These examples are borrowed from [WW99]. For mor e det ails and further examples, we refer to [WW98 , WW99] .
Fig. 1.25. The flux of t he solut ion on a rot ating geometry
We st art with a t ime-dependent problem. For st andard conforming discretizat ion schemes, remeshin g is very often required afte r each time-st ep. In particular in 3D, this can be very expensive and inefficient . Using mortar techniques, frequ ent remeshin g can be avoided, even in t he case of a rotating geomet ry and for an arbit rary ratio between th e meshsize and t he time-st ep. In this example, th e dom ain [l is not simply connecte d, and is decomposed into two subdom ains , where the int erior one rotat es with a fixed angular speed; see Fig. 1.25. The int erface between t he int erior and the outer sub domain is a circle, and we use discontinuous piecewise constant Lagr an ge multipliers. At t he outer boundar y, homo geneous Dirichlet condit ions are given whereas at t he inn er homogeneous Neum ann condit ion are assumed. The elliptic pr oblem und er consideration is the Pois son problem with a constant right hand side . The second exa mple shows an application of the coupling of th e primal and du al tec hniques as pr esent ed in Sect . 1.4.1. Th e impl ementation is based on the equivalent non conforming approac h given in Subsect. 1.4.2. We consider a simple flow probl em in het erogeneous media , mod eled by Dar cy's law for a pr essure pot enti al u. and t he flow a\lu . The dom ain is decomposed into several subdoma ins, where [ll represent s a simply connecte d polygonal channel region ; see Fig. 1.26. On [ll , we use non conforming finit e elements; t he ot her subdom ains are discret ized by st andard conforming finit e elements of lowest order. We choose the diffusion par amet ers al := 0.001 in [ll , and a2 := 1 elsewhere . Inflow and out flow bounda ry condit ions for th e ent rance and exit of the channel region are used and homo geneous Neuma nn boundary condit ions elsewhere . On [ll a finer mesh is used , and t he triangulations do not match at t he int erfaces. This example has been inspired by t he cover figure of [BS94].
1.5 Numerical Results
63
Fig. 1.26. Initial mesh for the channel domain (left) and flux (right)
In the next example, we apply the mortar technique to a linear elasticity problem. The Laplace operator is replaced by the following variational problem: Find Uh E Vi, such that
where the Lame constants P'k, Ak depend on the subdomains. The usual notations are used, and Vi, is a suitable constrained space in the sense of mortar techniques. We consider a composite material constructed from large bricks of hard steel [li, 2 :::; i :::; K, joined with thin layers of a less hard material [l1; see Fig. 1.27. The following parameters are used: Ak = 110743, !J>k = 80193, 2 :::; k :::; K, and Al = 135671, !J>1 = 67837. Our choice of this geometry has been inspired by the picture of bricks and mortar often used in introductions to mortar finite element methods. I I
I I
II
I
I
II
Fig. 1.27. Deformation of the composite (left) and zoom of a thin layer (right)
64
1. Discreti zation Techniques Based on Dom ain Decomposit ion
In Sub sect. 1.5.1, we consider t he discreti zat ion err ors in t he energy norm and t he L 2 -nor m for the mor t ar methods describ ed in Sect. 1.2. We compare t he four different Lagr an ge multipli er space s introduced in Subsect. 1.2.4. Sub sect ion 1.5.2 concerns th e num erical performance of a non-optimal mortar method , for which th e best approximation property of the const rained space is violat ed . T he a pri ori est ima tes in t he energy norm are only of order h 1 / 2 . Nevert heless, we find t hat asymp t otic ally t he discretiz ation err ors are almost the sa me as for the other mor t ar methods. In Subsect . 1.5.3 , the influence of the choice of t he non-mort ar side is considered. Uniform and adaptive refinement techniques are used to illustrate t he importan ce of a suitable choice of t he mortar side in t he case of discontinuous coefficient s and uniform refinement techniques. A post eriori error est imators which include the weight ed L 2 -nor m of t he jump on th e non-mort ar sides generate for both sit uations optimal trian gulati ons. The adapt ive refinement in the neighb orhood of t he int erface depend s strongly on the choice of t he mort ar sides. Finally in Sub sect . 1.5.4, we study t he influence of t he jump in t he coefficients on t he ada pt ive refinement pro cess. An essent ial tool for any efficient num erical solut ion pro cess is also a good iterative solver. In t his section, we do not discuss an d ana lyze these mat t ers, bu t not e th at different efficient solvers are developed in Chap . 2. Here, we remark only t hat well known methods like multi grid t echniques, iterative substructur ing methods , and hierar chical basis pr econditioners have been exte nded to the mor t ar set t ing. In the experiments reported here, we have used a multigrid method for saddle poin t problems as our iterative solver . This approac h has been applied in [WW98 , WW99], and fur ther analyzed in [BD98, BDW99, BraOl]. 1.5.1 Influence of the Lagrange Multiplier Spaces
We now present some num eri cal results in 2D illustrating t he discreti zation errors for different mortar set t ings. We recall t hat in t he standa rd mort ar approac h t he Lagran ge mult iplier space M}~ is used , whereas alte rnat ive Lagrange multiplier spaces M~ , 2 ::::; l ::::; 4, are proposed in Subsect . 1.2.4. Using t he same not at ions as before, uh denot es the mortar finit e element solution associate d with t he Lagran ge multiplier space M~. We recall that t he unconstrain ed product space is t he same for all four mortar solut ions. The definitions of M}~ and Mt are based on du al basis functions. Furthermore, t he elements in M~ and MK are piecewise linear , whereas t hose in Ml. and M t are piecewise constant. Pi ecewise linear conforming finit e elements are used on each subdomain. A comparison of th e discreti zati on err ors of u~ and can also be found in [WohOOa] for Examples 1-3. In th is subsection, we use only uniform refinement . St ar ting with an initial t riangulation To , t he t riangulation Ti on level l is obtained by uniform refinement of Ti- l ; each element of Ti-l is decompos ed into four congruent sube lements.
uy,
1.5 Numerical Results
65
The discretizations discussed in Sect. 1.2 are compared for the following four examples. Example 1 is given by: - Liu = f on (0,1)2, where the right hand side f and the Dirichlet boundary conditions are chosen so that the exact solution is (exp( -500s) -1) (exp( -500t) -1) (exp( -500yy) -1) (1- 3r)2. Here, S := (x - 1/3)2, t := (x - 2/3)2, xx := (x - 1/2)2 ,yy := (y - 1/2)2 and r := xx + yy. The isolines of the solution and the initial triangulation are given in Fig. 1.28.
Fig. 1.28. Decomposition into 9 sub domains and initial triangulation (left) and isolines of the solution (right), (Example 1)
The domain is decomposed into nine subdomains, defined by D i j := ((i1)/3, i/3) x ((j -1)/3,j /3) , 1 ::; i,j ::; 3, and the triangulations do not match at the interfaces. We observe two different situations at the interfaces; the isolines of the solution are almost parallel to (J[ln n 8D 1 2 whereas at 8D n n 8D 2 1 the angle between the isolines and the interface is bounded away from zero on a large part of the interface. Where the isolines are orthogonal on the interface, the flux vanishes. We recall that the discrete Lagrange multiplier is an approximation of the flux. Table 1.1. Discretization errors in the L 2-norm, (Example 1)
~ # elem I Ilu - u~ 110 I Ilv. - v.~ 110 I Ilv - n~ 110 I lin - ·ltt 110 2.021163e-0 2.021306e-0 2.021196c-0 2.02129ge-0 72 0 1.017372e-1 1.014502e-1 1.014460e-1 1.013067e-1 1 288 1.166495e-1 1.166435e-1 1.166459c-1 1.166458e-1 1152 2 9.482530e-3 9.476176e-3 9.478390e-3 9.476248e-3 4608 3 2.802710e-3 2.79780ge-3 2.800444e-3 2.797812e-3 4 18432 7.130523e-4 7.121334e-4 7.126101e-4 7.121334e-4 5 73728 294912 1.789436e-4 1.788082e-4 1.788774e-4 1.788080e-4 6
In Table 1.1 and Table 1.2, the discretization errors are given in the £2_ norm and in the energy norm, respectively. The columns are ordered in the following way: Columns 3 and 4 show the results for the mortar solutions
66
1. Discreti zati on Techniques Based on Domain Decomposition
where t he Lagran ge multipli er spaces are based on piecewise linear functions. In Columns 5 and 6, we find t he results in th e case of piecewise constant Lagran ge multiplier spaces . Furthermore Columns 4 and 6 correspond to t he choice of a bior thogonal basis. The observed asympto tic rates confirm the theory. We find t hat t he energy error is of order h whereas th e erro r in the L 2 -nor m is of order h 2 • T here is no significant difference in the accuracy between t he different mortar algorit hms on any level neither in t he L 2 -nor m nor in t he energy norm. On level 6, t he difference in th e accuracy between t he best u~ and worst u~ mortar solution is less t han 0.08% in th e L 2 -nor m and less t ha n 0.04% in the energy norm . The influence of the choice of the Lagran ge multipli er space on th e accuracy of t he solution is negligible. Table 1.2. Discreti zation errors in t he energy norm , (Example 1)
~ # elem . 0 1 2 3 4 5 6
72 288 1152 4608 18432 73728 284912
I Il u-uklll
1.147900e+1 3.042101e-0 1.945246e-0 1.114075e-0 5.928275e-1 2.981975e-1 1.492382e-1
Il u - uxlll
Illu- uUl
Il u - uUl
1.147984e+1 3.034778e-0 1.946163e-0 1.113506e-0 5.923121e-1 2.98015ge-1 1.491841e-1
1.147918e+1 3.036952e-0 1.945328e-0 1.113775e-0 5.92591ge-1 2.981087e-1 1.492114e-1
1.147980e+1 3.033598e-0 1.94616ge-0 1.113507e-O 5.92311ge-1 2.980157e-1 1.49183ge-1
In our second exa mple, we consider t he unit squa re with a slit , n := (0,1)2\ [0.5,1 ) x {0.5}, decomposed into four subdomains; see Fig . 1.29. Here, we have no H 2-regularity and a O(h) and O(h 2 ) behavior of t he discreti zation erro rs in t he energ y norm and L 2 -nor m , respectively, cannot be expected. The right hand side f and the Dirichlet boundar y condit ions of - Llu = fare chosen such t ha t t he exact solution is given by (1- 3r 2)2r l/2 sin (1/ 2¢), where x - 1/ 2 = r cos ¢, and y - 1/2 = r sin ¢ . The solut ion has a singularity in t he center of the dom ain .
.ffA ~ I. \\\~
~~ ~
:::::
Fig. 1.29. Decomposit ion into 4 sub domain s and init ial triang ulation (left) and isolines of the solution (ri ght) , (Example 2)
1.5 Numerical Results
67
The discretization errors are given in Tabl es 1.3 and 1.4. In this case, we observe a significant difference in t he performance of the different mortar methods. The discretization errors in th e £2 -norm of th e alte rnative mortar methods with M t and M~ as Lagrange multiplier spaces are asymptotic ally better th an those of th e others. It seems that th e biorthogonal Lagr ange multiplier spaces M t and M~ provide a better approximation in this special case without full regularity. We not e that continuity of th e flux is only guar anteed if th e weak solution is in H2+€(n), t: > o. As in Exampl e 1, the discretiz ation errors measured in th e energy norm are comp ar able. T he st andard mortar approach u~ and u~ give slightly better results than u~ and ut . Table 1.3. Discr eti zation erro rs in the £ 2-norm , (Ex ample 2)
~ # elem . 44 176 704 2816 11264 45056 180224
0 1 2 3 4 5 6
I Ilu -
u~llo
4.896283e-2 1.651238e-2 4.488552e-3 1.254716e-3 3.878438e-4 1.401538e-4 5.883500e-5
I lIu - uillo I Ilu4.861265e-2 1.619017e-2 4.281367e-3 1.125460e-3 3.04604ge-4 8.68066ge-5 2.649174e-5
u~ l l o
4.882552e-2 1.637565e-2 4.39418ge-3 1.190041e-3 3.43201ge-4 1.109596e-4 4.153425e-5
I lIu -
u~ lI o
4.861265e-2 1.619057e-2 4.281791e-3 1.125644e-3 3.047055e-4 8.68675ge-5 2.653024e-5
Table 1.4. Discreti zation err ors in the energy norm , (Ex ample 2) level 0 1 2 3 4 5 6
I#
elem 44 176 704 2816 11264 45056 180224
I Illu - uUl 6.000955e-l 3.55327ge-l 2.045833e-l 1.23293ge-l 7.824813e-2 5.184650e-2 3.536026e-2
Illu - uilll
Illu-uUl
tllu-uU
6.050778e-l 3.584246e-l 2.069586e-l 1.252113e-l 7.975380e-2 5.29837ge-2 3.619496e-2
6.015198e-l 3.563046e-l 2.053318e-l 1.238978e-l 7.872322e-2 5.220587e-2 3.562424e-2
6.050778e-l 3.584008e-l 2.069517e-l 1.25205ge-l 7.974960e-2 5.298063e-2 3.619266e-2
Our next exa mple illustrates the influence of discontinuous coefficients. We consider t he diffusion equation -diva\7u = f , on (0,1)2, where the coefficient a is discontinuous. The unit squ ar e n is decompos ed into four non overlapping subdomains ni j := ((i - I)/2, i/2) x ((j - I)/2, j/2), i,j E {I ,2} , as in Fig. 1.30. The coefficients on th e subdomains are given by all = a22 = 0.00025, a12 = a21 = 1. The right hand side f and the Dirichlet boundary conditions are chosen to match a given exact solution, (x - 0.5)(y - 0.5) exp(-1O((x - 0.5)2 + (y - 0.5)2))/a . This solution is continuous with valu e zero at t he inte rfaces. Furthermore, t he jump of th e flux, [a\7u· n]' vanishes on t he interfaces. Because of th e discontinuity of t he coefficients, we use a highly non-matching triangul ation at the int erface; see Fig. 1.30.
68
1. Discreti zation Techniques Based on Domain Decomposition
Fig. 1.30. Decomposition into 4 sub dom ains and initial tri an gulation (left) and isolin es of t he solu tion (right) , (Ex ample 3)
The discreti zation err ors in the £ 2-norm and in t he energy norm are given for the differ ent mortar discr etiz ations in Tabl e 1.5 and Tabl e 1.6, resp ectively. We observe an O(h 2 ) behavior for the discretization err ors in t he £ 2-norm , and t hat t he energy err ors are of order h . Table 1.5. Discreti zation errors in t he L 2-norm , (Example 3)
~ # elem . 0 1 2 3 4 5 6
I lIu-
68 272 1088 4352 17408 69632 278528
u~ lIo
3.184810e-0 9.416096e-1 2.42556ge-1 6.093936e-2 1.52447ge-2 3.811271e-3 9.527881e-4
I Ilu - uXllo I Ilu 2.981474e-0 9.358117e-1 2.431694e-1 6.103994e-2 1.52548ge-2 3.812137e-3 9.52856ge-4
u~llo
3.113584e-0 9.399402e-1 2.42725ge-1 6.096660e-2 1.524751e-2 3.811503e-3 9.528064e-4
I lIu -
u~llo
2.981474e-0 9.358117e-1 2.431694e-1 6.103994e-2 1.52548ge-3 3.812137e-3 9.528568e-4
Table 1.6 . Discreti zation errors in the energy norm , (Ex ample 3)
~ # elem. I 0 1 2 3 4 5 6
68 272 1088 4352 17408 69632 278528
Illu -
u~ 1 1
1.17388ge-0 6.115732e-1 3.083728e-1 1.545031e-1 7.72922ge-2 3.865144e-2 1.932641e-2
Il u-uUl
Il u-uUl
Illu-uU
1.19925ge-0 6.18743ge-1 3.094938e-1 1.546515e-1 7.731113e-2 3.865380e-2 1.932670e-2
1.181552c-0 6.133283e-1 3.086346e-1 1.545374e-1 7.729663e-2 3.865198e-2 1.932648e-2
1.19925ge-1 6.18743ge-1 3.094938e-1 1.546515e-1 7.731113e-2 3.865380e-2 1.932670e-2
As in Example 1, t here is only a minimal difference in the performan ce of t he mort ar approa ches with conforming PI -elements on t he subdomain s. Again t he numerical result s confirm the t heory. On level 6, the difference in t he accuracy between the best u~ and worst u~ mortar solution is less t han 0.008% in the £ 2-norm and less t han 0.002% in t he energy norm.
1.5 Numerical Results
69
In the last example in this subsection, we combine discontinuous coefficient s and a weak solution with a singularity. The unit square is decomposed into two subdomains. D l is a L-shape dom ain and D z := D \ D l ; see Fig. 1.31. We have no HZ-r egularity, and no O(h) and O(h Z ) behavior of the discretiz ation err ors in t he energy norm and in the £ z-norm, respectively, can be expe cte d . The Dirichlet boundary conditions of -divaVu = 0 ar e chosen so that the exa ct solut ion is given by r " sin (a + 1) on D l and by f3rOi sin(a + z) on Dz. Here, the par ameters are x - 1/2 = r cos , and y - 1/2 = r sin e, a := 0.6675 , 1 := (1 - 0.75a)1r, z := (2 -1.75a)1r , f3 := sin( d/ sin(21ra + z ), al n 1 := 1 and al n 2 := - t an (0.25a1r)/ t an(0.75a1r) . The solution is continuous and non zero at th e interface, and t he normal derivative, but not the flux , has a jump at the int erface.
Fig. 1.31. Decomposition into 2 subdomains and initial triangulation (left) and isolines of th e solution (right) , (Example 4)
Tabl es 1.7 and 1.8 show the discr etization err ors for the different Lagrange multiplier spaces. As before in Ex amples 1 and 3, the influence of the Lagrange mul tiplier space on the discreti zation errors is negligible. From the beginning, the err ors in t he energy norm are almost t he same in all four cases . The £ z-norm is more sensitive to t he choice of the Lagran ge multiplier space. But even for this norm , t he difference in the accuracy is smaller than 0.1 % afte r two refinement ste ps. Table 1. 7. Discretization errors in the L 2-norm , (Example 4) ~ # elem 60 0 1 240 2 960 3840 3 15360 4 61440 5 245760 6
I
Ilu - uk110 6.113544e-3 2.402816e-3 9.606488e-4 3.851735e-4 1.546044e-4 6.194618e-5 2.475497e-5
I
lIu - u~lIo 6.154868e-3 2.416258e-3 9.610276e-4 3.851173e-4 1.545783e-4 6.193827e-5 2.475282e-5
I
lIu - u~lIo 6.124583e-3 2.406854e-3 9.607471e-4 3.851625c-4 1.545983e-4 6.194430e-5 2.475445e-5
I
lIu - u~lIo 6.154868e-3 2.416258e-3 9.608221e-4 3.850316e-4 1.54545ge-4 6.192830e-5 2.475004e-5
70
1. Discretization Techniques Based on Domain Decomposition Table 1.8. Discretization errors in the energy norm, (Example 4)
level I # elem. I Illu - u~111 1.41006ge-1 0 60 240 8.587308e-2 1 5.262487e-2 2 960 3.251092e-2 3840 3 2.021560e-2 4 15360 61440 1.262635e- 2 5 7.909156e-3 245760 6
Illu-uUl
1[lu-ua
1.41009ge-1 8.587284e-2 5.262505e-2 3.251115e-2 2.021575e-2 1.262645e-2 7.909221e-3
1.410075e-1 8.58728ge-2 5.262490e-2 3.251097e-2 2.021563e-2 1.262637e-2 7.909171e-3
1[lu - u~1II 1.41009ge-1 8.587284e-2 5.262526e-2 3.251136e-2 2.021590e-2 1.262655e-2 7.909282e-3
In Examples 1, 3, and 4 the difference in the accuracy for all four uL 1 :::; i :::; 4, is smaller than 0.1% on the finest refinement levels. Only in Example 2 can a significant difference be observed. In that case, the errors in the L 2 -n or m differ by more than 100% between u~ and u~. Additional test examples with singularities show that no clear pattern can be observed. The following two figures illustrate the discretization errors given in Tables 1.1-1.6 for u-ul, (standard) and u-u~ (dual). In Fig. 1.32, the errors in the energy norm are visualized whereas in Fig. 1.33 the errors in the L 2 -n orm are shown. In each figure a straight dashed line is drawn below the obtained curves to indicate the asymptotic behavior of the discretization errors. E
Example 1
g
t':~~~~~ '~0'1L
w
100
Example 3 standard
dual
y==7/sqrt(x)
~
_.~--
•
§
£ 0.1 J
1000 10000 100000 Numberof elements
Example 2
E
c
100
1000 10000 Numberof elements
Fig. 1.32. Discretization errors in the energy norm versus number of elements
In Examples 1 and 3, we can observe the predicted order h for the energy norm and the order h 2 for the L 2 - n or m almost from the beginning. In these two examples only one plotted curve for the standard Lagrange multiplier space M~ and the dual space M~ can be seen, since the numerical results are too close. In Example 2, where we have no full H 2 -regularity, the asymptotic behavior starts late. We observe an O(h 1 / 2 ) behavior for the discretization errors in the energy norm for both mortar methods. During the first refinement steps the error decreases more rapidly than later. For the L 2 -n orm the asymptotic rate is O(h 3 / 2 ) . Moreover, it seems to be the case that the Lagrange multiplier space M~ performs asymptotically better than the standard one. However, this has not been observed for some other examples without full regularity. The discretization error in the L 2 -n orm is
1.5 Numerical Results
71
more sensitive to the choice of the Lagrange multiplier space than the error in the energy norm. Example 1
~
~
£;
Example 2 E
g
0.1
~
0.01
£; c
0.1
.~
LiJ 1.-05
1000 10000 100000 Numberof elements
standarddual
E
g
y:::150fx
~
0.001
LiJ 0.0001 100
dual
V=O.2Jx"
0.01
': 0.001
e
Example 3
s!andard~
0.1
s:
c
0.01
UJ
0.001
e
0.0001 100
1000 10000 100000 Numberof elements
100
Fig. 1.33. Discretization errors in the L 2-norm versus number of elements
We remark that the theory does not make any statements about the constants in the a priori estimates. However, the numerical results show that the discretization errors for the different Lagrange multipliers are almost the same. Thus from the point of accuracy, there is no preferable Lagrange multiplier space. 1.5.2 A Non-optimal Mortar Method To get a better understanding of the situation at the interface and the role of the Lagrange multiplier, we now consider a non-optimal discretization. Here, we understand non-optimal in the sense that no order h a priori estimate can be obtained. Such a discretization scheme can easily be constructed if we use different Lagrange multiplier spaces on the two sides of the interface. We like to use a biorthogonal basis on the non-mortar side, and nonnegative Lagrange multiplier basis functions on the mortar side. We start with the spaces M~ and M~ and define the constrained space Vh by
~ := {v E Xhl. Vh
I
(vln n <=
)
2 'l/Ji3 - vlnn( = ) 'l/Ji) do
= 0, 1 ::; m ::; M,
.} 2 ::; N m
,
'Ym
where'l/J; and'l/Jy are defined as in Subsect. 1.2.4. We recall that 'l/J; and 'l/Jy are both associated with the same vertex on 'Ym. Using the same tools as discussed in detail in Sect. 1.1 and 1.2, it can be verified that the consistency error is of order h but not the best approximation error. For the best approximation error, we find only (1.65) We do not prove this result because it is of no theoretical interest. However, it can easily be seen that no optimal result can be obtained.
72
1. Di screti zation Techniques Based on Domain Decompositi on
Let us assume that t he trian gulat ion is not uniform in t he sense t hat at least one vertex Xi at an int erface "1m , 1 ::; m ::; M , not adjacent t o the endpoint s of "[m , is not at the same t ime at t he midpoint tn; of t he support of 'l/Jf; see Fig. 1.34.
Fig. 1.34. Vertex
Xi
and midpoint
m i
do not coinci de
Then , for v E PI (D) , we obtain V(Xi ) =
J
Isupp1 'l/J21 i
V'l/Ji3 da,
I~
If v is not constant along "1m, t hen V(Xi ) i- v (mi ) since by assumpt ion Xi im i . From the definiti on of Vh , we find that PI (D) rt Vh . Here, we have assumed t hat no Dirichlet boundary condit ion is imp osed on th e product space X h . In the case of such a t riangulat ion, PI (D) is not a sub space of Vh , and no a priori bound in terms of hlul2 holds. Based on this obser vat ion one might get t he idea of replacin g 'l/Jf in the definition of Vh by .(f;i, where .(f;i is equal one in a circle with mid point Xi and a radi us depending on t he local meshsize, and zero outs ide. From t he point of view of t he ap proximation quality t his creates an issue. In fact , such a modificati on saves th e best approximation pr operty but t he optimal order of the consiste ncy error is t hen lost unless we again have a very special t rian gulati on .
Fig. 1.35. Sp ecial t ria ngulations on the int erface
The approximat ion propert y requires t hat Xi is t he midpoint of t he support of .(f;i' and t he analysis for the consistency erro r requires that the sum
1.5 Numeri cal Results
73
over t he basis functio ns on t he interfaces equa ls one. Figur e 1.35 shows two exa mple of t riangulations for which consiste ncy and approximation property can be satisfied at the same time. The circles in Fig. 1.35 indicate th at t he verti ces are the midp oints of t he supports of t he corres ponding basis functions. We remark that only t he midp oint s of t he circles are vert ices. At the left , t wo adjacent circles int ersect at the midpoint of t he edges. This is not t he case on t he right , there th e ratio of the length of two adjace nt edges , not sharing an endpoint of "[m» is t he same but not equa l to one. Only if all triangulat ions associated with t he non-mortar sides have a st ructure as indicated in Fig. 1.35, has the constrai ned space Vh with replaced by :(j;i an order h approximat ion property. If t his condit ion is violated , we cannot, according to t he t heory, expect an order h a pri ori estimate of the discreti zation error in the energy norm . The du ality of the requi rement s for t he approximation pr operty and t he consiste ncy error indi cates t hat t he use of different spaces as test and trial space could be an issue for this problem : Find Uh E i\ such t hat
,¢;
A similar sit uation is st udied in [CLM97]. In t hat pap er , one Lagrange mult iplier space is used but t he exact int egr al over the int erfaces is replaced by num erical quadrature. One quadrature rule is based on t he mesh on t he nonmort ar side and t he ot her on the mesh on t he mortar side. Then , aga in th e ap proximation property and t he consistency error are optimal only for very special tri an gulat ions. The idea is now to use a variationa l formulation where test and t rial spaces are different . No a priori est imate s for t he discret ization error are proved . In par ti cular, the well-posedness of t he variational problem is an open question . However, num erical results indicate t hat it might be an order h method; see [CLM97]. The following two tables show the discreti zation errors for this nonopt imal approach for t he examples alrea dy discussed. The initi al trian gulati ons do not have the required form in any of t he Examples 1- 4. However , if we compare t he num erical results with t hose obtained in Subs ect. 1.5.1, we get almost th e same accuracy. Table 1.9. Discreti zat ion errors in t he £ 2-norm for Examples 1-4 level 0 1 2 3 4 5 6
I
Ex ample 1 2.021200e-0 1.019843e-1 1.170394e-1 9.532570e-3 2.801035e-3 7.115364e-4 1.7860 lOe-4
I
Example 2 4.855250e-2 1.650063 e-2 4.380014e-3 1.166031e-3 3.234870e-4 9.563012e-5 3.041376e-5
I
Ex ample 3 2.981412e-0 9.358003e-1 2.431747e-1 6.104268e-2 1.525571e-2 3.812352e-3 9.529113e-4
I
Ex ample 4 6.530118e-3 3.296302e-3 1.158950e-3 4.268715e-4 1.628104e-4 6.351176e-5 2.504495e-5
Iu-
3x + 2y 9.961053e-3 7.211486e-3 1.990688e-3 5.424928e-4 1.454180e-4 3.824237e-5 9.944375e-6
1. Discretization Techniques Bas ed on Dom ain Decomposition
74
Table 1.10. Discreti zat ion errors in the energy norm for Ex amples 1-4 level 0 1 2 3 4 5 6
I
Example 1 1.14 7916e+1 3.029716e-O 1.953580e-O 1.117625e-O 5.928700 e-1 2.979264 e-1 1.491143e-1
I
Example 2 6.04539ge-1 3.613288e-1 2.076923e-1 1.254030 e-1 7.98095 3e-2 5.300016e-2 3.619913e-2
I
Example 3 1.199261 e-O 6.187424e-1 3.094936e-1 1.546515 e-1 7.731113e-2 3.86 5380e-2 1.932670 e-2
I
Ex ample 4 1.410401e-1 8.591741e-2 5.264523e-2 3.252204e-2 2.022203e-2 1.263017 e-2 7.9114 74e-3
I u = 3x + 2y 1.806513 e-1 2.085708e-1 9.902825e-2 5.02021Oe-2 2.521881e-2 1.256562e-2 6.231056e-3
In Ex amples 2 and 3, t he initi al triangulations on t he non-mortar side are almost of the required form . The only except ions are th e vertices adjacent t o th e endpoints of t he non-mortar sides. Only in Ex ampl es 1 and 4, are the t riangulat ions not as close to t he required form. In Ex ampl e 1, we get minimal bet t er results , and in Ex ampl e 4, we obtain minimal worse results than before. We again observe t hat t he £ 2-err or is mor e sensit ive to t he choice of t he constraints t han th e energy err or. These num erical results would indi cat e t hat the discreti zation error in the energy norm is of ord er h and in th e £ 2_ norm of ord er h 2 . The effect of th e non-optimality of t he method is shown in Column 6. We use the Lapl ace operator, select u = 3x + 2y as exac t solut ion, and choose t he sam e decomposition into subdomain s as in Example 1 but th e triangulations on t he non-mortar sides are mor e irr egular . An optimal method would yield zero as discreti zation error on each level which is obviously not t he case for t his setting. 1.5.3 Influence of the Choice of the Mortar Side
The following num erical results show t hat t he choice of t he non-mortar sides and t he appropriate refinement at the interfaces are quite crucial in the case of discont inuous coefficient s. The energy norm as well as the £ 2-norm are considered for two different sit uations. Uniform and ada pt ive refinement t echniqu es are used to illustrate the influence of t he choice of the mortar side. In t he case of an adaptive scheme, we start with a conforming triangulat ion, whereas in t he case of uniform refinement , we use a highly non-m at ching init ial triangulat ion. We consider -div( a\7u) = f on fl . The coefficient a is constant on t he different subdomain s and has a jump across t he int erfaces. We consider only examples where the right hand side f does not reflect t he jump in the coefficient a. For examples where f reflects the jump, t here would be a principal difference in the results. In Situation I, t he non-mort ar side is defined where t he coefficient a is smaller whereas in Situation II , t he non-mortar side is associated with the lar ger coefficient. We st art with an example of highly nonmat ching t riangulat ion and st rongly discontinuous coefficients . Figur e 1.36 shows t he decomp osition into subdomain s, the non-mat ching t riangulat ions
1.5 Numerical Results
75
and the isolines of the numerical solutions for Situations I and II. The coefficient a has the same value on the inner and outer sub domain and is smaller in the middle one. Although in both situations, the same triangulation is used, two completely different results are obtained. A good approximation can be found only in Situation I when M h is of higher dimension. The constrained space Vh is of higher dimension in Situation II, but then, the approximation is incorrect; see [HIK+98, BDL99].
Fig. 1.36. Triangulation (left), solution for Sit. I (middle) and Sit. II (right)
In the rest of this subsection, we focus on the influence of the choice of the non-mortar side. We study this influence in Examples 5 and 6 for adaptive and uniform refinement strategies. To start, we briefly discuss some aspects of an adaptive refinement algorithm. For standard conforming discretizations, different techniques for the construction of efficient and reliable local a posterior error estimators are well known. Residual based error estimators or hierarchical basis estimators based on a higher order discretization scheme are very often used. We refer to [Ver96] for an excellent overview and introduction to the basic concepts. Both residual and hierarchical error estimators have been adapted to the mortar settings [BH99, PS96, Woh99a, Woh99c]. The main problem is to take care of the non-nestedness of the nonconforming finite element spaces. It turns out; see Lemma 1.7 in Sect. 1.2, that an appropriate measure for the nonconformity is a weighted L 2 -n orm of the jump. In particular, it can be shown that an upper bound for this weighted L 2 - n or m is given by the discretization error in the energy norm. For each element, with an edge on a non-mortar side, the jump term
is part of the elementwise contribution of the error estimator. Here, a e stands for the coefficient on the non-mortar side. In the case of standard conforming triangulations, certain elements are marked by the error estimator and refined, and the refinement rules create a conforming triangulation in each adaptive step. In contrast to this, no additional rules control the adaptive refinement at the interfaces in the mortar settings. The only information transfer between the sub domains is obtained
76
1. Discretization Techniques Based on Domain Decomposition
by the local contributions of the error estimators. A local error estimator which does not reflect the jump can therefore not guarantee an appropriate refinement at the interface. The error in the Dirichlet boundary conditions at the interfaces is measured by the jump term of the finite element solution, while the difference between the discrete flux and the discrete Lagrange multiplier a'\l'Uh· n - Ah controls the error for Neumann boundary conditions. A detailed discussion of a posteriori techniques for mortar finite elements can be found in [BH99, PS96, Woh99a, Woh99c]. Here, we choose a mean value strategy to control the adaptive refinement process. An element T E Ti on level l is marked for refinement in the next step if the local contribution of the error estimator TIT satisfies 2 TIT
2': (J N1 'L"
2 TIT' ,
I T'ETi
where Ni is the number of elements in Ti, and we set (J = 0.95. The idea behind this is to equilibrate the error per element and to obtain a prescribed accuracy at a minimal cost.
Fig. 1.37. Isolines of the solution (left), decomposition into 5 subdomains (left middle), level difference 2 (right middle) and level difference 3 (right) of the initial triangulation, (Example 5)
Example 5 illustrates the influence of the choice of the Lagrange multiplier. We consider the diffusion equation -diva'\l'U = t , on (0,1)2, where the coefficient a is discontinuous. This example is discussed in detail in ['Voh99a], where hierarchical error estimators for mortar methods are studied. The unit square [l is decomposed into five sub domains as shown in Fig. 1.37. The coefficients in the sub domains [li are given by a5 = 5000, ai = 1, i E {I, 2, 3, 4}. The right hand side f and the Dirichlet boundary conditions are chosen to match an exact solution, 'U(Xl' X2) = l/a sin(31rxd sin(31rx2) 2::,j=l exp(-800(xj - i/3)2). This solution is continuous and [ai'\In· nlJ is equal to zero on the interfaces. The isolines of the solution are shown in Fig. 1.37. We now consider the two different possible choices of the mortar sides separately. In Situation I, the discrete Lagrange multiplier space is associated with the triangulations given on [ll, [l2, [l3 and {/4, whereas in Situation II, the triangulation for the Lagrange multiplier space is inherited from the one on [l5.
1.5 Numerical Results
77
Figure 1.38 shows the influence of the choice of the Lagrange multiplier on the adaptive refinement process for Example 5; the adaptively refined triangulation on Level 3 and Level 4 are given for both situations. In Situation I, we observe a sharp interface between the different subdomains. However this is not the case for Situation II, where we obtain a triangulation which tends to be more conforming at the interfaces. Furthermore, more nodes are generated on the side where a is larger. The additional refinement is a consequence of the choice of the non-mortar side.
Fig. 1.38. Situation I (left) and Situation II (right), (Example 5)
We recall that the dimension of the Lagrange multiplier space is given by the numbers of edges on the non-mortar side minus one. A sharp interface in Situation II would lead to poor approximation properties for the Neumann boundary condition. Thus, we have to adapt the triangulation along the interface on the side where a is larger, and increase the dimension of the Lagrange multiplier space; see the right part of Fig. 1.38. However, the number of these additional elements next to the interface can be neglected asymptotically because they are associated with a ID interface problem. In Situation II, the jump of the discrete solution plays an important role in the definition of the error estimator, since it controls the nonconformity of the mortar finite element solution. Without this term the error estimator would fail. In Situation I, it could be neglected and no significant difference can be observed in the adaptive triangulations. We point out that the adaptive refinement on both sides of the interface in Situation II is not enforced by any refinement rules but only by the local contributions of the error estimator. Comparing the true error, in Table 1.11, we find that the performance is asymptotically the same in Situations I and II, i.e., the number of elements, to obtain a given accuracy, is asymptotically almost the same. We introduce X :=
J]Ii:; Illu - uhlll
as measure of the performance. In fact, X gives a rough idea of how many elements NT are required to obtain a given accuracy; the smaller X, the fewer elements are required. The ratio in Situation II between the number of additional elements in the neighborhood of the interfaces and the total number of elements tends to zero in the adaptive refinement process.
78
1. Discreti zati on Techniques Based on Domain Decomposit ion Table 1.11 . Effect ivity index ( and performan ce X, (Exam ple 5)
#
elem . 144 232 456 1016 2632 6088 12880 3101 2 57344 138184 240360 568780 972488
Situati on true err. 1.771e-0 1.285e-0 6.302e-1 3.794e-1 2.30ge-1 1.605e-1 1.024e-1 6.78ge-2 4.600c-2 3.156e-2 2.228e-2 1.525e-2 1.103e-2
I ( 1.674 1.301 1.244 1.19 2 1.155 1.140 1.159 1.170 1.226 1.179 1.244 1.199 1.252
#
X
21.25 19.57 13.46 12.09 11.85 12.52 11.62 11.96 11.02 11.73 10.92 11.50 10.88
elem , 144 312 536 1112 1600 3688 4996 11905 19700 44962 88471 190545 428246
Situation t rue err. 1.771e-0 1.285e-0 7.78ge-1 4.563e-1 3.715e-1 2.155e-1 1.805e-1 1.100e-1 8.392e-2 5.430e-2 3.896e-2 2.535e-2 1.774e-2
II ( 1.996 2.572 1.388 2.826 1.258 1.913 1.223 1.404 1.241 1.271 1.240 1.248 1.235
X
21.25 22.71 18.03 15.22 14.86 13.08 12.76 12.00 11.78 11.51 11.59 11.07 11.61
In our experiment, we star t with a global conforming t riangulation . Each subdomain is decomp osed into 16 elements. In t he first refinement ste ps , we observe that X is considera bly lar ger in Situation II . However , t his difference vanishes asymptot ically. From Level 8 on, X can be seen to oscillate. Asymp totically it appears as if X is smaller for Situati on I t han Situation II , when t he level is even , while the opp osite is t rue on an odd level. This is also reflected in t he effectivity index ( which is defined by ( := est imate d err or
,
t rue error and which is a measur e of t he quality of the err or est imator. For a good est imator, it is required t hat ( tends asympto tically to a value close to one. In a second test setting for t he same example, we start with a highly non mat ching t riangulation at t he interface and use uniform refinement techniques. We work with level differences of two and three between the t riangulations on t he subdomains; see Fig. 1.37 for the initi al t riangulati ons. The choice of t he initial t ria ngulation is motivat ed by the following observat ion: Within t he ada pt ive refinement pro cess, we find in Situation I a level difference of approximately 3 at t he interface and t hus, rou ghly hd h 2 ~ ijada2. This reflects the fact t hat during the refinement process t he erro r in t he energy norm per element is equilibra te d. In Tabl es 1.12 and 1.13, t he discretization erro rs for th e two different situati ons are given. Table 1.12 shows the discreti zation errors in the energy norm as well as in t he L 2 -nor m for Situation I and Situation II , if t he initi al trian gulation has a level difference of t hree at th e int erface; see t he right picture in Fig. 1.37. In Situati on II , t he results are significantly worse. Columns 6 and 7 give t he ratio between the err ors . For t he L 2 - nor m , t he rati o between t he errors in t he t wo sit uations is almost ten, even afte r four refinement st eps.
1.5 Numerical Results
79
The erro rs in Situation II are ext remely bad compared to those of Situation 1. For th e energy norm this ra tio is not as extreme and improves consid erable with an increasing number of nod es. Asymptoti cally, it seems to tend to one. Table 1.12. Discreti zation err ors in the case of a level difference 3, (Example 5)
#
elem . 4176 16704 66816 267264 1069056
Situati on I En- err or L"-error 2.8184e-1 1.2120e-3 3.480ge-4 1.4485e-1 7.073ge-2 8.2695e-5 3.5318e-2 2.0606e-5 1.7654e-2 5.1475e-6
Situat ion II En- error L"- error 1.0216e-0 3.9234e-2 4.4993e-1 8.7215e-3 1.5808e-1 1.5457e-3 6.2716e-2 3.0342e-4 2.2741e-2 4.2456e-5
Ratio En 0.276 0.322 0.477 0.563 0.776
IIII
L"
0.031 0.041 0.054 0.068 0.121
Table 1.13. Discreti zation err ors in the case of a level difference 2, (Example 5)
#
elem . 1104 4416 17664 70656 282624 1130496
Situation I L"-error En-error 4.5473e-3 5.5021e-1 1.2254e-3 2.8155e-1 3.0974e-4 1.4002e-1 7.7758e-5 6.9964e-2 3.4976e-2 1.9458e-5 1.7487e-2 4.8657e-6
Sit uation II L"-error En- error 1.126ge-0 3.9831e-2 4.974ge-1 8.3860e-3 1.503ge-3 1.9554e-1 8.6293e-2 2.9904e-4 3.7670e-2 4.4813e-5 1.7848e-2 7.0931e-6
Ratio En 0.488 0.566 0.716 0.811 0.929 0.980
IIII L" 0.114 0.146 0.206 0.260 0.434 0.686
The same type of results is given in Table 1.13, but now, the initial t riangulation has only a level difference of two at th e int erfa ce instead of three. The difference between Situations I and II is not as ext reme as in Table 1.12. Starting with a ratio for th e energy norm of approximately 0.5 on t he initial trian gulation, the ratio tends to one with an increasing numb er of refinement levels; on t he finest level, where we have mor e t ha n a million elements, the ratio is approximately 0.98. We observe t he same typ e of behavior for the L 2 -nor m . However , even more nodes are needed before the asymptot ic ratio of one is reached ; the L 2 -nor m depends more sensitively on the choice of the Lagran ge multiplier th an t he energy norm. Fi na lly, Fig. 1.39 displays the numb ers of Table 1.12 and Table 1.13. For Situation I, the correc t order of the discretiz ation scheme can be observed from th e beginning. All cur ves par allel t o y = 15/,jX and y = 3/ x have th e correct order in th e energy norm and the L 2 -nor m , respectively. It can be seen that the ada ptive refinement on both sides of the int erface for Situation II , shown in Fig. 1.38, is really necessary . A sha rp int erface between the subdomains makes sense only for Situation 1. In Situation I, there is almost no difference in th e accuracy obtained if a level difference of 2 or 3 is used . However in Situ ation II, a level differen ce of 3 yields much worse results . For t he energy norm , it appears th at asymptotically we get the same performance
80
1. Discreti zation Techniques Based on Domain Decomposition
for the different choices. However , the asymptoti c behavior starts very late and depends sensit ively on th e level difference. 0.1
r-----------""1
E o c
~
~
Q)
cQ)
0.00 1
Q)
-=
Q)
-=
.S:
g
.5
g
W
0.0001
la-05
UJ
1• •0£ ' -_ _
0.0 1 '---~-------.,j 10000 100000 18+06 1000
1000
Number of elements
~
100 00
~
__
~..J
1e+06
100000
Number of elements
Fig. 1.39. Error in t he energy (left) and in t he £ 2-norm (right) , (Ex ample 5)
For t he £ 2-norm t he observed phenomena are even more significant, and the obtained accuracy for a highly non-mat ching t riangulat ion depend s st rongly on t he choice of the non-mortar side. In th e case of a level difference 3, the asy mptotic ran ge will never be reached in pr actical relevant computat ions for t he £ 2-norm . Thus, workin g with highly non-matching triangulat ions and uniform refinement t echniques requires t he pr oper choice of t he non-mortar side. The error est imat or can deal with both sit uations . In Situation I, a sharp int erface will be generate d, whereas in Situ ation II ada ptive refinement will be observed on both sides of th e int erface.
Situation I - Situation II - ...-
40
'" 35
g
§ 30
o ~ 25
1000 10000 100000 10+06 Number of Elements
•
20 18 1000
70 ,........---
Situation I - level DiU2
~ itualion II- leve l DiU 2
<,
-_.•--
'"~
~
~
' . . . . ....-
10000 100000 10+06 Number of Elements
&.
60
50
-
--..------,
Situation l-level01t13 II • Level DiN3 -_ ...--
~S ituation
.,~~
40 30
'"
<,<,
10000 100000 10+06 Number of Elements
Fig. 1.40. Performan ce ada pt ive (left) and uniform refinement , level difference 2 (middle) and level difference 3 (right) , (Ex ample 5)
Figure 1.40 shows t he performance of the different refinement strat egies and sit uations. The adapt ive strategy has a considera bly better performanc e t han t he uniform strategy. For both situations t he perform an ce t ends asympt ot ically t o valu es between 11 and 12 in t he adaptive case . In Situation I, t he asy mptotic ran ge is reached earlier than in Situation II , and the performance is slightly better. The asy mptot ic behavior start s afte r a few refinement steps for both situations. In the middle and right part of Fig. 1.40, the perform an ce for the uniform case is displayed. Here, we observe a big difference between Situati on I and Situat ion II. We consider the cases of two (Case 1) and three
1.5 Numerical Results
81
(Case 2) level differences in the initial triangulations; see Fig . 1.37. If the mortar side is chosen appropriately, i.e., we are in Situation I, we obtain an almost constant performance from the beginning on. In Situation I, we obtain a performance of ~ 18.25 in Case 2 and a performance of ~ 18.59 in Case 1. The performance of Situation II , seems to tend asymptotically to the performance of Situation 1. But the asymptotic starts extremely late. In Case 1, the asymptotic is reached after five refinement steps. In Case 2, the asymptotic is not reached even for more than a million elements. The bad approximation property of the Lagrange multiplier in Situation II results in the poor performance . However, the approximation property of the Lagrange multiplier space is of order h~~~, where h non is the meshsize on the non-mortar side . In Situation II , h n on is considerably larger than the meshsize on the mortar side. The fact that the approximation order of the Lagrange multiplier space is higher than the approximation order of the finite element space yields that the influence of the choice of the mortar side vanishes asymptotically. But the asymptotic starts so late, that it will be not reached for many numerical applications. Using uniform refinement strategies, it is extremely important for the performance of the method to make the correct choice of the mortar side. In our last example in this subsection, we consider a different coefficient in each subdomain. It has been originally studied in [Woh99b]. As in Example 5, we find an essential difference between the two choices of mortar sides . In the case of uniform refinement, the obtained accuracy depends highly on this choice and in the case of adaptive refinement , the refinement at the interface reflects this choice .
o Fig. 1.41. Isolines of the solution (left) and non-matching initial triangulation (right) , (Example 6) The unit square is decomposed into four subdomains Di j := (i/2, (i + 1)/2) x (j/2 , (j + 1)/2), i ,j E {O, I} and alSl ii := aij where aoo := 1, alO := 250, aOl := 5000 and all := 10. The data are chosen to match the given solution u(x, y) = (x - 0.5)(y - 0.5) exp( -40(x - 0.5)2) exp( -40(y - 0.5)2)/a. The isolines of the solution are shown in Fig. 1.41. Figure 1.42 shows the adaptive refinement process for both situations on Level 4 and Level 5. We use a conforming initial triangulation with 8 elements. As in Example 5, we
82
1. Discretization Techniques Based on Domain Decomposition
obtain a sharp interface between the sub dom ains in Situation I wher eas in Situation II , adapt ive refinement is observed on both sides of the int erface.
Fig. 1.42. Ad aptive refinem ent for Situations I (left) and II (righ t) , (Example 6)
Finally, we compar e Situations I and II for a given trian gulation with a sharp interface between highly refined and unr efined subdomains. On each subdomain a uniform trian gulation is used ; see Fig. 1.41. The meshsizes in t he subdom ains reflect the ratio of t he coefficient on t he subdomain s. Alt hough t he const ra ined finit e element space has mor e degrees of freedom in Situation II, t he discreti zation err ors in the energy norm as well as in th e L 2 _ norm are worse compa red with Situation 1. Due t o t he bad approximation pr operty of the Lagrange multiplier in Situation II , th e consiste ncy err or of t he discr eti zation appears to play a significant role. Table 1.14. Error in the energy and L 2 -norm for Situations I and II in t he case of uniform refinem ent , (Ex ample 6)
#
elem . 170 680 2720 10880 43520 174080
Situation I L i- extct En- error 3.6023e-3 4.447ge-5 1.841ge-3 1.144ge-5 9.2500e-4 2.8676e-6 4.6427e-4 7.1992e-7 2.325ge-4 1.805ge-7 1.1637e-4 4.5265e-8
Situati on II L~-Error En- error 6.1857e-3 2.4307e-4 3.2997e-3 7.0047e-5 2.0115e-3 3.4793e-5 7.4883e-4 4.5648e-6 3.1681e-4 7.2615e-7 1.4665e-4 1.4541e-7
R atio En 0.582 0.558 0.460 0.620 0.734 0.793
IIII L~
0.183 0.163 0.082 0.158 0.249 0.311
In Tabl e 1.14, the erro rs in the L 2 -nor m as well as in the energy norm are given. Columns 6 and 7 show t he ratio of t he discreti zation err ors for the two different sit uat ions . As in Ex ampl e 5, the error in the L 2 -nor m dep end s in a more sensit ive way on th e choice of th e mort ar side t han t he err or in the energy norm. Asymptotic ally th e ratio seems to t end t o one. However , with mor e than 100000 elements t he ratio in t he L 2 -nor m is st ill only 0.311. With a sharp int erface and uniform refinement , satisfying results can only be obtained for Situation 1. Appl ying adapt ive refinement strategies, both sit ua t ions can successfully be handled , and two different types of refined triangulations will be generate d according t o t he choice of the mortar sides.
1.5 Numerical Results
83
1.5.4 Influence of the Jump of the Coefficients
Our last example in this section, reflects the influence of the jump in the coefficient on the adaptive refinement process. Adaptive refined triangulations are compared for different jumps in a. We consider -div(aV'u) = f on the unit square. The right hand side f and the Dirichlet boundary conditions are chosen for an exact solution u = exp( -1500 * (r 2 - 0.2)2) - exp( -3000 * (r 2 0.075)2) for a = 1, where r 2 := (x - 0.5)2 + (y - 0.5)2. Here, D is decomposed into nine sub domains D i j := (i/3, (i+ 1) /3) x (j /3, (j + 1) /3),0 :::; i, j :::; 2. The coefficient a is now chosen piecewise constant using a red and black ordering of the subdomains: a := ai on Di j if i + j even and a := a2 on Di j if i + j odd.
Fig. 1.43. Adaptive refinement on Level 4 and Level 5, (Example 7)
Here, the influence of the jump in the coefficients on the generated triangulations is of interest. Crouzeix-Raviart elements on D C R and conforming Pi-elements on D p " D = D p , U DCR, are coupled by means of piecewise constant Lagrange multipliers; as in Subsect. 1.4.2. The Crouzeix-Raviart elements are used in the sub domains where a = ai. Figure 1.43 shows the adaptivcly generated triangulations on Level 4 and Level 5 for the choice of ai := 1 and a2 E {l, 10, 100, 1000}. With increasing a2 the obtained triangulations tend to be more and more nonconforming at the interfaces between the subdomains. We recall that no matching condition is imposed for the triangulations of the different sub domains. Considering the situation at the interfaces in more detail, we observe that the jump between the meshsizes depends on the jump of the coefficient a. In the case of a2 = 1, we obtain an almost matching triangulation at the interfaces, whereas in the case of a2 = 1000 a highly non uniform triangulation
84
1. Discretization Techniques Based on Domain Decomposition
is generated. Roughly speaking, we find that the ratio of the local meshsizes hI/ h 2 is approximately aI/ a2. This is related to the mean value refinement strategy being used.
\!
Table 1.15. Effectivity index on
I
level 1 2 3 4 5 6 7 8 9
I#
elem. 144 528 1408 1864 4048 9032 21112 43944 100600
I <:
on fl p, 1.2121 1.1487 1.0323 1.1370 1.2573 1.2756 1.2763 1.3049 1.3105
flp"
I <: on
and fl
fl O R fl O R
I <: on
(az =
fl 1.8060 1.3543 1.1981 1.3065 1.4070 1.4189 1.4340 1.4609 1.4704
2.3357 1.4936 1.3847 1.4660 1.5403 1.5415 1.5784 1.6090 1.6076
1), (Example 7)
I err.
fl p
,! flo R I
1.106 0.965 1.042 1.033 0.992 0.996 1.009 1.026 0.974
Table 1.15 shows the effectivity index related to fl as well as to the two different discretizations. We use the simplified residual type error estimator for the Crouzeix-Raviart discretization which is proposed in Subsect. 1.4.3. In particular, we do not have to compute the jump of the flux across the edges. It is sufficient to evaluate the nonconformity of the finite element solution. For the conforming PI-elements, we apply a scaled residual based error estimator; see, e.g., [Ver96]. In both cases, we observe an overestimation of the error. The last column reflects the ratio of the error in the energy norm on the subdomains fl p , and flCR. Asymptotically, it is close to one alternating between values less and greater than one.
Fig. 1.44. Adaptive refinement on Level 6 for
a2
= 0.1 (left) and az = 0.01 (right)
Finally, Fig. 1.44 shows the triangulations on Level 6 for a2 E {0.1, 0.001}. Now, the non-mortar sides are defined on fl i j with i + j odd. As before, we observe a sharp interface between the different subdomains.
2. Iterative Solvers Based on Domain Decomposition
This chapter concerns iterative solution techniques for linear systems of equations arising from the discretization of elliptic boundary value problems. Very often huge systems are obtained, with condition numbers which depend on the meshsize h of the triangulation, which typically grow in proportion to tc:", Then, classical iteration schemes like Jacobi-, Gauf3-Seidel or SOR-type methods result in very slow convergence rates. Figure 2.1 shows the convergence rates and the number of iteration steps versus the number of unknowns, for a simple model problem in 2D. In the left, the convergence rates are given, and in the right the number of iteration steps to obtain an error reduction of 10- 6 are shown. For the Jacobi and the Gauf3-Seidel method, the asymptotic convergence rates are 1- O(h 2 ) . The optimal SOR-method is asymptotically better and tends with O(h) to one. However, the optimal damping parameter is, in general, unknown. The number of required iteration steps reflects the order of the method. For the Gauf3-Seidel and the Jacobi method, the number of required iteration steps grows quadratically with one over the meshsize. In case of the optimal SOR-method, the increase is linear. Moreover, the numerical results show that the Jacobi method requires two times the number of Gauf3-Seidel iteration steps. In case of the optimal SOR-method 1 Jacobj ..y");x",,,,.'H"'.'r'"+""~"""">+<j 0.95 ' / .ft<3auss-Seidel _"4-~-
if
0.9
~4-"''''-
00'
~§ ~::51/a'''!
15 5000
0.55 0.5
0 3000
opt. SOR
g; 0.65 o15 06 :"
j
~2000 E
/i
~1000
L
o
.~J~a~co~b:'..i_ ----.J
.)4000
0.75
l? 0.7
7000p ., ~ opt.SOR 6000 , .; ~, Gauss-Seidel
~_~_~_~_
10
20
30
40
50
Number of unknowns n2
60
00
20
40
60
Number of unknowns n2
Fig. 2.1. Convergence rates and number of iterations
Nonlinear iteration schemes as the conjugate gradient method give better results with a convergence rate depending on the square root of the condition number of the preconditioned system. Figure 2.2 illustrates the B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition © Springer-Verlag Berlin Heidelberg 2001
86
2. Iterative Solvers Based on Domain Decomposition
quality of a preconditioned conjugate gradient methods in 2D. A hierarchical basis method and the BPX-preconditioner are test ed ; we refer to [BPX90 a, BPX90b, Yse86, Yse93] and Subsect . 2.1.1. If a conjugate gradient method is applied without preconditioner , the number of iteration ste ps to obt ain a given accuracy is inversely proportional to the meshsize. In the case that a preconditioned version is used , t he required number of iteration st eps can be much smaller, and is in the best case independent of the meshsize. For the hierar chical basis method, we observe a logari thmic growt hs in 2D and the BPX-preconditioner results in an optimal method .
'" c
.Q
"§
.~ C:n o
... Q)
.0
E
z"
900 800 700 600 500 400 300 200 100
r--~~~~~~--~-...,-.,
without precond.HB-precond . BPX-precond.·····
°1~O-==!l=~::...c.._~=--~~,-,:-~~
100
1000
10000
Number of unknowns n
100000
Fig. 2.2. Convergence rates of pr econditi on ed cg-m ethod
Here, we focus on the construction of efficient iterative solvers for non standard discr eti zation schemes. In par ti cular, vector fields discret ization te chniques such as Raviart-Thomas finit e element s, Nedelec finit e elements, and mortar methods are considered. Well established t echniques for Lagrangian Pi-element s such as iterative subst ruct ur ing or multigrid methods are modified and ada pted to t hese special situations. In a first part , we consider t he abstract theory of Schwar z methods including examples of addit ive and multiplicative variant s. These techniques provid e a powerful tool for the efficient iterative solution of the huge syst ems of equations. We focus on precondi tioned conju gate gr adient methods where the preconditioner is built from t he solution of subproblems of less complexity and which are eit her related to a decomposition of the geomet rical domain into sub domains or a hierar chical splitting of the finit e element space into subspa ces. Our second main concern is the const ruction of special multigrid methods for the dom ain decomposition techniques introduced in Chap . 1. A Dirichlet-Neum ann and two different multigrid algorithms for the mortar method are studied. We can int erpret e the Dirichl et-Neum ann method as a block GauJ3-Seidel preconditioner for the unsymmetric mort ar formul ation. The first of the propos ed multigrid method is based on the new positiv e definite mortar formul ation on t he un const rained product space whereas the second one works with the saddle point formul ation.
2.1 Abstract Schwarz Theory
87
2.1 Abstract Schwarz Theory In this section, we give only a brief overview of th e genera l framework of Schwar z methods. Many classes of preconditioners for large linear syst ems of equations arising from the discretiz ation of partial differential equat ions have been ana lyzed within this framework ; see, e.g., [BS94, Le 94, QV99, SBG96 , Wid99] and t he references th erein . Applications ar e particularl y well developed for conforming finite element approximations of elliptic probl ems. For details, we refer to [BPWX91 a , CM94, DW95 , Osw94, SBG96 , Xu92 , Yse93]. Technica l to ols for establishing upp er bounds for th e condit ion numb er are given and bri efly discussed in this sect ion. We outline exa mples such as a twolevel overlapping method [Breaa, DW94], an iterative subst ruct uring method [BPS86b, BPS89, XZ98], and a multilevel method [Ban96 , Yse86, Yse93, Xu92] for standa rd conforming piecewise linear finite elements. T he possibly earliest domain decompos ition method appears to be t hat of Herm ann A. Schwarz [Sch9a]. He int roduced an alternating method to prove t he existence of harmonic exte nsions on domains with nonsmooth boundar ies, more th an one hundred years ago. Figur e 2.3 shows the decomposition of such a domain into a circle and a rectangl e; t his ty pe of decomposition is used in the origina l work of Schwar z [Sch9a] to illustrat e his idea. A sequence of harmonic fun ctions is const ructed in each subdomain and is shown to converge; the convergence rate depends on t he overlap . The origin al proof was based on t he maximum principle. We refer to [QV99] for a convergence proof of t he alte rnating Schwar z method based on th e vari ati onal framework ; this approach is introdu ced in [Lio88].
Fig. 2.3. Decomposition used by Schwarz in his original work
Here, we briefly review the idea of Schwar z in t he context of an ellipt ic operator L . Let D l be a circle and D 2 be a rectangle as shown in Fig. 2.3. T he int ersection between Dl and D2 is denoted by D* . To prove the existe nce of a functi on satisfying
Lu = I, U
= g,
in D , D := D l U D 2 on oD ,
,
we pro ceed as follows: Assuming the existe nce of solut ions of the elliptic equation on th e subdom ains D l and D2 for given suitable boundary conditions ,
88
2. Iterative Solvers Based on Dom ain Decomposition
we obtain a sequ ence alternatingly updated on ftl and ft 2 . Each iteration st ep consists of two half ste ps, associat ed with the two subdomains
Lu f
= I,
uf = g,
Un 1 -
Un - l
2
Lu'2 = i . u'2 = g, u'2 = u f,
,
in ftl , on L o := 8ft n 8ftl , on L 2 := 8ftl \ 8ft , in ft 2 , on L 3 := 8ft n 8ft2 on t., := eo, \ 8ft
,
We remark that the Dirichlet boundary condit ions on L l and L 2 ar e obt ain ed by t he solution on ftl and ft 2 in the pr evious half step, resp ectively. This classical Schwarz method can also be rewritten in a variational form , and the error propagation operator can easily be given in terms of two projection operators. We will not consider any further det ails and inst ead refer to [SBG96 , QV99 , Wid88 , Wid99] . In the following, we consider only the vari ational formul ation of general Schwar z methods. The st arting point is the vari ational problem
a(u , v) = j(v),
v EV ,
(2.1)
where V is a finit e element space , and the bilinear form a(·, ·) is associate d with a selfadjoint, ellipt ic operator. Each Schwarz method is then based on a suitable decomposition of th e finite dimensional space V into subspaces
and on projection-like operators Ti , a :S i :S N , mapping V onto these subspaces . This decomposition does not have t o be a dir ect sum. Typically, the subspaces are relat ed to a sequence of nest ed t riangulat ions or with basis fun ctions having support in different subdomains. To obtain quasi-optimal results in the second case , very often , requires th e use of a coarse global space. In the following two subs ections, we briefly discuss the additive and multiplicative Schwar z method, and basic tools to establish bounds for the condit ion number and the error propagation. 2.1.1 Additive Schwarz Methods
The addit ive Schwar z vari ant is an important type of Schwarz methods. It provides a new opera t or equa t ion which can be much bet t er conditioned t han the original discret e elliptic problem . Very often , the arising syste m can be solved efficient ly by the conjugate gra dient method. The quasi-proj ection operat or T; : V ---+ Vi , is defined by means of an additional symmetric positiv e definit e bilinear form ai( ·, ·) on Vi x Vi
2.1 Abstract Schwarz Theory
o'i(Tiw , v ) := a(w, v ),
89
v E Vi .
In the case that we choose o'i(', .) = a(· , ·), the operator T i is th e ort hogonal projection onto Vi with respect to the bilinear form a(' , -). The additive Schwar z opera tor is given by N
Tadd :=
2: T i , i= O
and t he variational problem (2.1) can be rewritten as TaddU = g, where the righ t hand side 9 is defined as 9 := 2:;:0 gi with
o'i(gi, v) := f( v) ,
v E Vi
The right hand side 9 is chosen so that th e new problem has the same solution as the original one. Using the bilinear form a(', ') as inner product, we can apply th e conjugate gradient method to the preconditioned problem. Then , the convergence rate in th e energy norm can be bounded in terms of the condition number of Tadd . An est imate for th e small est and largest eigenvalue of T add is given by t he following lemm a; see [SBG96, Sect . 5.2]. Lemma 2.1. Let us assum e that for each v E V there exists a representati on,
v=
2:;:0 Vi,
Vi E Vi , such that N
2: o'i(Vi ,Vi) :s; cga(v , v )
(2.2)
i=O
Then, the operator T add is invertible and a lower bound of a(Tadd v , v) is given by C 2 a(v ,v ) :S; a(TaddV,V), v E V .
o
Moreover if the bilinear form a(·, ·) is bounded by a constant tim es o'i ("' ) on the range of t; a :s; i :s; N , i.e., a(Tiv , Tiv) :s; and if there exist constants
e ij
=
W
o'i(Tiv, Tiv) ,
eji ,
1
VEV ,
(2.3)
:s; i , j :s; N , such that
then, an upper bound of a(TaddV, v ) can be given in terms of a(v , v) and the spectral radius p(£) of the mat rix E := { e ij } f,j = l : a(TaddV, v ) :s; w (1 + p(£)) a(v , v ),
vEV .
90
2. Iterative Solvers Based on Dom ain Decomposition
P roof. For convenience, we review the proof and refer to [SBG96, Sect. 5.2, Lemma 3] for a mor e det ailed discussion . We start with the upp er bound and consider t he operator norm IITilia
'= liT-liZ a' t
sup
vE V
a(Tiv ,Tiv) ( ) . a v,v
Using t he definiti on of t he operator T; and assumption (2.3) , we find
a(Tiv, Tiv ) :Swai (Tiv, Tiv ) =wa(v,Tiv) :s wa(v , v) ~a(Tiv,TiV) ~ , and t hus IITilia :S w. The upp er bound of t he norm (2.4) impl y
a(
N
N
t=l
t =l
2: T iv ,2: Tiv
)
N
= _ ~ a(Tiv ,Tjv) t ,J=l
N
:S ,2:
and assumpt ion
N I l
t ,J= l
Eij
a(Tiv ,Tiv) 'ia(Tjv ,Tjv) 'i
I l
N
Eija( v ,Tiv) 'ia(v ,Tjv) 'i :S wp(E) 2: a(Tiv , v) i,j = l 1 ( N N ) ~ i =l :S w p([) a(v,v )'i a i~ Tiv , i~ t:»
:S w
2:
IITi li a
Adding To complete s t he proof of t he upp er boun d N
a(TaddV,v )
= a(Tov , v) + a( LTiV, v)
:S w(l
+ p([))a(v , v )
i= l
To prov e the lower bound, we use t he decomposition of assumpt ion (2.2)
o We remark t hat Eij :S 1 and t hus p([) :S N . Very often , an upp er bound for p([) ind epend ent of t he number of subspaces can be given. Thus, for man y int eresting ap plications, it is a routine mat t er to obtain a good bound for th e lar gest eigenvalue of T add . For multilevel t echniques, a strengthened Ca uchy- Schwarz inequ ality plays an imp ortant role, and very ofte n we find Eij :S C q-li- j l with q < 1. Then , p([) :S C is bounded independent ly of the number of subspace s; see, e.g., [Yse86]. In the case of overlapping domain decomp osition methods, a coloring argument is used. Each subspace is associated with a color such th at the subspaces Vi and Vj are a-ort hogonal, i.e., Ei j = 0, if t hey have t he sa me
2.1 Abstract Schwarz Theory
91
color. Then, the spectral radius p(E) is bounded by the minimal number of required colors; see Fig. 2.4. All subspaces having the same color can be grouped together into a class of subspaces. The number of non zero entries in each row of E is bounded by the number of classes. Moreover, each class can be regarded for the analysis and the implementation as one subspace; see, e.g., [SBG96].
Fig. 2.4. Coloring of sub domains into four classes
If aie,,) = a(',') restricted on Vi x Vi, (2.3) holds with w = 1. The delicate point in the analysis of an additive Schwarz method is often to find an optimal constant Co which measures the stability of the decomposition. A classical example of an additive Schwarz method is the Jacobi method. Here, each subspace is one dimensional and the number of subspaces is equal the dimension of V. In the following, we briefly review three typical situations of decompositions of V into subspaces in the case of piecewise linear Lagrangian finite elements. Many Schwarz variants have been designed and analyzed for this standard case in 2D and in 3D. Recently, they have also been generalized to higher order elements including spectral and hp-methods [Cas97, GC97, GC98, Pav94a, Pav94b, PW97], elliptic systems [PWOOa, PWOOb], and vector field discretizations [HTOO, TosOO, TWWOO, WTWOO]. Our first example is a two-level additive method with overlap. The domain [2 is connected with two triangulations, a macro-triangulation DI and a fine triangulation Th. In addition, [2 is decomposed into overlapping sub domains f2i . For simplicity, Fig. 2.5 shows the special case where each sub domain is the union of elements of the fine triangulation.
Fig. 2.5. Macro-triangulation and fine triangulation on
n
92
2. Iterative Solvers Based on Domain Decomposition
Each sub domain does not have to be the union of elements in the fine triangulation, and the decomposition into subdomains and the macrotriangulation do not have to be related. We assume that the diameters of the elements of the macro-triangulation are roughly the same as the diameters of the adjacent sub domains, and that both, elements and subdomains, are shape regular. The minimal width of the overlapping region of two adjacent subdomains is denoted by 6, and we call 6/ H the relative overlap. Then, Va := VH is just the standard conforming finite element space associated with the macro-triangulation, and the subspaces Vi, are given by Vi:={vEVlsuppvcf?;},
l:::::i:::::N,
where N is the number of subdomains. The degrees of freedom of Va are marked by filled circles in Fig. 2.5. Defining T, as orthogonal projection with respect to the bilinear form a(·, '), the additive Schwarz method yields a condition number bounded linearly by the inverse relative overlap
see [DW94, Wid99]. It has been shown only recently in [BreOO] that this estimate is sharp if the width of the overlap is bounded by ch. We remark that the results holds also for the 3D case. The next example is an iterative substructuring method for PI-Lagrangian finite elements; see [BPS86b, BPS89]. In [BPS89J, a wirebasket algorithm in 3D has been proposed. Here, the condition number of the additive operator is bounded by a polylogarithmic factor. Three different types of subspaces define the operator T ad d ; see Fig. 2.6. A coarse subspace is necessary to obtain bounds for the condition number independent of the number of subdomains. As in the case of the overlapping variant, the coarse subspace is the finite clement space associated with the macro-triangulation, i.e., Va := VH.
Fig. 2.6. Decomposition of V into three types of subspaces
In contrast to the overlapping case, we now consider a non-overlapping decomposition of f? into sub domains. The elements T of the macro-triangulation define the subdomains, and as before we set
2.1 Abstract Schwar z Theory
VT := {v E V
I supp v e T},
93
T E IH .
Since t he coarse elements are non-overlapping the sum over t he VT is direct and a proper sub space of V . We add a third type of subspaces, which are associated with t he individua l edges of the coarse t riangulation VE := {v E V
I supp v C 1'1 UT2, a(v,wT ) =
0, WT E VT , T E
lId ,
where aT I n aT2 = E E £H , and £ H is the set of edges of t he macrot riangulat ion . The dimension of VE is given by the numb er of fine edges e C E minus one. Each element in VE is uniquely defined by its values at the vertices on the macro-edge E , and is obtained by a discrete harmonic exte nsion. Using th e exac t pro jections onto th ese subspaces, t he iterative substructuring method provid es a polylogari thmic upp er bound for th e condition numb er of the additive Schwar z method of t he form I';;(Tadd)
:s: c (1 + (log ~ )2) ;
see [BPS86 a, BPS86b] . Recentl y, it has been shown in [BSOO] t ha t this bound is sharp. In t he case of PI -Lagrangian finite elements, t here is an essent ial difference between the 2D and t he 3D case. To obtain quasi-optimal result s for the 3D case, t he coarse space Vo has to be modified in t he case of highly discontinuous coefficient s. The pr ojecti on onto the finite element space VH is replaced by a different suitable low dimensional problem. One of t he successful methods is based on a wirebasket type space and a quasi-projection is selected; see [BPS89]. We refer to [DSW94], for different possibles choices of Vo and To. In Sub sect . 2.2.2, we establish an iterative substructur ing method for Raviart- Thomas vector fields in 3D. As we will see, there is a prin cipal difference between the result s for t his finite element space and t hat for t he standa rd Lagran gian finite element space . Our last exa mple is the well known hierar chical basis method in 2D int roduced in [Yse86]. Here, th e subspaces are not associated with geomet rical subdomains but with a nested sequence of t ria ngulations , Ti , 0 :s: l :s: N. The t riangulat ion Ti+1 is obtained from Ti by decomp osing each element into four congrue nt subelements . Now, th e sub space Vi is associate d with the t riangulation Ti and defined by
Vi
:= V7i \ V7i_l '
O:S: i :s: N ,
where V7i, 0 :s: l :s: N , is th e st and ard finit e element space associate d with t he trian gulati on Ti, and VT_l := 0. Using this not ation , we have V = VTN' In cont rast to the first example, these spaces form a dir ect sum. The filled circles in Fig. 2.7 illustrate th e nodes of t he finite element space V . Each circle havin g t he same size is a degree of freedom of the same Vi , the lar ger t he diamet er is the smaller th e index i . The lar gest circles belong to Vo and t he smallest to VN .
94
...
2. Iterative Solvers Based on Domain Decomposition
....
•• o-~
..'"
III" -""
..#1
JiI1h
'*
...
1:
... I "I'
....
i
'IIII'
Fig. 2.7. Hierarchical decomposition of the nodes of V
The implementation can be based on a quasi-projection. One possibility is to define the modified bilinear form ii (-, .) by Ci(Vi' Wi) :=
L Ctj (3j a(¢;, ¢;)
,
j
where ¢j are the hierarchical nodal basis functions of Vi, and Vi := Lj Ctj¢;, Lj (3j¢;. Then, a straightforward computation shows that the quasiprojection T, is given by
ui, :=
This can be also interpreted as a decomposition of Vi into one dimensional subspaces associated with the nodal basis functions
¢;
Vi
= Lspan j
¢j =: L~i j
In this case, the exact projections P] onto the one dimensional spaces ~i are given by i a(v,¢;) i Pjv = a(¢;, ¢j) ¢j , and we find T; = Lj case
Pj.
The following quasi-optimal result holds for the 2D
see [Yse86, Ban96], where N is the number of refinement levels. In the 3D case, only an exponential bound in N can be obtained. For all three examples, the upper bound for the norm of Ta d d is bounded independently of the number of subdomains. A coloring argument is used to bound the spectral radius of E in Lemma 2.1 for the two first examples. In the case of the hierarchical decomposition, the constant upper bound is based on a strengthened Cauchy-Schwarz inequality. It can be shown that
2.1 Abstract Schwarz Theory
· vt·)! a(v·J ' vJ·)! , a(v t,· vJ·) _< C 2- li - j 1a(v"
v t· E
Vi , V J·
T T.
95
E VJ · '.
see, e.g., [Yse86]. In Subs ect . 2.2.3, we show that the same qualit ative bound can be obtain ed for Raviart- Thomas finite elements even in t he 3D case. 2.1.2 Multiplicative Schwarz Methods
The secon d imp ortant famil y of Schwar z methods is th e multiplicative one. A t heoretical ana lysis of t his ty pe can be found in [BPWX91b, CW92 , CW93 , SBG 96, Wid99] . Ex amp les are given by multigrid methods; see Subsect. 2.1.3. The multiplicative Schwarz vari ant is defined by N
Tmul := Id -
II (Id i=O
Ti )
.
In cont rast to the additive type, t he definition depends on the ordering of the subspaces . A classical example of a multipli cative Schwar z method is t he well known GauE-Seidel method. Using t he multiplicati ve vari ant as an iteration method, the error propagation opera tor can be written as N
E rnul =
II (Id i=O
Ti )
The following lemm a gives an upp er bound for th e norm of Ernul, a proof of which or quit e similar result s can be found in [BPWX91 a, SBG96 , Wid99]. Lemma 2.2. Under the assumptions of Lemma 2.1 and w < 2, the energy norm of the error propagation operator of the multiplicative Schwarz method is bounded by 2 -w IIEmultlla :::; 1- (1 + 2w2p([ )2)c g ,
where w:= max(l , w). Proof. To obtain an upp er bound for th e norm of t he erro r prop agation operator Ernul, we setEi := (Id - T i ) .. . (Id - To), 0 :::; i :::; N, and E_ 1 := Id and define H; by R ; := (2 - Ti)Ti . Then , we have the identi ty
Here, t he t ranspose is taken with respect to th e bilinear form a(·, .). Summing over all i and using a telescopic cancellation yield
E~ul Emul
= Id -
N
LET-IRiEi-l i=O
96
2. Iterative Solvers Based on Domain Decomposition
Und er the assumption w < 2, the operators R ; are symmetric and positive definit e. Moreover by mean s of IITill a :S w, we find R; 2:: (2 - w)Ti and thus N
E~uIEmul :S Id - (2 - w)
L E[I TiE i-l
.
(2.5)
i=O
Using t he identity E j + TjEj- 1 = E j- 1 , 0 :S j :S N , we obtain by summing over j and canceling common terms i- I
Id = E i -
1 + LTj E j - 1 ,
1 :S i :S N
j=O
Now, this identity can be used to obtain an upp er bound for a(Tiv , v) . The assumption (2.4) and the bound for IITill a tog ether with the Schwarz inequality yield
a(Tiv ,v )
i- I
= a(Tiv , E i- 1 v ) + a(Tiv, Tov) + 1:= a(Tiv, TjEj- 1 v) j=1
:S
a(Tiv, v) ~ (a(TiTov ,Tov) ~ + w
:S a(Tiv, v) ~
t €ija(TjEj-lV,Ej-l V) ~)
J=1
(a(TiTov,Tov)~ + w f €ija(TjEj-lV, Ej-l V)~ )
a(Tiv ,v) :S 2a(TiTov ,Tov) + 2w2(
f
J=1
J=1
€ij a(TjEj_l V, Ej_lV)~ ) 2
.
Here, we have used th at €ii = 1, €ij 2:: 0, and th e definition of W. Fin ally, we sum the last inequality from i = 1 to N and use a((Tadd - To)Tov,Tov) < w p(£) a(Tov , Tov). Then , p(£) 2:: 1 gives
a(Tadd v , v ) = a((Tadd - To)v , v ) + a(Tov , v ) N :S (1 + 2w2p(£))a(To v , v ) + 2w2p(£) 2 1:= a(TjEj_ 1 v , E j- 1 v )
:S (1+2w2p(£)2)
N
j= 1
1:= a(TjEj_ 1 v ,Ej- 1 v)
j=O
.
Now in term s of (2.5) and t he lower bound of Lemma 2.1 , th e norm ofthe err or propagation operator of the multiplicative Schwarz vari ant can be bounded by 2
2- w .
IIEmultll a:S 1- 1 + 2w2p(£)2 ;~[
a(TaddV, v) 2- w a(v , v) :S 1- (1 + 2w2p(£)2)Cg D
Obviously, t he bound is of int erest only if w < 2. If we use the exact projecti on for Ti , w = 1. Otherwise a simple rescaling of the projection-like operators T;
2.1 Abst ract Schwarz Theory
97
will always yield w < 2. However, a rescalin g of some T; also influences the constant Co in Lemma 2.1. We note that t he multipli cative vari ant of a Schwar z method , very oft en , results in fewer iterations t o obtain a given accuracy th an t he correspond ing addit ive method. However , additive methods can be very oft en mor e eas ily par allelized . In cont rast t o t he addit ive variant , th e multiplicative one is in general non- symmetric. A symmetrized form can be defined by T m ul + T;;:ul ' where T;;:ul is t he adjoint operator or by
In the case t hat the exac t projection is used for To, we can use (Id - T O)2 = (Id - To) to eliminate one ste p. Many other vari ants , such as hybrid schemes, can be obtain ed by combining addit ive and multiplicative components. As an example of a multiplicative Schwarz vari ant, we mention the V-cycle for standard Lagran gian finit e elements [Bra93, Osw94]. The corresponding addit ive variant is the well known BPX-preconditioner [BPX90 a]. We refer to [G095] for an abst rac t t heory for addit ive and multiplicative Schwar z methods. A different way t o analyze a multigrid method , in particular W- cycles, is discussed in t he following; we will give a brief introduction t o multigrid methods in Subsect . 2.1.3. In Sect. 2.3 and Sect . 2.5, we focus on two multigrid methods for mor t ar finit e elements . To prove level ind epend ent convergence rates, we do not use th e Schwar z th eory but inst ead ideas discussed in [Hac85]. 2.1.3 Multigrid Methods Multigrid methods provi de optimal iteration schemes for t he solut ion of systems of equations arising from t he discreti zation of ellipt ic boundary valu e pr oblems. The computational cost t o reac h a given accuracy is linear in th e number of unknowns. For a general introduct ion to t hese methods and relat ed t echniques, we refer to [BH83, Bra93, BS94, BY93, Hac85, McC87, Osw94]. There are two approaches of establishing bounds for the convergence rate of a multigrid meth od . One uses th e abst rac t fram ework of multiplicative Schwarz method s and can be found for exa mple in [BS94, Br a93]. We do not discuss t his ap proac h here. In our analysis, we instead follow the arguments in [Hac85] and establish suitable approximat ion and smoot hing properties; a level ind ependent convergence rat e for the W-cycle can then be shown only if t he number of smoot hing ste ps is lar ge enough. The addit ive vari ant of a Schwarz method defines, in general, a pr econditioned syste m and Kr ylov subspace methods, e.g., a conjugate gradi ent method , can be used as accelera t ors. In cont rast t o this , the multiplicative form itself provides an efficient iterative solver . Here, we briefly discuss the bas ic ingredients of a multigrid method. It consist s of smoot hing steps and correct ion steps in lower dimensional finit e element spaces . The smoot hing
98
2. Iterative Solvers Based on Domain Decomposition
part damps the oscillatory component of the error whereas the correction is associated with a problem on a coarser mesh. Multigrid techniques are motivated by the following observation: Classical iteration schemes restricted to the subspace of high frequency show a fast convergence. The subspace of low frequencies cause the observed slow convergence. Figure 2.8 illustrates the smoothing effect of a Gaufi-Seidel method. After a few smoothing steps, the error on the fine mesh can be accurately represented on a coarser mesh.
4
o
o
0
0
Fig. 2.8. Effect of one symmetric Gaufi-Seidel smoothing step
Similar to the multilevel method described in Subsect. 2.1.1, a multigrid method is often associated with a nested sequence of triangulations and corresponding finite element spaces Vk , 0 :S k :S j. We restrict ourselves to the case of nested spaces, Vo C VI ... C Vj, and refer to [BDH99b, Bre89, BV90, Osw94] for a discussion of nonconforming situations. The iterative solver for Azuz = fz, on level l, is defined recursively. It depends on the choice of the transfer operators Vi ----+ Vi-I, and ILl: Vi-I ----+ Vi, 1 :::; l :::; i, and the smoothing operators G l , G z
1;-1 :
ur+
1
:=
MG(l, ul' [i),
v ~0 ,
where up is the initial guess. On level zero, MG(O, u~, fo) is defined by MG(O,u~,fo) := AOlfo. The multigrid operator MG(l,ul,fz), l > 0, is given recursively and consists of three steps - presmoothing, error correction and postsmoothing. The correction step depends on the transfer operator. • Presmoothing step: For 1 :S i :S
Zzi
:=
Zii - I
Tnl,
let
+ G-1 1 (1Z-
A ZZzi-I) ,
where Tnl ~ 1, G l is a suitable smoothing operator and • Correction step: dz := fi - Azz;'" - Defect restriction: di- l := I;-ld i
(2.6)
z? := ul .
2.2 Vector Field Discretizations
99
- Coarse grid correction : For 1 ::; j ::; p , let j . - MG (l j-1 d ql1 .- 1 , ql-1 ' l-l ) ,
where q?-l := o. - Prolongation of the correction : Z;"l +l := • Postsmoothing step: For m1 + 2 ::; i ::; m1 1 z (nt Zli := zli- 1 + G-
Z;"l + IL l qf- 1 + 1 + mz, let - A lZli- 1) ,
wher e mz ~ 0 and G z is a suitable smoothing op erator. Finally, a multigrid step is defined by M G (l, u'(, Il) := z;"1+m2+1 . T he case p = 1 is called the V-cycle and the case p = 2 t he W -cycle. In 1 pr acti ce, t he rest ricti on is, very ofte n, chose n as t ranspose of t he prolongat ion IL l ' and in the case of nest ed spaces VI- 1 C VI the natural injection is taken for IL l. The easiest typ e of a smoot her is a damped Rich ardson met hod where the damping fact or has to be smaller t han one over the lar gest eigenvalue of A l . Symmet ric Gaul3-Seidel methods and incompl et e LV-fact orizations pr ovid e more robu st smoot hers. The choice m 1 = m z , G1 = and IL l = (IL 1)T yields a symmet ric it erative solut ion scheme . One possibility t o prove level ind epend ent convergence rate s for W- cycles relies on suitable approximat ion and smoot hing properties [Bra97, Hac85). The analysis of the mult igrid case is based on the two gri d case and a perturbation argument . In t he two grid case , approximat ion and smoothing prop erties yield that t he convergence rates are ind ep endent of t he meshsize provided t hat t he numbe r of smoothing ste ps is large enough. Additionally, a stability est imate for t he smoot hing scheme is required for t he multigrid analysis.
11-
Gr,
2.2 Vector Field Discretizations In t his section, we consider a boundar y valu e pr oblem for vect or fields , associate d with t he divergence op erator . We focus on t he 3D case and t he well known Ravi art-Thomas finit e elements; see, e.g., [BF91). Applications of t hese vect or fields can be found in [AFW97) . In Sub sect . 2.2.2, we develop an it er at ive substruct ur ing method for this class of finit e elements and in Sub sect . 2.2.3 , one based on a hierar chical basis. We not e t hat t here are several inter esting differences when compared to the H 1 -case. We consider the following second ord er partial differential equat ion for vector fields
Lu := - grad (a divu)
+B
u
tr- n
=f, = 0,
in n , on an
(2.7)
100
2. It erative Solvers Based on Domain Decomposition
where D is a bounded polyhedr al domain in IR3 of unit diamet er , and nits outward norm al. We assume that f E (L 2 (D)) 3, th at the coefficient matrix B is a symm etric uniforml y positive matrix-valued function with bi j E L OO(D) , 1 :::; i,j :::; 3, and t hat a E L OO(D) satisfies a ~ ao > 0 almost everywhere . T he weak formul ation of problem (2.7) is defined in a vector valued Hilbert space of which (HI (D)) 3 is a pr oper subspace. Applying Green's formul a, we find t ha t H (div j D) is a suitable space
equipped with the inner product (., ')div (W,V)div :=
!
w·vdx+
[J
!
divwdivvdx
[J
T he corresponding norm is defined by Ilwllaiv := (w , W)div. The boundar y condition in (2.7) will be imposed on t he space. A trace theorem given in [BF91] shows t hat the normal component, v -n , of any vector v E H(div; D), on t he boundar y aD , belongs t o t he spa ce H- I / 2(oD) . The subspace of vectors in H( div ; D) with vanishing normal component on aD is called H o(div ; D). We are now prepared to formul at e the weak vari ational form of t he boundary value problem (2.7) : Find u E Ho(div ; D) such t hat a(u ,v) = !r.vdx ,
VEHo(div j D) ,
(2.8)
[J
where t he bilinear form a(·,·) is given by a (w, v ) := !(adivwdiVV+BW.V)dx ,
w ,vEH(div ; D).
[J
An energy norm defined by IIIwlll 2 := a(w , w) is associate d with the bilinear form a(·, -). In parti cular , t his energy norm is equivalent to the Hilbert space norm II . Ii div. The equivalence constant s depend on th e coefficient s and can be quit e lar ge or quite small. Before defining t he Raviart- Thomas space of lowest ord er , we consider t he trace of an element in H (div ; D) in more detail. Throughout t he rest of t his sect ion, we will work with scaled Sobolev norm s; the size of the domains is reflected in weight factors. Given a bounded open Lipschitz dom ain D CD, with a boundary aD and a diameter HD, let 1·l s;D denote t he semi norm of the Sobolev space H S(D). Then the scaled norms are defined by 2 2 1 2 11 4>l k D = 14>II ;D + HD211 4>IIO;D'
1
4> E H (D) ,
2.2 Vector Field Discret izati ons
101
and 2 2 1 2 11 ¢11 1'<>D = 1¢l l. 11 ¢llo·,aD, aD + -H 2 'v 2' D
1
¢ E H 'i (8D ) .
In t he case t hat D = rl, we will drop the reference to the region. In contrast to t he pr evious sections, the H- l/2-norm will from now on reflect t his scaling, and it is defined by Ilw , nll _l2 'aD := '
sup
1 4> E H 2 ( tl D ) 4>'1'0
(w · n, ¢) 11 ¢11 !; aD '
wE H (div ;rl) ,
where (-, .) represe nts t he du alit y pairing between H - 1 / 2( 8D) and H 1 / 2( 8D) . T he following lemm a provides a basic est imate for th e trace of an element in H(div ; rl) . The proof is quite elementary and is based on Green 's formula and a scaling argument . Lemma 2.3. Th ere exists a cons tan t C , which is in depen den t of th e diameter of D but depen d on the shape regularity of D , such that for w E H (div ; rl) Ilw , nll ~ ! ; aD :::; C (1I wI1 6;D + H bll div w!l6;D) . We will use t he well known Raviart- Thomas spaces for th e discreti zation of (2.8). T he finit e element approximation is given on a t riangulat ion Th , t he element s of which are denoted by T . The set of int erior faces and edges of t he trian gulations Th is called :h and [ h , respecti vely. 2.2.1 Raviart-Thomas Finite Elements Our st udy concerns t he lowest order Raviart- Thomas finite elements for t he discrete ap proximation of (2.7); see [BF91]. In Sect . 1.4, t his space was already used to define a mort ar finite element discreti zation. Here, we work in 3D and t he definiti on of t he degrees of freedom is essent ial, and we th erefore consider t his space in more detail in t he rest of th is subsect ion . T he globa l Raviar t-Thomas finite element space RT(rl ; Th) is defined by mean s of t he local ones
RT( rl ; Th) := {w E H(div; rl)1 wI T E RT(T ), T E Th} , where RT(T) stands for t he local Raviart-Thomas space . In the case of a hexahedr al triangulation , where t he elements are cubes , t he local space has dimension six, while in t he case of a simplicial triangulat ion, where t he elements are tetrah edras, t he local space is four dimension al. For a cube with sides parallel to t he coordinate axes, RT(T) is given by
102
2. It erative Solvers Based on Dom ain Decomposition
In t he case of a tetrahedr a, RT(T) is defined by
RT(T) :=
+ f3 X ) + f3y , a3 + f3z
a 1 (
(2.9)
a2
The degrees of freedom of RT( D ; T,.) are given by the averages of th e norm al compo nents over the faces F of th e triangulation:
AF(W) :=
I~I
J
w · n dC7
F
Here, IFI is t he area of th e face F and t he direction of the normal can be fixed arbit ra rily for each face. This formula also defines th e natural int erp olation operator from H(div ; D) onto t he space RT(D ; T,.) . We not e t hat t he norm al component of any Raviar t- Thomas vecto r field is constant and continuous across each face, see Fig. 2.9.
Fig. 2.9 . Local degrees of freedom of a lowest order Raviar t-Thomas vect or field
We define the subspace of vectors with vanishing norm al components on the boundar y of D by
Vi, := RT(D j T,.) n Ho(div ; D) As in t he case of Lagran gian finite elements , t he L 2 -nor m of t hese discrete vect or fields can be bounded from above and below by mean s of t he values of t heir degrees of freedom . For W E RT(T) , we have C
L (H; AF(W)f
F e8T
: :;
Ilw11 6;T :::;
C L (H; AF(W)f Fe8T
'
(2.10)
where H F is the diameter of t he face F . The equivalence constants depend on the aspect ratio of t he elements, bu t do not depend on the diameter. Moreover , t he following inverse estimate holds: 1 IIdivwllo;T :::; C HT II wll o;T,
w
e RT(T)
2.2 Vect or Field Discreti zation s
103
These bounds can easily be shown by using t he affine equivalence of the elements of t he t riangulations and t he finite dimension of th e local spaces . We not e t hat relativ ely few st udies exist of domain decomp osition methods for H(div ; n) and H(curl ; n) in 3D; we refer to [CPRY97, HTOO, TosOO], for two-level overlapping methods, to [AFWOO, BDH+99a, Hip96, Hip97, Hip98] for multilevel and multi grid methods, and to [AV99] for a study of an iterative sub structuring method in H(curl ; n). We refer to [AFW97 , Bre92, EW92, Mat93 a, Mat93b] and to the references therein , for some Schwarz methods for problems in H (div ; n) in 2D. A certain class of multilevel methods for the mixed approximation of th e Laplace equation is discussed in [Sar94] . Here, we const ruct a quasi-opti mal it erative subst ruct ur ing method as well as a hierarchical basis method for t he 3D case. 2 .2.2 An Iterative Substructuring Method
In this subsection, we int rod uce and analyze our iterative subst ruct ur ing met hod, originally described in [WTWOO] . For a quite general introduction to dom ain decomp osition methods for vector fields discretizations , we refer to [Tos99]. We restrict ourselves to the case of a hexahedr al triangulation Tit which is obt ain ed by qu asi-uniform refinement from a coarse macro-tri angulation TIl ' From now on t he generic elements, faces and edges of Tit are denot ed by t, t , and e, and t hose of TIl by T , F , and E , respectively. The set of int erior faces of the macro-tri angulation TIl is called :FII . We use a simplified notation for the ratio of the meshsizes between macro and fine triangulation. In t he context of elementwise est imates, H / h denot es the ratio of t he local meshsizes whereas in globa l bounds, H / h stands for th e maximum of t he local rati os. As we have seen in Sect . 2.1, t he first ste p towards th e introduction of an addit ive Schwarz meth od is to define a set of subspaces; see Lemm a 2.1. As in t he 2D case for standa rd Lagrangian finite elements, we introduce t hree different typ es of subspaces called VII , VF and VT. To obtain scalabl e bounds, we cannot avoid th e use of a globa l space. But in contrast to t he standard Lagr angi an finite element s in 3D, th e low dimensional Raviar t-Thomas space associated with the macro-triangulation, VII := RT(n ;TII) nHo(div ;n) , can be used for this purpose while at the same ti me the constant in t he estimate of the condit ion number does not depend on the ju mp s of the coefficients . The local spaces V'r , T E TIl , are associated with the elements, also called substruct ures, of t he macro-t rian gulation. Each element in VT ,
V'r := {v E Vi, I supp veT} ,
T E TIl ,
has sup port contained in T. Thus, the projections onto VT are orthogonal. The t hird t ype of subspaces is associate d with t he faces F E :FH of the macro-t riangulation. T he supports of its elements are contained in two substructures T 1 and T 2. For each interior face F E :FII , th ere are t wo elements
104
2. Iterative Solvers Based on Domain Decomposition
T 1 and T 2 E TH such t hat F = aT1 n aT2, and we set '1'F := '1'1 U'1'2. The face spaces are defined by VF
:= {v E
Vh
I a (v , w)
= 0, w
s V T1 + V T 2 ,
suppv C '1'F }
We not e t hat an element v E VF is defined uniquely by its valu es v . n on F. Now, we decomp ose 11" into t he coarse space VII , the face spaces VF , F E FH ' and t he int erior spaces V T , T E TH, and we observe that the coarse space VH is contained in t he union of the face and int erior spaces. The decomp osition, Vh=VH
+
L
VT
+
T ETH
L
VF
,
(2.11)
FEFH
on which our iterative sub structuring method is based , is therefore not a direct sum. Fi gur e 2.10 illustrat es t he local decompositi on of Vh into t he t hree different t ypes of subspaces. Each type of subspace is symbolized by t he faces which arc associate d with the degrees of freedom.
/:
/.
/ : / i / : / : :
::v ·('t"-.:+:~·l- - ._;('t-... : //;:--1-- ,::·t:·::f·· ) /-i- .. i7 •.. .. )/:_- -)L .. .. V /',
Fig. 2.10. Decomposition of Vh into t hree types of subspaces For simplicity , we restrict our selves t o t he case that t he exact pr ojections onto the sub spaces arc used to define the additi ve Schwar z method. According to Lemma 2.1, the proof of an upper bound for t he norm of Tad d is elementary. It is based on a coloring argument, and t he coefficient s a and B do not enter int o t he bound. The crucial par t is t o find a decompositi on of v E Vh such that (2.2) holds wit h a Co as small as possible. A bound for t he corresponding multi plicat ive vari ant follows from Lemma 2.2. Ob serving t hat V h = L:TETH V T + L: FEFH V F is a dir ect sum, the decomposit ion v
=VH
+
L
TE TH
VT
+
L
vF ,
F EFH
with V H E V H , V F E V F , and V T E V:r , is unique as soon as VII has been fixed . A first ste p towards t he pr oof of a sharp bound for Co is t o find a suitable v H E VH . In the following subs ubsection, we consider a standard int erpolant PH ont o t he global coarse subspace. By mean s of th e discret e norm equivalence (2.10) , we will establish stability bounds in t he £ 2_ and
2.2 Vector Field Discretizations
105
the H (div ; D)-norm. A detailed discussion of this operator can be found in [WTWOO]. 2.2.2.1 An Interpolation Operator onto V H. An important role in the analysis of our additive Schwarz method is played by the interpolation PH onto the global coarse subspace VH . This interpolation operator is defined in terms of the degrees of freedom of VH by
Ap(PHV):=
I~I /
v· u da,
FE
hi .
F
We note that the stability estimates for the interpolant PH will enter into our estimate of the constant Co. As we will see, PH is not uniformly stable in h. The following lemma can be found in [WTWOO] and gives a bound for the interpolant PH in terms of the ratio H / h. Lernrna 2.4. There exists a constant C, which depends only on the aspect ratios of T E TH and the elements of TI", such that for all v E Vh,
(2.12)
Iidiv (PHv)[16;T ::; Ildivvl16;T , IlpJIv116;T
< C ( (1 + log ~) Ilv116;T + H}lldivvI16;T)
(2.13)
Proof. For a better understanding of the techniques, we review the proof given in [WTWOO]. By a simple computation and the use of Green's formula, we find that (div (pJIv)) IT is constant and
where IIH is the L 2-projection operator onto the space of constants on T E TH; see [BF91, Sect. III.3.4]. Inequality (2.12) follows immediately. The proof of (2.13) uses Green's formula, the norm equivalence (2.10), and a partition of unity very similar to the one given in [DSW94] for the simplicial case. Consider a face F c BT, and note that it is decomposed into non-overlapping faces of the fine triangulation; see Fig. 2.11.
H
Fig. 2.11. Decomposition of F
106
2. It erative Solvers Based on Domain Decompositi on
Number t hese faces so th at h , 1 :S i :S nF , have at least one vertex on an edge of F ; see Fig. 2.11, and let II ,12 ,13 ,14 be t he faces t hat contain a corner point of F . Let t, C T , be t he associated element s. We remark t hat, by ass umption, t he t riangulation restrict ed to t he face is qu asi-uniform, and t hus nF :S C( H l h) . Let 1'JF be a cont inuous , piecewise t rilinear fun cti on defined on T , which vanishes on aT \ F and is equal to one at all int erior mesh point s of F. It equa ls one in the grey shadowed region in Fig. 2.11. The exte nsion of 1'J F t o t he int erior of T has values between zero and one, and t he absolute valu e of it s gradient is bounded by C I max (r , h), where r denotes t he dist an ce to t he wirebas ket of T . The wirebasket of T is t he union of t he twelve edges of T. We refer to [DSW94] for an explicit construction of such a function for a simplex; t his constru ct ion can easily be adapted to the cubic case. The followin g upper bound for t he HI-semi norm and t he £ 2-norm can then be est ablished I1'J Fl tT
:S CHT (1 + log ~),
II1'J F I16;T :S CH'f .
(2.14)
Using (2.10), it is sufficient t o bound t he absolute valu e of AF(PHV) to get an upper bound for t he £ 2-norm of PHV. Appl ying t he definit ion of PH and Green 's formula, we obtain
where f3i = 3/4 for 1 :S i :S 4 and f3i = 1/2 for 5 :S i :S t he £ 2-norm of PHV can be bo unded by H TllpHV I1 6'T
,
:S C 2:= r cer
(fT(
+ C F~T
n.j»:
(1'J F div v + grad 1'J F . v ) 4
Thanks to (2.10),
dX) 2
n
) 2
t i~ IlilA!i (v) + ~ i~ IhlA!i (v)
By mean s of (2.14), we find an upp er bound for t he first sum on t he right han d side (2.15) Applying (2.10) again and keepin g in mind t hat nF is bo unded by C'Hl li , we find an upper bound of t he second te rm
2.2 Vect or Field Discreti zati ons
C
L
FeaT
L 11i1 2(Af i (v)) 2 :::; CHT II vll 6;T
107
np
nF
(2.16)
i= 1
The upp er bound (2.15) and inequality (2.16) finally give
IlpH v ll 6;T :::; CHfildivvll~ ;T + C
(1 +
log
~) IIvll~;T
0
We can obtain a similar est imate for the energy norm on each subst ruc t ure J B
(PHV) ·(PHV)dx:::;C ;~ (1+10g~) J
l'
Bv ·vdx
l'
+ CHf
l TJa divv divvdx aT l'
Here, aT is t he minimum of a( x ) on T, and (31' and IT satisfy
Thus, the const ant in t he corresponding global est imate dep ends on the ratio of the coefficients B and a on individual sub structures, but not on the jumps of t he coefficients between the subst ruc t ures . Remark 2.5 . Th e interpolati on operator PH is logarith mically stable in th e II . IIdiv -norm . Th is resu lt holds fo r the 3D case as well as in the 2D case; see [TWWOO). In contras t, th e nodal in terpolant onto contin uous finit e element spaces has a norm which grows as (H/h)I / 2 in the 3D case; see [DSW94j. Th is is an essentia l difference between the fa ce based R aviart -Thomas finite eleme nts and th e st andard ver tex based Lagrangian finite elements. 2.2.2.2 An Extension Operator onto V F. T he second basic tool in t he analysis of our it er ative subst ructur ing method is an exte nsion op er ator from Wh(F) onto VF. Here, Wh(F) is t he finit e element space of piecewise constants on f C F . The idea is analogous to a result for the st andar d finit e element case , in which the HI-norm of a discret e harmonic function is bounded by t he H 1 / 2-n orm of it s t race on the boundary. The following lemma shows t hat t he exte nsion op erat or is uniquely defined and stable. Lemma 2.6 . Th ere exis ts a unique extension operato r Ji F : Wh(F) ----+ VF satisfyin g (Ji F/-L) . n ip = /-L , /-L E Wh(F) . Furth ermore if JF /-L da = 0, we have th e follo wing sta bility estim ate
where /-L is exte n ded by zero on
aT \ F.
108
2. Iterative Solver s Based on Domain Decomposition
Proof. The degrees of freedom of VF are exactly the normal components on f C F , which are constant on each face f . Thus the trace space of norm al compo nents of VF is exac tl y Wh(F) . Therefore, the existe nce and uniqueness of such an extension operator follows from the definition of VF. The pr oof of t he stability est imate is based on a lemma established in [WTWOO] , where a stable exte nsion to a divergence free vector field is const ructed for f-t E Wh(F) with I F u da = O. Here, we sket ch the ideas of t his proof and refer to [WTWOO] for t he det ails. The starting point is a Neumann boundar y valu e problem for t he Laplace operator on T , where FeaT . We extend f-t by zero on aT \ F . The extension is still denoted by f-t . Then , the Neumann boundar y value problem -Llu = 0,
in T ,
au an =f-t ,
on aT
(2.17)
has a solut ion . Moreover, we obtain a unique solution of (2.17) by imposing the addit ional condi tion u dx = O. An elementary regularity result gives an upper bound of t he H -norm of th e solut ion in te rms of t he boundar y condition, i.e., luh ;T S; C IIf-t II-l!2;8T' The resulting vector field grad u is in H(div ; T) . In a second ste p, we apply the natural int erpolation operator onto t he Raviar t- Thomas vector fields, and denote the result by v /l' The stability, Ilvltlldiv;T S; ClukT , is a consequence of an approximation property and an inverse est imate . By const ruct ion v It is divergence free, and it has the same normal t rac e as 1-lFf-t on D'I', Thus, 1-lFf-t, restricted to T , can be writ t en as 1-lFf-t = v u + VT , where VT E VT . Keeping, t he definiti on of VF in mind , we find t ha t
Ir
where aT(-,' ) is t he restriction of a(', ') on T . 0 As already mentioned , an important step in the pro of of a sha rp bound for Co is t he choice of V H . Defining VH := PHV , we find
and t hus, by means of Lemm a 2.6, we get an upp er bound for t he norm of V F in terms of t he norm al component s of v - PHV restricted to F
where f-tI F = ((v - PHV) . n) IF and f-t = 0 on aT \ F . To apply Lemm a 2.1 , an est imate of t he H -l! 2-norm of f-t in te rms of the norm of v is required . We conclude this par agraph by proving a decomp osition lemm a for piecewise const ant fun ctions on t he boundar y of a subst ructure . We recall that t his space is exactly t he trace space of normal component s of Raviart-Thomas finite elements.
2.2 Vect or Field Discreti zat ions
109
The following lemm a is established in [WTWOO] . Here, we briefly discuss t he main ideas and omit some technical det ails . In the following each element f.L F in Wh(F) is extended by zero onto aT \ F and is st ill denoted by f.LF. The subspace of Wh(F) with zero mean value on F is called WO;h(F) := {f.L E Wh(F ), JF f.L do = O} , and the six dimensional space on aT containing all fun ctions which are constant on each face F is denoted by W H (aT) . Lemma 2.7. Let T be in TH' let f.LF E WO ;h(F) , FeaT , and let f.L := L- FC[)T f.LF · Th en, there exists a constant C, in depen dent 01 f.LH E WH(aT)
and h such that
2 ). IIf.LFII_2!;[)T :s; C (1 + log hH) ((1 + log hH)!If.L + f.LHII_2! ;[)T + 11f.L11_!;[)T (2.18) Proof. By definiti on , t he mean valu e on aT of f.LF is zero. Applying a Poin care-Friedri ch's type inequ ality, we obtain
sup 1 E H'i([)T)
where c is t he mean valu e of¢ on aT. The essent ial idea is now t o replace t he supremum over H 1 / 2( aT) by t he sup remum over a suitable discret e space . We int roduce a space of bubbles on aT by
where Ai , 1 :s; i :s; 4, are t he bar ycentric coordinate funct ions of the face We also use t he space of conforming bilinear finit e elements on aT Sh(aT) := { ¢ E C(aT)1 ¢I! E
QU) , 1 caT}
I·
,
where QU) is t he space of bilinear funct ions on I. Let P : H 1 / 2 (aT ) --+ Bh (aT ) + Sh(aT) be a uniform H 1 / 2 -st able operat or which sat isfies
J
J
F
F
¢ f.LF do =
P ¢ f.LF da,
f.LF E Wh(F) , r c
er .
(2.19 )
For t he existence of such an operator, it is sufficient to construct one example. We set P ¢ := ¢S+¢B , where ¢s := Ps ¢ is given by a st andard locally defined H 1 / 2 -st able quasi-projection opera t or Ps ont o Sh(aT ) j see, e.g., [SZ90], and ¢B E B h(aT) is defined by
J
J
f
f
¢B do =
(¢ - ¢s ) da,
1c
et .
110
2. Iterative Solvers Based on Dom ain Decom positi on
Then , P satisfies (2.19) by construction. Fur th erm ore, t he Hl/2-semi norm of PcP can be bounded by IP¢>l l/2;8T ::; l¢>sI1/2;8T + I¢>BIt/2;8T. If t he quasiproj ection operator Ps satisfies an approximation prope rty, we find for the second term
which yields t he H 1 / 2- st ability of P. Moreover, the spaces Sh(8T ) and B h (8T ) satisfy an orthogona lity relation, and we find ('\l'l/Js , V'tPB)O;8T = 0, 'l/Js E Sh(8T ), 'l/JB E B h(8T). Then , an int erp olation argument and an inverse esti mates yield
These prelimin ar y considerations guarantee the following norm equivalence
Now, the supremum over H 1 / 2 (8T ) can be modified . In particular , we can replace t he cont inuous spa ce H 1 / 2 (8T ) by t he discret e spaces Sh(8T) and B h (8T ), and we find t he following upp er bound
II /1F II -1 '8T < C 2'
-
sup 1
t/JEH2(8T) ,p;econ st.
(
(/1 F, P ¢»
< C
1¢>I I '8T -
sup 4>EBh(8T) 4> =/:0
sup
4>ESh(8T)+Bh(8T)
2'
¢ ;econ st .
(/1F, ¢»
1¢>!I.8T 2
1
(2.21)
(/1F, - ¢» -+ 1¢>I I ;8T 2
sup 4> E Sh(8T)
(/1 F, ¢») .
1¢>II;8T 2
The first inequ alit y is a consequence of (2.19), whereas th e second follows from t he st abili t y in t he H 1 / 2- semi norm of P . Finally, t he last inequality follows by using (2.20). We next consider t he two term s on t he right hand side of (2.21) separ at ely, and start with the first te rm. A local inverse est imate and an interp olati on arg ument shows t hat t he rest riction ¢>F of ¢> E B h (8T ) to F is H 1 / 2- st able,
where ¢>F := ¢> on F and ¢>F := 0 on 8T \ F . Using t his bound, we find t hat for ¢> E B h(8T)
1(/1F, ¢»
I = 1(/1, ¢>F) I < 11/111- ~ ; 8T II ¢>FII ~ ; 8T I¢>I ~ ;8T I¢>I ~ ; 8T I¢>I ~;8T 11/111-1'8T I¢>F!I.8T ::; C < C I I /1 I I-~;8T 1¢II '8T 2
2'
2'
(2.22)
2.2 Vector Field Discretizations
111
Unfortunately, the second term in (2.21) cannot be bounded as easily. We start by defining for each ¢ E Sh(aT) a weighted average c¢ C¢./ {}F dO":= F
.I
h({}F¢) do .
F
Here, '1') F is given in the proof of Lemma 2.4, and h is the nodal interpolation operator onto Sh(aT). Then, the supremum in the second term on the right hand side in (2.21) can be replaced by sup ¢ ESh (8T ) 1'¥:const.
(MF, ¢) 1¢ll'BT 2'
sup
(2.23)
¢ E S h (8 T ) ¢#const.
i.e., we need only consider functions ¢ which have a zero weighted mean value on F. In the last step, we have used the following norm equivalence
This is a Poincare-Friedrich's type inequality, and can be easily proved by contradiction. We can now apply techniques well established for the standard Lagrangian finite elements in 3D; see [DSW94]. In general, ¢ E Sh(aT) cannot be decomposed in a uniformly stable way into face and edge contributions. Therefore, we start by decomposing ¢ into a sum of contributions ¢F supported on individual faces FeaT, and ¢w supported in a neighborhood of the wirebasket which is one clement wide; see Fig. 2.12.
Fig. 2.12. Neighborhood of the wirebasket
By means of this decomposition, we can rewrite ¢ as
¢=
L
¢F +¢w .
(2.24)
FeBT
The H 1 / 2 -semi norm on aT of ¢w can be bounded by means of an inverse estimate in term of its one dimensional L 2-norm over the wirebasket W
112
2. It er at ive Solvers Based on Dom ain Decom posit ion
Upper boun ds for Il cPwll& ;w and Il cPFll i/2;8T are established for t he simplicial case in [DSW94]. Following [DSW94, Lemm a 4.3 and Lemma 4.5], and observing t hat t he H I-norm on T of a discrete harmonic function can be bounded by t he H I / 2 -norm of its t race on aT, it can be shown t hat 2 IlcPwllo ;w
H) IlcPll2!;8T
~ C 1 + log h (
(2.25)
'
Il cPFII! ;8T ~ C(1 + log ~ )21IcPll ! ;8T .
(2.26)
The proofs in [DSW94] are for t he simplicial case, but they can be carr ied out in exactly t he same way for the hexahedral case and det ails are t herefore omitted. We find, by using t he splitting (2.24) , t hat
(/-LF, cP) =
L
(/LF, cPp) + (/-LF,cPw ) = (/-L,cPF ) + (/-LF, cPw ) .
(2.27)
Pe8T
Since h (7'J FcP) = cPF = t hat cq, = 0, we obtain
h(7'JFcPF ) on aT,
and since we can always assume
and Cq,F = O. The first term on t he right side of (2.27) can be bounded by mean s of (2.26):
For each
cPw , t here
cPB E B h (aT)
is a uniqu e
JcPw do JcPB da, =
f
f
such that
cPB =
E r.;
F .
f c
0 on aT \ F and
f
Moreover , t his mapping is locally uniformly stable in th e £ 2-norm on aT; Il cPBIIO;j ~ CllcPw llo;j. An inverse estimate toge t her with the definition of cPB easily yield
Il cPBII! ;8T ~
C ~ llcPB I I&;8T ~ C~ l lcPw ll&;8T ~ C Il cPwll&;w
By mean s of t his bo und and (2.25), we finally obtain
I(/-LF, cPw )1= I(/-LF, cPB )1= I(/-L, cPB )1~ CII/-LII-!;8TllcPw llo;w ~ C (l
H ! + log h) 211/-L11- ! ;8TllcPll! ;8T .
(2.29)
The pro of is complete d by combining (2.21), (2.22), (2.23), (2.24), (2.28), and (2.29). 0
2.2 Vector Fi eld Discreti zations
113
2.2.2.3 Quasi-optimal Bounds. We ar e now able to formul at e the cent ral resul t of t his section. In t he following theorem , we specify a suitable decomposit ion of v E Vh and give an upp er bound of t he constant Co in Lemma 2.1. This result was originally given in [WTWOO].
Theorem 2.8. For each v E Vh, there exis ts a decomposition v = v II +
L: VT + L: TE TH
vp ,
P EFH
corresponding to (2.11) , such that a(VH,VH ) +
L: a(VT ,VT) + L: a(vp ,VP) :::;C(1+1 0g~ra(v ,v) ,
TETH
PEFH
with a constant C, independent of h, H, and v . Proof. The choice VH := PHV and Lemm a 2.4 yield
where aTC· ) is the restriction of a (' , ') on one substructure T . Then , Vp is uniquely given as t he discret e harmonic exte nsion of J.Lp := (v - VH ) . n lF' The minimization property of 1-l p gives
see Lemm a 2.6. In a final step, we apply Lemm a 2.7 and bound IIJ.Lpl l- 1/ 2;8T in te rms of aT(v, V)1/ 2. Setting J.L H := VH . n , we obtain
Lemm a 2.3, the trian gle inequ ality, and Lemm a 2.4 yield
and t hus by applying Lemm a 2.3 to Ilv . n ll-l / 2;8T' we find aT(Vp,
H
vr ) :::; C (l + log h)
2
aT (v , v ) .
An upper bound for aT(vT , VT) is now an easy consequence of the trian gle inequ ality. Finally, t he global upp er esti mate is obtained by summing t he local ones over the elements of t he macro-trian gulati on. 0
114
2. Iterative Solvers Based on Domain Decomposit ion
Remark 2.9. Th e cons tant C in Th eorem 2.8 depends on the coeffici ents but not on the m eshsize. Since all estim ates are done locally on ind ividual su bstructures, it can be shown by a more careful analy sis that C is in dependent on the jumps of th e coeffi cients. However, it depends lin early on
Using exac t orth ogonal projections onto t he sub spaces, t he addit ive Schwar z operat or Ta d d , defined by t he decomposition (2.11), has a condition number which is bounded logari thmically in t erms of the ratio H / h K: (Tad d )
::;
C (1
+ log ~) 2
,
where th e constant C does not depend on the jumps of th e coefficients. This is an easy consequence of Lemm a 2.1, Theorem 2.8, and a coloring argument. 2.2.3 A Hierarchical Basis Method In t his subsect ion, we introdu ce a hierarchical basis method for Ravi artThomas finit e elements in 3D. We obt ain th e same qualit ative upp er bound for t he condition number of t he additive Schwar z meth od as for standa rd Lagr angian finit e elements in 2D; see, e.g., [Ba n96, Yse86, Yse93]. We recall that t his result does not hold in 3D for th e Lagran gian finit e elements. The starting poin t for the definition of our hierar chical bas is method is a nest ed sequence of ad ap tively generate d simplici al t riangulat ions , 70,' " Tj. We can use some standard refinement rules gua ra nteeing t hat t he t riangulations form a famil y of shape regular and locally quasi-uniform triangulations; see, e.g. , [Bey95]. The sets of int erior faces are denoted by Fi , o ::; I ::; j , and the associated Ravi art- Thomas finite element spaces are called VI , 0 ::; I ::; j. They form a sequence of nest ed spaces satisfying Vo C VI C ... C 10 C Ho(div ; rl). The local Ravi ar t-Thomas spaces with four degrees of freedom per element ar e defined by (2.9), and th e global space has one degree of freedom per int erior face. To define our Schwarz method, we consider two different types of decompositions - a horizontal and a vertical. The horizontal decomposition is based on the hierarchy of finit e element spa ces whereas th e vertical one reflects a Helmh olt z-typ e decomposition of the vector fields. This ty pe of split t ing has already been used in [HW97, Woh95] for the 2D case . We also note that a hierar chical basis pr econdi tioner for the saddle point pr oblem ari sing from t he mixed finite element discretiz ation of an ellipt ic second order operat or has been introduced in [Woh95]. Similar decomp ositi on techniques are used in [Hip96 , Hip97] t o construct an optima l multigrid method for Raviar t- Thomas finite elements in 3D based on a uniformly refined sequence of t riangulat ions .
2.2 Vector Field Discretizations
115
Efficient preconditioners and multigrid techniques are discussed and analyzed in [AFW97, AFW98, AFWOO]. The natural interpolation operator PI : Vj ---+ Vi, 0 ::; l ::; i . will play an essential role in the analysis of our hierarchical basis method. For the convenience of the reader, we recall the definition already given in the previous subsection AF(PI(V)) :=
I~I
J
v· nda,
FE:FI ,
F
wher e AF( ') are the degrees offreedom as given in Subsect. 2.2.1. We remark that Pj = Id, and we define P-l := O. 2.2.3.1 Horizontal Decomposition. We define a family of subspaces by
o::; l ::; i , of Vj
Then, obviously Vo = Vo and Vi C Vi and Vi n Vk Furthermore, each element v E Vj can be written as ~
~
j
0, 0
<
Vi,
l =J k ::;
i-
j
v = l:)PI - PI-l)V = : LVI, 1=0
1=0
and thus j
Vj =
LVi
(2.30)
1=0
is a direct sum. This decomposition already defines an additive Schwarz method if the exact projections are used . In a first step, we show that the constant Co in Lemma 2.1 can be bounded quadratically in j and in a second step that the spectral radius of [ in Lemma 2.1 is bounded by a constant independent of j . The following lemma gives an upper bound for Co and can be obtained easily from Lemma 2.4. Lemma 2.10. There exists a constant independent of j such that a(VI, VI) ::; C(l
+ l)
a(v, v),
0::; l
::; j .
Proof The proof is based on Lemma 2.4. Using the triangle inequality, we find a(vt,vI)::; 2(a(Plv,Plv) +a(PI-lv,PI-lV)) .
In the case of the iterative substructuring method , we assumed that the fine triangulation is obtained by quasi-uniform refinement of a coarse triangulation. Thus before we can apply Lemma 2.4, we have to make some modificai; ... f; tions. We introduce a fictitious sequence of nested triangulations
To,
116
2. Iter ative Solvers Based on Domain Decomposition
such that each element in Ti , 0 ::; I ::; i . can be written as a union of element s An explicit const ruct ion for the 2D case is given in [Woh95]. Figur e in 2.13 illust ra tes t he const ruction of in 2D. The first row in Fig. 2.13 shows t he sequence Ti and t he second row t he corres ponding sequence 'i;
t;
Ti
Fig. 2 .13. Constructio n of t he sequence
f;
in 2D
To obtain i; we refine each element, T E 10 , I t imes quasi-uniforml y. Here, regul ar refinement int o eight subelements as well as refinement based on bisecti on can be ap plied; see, e.g., [Bae91, Bey95]. We insist t hat a refinement step based on bisect ion is carr ied out at most one t ime during t he I ste ps, and t hus the shape regulari ty of th e t riangulations is preserved . Then , each which is a subset of T E 10 has a diamet er bounded from above element in and below by constants ti mes 2- lHT , where H T is t he diam et er of T , and log(H/h) is bounded from above by C l. We remark t hat each element in Ti , which is not at t he same t ime an element in i; is also an element in '0. We assoc iate with i; t he Raviart-Thomas finit e element spaces ~ , and find by const ruc t ion t hat Vi c ~ and moreover that PlY = PlY for v E 11; . Here, PI is the natural interpolant ont o ~ . Observing t hat satisfies t he assum pt ion of Subsect. 2.2.2, we can apply Lemm a 2.4 and obtain
Ti
Ti
a(v l' VI) ::; 2(a(Plv, PlY) + a(Pl-l v , Pl- l v)) ::; C (l
+ I) a (v , v) . o
App lying Lemm a 2.10 and summ ing over the refinement levels, we obtain an upper bound for Co j
L a(vl ' VI) ::; C(l + j2 ) a (v, v)
(2.31)
1=0
To prove an upper bound for t he condit ion number , we have to consider
E in more det ail since establishing a sharp bound for p([) is not as easy as in case of overlapping or it erative subst ructur ing meth ods. To get an upp er bound for p(E), which is independ ent of t he number of refinement levels, we have t o show a suitable st rengt hened Cauchy- Schwarz inequality.
2.2 Vect or Fi eld Discreti zations
117
Lemma 2.11. Th e foll owing Cauchy-Schwarz inequalit y holds
Proof Without loss of generality, we assume that l > k . We consider one eleme nt T E T,. at a ti me , and observe that v klT E R/(T). In a first st ep , we show that t he divergence of Vk and of VI are orthogonal with resp ect t o the £ 2-scalar product
J
div v. div vs d»
=
T
J J
div(Plv-PI_ lv)divvkdx
T
(III (div v) -
=
tu., (div v)) div v» dx
= 0
T
Her e, III and lIl- 1 are the £ 2-projecti ons onto t he space s of piecewise constants associate d with the t riangulat ions 11 and 11-1 . To get an upper bound for (VI , V k) d iv ;T , we have to consider in a second step t he cont ribut ions of t he £ 2-term. Observing t hat v klT can be written as a gradient of a qu ad ratic function wit h mean value zero, VkIT = grad ¢, ¢ E P2 (T) , we find by applying Gr een 's formul a
J
VI ' v» dx
J
=-
div v, ¢ dx
T
+
T
By t he definitions of VI and
J
¢ VI . u da .
8T
Ti-1, we get the following orthogonality relations
J
div v,
fil - 1¢ dx
= 0 ,
T
where fi l - 1 is t he £ 2-proj ecti on onto the space of piecewise constant s assoand ciate d with t he t rian gulat ion
Ti-1 ,
Jfi
l- 1 ¢ VI . ti da
=0
8T
By mean s of these orthogonality relations and t he discret e norm equivalence (2.10), we obtain the followin g upper bound
J
J
T
T
VI ' v» dx = -
divvI(¢ - n -1 ¢) dx
:::; C2 -(l- 1- k)
+
J
(¢ -
8T
fil- 1 ¢ )VI . u da 1
(hTlldivvdlo;TllvkIIO;T + hj,llvl' nllo;8Tllv kllo;T)
:::; C2- (I - k)/ 2(\Idiv vdlo;T + \Ivdlo;T)llv k\lo;T .
o
118
2. It erative Solvers Based on Dom ain Decomposition
An upper bound for p(£) can be now derived by using the equivalence between t he energy norm and the Hilb ert space norm and Lemm a 2.11. It is bounded by a constant ind ep end ent of t he number of refinement levels
p(£)
~
C .
Combinin g t he est imate for p(£) and Lemma 2.10 with Lemma 2.1, we find that the condit ion number of this addit ive Schwarz method with exac t projections onto t he subspaces is bounded quadrati cally in j
The computat ional cost for t he applicat ion of t he exac t pr ojection is in pr act ice t oo high . Thus, the exac t pr oj ecti on onto the subspaces will, very oft en , be repl aced by qu asi-projecti on operat ors Ti onto Vi.:- We define our qu asipr oj ecti ons Tl in terms of a vertical decomp osition of Vi , 1 ~ 1 ~ j , and show t hat t he condit ion number is st ill bounded qu adrati cally in terms of j . In particular , we prove that (2.3) is satisfied with w ind epend ent of j . 2.2.3.2 Vertical Decomposition. We decompose t he global space Vi int o local low dim ension al spaces which are assoc iate d with t he faces and elements of the triangulati on on level 1- 1. This split t ing is motiv at ed by a Helmholt ztyp e decompositi on . In case of a Helmholt z decompositi on , t he space would be writ t en as a dir ect sum of a divergence free subspace and it s or thogonal complement. Here, we decomp ose Vi int o a sum of divergence free subspaces plu s a kind of qu asi-orthogonal complement which is called, in the following, a surplus space. We not e t hat t he divergence free Raviart-Thomas vect or fields can be obtain ed from Ned elec finit e elements; see [Ned82]. The family of Nedelec finit e elements form a subspace of t he Hilb ert space H( curl ; n )
We work wit h the Ned elec finit e elements of lowest order which are defined by
N V(n ; Tt) := {q E H(curl ; n)1
qlT E N V (T ), T E Tt} ,
0 ~ 1~ j ,
wher e N V(T) stands for t he local Nedelec space. In t he case of a simplicial triang ulation, t he local space has dim ension six
NV (T) :=
(~~: ~~~) (Y3
+ (J3 Z
.
The degrees of freedom of t he global space are given in t erms of t he t an genti al components on t he edges
Ae(q)
:=
I~I
J
q . tds ,
e
2.2 Vector Fi eld Discret izati ons
119
where e is an edge of the triangulation Ti, and th e directi on of t depends on t he dir ection of n. We now int roduce, for each face F E F I-I which is not an element of FI , a subspace of at most three degrees of freedom
NVI ;F := {q E N V (D;Ti)1 Ae(q) = O, e
et F}
see Fig. 2.14. Moreover , we have curlNV (D;Ti) C lIi. F
Fig. 2 .14. Different refinement techniques on F E FI- 1
We define our divergence free Raviar t- Thomas subspaces associated with t he faces in te rms of t he Nedelec spaces N V I;F by
Vi ;F := curl NVI ;F . It can be easily seen t hat Vi ;F is a subspace of Vi supp orte d in two elements of Ti- I ' An element VI E lIi is also in Vi if and only if VI . n do = 0, for F E F I-I . Let V = cu rl q with q E N V I;F, th en
IF
f
v . n do
=
f
q . t ds
= 0,
F E FI-I
of
F
The sur plus spaces Vi ;T , T E Ti-I \
Vi ;T := {v E
Ti , will be defined
Vii
locally by
supp veT} .
T he dimension of Vi ;]" is given by the number of faces F E FI which are in th e interior of T ; it is bounded by eight . We point out that Vi ;T can also contain a divergence free element. It has such a one dimensional divergence free subspace, in t he case t hat T E Ti- I is refined in such a way t hat it has an interior edge; see Fig. 2.15. We can now define our vertical decomp osition in te rms of t he local divergence free face spaces Vi ;F and t he element spaces Vi;T . We find t he following direct sum repr esent ation for Vi
lIi =
L
FEFl _l\F,
Vi ;F (fJ
L
Vi ;T '
(2.32)
TE 'Ti-l \ 'Ti
To define our quasi-projection operators, we have to specify a new bilinear form 0,(". ) on Vi x Vi , It is given by means of t he dir ect splitting (2.32) and t he origina l bilinear form a(' , .)
120
2. Iterative Solvers Based on Domain Decomposition
Fig. 2.15. Adaptive refinement such that ~;T contains a divergence free element
a(v[, v.) :=
L
aiv »; v»)
L
+
a(vT' VT) ,
TETi-l \ t.
PCF'-l \:F,
where according to (2.32) v: := 2:PE:F,_,\:F, Vp + 2:TETi_l\Ti VT· The following lemma shows the equivalence of the bilinear forms and a (', .) when restricted to Vz x Vz.
aC')
Lemma 2.12. There exist constants c and C such that
Proof. We start by proving the lower bound. It can be obtained easily by a coloring argument. We define vp := 2:PE:F'_l\:F, v tr, VT:= 2:TETi_l\Ti VT, and find
a(v[,vz) = a(vp+VT,VP+VT) ::; ~a(vF,vF)+5a(vT,vT)
::;5(
2:
FE:F'_l\:F,
a(vp,vp)
+
2:
TETi-l\Ti
a(vT,vT)) =
5a(v[,vz)
The proof of the upper bound relies on the norm equivalence (2.10). Let v r be an element in Vi;p, then a(vp, v r )1/2 is equivalent to its L 2-norm : 2:
FE:F'_l \:F,
a(vp, vr-)
2: 2: Ifl~ Ilvp . nll~;f PE:F'_l \:F, fE:F,
= C
2: 2: FE:F'_l \:F, ifCF e r,
If I ~ 'lv n116;f
< C 2: Ifl~llv.nI16;f < Ca(v,v) fE:F,
By means of the triangle inequality, the following upper bound for a(vT' VT) can easily be established
L TETi-l\Ti
a(vT,vT)=a(v-
L PE:F'_l\:F,
VP,v-
L PE:F'_l\:F,
vp)::;Ca(v,v) D
2.2 Vect or Fi eld Discretizations
121
We define now our hierar chical basis method in terms of th e decompo sition (2.30) , the quasi-projections TI , 1 :::; l :::; j , given by
and t he exa ct projection To onto Vo with respect to t he bilinear form a(·, .). Then , the cent ral result of th is subs ection follows from Lemm a 2.1, (2.31), Lemm a 2.11, and Lemm a 2.12. The hierar chical basis method yields a quasiopt imal preconditioner. Theorem 2.13. Th e conditi on number of the additive Schwarz m ethod is bound ed by
T he exac t pro jection has to be applied on the coarsest level, but on t he finer ones we have to solve only local subproblems. For each face, F E FI -1 \ F I, we have t o solve a problem of dimension at most three, and for each element , T E Ti-1\ Ti , a problem of dimension at most eight . The comp utat ional effort can be further redu ced. The stiffness matrix associate d with one face is equivalent to a mass matrix and therefore can be replaced by its diagonal. This is, in general, not the case for the st iffness matrices associated with if t he nod al basis vector fields are used . However , if a basis transformat ion is performed , and t he divergence free element is spanned by one basis fun ction, we can work with t he diagonal of the stiffness matrix.
Vt;T
Remark 2.14. W e note that th e theoretical result s of Subsect. 2.2.2 also hold for simplicial triangulations and th e ones of Subsect. 2.2.3 for hexah edral trian gulation s. 2.2.4 Numerical Results
In t his subsect ion, we report on num erical results illustrating th e performan ce of t he proposed iterative substructuring method . Although t he analysis was carried out only for the 3D case in this cha pter , we also include some results for the 2D case . The results for t he 2D case can be found in [TWWOO], and t hose for the 3D case in [WTWOO] . For further num erical results including two-level overlapping and Neum ann-Neumann methods, we refer to [Tos99]. We use the ite ra tive subst ruc t uring method defined by t he decomposi ti on (2.11) and var y t he meshsize of our uniform coarse and fine triangulations. T he fine trian gulat ion T" consists of n 2 elements, with h = lin . We observe t he theoret ically predicted logarithmic behavior of the condition numb er. In a second test case, we consider the influence of the coefficient s a and B , where t he matrix B is given by B := bId. We refer to [SBG96], for a genera l discussion of pr acti cal issues concerning Schwarz methods.
122
2. Iterat ive Solvers Based on Dom ain Decomposit ion
2.2 .4.1 The 2D Case. We start wit h t he case a = b = 1. The following table shows t he est imated cond ition number of t he additive Schwarz operat or as a function of t he dimens ion of t he fine and coarse meshes. In additio n, t he num ber of conjugate gradient it erat ions requ ired to obtain a redu ct ion of t he residu al norm by a fact or of 10- 6 is given . The columns of Table 2.1 provide t he condit ion numbers for fixed ratios of t he meshsizes of t he coarse and fine t riangulations. T able 2.1. Condition number and number of cg-iterations, (in par ent heses)
I H lh n= 32 n=64 n=128 n-192 n-256
32
16 20.23 35.94 36.68 36.71 36.66
-
26.27 (11) 46.83 (20)
-
47.80 (18)
11 20) 18 17 17
8 26.50 (20 27.16 (21) 27.06 17 27.00 17 26.97 16
4 19.10 19.00 18.92 18.90 18.89
20 17 16 16 16
2 12.86 (17) 12.90 (16)
x x x
For a fixed rat io H / h, the condition num ber is quit e insensiti ve t o the dim ension of t he fine mesh. The numb er of it erati ons varies slowly wit h H / h and our result s compare well wit h t hose for finit e element approximations of t he Lap lace equation; see, e.g., [SBG96]. We find in Table 2.1 two values which are considerably smaller t han th e ot hers in t he sa me column. T his is t he case of n = 64, H / h = 32 and n = 32, H / h = 16 which correspond to a partit ion int o 2 by 2 subregions . The reaso n for t he smaller conditio n num bers of T ad d is a sma ller nor m of t he additive Schwarz operator in t hose cases . We recall t hat this bound is obtained by a coloring argument. The largest eigenvalue of T ad d is boun ded by 5 in all t he cases in Tabl e 2.1, except for n = 32, H'[h. = 16 and n = 64, H/h = 32, when it is bounded by 3.
45
Q;
40
.0
35
:::>
30
g
25
E
c: c:
'6
§ o
15
• calculated condition number - least square fitting
H/h
Fig . 2.1 6 . Condit ion nu mb er (asteris k) and leas t-square second order logarit hmic po ly nomial (solid line)
2.2 Vector Field Discr etizations
123
In Fig. 2.16, we plot the results of Tabl e 2.1 to gether with the best second order logarithmic polynomial least-square fit . The relative fitting err or is about 1.8%. Our num erical results are th erefore in good agreement with our theor etical bound and also confirm that our bound is sharp. Table 2.2. Condition number and number of cg-ite rat ions, (in par entheses)
I ttt:
b=O .OOOOl b-O.OOOI b-O.OOl b=O .Ol b=O .l b=l b- 10 b-100 b=1000 b=10000 b=100000
32 3.87 10 3.87 10 16.9 11 46.9 (14 46.9 14 46.8 20 45.3 22 40.8 25 29.8 24 17.4 18 9.41 14
16 4.68 13 36.3 16 36.5 16 36.7 17 36.7 17 36.7 18 36.4 22 34.8 23 28.4 23 17.3 17 9.37 14
8 4.86 13 26.2 16 27 (16) 27.1 16 27.1 17 27.1 17 27 (18) 26.7 20 24.5 21 16.8 18 9.3 (14
4 4.92 13 13 (15) 18.7 16 18.9 16 18.9 16 18.9 16 18.9 17 18.9 19 18.4 19 15.3 17 9.15 14
In Tabl e 2.2, we show some results when th e ratio of th e coefficient s b and a is changed. The est imated condit ion number and the number of iterations
are shown as a fun ction of H/h and b, for a fixed value of n = 128 and = 1. We observe t hat the condit ion number t ends to be ind epend ent of the ratio H/h when t he ratio bl a is very sma ll or very large. This behavior is not fully covered by th e t heory, but an analysis of th e limit cases b = 0 and a = 0 is carried out in [TWWOO] ; it can be shown t hat there is a uniformly st abl e decom positi on for v E Vh for these two limit cases. In particular , t he £ 2-st ability of our decomposition is gua ra ntee d. We recall th at in 2D, th e qualitative and quantitative results for our it erative sub st ructur ing method for the lowest ord er Raviart -Thomas finit e elements are exac tl y the same as for Maxwell's equat ion discreti zed by Nedelec finite elements. We remark t hat when Maxwell's equa t ions are discr etized with an impli cit time-scheme, th e time step is related t o the ratio b/a which is in general very lar ge. Our it erative sub structuring method th erefore appears quit e at t rac t ive for th e solut ion of linear syst ems arising from t he finit e element approximation of time-depend ent Maxwell's equat ions. a
2.2.4.2 The 3D Case. In this subsubsection, we present the sam e type of num erical results for t he 3D case. The condit ion numbers for the addit ive Schwarz method are given for different diamet ers of t he coarse and fine meshes, and coefficients a and b. In Table 2.3, we show the est ima te d condition number and the number of conjugate gr adi ent it erations in order to obtain a reduct ion of the residual norm by a factor 10- 6 , as a fun ction of t he dimensions of th e fine and coarse
124
2. It erative Solvers Based on Domain Decomposition
meshes. The condition number remains bounded ind ependently of n for a fixed ratio H / h. It is slowly increasing with H / h. We observe that the number of it er ations varies slowly with H / h and n . The largest eigenvalue is bounded by 7 in all the cases in Tabl e 2.3, except for n = 8, H/h = 4 and n = 16, H / h = 8; the latter cases correspond to a partition into 2 by 2 by 2 subregion s and, consequent ly, t he bound for the largest eigenvalue is 4. Table 2.3. Condition number and number of cg-iterations, (in parentheses)
I tttt: I n=8 n=16 n-24 n=32 n=40 n=48
8
4
2
19.46 16 32.78 (27 33.48 27 35.50 27 36.47 28
13.28 14 23.26 24 25.55 (26 26.01 26 26.08 25 25.91 22
15.15 (22) 17.37 (24) 17.43 (21) 17.42 (21)
-
-
In Fig. 2.17, we plot the results of Tabl e 2.3 together with t he best second order logar ithmic polynomial least-square fit . The relative fit ting err or is about 4.4%. Our numerical resul ts are t herefore in good agreement with our t heoret ical bound. Moreover , we observe t hat the given theoretic al bound is sharp. A more careful ana lysis shows that an example can be explicite ly const ructe d such t hat the bound is sharp.
Q;
-g
35 30
::J
~
25
~
20
o
c
o () 15 10
* calculated condition number - least square fitting
5 H/h
Fig. 2.17. Condit ion number (asterisk) and least-square second order logarithm ic polynomial (solid line)
In Tabl e 2.4 , we show some results when t he ratio of t he coefficients band a changes. For a fixed valu e of n = 24 and a = 1, t he est imate d condition
number and the number of it erations are shown as fun ctions of H / h and b. The condit ion number te nds to be bounded ind epend ently of t he ratio H / h when the ratio b/a is very small or very lar ge. Gener ally, t he numerical results for t he 2D case and t he 3D case show t he sam e qu alitative structure.
2.3 A Multigrid Method for the Mortar Product Space Formulation
125
Table 2.4. Condition number and number of cg-ite rations, (in parentheses)
I tttt:
b-le-09 b-le-08 b=le-07 b=le-06 b-1e-05 b-O .OOOI b=O .OOl b=O .01 b=O .l b- 1 b- 10 b=le+02 b=le+03 b=le+04 b-1e+05 b=le+06
8 4.00 4.00 4.00 4.00 17.5 29.5 30.9 32.3 32.6 32.8 30.0 23.6 14.4 8.42 6.75 6.72
10 10 10 10 11 12 15 20 22 27 29 26 21 16 14 14
4 5.81 5.81 5.82 21.0 25.0 25.0 25.3 25.4 25.5 25.6 23.4 20.4 14.1 8.57 6.98 6.91
16 16 16 18 19 19 21 22 25 26 26 25 22 17 15 15
2 6.29 15 6.29 15 6.29 15 6.29 15 16.1 18 17.1 18 17.2 18 17.2 18 17.4 20 17.4 21 17.1 23 15.1 22 12.6 19 9.43 17 7.92 17 7.80 16
2.3 A Multigrid Method for the Mortar Product Space Formulation In t his section, we consider a multigrid method for mortar finite element discreti zations. Recently, a lot of work has been done on efficient iterative solvers for t he linear syste ms that arise from t hese discretiz ations; see, e.g., [AK95, AKP95, AMW96, AMW99 , BD98 , BDL99, BDH99b, BDW99 , CW96 , Dry98 a, Dry98b , EHI+98, EHI+OO, GPOO, HIK+98, Kuz95a, Kuz95b, Kuz98 , KW95 , Lac98, Le 93, LSV94, WW98, WW99] . All these techniques are eit her based on the positive definite vari ational problem on th e const rained non conforming space or on t he equivalent saddl e point problem. In the first case , the mortar projection ty pically has to be applied in each iteration or even in each smoot hing ste p, and this requires the exact solution of a mass matrix syste m. In t he saddle point formul ation, a Schur complement system plays an essent ial role. Efficient iterative solvers for the nonconforming formul ation (1.3) or an equivalent int erface problem have been developed. Additiv e Schwar z methods including iterative substruct ur ing, overlapping and Neumann- Neumann typ e algorit hms as well as multigrid methods have been discussed and an alyzed ; see, e.g., [BDH99b , CDS98, Dry96 , Dry97, Dry98a, Dry98b, GPOO, Le 93, LSV94]. Here, we combine t he idea of dual basis functions for th e Lagran ge multiplier space with standa rd multigrid techniques for symmet ric positive definite syste ms. The new mort ar formulation, analyzed in Sect. 1.3, is t he point of departure for our multi grid method. Thi s approac h was originally studied in [WK99] and has been extended to linear elasticity problems in [KWOOb].
126
2. Iterative Solvers Based on Dom ain Decomposition
In parti cular , we do not have t o solve a modified Schur complement syst em exac t ly or iteratively, nor do we have to solve a mass matrix syst em in each smoothing step. Here, we assume t hat To , 1i ,. .. ,1j form a nest ed family of quasi-uniform and shap e regul ar t riangulat ions satisfying h l - 1 = 2hl , 1 ~ I ~ j. The associate d unconstrain ed pr oduct spaces, denot ed by Xi , are then nest ed. FUrt hermore , we assume full H 2 -regularity of th e ellipt ic problem . We pr esent our multi grid analysis in two different forms. In t his secti on , we work with a matrix setting and choose scaled nod al basis functions for the vecto r representation . The multigrid analysis in Sect. 2.5 is given in th e abst rac t operator not ation. Here, we focus on a multigrid method for mortar finit e element s using du al basis spaces for t he definition of t he Lagran ge multiplier. We use neither t he non conforming formul ation, (1.14), nor t he saddle point formulation , (1.15) , but inst ead the positive definit e one, (1.43) , on the product space introduced in Sect. 1.3. The start ing point of our analysis is the modified syste m on the pr odu ct space. The main difficulty in solving the linear syste ms arising from mor t ar discretizat ions is the handling of t he const raints at th e inte rfaces . Using the approach pr esent ed in Sect. 1.3, multigrid methods can be applied dir ectly to t he positive definit e formul ation (1.43) . In cont ras t t o the const ra ined nonconforming case, the un constrain ed product spaces associate d with a nest ed sequence of t riangulat ions are nest ed . However , using t he st andard injection to define the rest riction and pr olongation would not provid e a suitable approximation property. We define level depend ent bilinear form s and suitable tran sfer operators in Sub sect. 2.3.1. By means of the a pri ori est imates (1.22) and the definit ion of t he level depend ent bilinear form , we establish an approximation property in Sub sect . 2.3.2. In Subsect. 2.3.3, we consider a general class of smoot hing operators for which smoothing and stability pr operties will be shown. Sub sect . 2.3.4 concern s a simplified impl ement ation of the smoot her. Finally in Subsect . 2.3.5, num eric al results illustrat e th e performances of Vand W- cycles. 2.3.1 Bilinear Forms
We set dim X, =: n l and denot e the Euclidean scalar product in IRnl by (', .). Furthermore, we use the same not ation for VI E X; and it s vector representation VI E IRn/ with respect to the standard nodal basis in 2D and t he by h 11 / 2 scaled nod al basis in 3D . Using these nod al basis fun cti ons and this not ation yield that the L 2-norm of VI E X, is equivalent in 2D and in 3D to hi ti mes the Euclidean vector norm , Ilvlil := ( VI , VI)1 / 2 , of VI E IRn/
cIlvdla ~
hi
Ilvlil
~ C
Ilvdla .
Since, we are working with t he vari ational form on the un constrain ed product space no norm has t o be specified for the Lagran ge multiplier.
2.3 A Multigrid Method for the Mortar Product Space Formulation
127
We recall th at the linear form g2(-) defined in (1.41) is given in terms of t he bilinear form b(·, .), t he Lagrange multiplier nod al basis functions, and a t rivial exte nsion by zero. The st ability constant of this ext ension depends on the meshsize. As a result , t he linear form g2(-) depends on the level l. We define our level depend ent bilinear form iil (-, ·) in terms of the original bilinear form a(·,·) and g2 (·) by
Using Proposition 1.15 and an inverse estimate , it is easy to see t hat an upp er bound for iii (v,v) is given by the squa re of a mesh dependent norm (2.33) We observe that v - g2(V) E Y. Then , t he uniform ellipt icity of a(· , ·) on Y x Y yields a lower bound for iii (v,v) iii (v,v) ~
C(llv- g2(v)lli + Ilg2(V)IIi)
~
Cllvllr.
K
vE
II Hl([h)
. (2.34)
k= l
Workin g with th e bilinear form ii j ( · , ·) on X, x Xi , 0 ::; l ::; i . we cannot expect an approximation prop erty with a constant growing slower than 2j - l . Inst ead of usin g the bilinear form ii j (. , . ) on all levels, we consider the bilinear form iil(-,·) on levell . The stiffness matrix Al is associate d with this bilinear form, and the vari ati onal problem (1.43) can be written as AlUI
=11 ,
on level l . In compa rison with formula (1.48), we have added t he index l to ind icate the level dependency. In the next subsection, we show that a suitable approximation prop erty holds for the finite element spaces X, if t he level depend ent bilinear forms ii i (., .) are used and if the defect has a special structure. To define our multi grid method, we need to specify rest riction and pro longation matrices. Let us first consider the standard grid tran sfer operators to see why they do not meet our requirement s. They are denot ed by If-I : jRn l ---+ jRn 1- 1, and by IL l : jRnl- l ---+ jRnl . As usual , IL l is chosen as t he matrix repr esent ation of the natural injection of X l - l in X; with respect to t he specified nod al basis, and th e restriction If-l is defined by If-l := (ILl) T . Thus, we have
Wr
Let Wl-l be t he solut ion of the defect problem in Xl - I, i.e., A l - l Wl- l = let dl - l := If - l dl E Xl -I. Then , even if dl = 0, in genera l
dl - l , and
128
2. Iterative Solvers Based on Domain Decomposition
W?-:.l dl - 1 i:- O. We recall that WI scales the values on the non-mortar sides; see Subsect. 1.3.3. As a consequence, the constraints are in general not satisfied on levell - 1, i.e., BT-1Wl-1 i:- O. In the next subsection, we show our approximation property only for the special case Wl~l dl- 1 = 0, and thus we cannot use If-1 as a restriction operator. These preliminary remarks show a need for a modified restriction operator (lrnod)~-l which should satisfy W;C 1 dl - 1 = 0 if W{ dl = O. Our new transfer operators are defined in terms of the local projection Pl := BIW{
(2.35)
and are based on the decomposition Id = (Id - Pd + Pl. The support of PIVI , VI E Xl, is contained in the union of small strips of width hI located on the non-mortar sides of the interfaces; see Fig. 2.18. We note that W{ (Id - PI) = 0, and we define (Irnod)~-l := (Id - Pl-r) If-1 (Id - PI)
+
Pl_ 1If- 1Pt
and (Irnod)L1 := ((Irnod)~-l)T. The coarse grid correction Wl-1 is now given as the solution of ii l - 1 (Wl- 1, Vl- r) = (dl- 1,Vl-r)O,
VI-1 E Xl-I,
(2.36)
where d l - 1 := (Irnod)~-ldl, and dl is the residual on level l. Thus, if the iterates on level l satisfy the constraints, the correction Wl-1 will satisfy the constraints on levell - 1, i.e., BT-1Wl-l = O.
Fig. 2.18. Structure of the support of Pn»
In the case that the residual dl on levell satisfies W{ dl = 0, we can compute d l - 1 by d l - 1 = (Irnod)~-ldl' where (Irno(d~-l is the simplified restriction operator
(2.37) If the correction WI-Ion levell - 1 satisfies Br_l Wl-1 = 0, the prolongation
operator (Irnod)Ll can be replaced by its simplified form: I
T
I
(Irnod)l-l := (Id - PI ) 11- 1 .
(2.38)
2.3 A Multigrid Method for the Mortar Product Space Formulation
129
Proposition 2.15. Under the assumption BT-l WI-l = 0, we have th e following norm equiv alence
IlwI- lll ::; II(Imod)Llwl-lll ::; c IlwI- lll . Proof. The upp er bound follows from IIPI-lil ::; C and th e st ability of ILl ' c
Observing that Id - ~T modifies only t he nod al values at the vertices in the interior of t he non-mortar sides, we find II(Imod)Ll wl-lll ~ IIILlwl-lIIR ~ IlwI-lIIR where II . IIR denot es the Euclidean norm without the component s asso ciated with t he int erior of the non-mo rtar sides. Using the assumpt ion Bl~l WI-l = 0, we find that IlwI-lll ::; C IlwI-lIIR. 0 As we will see in Subsect. 2.3.3, our special class of smoot hers satisfies I- l = 0 by construction, and we can th erefore use l = 0 and BT-l W (Imod):-l and (Imod)Ll in th e implementation. Our multigrid method is then defined in terms of the level dependent bilinear form al(',.) and the simplified transfer operators. In the next two subsections, we est ablish suitable approximation and smoothing pr operties.
Wrd
2.3 .2 An Approximation Property
An essential tool for establishing level independent convergence rates for t he W- cycle is a suitable approximation property. The saddle point problem (1.15) form s the starting point of our ana lysis. We establish an approximation property in t he £ 2-norm, based on the a priori estimates for t he saddle point probl em (1.15). In [WohOOb], an approximation prop erty for the saddle point problem on level l was shown if the right hand side is orth ogonal to th e space X I - l x M I - l with respect to a level depend ent bilinear form . Thi s result was established for the standard Lagr ange multiplier spa ces which are nest ed. We cannot apply thi s result in our setting since t he Lagran ge multiplier spaces do not have to be nest ed; in particular , t he examples satisfying (1.32) present ed in Sub sect . 1.2.4 are non-n ested , i.e., M I-l et MI . Instead , we establish a weaker result for the mor e general situation of non-nest ed Lagrange multiplier spaces . In th e case of standa rd Lagrange multiplier spaces and t he st andard prol ongati on , we refer t~ [BD98, BDW99J Let WI and WI- l be the solution of Aiwi = dl and A I - l W I-l = dl - l , respectively, where the stiffness matrices and dl- l are defined in th e previous subsection. The following lemma gives an upp er bound for WI- (Imod)L l WI-l in t he Euclidean vector norm. Lemma 2.16. (Approximation property) Und er the assumption th at d l = 0, th e follow ing upper bound s hold
Wr
Ilwl -
IL l wI- III::;
C Ildlll ,
Ilwl - (Imod)Ll wI-III ::; C Ijdtll with a cons tant C in depen dent of th e refin em ent level l .
130
2. Iterative Solvers Based on Domain Decompositi on
Proof We start with t he first inequ ality. Observing that WEI d l -
l
=
W;~:' l(Id - PI_dIf -ldl = 0, we find t ha t BT wl = BT-l wl- l = O. To prove t he asserti on, we consider relat ed saddle point pr oblems. Defining Al := Wt(dl - A lwl ) and AI-l := W;~_l(If- ldl - AI- l WI- d , we find that (WI , AI) and (WI-l,Al-d solve the sa ddle point pr oblems
respectively. We not e t hat the right hand side of t he sa ddle point problem on levell - 1 is defined in t erms of the st andar d restriction If-I . Associated with dl is a un iqu e f d E XI C LZ(D) such t hat for VI E X I
and such that hi Ilfdllo ::; C IlddJ. We now define a conti nuous vari ati on al problem in t erm s of fd by: Find W E HJ (D) such t hat
a(w, v) = (fd,v )o,
v
E
HJ(D) .
Then , WI E Vi and WI-l E Vi - I are t he corres ponding non conforming mort ar finite element approximations of t his vari ational problem on levell and l- 1, resp ectively. Using the full H Z-regulari ty and t he a pr iori est imates in t he LZ-norm, we obt ain Il wl - I Ll wI-I II ::; ~ II WI - wI- l llo ::; ~ (11 wI - w llo
+ Il w - wI- l llo)
::; Chtllwllz ::; Chtllfdllo ::; C Ildtil The second asse rtio n is based on t he t riangle inequ ality
To estimate t he second term, we observe t hat pt is t he algebraic representation of 9z(-) on levell. Using the definit ion of the linear form 9Z ('), we find in te rms of Lemma 1.7, Propo sition 1.15 and t he a pri ori est imates for the energy norm
C z IJPtIL l wl- lll ::; hf'119Z(WI-dI15
C
M
< hi m~1 1 1 [WI -l ] 1 15;'"Ym
::; C Il wI- l - w lli < C hfllwll ~ ::; Clldtll z
0
In t he case t hat t he Lagran ge multiplier spaces are non-nest ed , we cannot writ e WI - WI-l E X, as t he solut ion of a sa dd le point problem on Xi x MI. T he approximation property t hen has t o be established in te rms of IlwI - wllo an d Il wI-l - wllo, where W is a suitable element in HJ( D). We point out that dl = O. t he approximation prop erty is only shown for the special case t hat We t herefore have to use special types of smoothers t o satisfy t his condit ion.
wt
2.3 A Multigrid Method for the Mortar Product Space Formulation
131
2.3.3 Smoothing and Stability Properties In addit ion to the approximat ion property, a suitabl e smoothing property has to be established to obtain level ind epend ent convergence rat es. To measure t he smoothing effect , IIAl eil1 has to be bounded by Il e?ll, where ei is the it er ation err or in the mth-step and e? t he initial err or. In cont rast to the two grid convergence analysis, approximat ion and smoothing prop erties are not sufficient t o obtain a level ind epend ent convergence rate in the full multigrid case . This is so since the st abili ty constant of t he smoo thing op erator in the II . II-norm will also ente r into the est imates. In t his subsection, we const ru ct suitable smoot hing op erators for t he syste m on the un constrain ed product space. In cont ras t to the other sect ions, here we use the symbol A t o denote eigenvalues . We focus in part icular on a special structure of t he smoot her yielding it er at es which sat isfy the constraints at the int erface. As a consequence, we obtain a residual satisfying di = 0, m ~ 1, for any numbe r of smoothing ste ps. Only in this case, are we in the setting of Lemma 2.16 , which concerns t he approximat ion property. To define a suitable smoot her for our multigrid method, we start with t he 2 x 2 blo ck decomposition of Al introduced in Sect . 1.3 . We recall that the first blo ck is obtain ed by grouping t he vectors having ind ex 0, 1, or 2 t oget her. Using uk := (u5' ,uf, un , UN := U3, (1.48) can be rewritten as
wt
For simplicity, we use the level ind ex l only for t he glob al matrix and vector sym bols, e.g., the st iffness matrix Al and the solut ion UI , but not for subblocks, e.g., A RR. According to (1.49) , t he blocks are given by A RR = T T AT A A RR + 2MANNM - MA NR - ARNM , A NR = A RN = MA NN and AN N = AN N. Here, M is t he scaled mass matrix given in Subs ect. 1.3.3 as M = MRD - 1 . Thus, A NN is the symmetric positive definit e submatrix of Al asso ciated with t he int erior degrees of freedom on t he non-mortar sides. It s eigenvalues are between c and C.
Proposition 2.17. The matrix A RR is symmetric positive definite with eigenvalues bounded from below by chr and from above by a constant C, and IIARRII ~ c. Proof. To obtain an upper bound for the eigenvalues of ARR , we start with the definition of a(-, ') , observe (2.33) and use an inver se est imate
xkAnnxn = ii(xn , XR ) :S C ~21IxRII~ :S CllxRI1
2
I
T he lower bound is an eas y consequence of (2.34) , and the bound for the norm can be obtained by choosing one interior nod al basis fun ction as tes t 0 fun ction in t he definition of the norm.
132
2. Iterative Solvers Based on Domain Decomposition
Let G R be a symmetric, positive definite smoothing operator for fying
ARR , satis(2.39)
Here, a(T) denotes the spectrum of the operator T. We then define our iteration matrix Gil as a 2 x 2 block matrix by (2.40)
The coefficient a is defined by a := 0.5 .A;:;:;~x;N' where Amax ;N is the maximum eigenvalue of ANN. Now our iteration scheme is given in terms of Gil. The mth-iterate zi E XI is defined as ZIm := zim-1
+ G-I 1(dI
-
A'IZIm-1) ,
(2.41)
where dl stands for the right hand side of the system Aizi = dl which has to be solved, zi the iterate in the mth-step, and z? the initial guess. Each smoothing step can easily be performed provided that the application of is cheap.
G;'/
Remark 2.18. A possible choice for GR is a damped Richardson method, i.e. , GR := /1. Id, where 0 2: f.L 2: Amax;R' and Amax;R is the maximum eigenvalue of ARR . Then, GR satisfies (2.39). The stability constant of the iteration (2.41) depends essentially on the condition number of the operator Gil . The following lemma shows that the algebraic properties of Gil are inherited from G'i/ . In particular, the condition number of Gil is bounded by that of GR.
Lemma 2.19. Under the assumptions (2.39) , Gil defined in (2.40) is a symmetric and positive definite operator satisfying cId :S GI :S Old with constants c, 0 independent of the meshsize. Proof. The symmetry of Gil can be seen in formula (2.40) . There remains to estimate the eigenvalues of Gil. A straightforward computation shows that Gil is spectrally equivalent to its block diagonal matrix
c1xl'(diag Gi 1)xI :S xTGi 1xI:S 01xl'(diag Gi 1)xI , where diag Gil is the block diagonal matrix of Gil , and C1 := (5 - VU)/4, 0 1 := (5 + VU)/4. Considering diag Gil in more detail , we find by means of IIMII :S 0 and (2.39) the following upper bound
2.3 A Multigrid Method for the Mortar Product Space Formulation
133
xT(diag Gil )Xl :::; C IIGnl11 (11 xR11 2 + IIMxNI1 2 + exll xNl12) lllll 2 xtll , :::; C IIGn where xT := (x'k , x 'J:,) . In the first inequ ality, we hav e used the fact that IIGRII :::; C and in the last one that ex is bounded. By using that ex ~ c and (2.39) , we can give a lower bound for xT Gl lxl in t erms of IIGRII xT(diag Gi l ) Xl ~ IIGRII-
l
(11 xR11
2
+
ex Il xN11
2)
~ C IIGRII-lll xtlI2 .
0
We remark that, in general , an inequality of the form IIMxNl1 ~ c IlxN11 with c > 0 do es not hold . With Gil positive definit e, we st ill have to show that G, provides a good approximat ion of the high frequency part of Al in order to obtain a smoot her. We recall that t he modified restriction introduced in Subsect . 2.3.1 guarantees that the right hand side of (2.36) on levell - 1 has the form dT-l = (d'k , 0), if t he defect on t he previous level has this form. Lemma 2.20. (Smoothing property) Und er th e assumptions B Tz? = 0, Wrd l = 0 and (2.39) on GR , we obtain
IIA I el ll :::;
C Il e?ll ,
m
where el := zi - zl' . Furth ermore, each itera te zl' sat isfi es BT zl' =
o.
Proof We st art by proving BT zl' = 0 by induction. Assuming that T B IT zm I -- 0 , we find for B I z Im+l B IT zlm+ l = B IT zlm
+ BTG-lA ' I elm I I
' I elm . = B ITG-lA I
Considering B T Gi l Al in mor e detail , and observing that Wr dl = 0 gives BTZl = 0, we obtain M T e7i + eN = tr:' BT el = 0 and thus (A RR A RN) ( e7i) B ITG I- l A' I elm -_ B ITG-l I A' A' em NR NN N
=
BT
C:::R' 1MT~~~~M+ald)(ARRe~ : ARNe~)
_ B IT (
-
=
_
(M RT -
Id MT ) G-l(A R ' RR em R
+ A'RN emN )
, ' ) DM T) G -R1 ( ARRe7i + ARNeN = 0 .
Here, we have used A NR = A NNMT . Now, the assumpt ion B T z? = 0 yields t he assertion B T z;n+l = 0 for m ~ 1. In a second st ep , we prov e t he smoot hing prop erty of the it eration (2.41). As shown in Lemma 2.19, t he op er ator Gil is symmet ric and positive definit e. l 2 T hen, Gi / is well defined and AIel can be rewritt en as
134
2. It erative Solvers Based on Domain Decomposition
Since the norm of a symmetric matrix is bounded by its spect ral radius , it is sufficient to est imate the eigenvalues of G~1 /2 AIG~ 1 /2
II Ale111 ~ IIGill
m~ .
sE cr(G 1
Ad
18(1 - 8)mlll e?11
We therefore consider t he eigenvalue problem Gil Al Xl = A Xl, xT (X~ , X];;. ), and prove 0 ~ A ~ 1. To start , we give the block st ruct ure of
Gi 1A l
2
NN
R
+ aANN
Here, we have used A RN = MANN . Obviously, we find that A > 0 since Gil and Al are symmet ric and positive definit e. Let us first assume XR =J O. Then since Gi l Al is a lower blo ck trian gular matrix , A is an eigenvalue of Gil Al with the eigenvect or Xl only if XR is an eigenvect or of the lower dim ensional problem
By usin g t hat x~GRxR get
2: x~ ARRx R follows from (2.39) and
A NN > 0, we
and t hus A < 1. Let us n-;w consider t he second case x T = (0, x];;. ), i.e., XR = O. Then , t he eigenvalue pr oblem Gi l AI XI = A Xl redu ces t o t he following eigenvalue problem on a smaller space
To prove A ~ 1, we use t hat Al is posit ive definit e. We set wT ~ 1 T ~ 1 T T~ ((AFlRMYN) , ( - A ]:m YN) ) , then w I AI WI 2: 0 yields T
T
~
l
T~l
yNM AFlRMYN ~ yNA]:mYN .
Using t he ass umpt ion (2.39) on G R and the definition of a, a straightforward calculation shows T
~-1
1
T
T
-1
1
T
T
~-1
T
A yNANNYN = 2YN M G R M YN + aYNYN ~ 2YNM A RRMYN
1
T
~-1
+ 2YNANNYN <
T
~ -1
YNA NN YN .
2.3 A Multigrid Method for the Mort ar Product Space Formulation
135
Since AN~ = A N~ is symmet ric positive definite, we find that A:S 1. Thus, t he eigenvalues of G-;1/2 AIG-;1/ 2 are bounded from below by zero and from above by one, and by means of Lemma 2.19
o In cont ras t to the algorit hms given in [BD98, BDW99], there is no need to solve a Schur comp lement syst em to gua ra ntee that BTzl = 0 in each smoot hing ste p. The action of o;' can be obtained easily by applying Gi/ . R emark 2.21. Th e secon d colum n of c;' is redundant in the application of the smoother and can be replaced by zero. Thus , the implementation of th e iterati on (2.41) does not require the value of a . Furth ermo re in Subsect. 2.3.5, we will show that G I , in (2.41), can be replaced by a GaujJ-Seidel smoother.
In addit ion to the smoot hing and approximation prop erty, th e II . 11stability of t he ite ra tion matrix Id -c;' Al is also required for the convergence of the W- cycle; see, e.g., [BS94, Hac85]. The stability is inherited from Gi/ just as t he smoothing prop erty. Lemma 2.22. (Stability estimate) Under the assump tions (2.39}, the iterati on (2.41) is stable and th e iterati on error er is bounded by
Il erll :S C Il e?ll,
m ~1 .
Proof. Since the iteration error er can be written as
er = (Id -
c;: AI)me? ,
we can bound lI erll by Il erll :S IIG-;1 / 2111IId - G-;1/ 2AI G-;1/ 2 1Im IIG:/
211I
Je?11
By using t hat a(G l l AI) C [0,1], we find t ha t Il erll :S J fl,(Gt} Il e?lI , and t he stability of t he iteration (2.41) follows immediat ely from Lemm a 2.19. 0 We can now formulate our main result for the multigrid method defined by t he smoot hing iteration , (2.41) , the coarse grid problem , (2.36) , and the modified prol ongation (Imod)Ll for th e coarse grid correc tion. Theorem 2.23. Under the assumptions BTz? = 0 and (2.39), th e convergence rat es for th e W- cycle are in depen dent of the number of refineme nt levels provided that the num ber of sm oothing steps is large en ough. P roof. Using our special type of smoot her gua ra ntees th at BTzl = 0 if = O. Then t he residu al dl satisfies dl = 0, and by applying our special restri ction operator (Imod)l- l , we find th at W?::.l dl- l = O. Thus the assump tions of Lemm a 2.16 are satisfied, and t he proof follows from the approximation property Lemm a 2.16, the smoothing prop erty Lemm a 2.20, and t he stability est imate Lemma 2.22. For det ails on th e genera l theory, see, e.g., [BS94, Th. 6.5.9] or [Hac85, Th. 7.1.2]. 0
BT z?
Wr
136
2. Iterative Solvers Based on Domain Decomposition
2.3.4 Implementation of the Smoothing Step So far the smoothing step (2.41) is based on Gil and AI . However due to the special structure of the residual, we can replace G I by a lower block triangle matrix. In addition, Al does not have to be assembled. Introducing the non-symmetric matrix
the smoothing iteration (2.41) can be simplified. For simplicity, we suppress the level index for the matrices if the level l is clear from the context. Proposition 2.24. Under the assumptions smoothing iteration
BT z?
= 0 and
Wr
dl
o the (2.42)
yields the same results as (2.41). Proof. A straightforward calculation, as in the proof of Lemma 2.20, shows that = 0 and that the components associated with the index N of the residual are zero. Thus, the action of the non-symmetric smoother in (2.42) is the same as that of the symmetric one given by (2.40) and (2.41). 0
BT zr
Remark 2.25. We note that (2.42) can be interpreted as an inexact block GaujJ-Seidel smoother for Anon· The inverse of ARR is replaced by G E/ . A different class of smoothers can be easily constructed by using postprocessing techniques . The point of departure for the introduction of a modified smoothing step is the following observation.
Lemma 2.26. Let
VI
E
XI
be a solution of
AsemiVI := (Id - PI)AI(Id - Pt)VI = (Id - PI)!I Then, UI := (Id - Pt)VI is the unique solution of (1.48) on level 1.
The proof is straightforward. We remark that A semi is a positive semi-definite matrix. Observing BTWI = Id, we find that the kernel of A semi is given by the range of WI. Furthermore, the rows of A semi associated with the interior nodes on the non-mortar side are zero. Our new smoothing iteration is now defined in terms of a good smoother G~~i for Ascm i (2.43)
2.3 A Multigrid Method for the Mortar P roduct Space Formulation
137
where G~~i is a suitable symmet ric pseudo-inverse of Gsem i and sati sfies (2.44) The following theorem shows that the smoothing step (2.42) can be replaced by (2.43) without losing optimal convergence rates. Theorem 2.27. Th e smoothing iterati on (2.43) guarantees level ind ependent convergence rates for the W- cycle provided that th e number of smoothing steps is large enough, and the assumptions (2.44) on G~~i are sat isfi ed and
BTz?=o.
The proof follows the lines of Subsect . 2.3.3. For det ails we refer to [KWOOa]. We remark that t he implement at ion of th e smoot hing iteration (2.43) can be based on the positive semi-definite matrix Asem i and one local postprocessing st ep.
z?
pnz?
Wr
=
Proposition 2.28. Under th e assumptions := (Id and dl 0, th e ite rates Zt defin ed by (2.43) can be obtain ed from Zt by the local postprocessin g st ep
Zt
:=
(Id - I f) zt ,
m
2
1 .
Proof The assertion can be easily shown by induction 1 (d A semiZIm-I) Zlm = Zlm- l + (Id - pT)GI se m i I -
o 2.3.5 Numerical Results in 2D and 3D
We pr esent some num erical experiments illustrating th e performan ce of the algorit hm. The t heoretical results for the W-cycle are confirmed by our experiments. Fur thermore, we show observed level independent convergence rates for t he V-cycle. We use piecewise linear Lagrangian finit e elements in 2D and t he du al space introduced in Subsubs ect. 1.2.4.1, as the Lagrange multiplier space. In 3D, we use hexah edr al t riangulat ions and t he piecewise bilinear simplified du al basis funct ions W introduc ed in Subsubsect. 1.2.4.2 . St andard uniform refinement techniques are applied on the tri angulations of the different subdomains; each element is decompo sed into four congruent subt rianglcs in 2D and int o eight subhexahedras in 3D. The num erical results are based on the proposed multigrid algorit hm. In particular , we use t he level depend ent restriction and prolongati on operators given by (2.37) and (2.38) , respectively. We compa re three different smoothing operat ors and th e influence of t he number of smoothing st eps on the convergence rates. In th e first
MK,
138
2. Iterative Solvers Based on Domain Decomposition
case, G R is a damped J acobi method , GR := /-L diag A R R , /-L = 5/4. Then, Theor em 2.23 yields a level independ ent convergence rate for the W -cycle provided that the number of smoot hing st eps is large enough . Additionally, we apply a symmetric and a non- symmetric Gau/3-Seidel smoother for GR , where t he unknowns U R are ordered lexicographically. The impl ement ation follows Sub sect. 2.3.4, and one smo othing st ep is realized in t erms of (2.42) . We re mark that the ordering in the block Gau/3-Seidel method (2.42) is impor t ant. Only if the unknowns U N = Us are considered afte r the unknowns Ul and U2 , will t he const raints at the int erfaces be sati sfied in each smoothing ste p . To te st our method , our mul tigrid start it erat e is set t o zero on each refinement level. In pr act ice, nest ed it eration in t erm s of the modifi ed prolongati on would be used t o define the start it er at e on the next level. The start it er ates w? for the smoothing st eps are defined as usual. They are set t o zero in t he pr esmo othing pro cess on level I < j and are defined in t erms of the actual it er at e and the prolongated defect correction for t he post smoothing step. Now, t he definition of our modified prolongation op erator yields t hat w? = 0 for all st ar t it erates.
BT 0 ,9
~
0.' ~
8e'
-
Gauss -Seide l
- - . --
Jacobi
0.7
0 .3 0.1
10'
I
..
Gauss-Seidel --.- - Jacobi
~
0.7
~
0 .5
t E:
8
10 2 10 3 10· 1 0~ Number of eleme nts
.f 8
0 .1
V (I , I)- cycle
- - .--
Gauss-Seidel Jacobi
0.7 0.5
0.3 0.1
10 1 101 10· 10 5 Number of elements
. 0.' ~
g
f
g
0 .3
10'
-
~
if I
8~
Gauss-Seidel
0.5 0.3 0.1
/"
- - --- - . - - - ~
10 2 103 10' l OS Number of elements
10'
W(I , I)-cycle
V( 3, 3)-cycl e
-
••••• J acobi
0.7
W(3, 3)-cycle
Fig. 2.19. Jacobi and symmetric Gaufi-Seidel smoother, (Example 1)
I 4 I 4 ~[;l' ~i [:J '4 ~l:J' ~i EJ' 4 0.3
0.3
a; ~
8
0 .2
iii
0. 1
8
10 1
~
0.3
0.3
0 .2
Qi
0.2
Cii
0.2
0 .1
8
0.1
~
0.1
E:
Number of elements
Number of elements
Number of elements
10 2 10 3 10· 10 s Numb er of elem ents
V(I , I)-cycle
V( 3,3)-cycl e
W( I , I)-cycle
W(3, 3)-cycle
10 2
10 3
10'
105
101
10 2
10 3
10'
l OS
10'
10 1
10 3
10'
105
10'
Fig. 2.20. Non-symmetric Caufl-S eidel smoother, (Example 1)
Figures 2.19-2.24 show t he convergence rat es of t he V- cycl e and the W cycle for the t hree different smo other s versus the number of elements. We consider Examples 1-3, introduced in Sub sect . 1.5.1. The results for the J acobi and t he symmetric Gau/3-Seidel were originally pr esent ed in [WK99]. In Figs. 2.19 , 2.21 and 2.23, we compare the J acobi and the symmet ric Gau/3Seidel smoother , whereas in Figs. 2.20, 2.22 and 2.24, the non-symmetric
2.3 A Multigrid Method for the Mortar P roduct Space Formulation
139
Gauf3-Seidel smoo ther is used. All smoot hers can be easily applied, and no ext ra work is required to satisfy t he const raints at the int erface. The two pictures on t he left in t he figures show t he results for the V-cycle, whereas t he two pictures on the right show the corres ponding results for the W-cycle. The number of pre- and postsmoothing steps is equa l m ; we provide results for m = 1 and m = 3.
. QI
~
~ ~ 8
-
Gauss-Seidel
-- . - -
Jacob i
0.5 0.3
/"" .../
O.l /~---
10'
.
0.9
e
0.7
~
0.5
~ 8
0.3
-
QI
0.7
10 2 10' 1 0~ 10 Number of elements
5
-- .--
0.1
i o'
....
Gauss-Seidel Jacobi
e
.
0.7
118>
0.5
8~
.....-
0.3
10 la' 10' 10 Numbe r of eleme nts
s
1
10
V(3,3)-cycle
Vel , I) -cycle
0.9
e
- • • --
;---
_
Gauss- Seidel Jaco bi
0 .7
t
/",
-
OIl
11
0.,/
y '
2
Gauss-Se idel --.-- Jacobi
0.5
i: 3
0.3
.r" 0. 1 . ,, ""
io'
10 l a' 10' l OS Number of elements 2
W(l , I)-cycle
10 10 3 10' lOS Number of elements 2
W( 3, 3)-cycle
Fig. 2.21. J acobi and symmetric Gau B-Seidel smoother, (Example 2)
~EJ" "D" i
I
~El" ~D"
~
0.3
Qj
0.2
iii
c~
0.1
8
10'
~
2
3
g
0.3
t
0.2
~
0.1
io'
10
2
10
3
10'
la'
g
0.3
t
0.2
§
0.1
3
Number 01elements
Nu mber of eleme nts
10 10 10' la' Number of elements
Vel , I) -cycle
V(3,3)-cycle
W(l , I)-cycle
10
10
10'
l OS
10'
2
0 .3
0.2
0. 1
10 1
10 2 10 3 10' l OS Number of elements
W( 3,3)-cycle
Fig . 2.22. Non-symmetric GauB-Seidel smoot her , (Ex ample 2)
We start with a compa rison of the num erical results for Examples 1 and 3; see Figs. 2.19, 2.20,2.23 and 2.24. Here, we have full regularity, and obtain level ind epend ent convergence rates for all test set tings, even in the case of th e V-cycle with just one pre- and one post smoothing ste p and as well as for t he non-symm etric Gauf3-Seidel smoot her. Increasing th e number of smoothing ste ps from one to t hree improves the convergence ra tes for both the W- cycle and V-cycle considera bly. The num erical results show that the convergence rates are approximately three tim es smaller for three smoot hing st eps than for one smoot hing ste p. Th e convergence rates for t he W- cycles are only minimally better than those for t he V-cycles. The results for the symmet ric Gauf3-Seidel smoot her are better th an t he results for t he J acobi smoot her and the non- symm etric Gauf3-Seidel method. However , one applicat ion of t he symmet ric vari ant is twice as expensive as one of t he non-symm etric. The non-symm et ric Gauf3-Seidel method yields considerably better results t ha n t he J acobi smooth er . Finally, we consider th e results for Ex ample 2 in mor e det ail. Figur es 2.21 an d 2.22 show t he observed convergence rates. Although we have no H 2 _
140
2. It erat ive Solvers Based on Domain Decomposit ion
regu lar ity, we observe level ind epend ent convergence rat es; see Figs. 2.21 and 2.22. In t he case of the non-symmet ric GauB-Seidel and t he J acobi smoot her, t he asympt ot ic phase starts lat er th an in the ot her exa mples. In creasing t he numb er of smoothing ste ps gives considera bly better results . Using t he symmetric GauB-Seidel vari ant , we obtain, even wit h just one smoothing st ep, convergence rates where the asympt ot ic can be observed from t he beginn ing ; see Fig. 2.21. R e mark 2 .29 . Mor e robust smooihers, e.g., of IL U type, can also be used, if th ey are mod ified so that th ey satisfy th e assumptions of Lem m a 2.16. A suitable mod ification can easily be carried oui in a local postpro cessing st ep. In this st ep, a scalar equati on is solved fo r each unkno wn in th e inte rior of the non-mortar sides; see Propositi on 2.28.
.. ~
t
8
Ga uss-Seidel .• ••• J acobi 0.7 0.'
."
0.3 0.1
10'
1/ 10'
..-_ ........-..... -... 10'
i
e'
0.3
10'
.
~
0.'
~
10'
Gauss-Se idel
.•••• Jac obi
0.7
8 10 '
-
0.9
~
........
....-- ---.--10'
10'
10'
~
10'
0.9
.. ~
g
0.1 10'
V(3,3)-cycle
Vel , l j-cycle
Gauss-S eidel
0.'
/' 10'
Number of elements
Number 01elements
-
••••• Jaco bi 0.7
y
....... ...- -- --.. 10'
10'
E-
Gauss-Seidel
--.- - Jacobi
0.7 0.'
~
10 '
-
0.9
..,.. - -.. ........... ......... 10'
10 '
10'
10'
10 '
Number of elements
Number of elements
W(l , I )-cycle
W(3 , 3)-cycle
Fig. 2. 23 . J acobi and symmetric GauB-Seidel smoother , (Ex ample 3)
~I G '4 '4 ~D '4 ~i [3" ~ D 0.3
g
0.3
0.3
g
Qj
0.2
0.2
Qj
0.2
0.1
8
0.1
~
~
0.2
~
~
u
0.1
8
0.1
o
10'
~
1 0~
10 3
10'
1 0~
Number of ele men ts
Vel , l j-cycle
10'
10 2 103 10' l Os Number of elements
V(3, 3)-cycle
10'
~
10 2
10 3
10'
10 5
Number of elements
W(l , I )-cycle
0.3
10'
10 2
10'
10'
10 5
Numbe r 01elemen ts
W(3 , 3)-cycle
F ig. 2.24. Non-symmetric GauB-Seidel smoot her , (Ex ample 3)
We point out t hat in our approach t he const raints can be sat isfied in each smoot hing st ep by applying a simple Gau B-Seidel smoot her. Thus, no addit ional work is necessary to obtain iterat es in the constrained space. The only difference, compared wit h a st andard multigrid algorit hm for symmet ric posit ive definite problems on nest ed space s, is th e choice of the tran sfer operat ors . In our case , we replace th e standa rd restriction operator I{- l by the level depend ent operator (Imod):- l = (Id - Pl-d I{- l as defined in (2.37), and th e modified pro longatio n given by (2.38) is used . Observi ng that Pl - 1 is not only sparse but t hat th e number of non zero ent ries is bounded by Cylri in 2D and by Cn 2 / 3 in 3D, we find that t he extra amount of work
2.3 A Multigrid Method for the Mortar Product Space Formulation
141
is considerably less than one smoothing step; here, ti is the number of unknowns. In particular, the modifications involve only a scaled mass matrix on the interface and can be carried out as a local postprocessing step, i.e., Id -
PZ-1
Id
0)
= ( MT 0
.
Finally, we consider two examples in 3D illustrating the performance of our multigrid method. Here, we use trilinear finite elements on hexahedrons and the simplified dual Lagrange multiplier space introduced in Subsubsect. 1.2.4.2. As in the 2D case, we compare the asymptotic convergence rates of the V- and W-cycles in case of one and three smoothing steps. We apply a symmetric Gaufi-Seidel and a Jacobi smoother.
Fig. 2.25. Triangulation and isolines for Example 4 (left) and Example 5 (right)
In Example 4, we consider a sandwich-like composite material. The domain [l := UT=l [li is decomposed into three hexahedrons [li := {(O, 1)2 X (Zi, zi+d} where Zl := 0, Z2 := 1, Z3 := 1.2, Z4 := 2.2. As model problem, we consider -diva\7u = 1 on [l where the coefficient a is piecewise constant, alS!i := 100, i = 1,3 and alS!2 := 1. Dirichlet boundary conditions are applied on the upper and lower part of the domain, u(x, y, z) = 1000 ((x - 1/2)2 + (y - 1/2)2)1/2 . (1 - y/3) exp( -10(x 2 + y2)) if Z = Zl or Z = Z4, and homogeneous Neumann boundary conditions are taken elsewhere. In the left part of Fig. 2.25, the nonmatching initial triangulation and the isolines at the interface are shown. _.- Gauss-Seidel Jacobi
~
0.5
E'
,.-.-
~
8
0.1 .' /~ _ _ ~
10'
-
. ___
10' 10' 10' 10' Number of elements
V(I, I)-cycle
Gauss-Seidel - ,-- Jacobi
l'!
0.7
~
0.5
E' ~
8
10'
2
l'!
11
~
~.:
10' 10' 10' 10$ Number ot elements
V(3,3)-cycle
Gauss-Seidel .-- Jacobi
Gauss-Seidel -.-- Jacobi
~
-~
-+-
.$
~
0.7 0.5
/.-_ .. ..-._---..--. 0.1 /~_~ .. _ _ ~
10'
10' 10' 10' 10· Number of elements
W(I, I)-cycle
2
l'!
0.7
~
0.5
E'
~
~c=4.. ~
10'
10'
10'
10'
10'
Number of elements
W(3,3)-cycle
Fig. 2.26. Jacobi and symmetric GauB-Seidel smoother, (Example 4)
The non-mortar sides are defined on the middle hexahedron and the dimension of the Lagrange multiplier space Mh(rm) is 16 on the initial trian-
142
2. Iterative Solvers Based on Domain Decomposition
gulation. As can be seen in Fig. 2.26, the performance in 3D is comparable to the 2D results. The numerical results confirm the theoretical ones. Asymptotic convergences rates which do not depend on the refinement level can be observed. Even for the V(I, I)-cycle, a constant asymptotic convergence rate is obtained. In Example 5, we consider a non-convex domain shown in the right part of Fig. 2.25. It is decomposed into three subdomains fl I := {(O, 1)2 X (0, I)}, fl 2 := {(I/3,2/3)2 x (I,2)), fl 3 := {(0,I)2 x (2,3)). Here, we are in the geometrical nonconforming situation. The non-mortar sides which are defined to be on fl 2 cover only a part of the adjacent mortar sides on fl I and fl 3 • Gauss-Seidel Jacobi
l!
0.7 0.5
"~ 10
0.7
~
0.5
"~
0.3
8
-
l!
10
10
10
Number of elements
V(I, I)-cycle
10
Gauss-Seidel .-- Jacobi
-."
~~---_. 10 10 10 Number of elements
-~ Gauss-Seidel .. ,-- Jacobi
2
e
0.7
~
0.5
~
0.3
" 8 10
V(3,3)-cycle
... - Gauss-Seidel
~
~
"~
- .-- Jacobi 0.7 0.5 0.3
----.
10 10 10 Number of elements
W(I, I)-cycle
W(3,3)-cycle
Fig. 2.27. Jacobi and symmetric GauB-Seidel smoother, (Example 5)
On the initial triangulation, the Lagrange multiplier space on each interface is one dimensional. We impose Dirichlet boundary values on the left and right faces of aflI and afl 3 , and set u(x,y,z) = 10 for (x,y,z) E {afli, i E {I, 3}, x E {O, I} }, elsewhere we impose homogeneous Neumann boundary conditions. No Dirichlet boundary conditions are given on fl 2 . Figure 2.27 shows the convergence rates for the Jacobi and the symmetric Gauf\-Seidel smoother. The performance is not as good as in Example 4, but the asymptotic convergence rates seem to be independent of the refinement level. Increasing the number of smoothing steps yields considerably better results. 2.3.6 Extensions to Linear Elasticity
In this subsection, we consider the deformation of a linear elastic body as model problem. The body in its reference configuration is identified with the bounded polyhedral domain fl C JRd, d = 2,3. The planar case d = 2 can be interpreted as a model for an infinite long cylindric bar, fl x JR. Then, the deformation of the body is described by a system of partial differential equations. The displacement field U of the body satisfies the following boundary value problem -O"ij(U),j U
O"ij(U)' nj
= f;, = 0, = Pi,
in on on
fl ,
rD
rF
, ,
(2.45)
2.3 A Multigrid Method for the Mortar Product Space Formulation
143
where n is the unit outer normal on the boundary of rl . The volume force is denoted by f E (L 2(rl ))d, and p E (L 2(r F ))d is th e surface traction. We denot e tensor and vector quantities by bold symbol s, e.g., T and v, and its components by Ti j and Vi, 1 :::; i , j :::; d. The partial derivative with respect to Xj is abbreviated with the index .i : Furthermore, we enforce the summation convention on all repeated indices ranging from 1 to d , and we denote by 6 i j the Kronecker symbol. The syst em (2.45) is obtained by th e equation of equilibrium, the strain-displacement relation and t he constit utive law. In t he case of a linear elastic material , the st ress tensor a depends linearly on t he infinitesimal strain tensor €(u) := 1/2(\7u + \7u T ) . The stress tensor a is given by Hooke's law O"ij(U)
:=
E i jl m U l ,m ,
where Hooke's t ensor E := ( Eijl m ) fj lm=l' E i jl m E LCO(rl) , is assumed to be sufficientl y smoot h, symmet ric and uniformly positive definite, i.e., E ijl m
=
E jil m
=
El m i j ,
1:::; i,j, k, l:::; d
,
and there exists a constant such th at for each symmetri c tensor , ~ij =
~j i ,
In the case of a homogeneous isotropic mat erial, Hooke's te nsor has th e simpl e form
°
where E > is Young's modulus and v E (0,1/2) is the Poisson ratio . Then , t he components of t he st ress tensor can be writ ten in terms of E , v and the infinit esimal st rain € te nsor as
Using the Lam e constants f-l and A, th e st ress tensor satisfies u(u) = 2 f-l €(u)
+ A tr(€(u))
0 ,
where the trace of t he strain tensor is given by tr(€(u)) := Ekk (U), and the Lam e constants ar e defined by
E
f-l := 2(1 + v) '
A:=
E v (1 + v)(l - 2v)
Here, in an abuse of not ati on we use the same symbol for one of the Lame constants as for the Lagr ange multiplier in the mortar setting. In the following, we use Poisson's ratio and Youn g's modulus to specify a material.
144
2. It er ative Solvers Based on Domain Decomposition
T he boundar y aD = TD U TF is decompos ed into two complementary parts, a Dirichlet par t T D with non zero measure and a Neumann part TN , TN n TD = 0. On TD t he displacement is set to zero and on TN the sur face t raction is given. Then, t he space of admissible displacement s H;(D) is a subspace of H 1(D) := (H1(D))d and is defined by
H ;(D) := {v E H 1(D) I vl r D = o} St arting with (2.45) , int egration by part s yields t he weak formulation. Let u E H; (D) be the solut ion of the following vari ational pr oblem
a(u , v) = f(v) , where f(v) := (v ,f)o;!t by
a(w , v)
:=
+ (v ,p)o;rF'
J
v E H;(D) ,
(2.46)
and the bilinear form a(·,·) is defined
E i jl m W i ,j Vl ,m
dx ,
W,
v E H 1(D) .
a
Associated with a(' , ') is t he energy norm I I · I I , IIIvll1 2 := a(v , v). Appl ying the Lemm a of Lax-Milgram , the well-posedness of (2.46) follows from the cont inuity of the bilinear form a(·, ·) and the linear form fO and the second Korn inequality ; see, e.g., [Bra97, BS94]. We refer to [Cia88, Gur81 , MH94, K0 88] for a general introdu ction to continuum mechanics and elasticity. The decomposition of t he domain into subdomains is chosen according to t he different materials or bodies. We consider two different sit ua tions. In t he first sit uation, we assume that th e different materials are glued to gether such t hat t he jump of the displacements in t angenti al and normal dir ection vani shes. In Sub subsect . 2.3.6.1, t he question of uniform ellipticity is add ressed, and in Sub sub sect. 2.3.6.2, num erical result s illustrate t he deformation of a body. We solve a simplified lineari zed form of a contact problem in Sub sub sect. 2.3.6.3. In t hat sit uation, the weak solution of t he conti nuous vari ati onal problem does not have to be in H; (D) . At th e int erface, arbit ra ry displacement s in tangential dir ection are admissible, bu t t he jumps of the displacement s in norm al direction have t o be zero across the int erface, i.e., [u-.n] =0. For th e discreti zation , we use t he same finite element spaces of ord er one as before, see (1.7) . The unconstrained vector valued finite element space Xh is defined as pro du ct space X h := (Xh )d. In the case of the coupling in t ang enti al and norm al direction , t he Lagrange multiplier space M j, := (Mh) d is also vector valued . Then , t he saddle point formul ati on of the mortar method can be defined in te rms of th e bilinear form b(·, .) M
b(v, p,) =
L ([vd, m =l
ft i) , = ,
v E X j, , P, E Mj, .
2.3 A Multigrid Method for the Mortar Product Space Formulation
a(Uh , v)
+ b(v , Ah) =
b(Uh ' JL)
f(v) ,
v E Xh , JL E M h .
= 0,
145
(2.47)
Here, the bilinear form a(·, ·) is extended to th e nonconforming space X h by replacin g the int egral over il by it s broken form ~~=l Jfl k' The second equation of t he saddl e point problem gua ra ntees the weak continuity of t he solution us , In ana logy to Sect . 1.1, the nonconforming space V h is defined as the kern el of the operator B T : Xh - 7 Mj, associate d with th e bilinear form bt-, ') ,
and we find V h = (Vh)d . Under the assumption t hat a(',') is uniformly elliptic on V h x V h , i.e., K
«(v .vi
> cllvlli := Lllvlli;flk'
VEVh ,
k=l
th e following variational pro blem has a unique solution: Find Uh E V h such t hat
a(Uh 'v)
= f(v) ,
v E
Vh .
(2.48)
In the next subsubsection, we address th e question of ellipticity. A uniform discrete inf-sup condition yields in combina tion with the ellipt icity of a(' , ') on the kernel of the operator B T th e uniqu e solvability of (2.47) . We refer to Sub sect . 1.2.3 for t he pro of of the inf-sup condition in the scalar case. Since Mj, and X h are product spaces , the inf-sup condition follows from the scalar case. Moreover , the positive definite syst em (2.48) is equivalent to the saddle poin t problem (2.47) . 2.3.6 .1 Uniform Ellipticity. Here, we consider the uniform ellipt icity of th e bilinear form a(·, .) on the constrained spa ce V h X V h . We start with the special case that Bilk n TD has a non zero measur e for ali I :::; k :::; K . Then , Korn's inequa lit y can be applied to each subdomain , and we find
a(v, v )
=
K
L ak(V,V) ;::: k=l
K
CLllvlli;flk = Cllvlli , k=l
where akh ·) stands for th e restriction of a(' , ') to t he subdomain ilk; see, e.g., [Bra97, BS94]. Thus a(·, ·) is elliptic on H ;(il) x H ;(il). We remark that C does not depend on the numb er of subdomains. Unfortunately, many interesting cases do not satisfy this assumpt ion . However for the uniqu e solvability of (2.48), it is sufficient to have the uniform ellipticity of a(·,·) on v ; x v».
146
2. Iterative Solvers Ba sed on Domain Decomposition
In t he scalar ellipt ic case, the kernel of th e corr esponding bilinear form is th e subspa ce of piecewise constant functions. It s dimension is given by the number of subdomains fh such that a[h n To is empty. In th e elasticity setting, the kern el is of higher dimension . The rigid body motions per sub domain define a three dimension al space in 2D and a six dimension al space in 3D. To get a bet ter feeling for the kernel of a(·, '), we consider the case of two unit squa res nl and nz in 2D with homogeneous Dirichlet boundar y condition on one side of anI nan and homogeneous Neumann boundary condition elsewhere. We set vl n! := and vl», := (3 (y - Ye, Xc - x)T, (3 :f. 0, where (x c, Ye)T denotes th e center of gravity of 'Y := anI n anz . Then, v E (Y) Z but a(v , v) = 0, and thus a(· ,·) is not ellipt ic on (Y) Z x (Y) Z, where the nonconforming space Y is defined by (1.6) . Moreover to obtain ellipt icity on V h X V h, the dimension of the Lagr ang e multiplier space has to be lar ger t ha n d. In t he following, we assume th at N m :::: 2 in 2D, and that in 3D the t riangulation at the int erface is a tensor product mesh with N m :::: 4. We recall that N m is t he dimension of Mh b m). Based on this observation, we define the nonconforming space
°
V H :=
{VEH~(n)1
![V] 'J.LdS=O 'J.LEMH b m) ,I"'5.m "'5.M} ,
,=
where MH b m) := (MH b m))d is a suitable te st space. If M Hb m) C Mh b m) t hen V h C V H, and to obtain t he uniform ellipt icity on V h X V h it is sufficient to show the ellipt icity on V H xV H. A natural choice for MH b m) is Pl b m) . Unfor tunat ely, none of t he considere d Lagrange multiplier spaces satisfy Pr b m) C Mh b m). We start with t he const ruction of a new macro Lagrange multiplier space MH b m) C M hb m) in 2D. Let t E [0,1] be a par ametrization of t he ID int erface "[m» i.e., x E ;:Ym if and only if x = Xo + t x(XN=+l - xo ), t x E [0,1]' where Xo and XN= +l are the two endpoints of "[ra- Then , we decompose t he set of int erior vertices {Xi 11 "'5. i "'5. N m} int o two disjoint subsets . Introducing "'m:= max{ I "'5. i "'5. N ml t X i "'5. 0.5}, we define a test functi on PH by
where 'ljJi is eit her t he nod al st andard basis function 'ljJt or the du al one 'IjJ[; see Subsubsect . 1.2.4.1. Figur e 2.28 shows the shap e of the test function PH. In t he left , PH is given for t he st andard Lagran ge multiplier space and in t he right for the du al Lagrange multiplier space . It is easy to see t hat t he mean value of PH is equa l zero for both choices. Now, we define in t erms of PH the macro space MH bm ) := span{ipH E P H} , where PH := {I , PH} in t he 2D case. In 3D, we work wit h t he simplified dua l basis fun ctions given in Sub subsect. 1.2.4.2. We use the tensor product
2.3 A Mult igrid Method for the Mort ar Product Space Formulation
147
st ruct ure to define PH := {l,tLk-,tL1I ,tLk-tL1I} , where tLk- and tL1I are th e one dimensional t est functi ons with respect to the two different dir ections. Thus for the 2D and 3D case , we have M Hbm) C M hbm) .
5/4
5/4
~H
\I,~ o
0.5
1/3
o
1
5/9
_
·1
_
----"'-,· 1
Fig. 2.28. Test fun cti on J..tH for standar d (left) and du al (right) Lagrang e multiplier
Lemma 2.30. Let v E V H body m otion , th en v = O.
an d v restric ted on
,
fh , 1 :s;
k
:s;
K , be a rigid
Proof. We start with a subdomain rho such that afh o n r D has a non zero measur e. Due to t he Dirichlet boundary condition, v restricted to t his subdomain is zero. On each adjace nt subdomain nl , we can write vl n, al + W I , where a l E ]Rd and
WI
y - Yc ) = (XI ( Xc - X
,
in 2D and 3D, respectively. The center of gravity of I'mo := anko n ani is denot ed by (xc, Yc) in 2D and by (xc,Yc, zc) in 3D. Then , the int erface condition yields that for all adjace nt subdomains, we find in 2D
J
(al
+ W I)
. ei
da = 0,
1 :S; i
:s; d
I'~o
where e, denot es the i-t h unit vector, and t hus al = O. Introducing in 2D VII E P1b mo) , VII( XO) = -VH(XN~+d = 1, see Fig. 2.28, a st raightforward computation shows t hat
J
J..tH vHdO'
I'~o
= a u bmo l
148
2. It erative Solver s Based on Domain Decomposition
We find the lower bound aH 2: 1/3 if Mh(')'mo) is the piecewise linear dual Lagr ang e multiplier space as discussed in Subsubsect . 1.2.4.1 , and an 2: 2/9 if Mh (')'mo) is the st andard Lagrange multiplier space and thus WI = O. In 3D , WI = 0 follows from the 2D situation and the tensor product st ruct ur e. We not e that the lower bound for an does not depend on the mesh on "'(mo' St arting from rh o, we can move to each other subdomain by crossing interfaces. 0 The following lemm a yields the uniform ellipt icity of a(' , .) on V h X V h . Lemma 2.31. Th e bilin ear form a(· , ·) is uniformly elliptic on V H xV H. Proof. The proof follows from th e definition of V H and th e following upp er bound for t he broken HI-norm M
Ilvll i :S C1a(v, v)
+ C2 L
L
J
'PII [V] dO'1I 2,
II
E
V
m=l
H~U:?) ,
(2.49)
where II . II st ands for t he Euclidean norm in jRd, and 0 < C1, C2 < 00 are suitable constants. The upp er bound (2.49) can be est ablished by cont radiction. We assume t hat (2.49) is not true for any constants 0 < C1, C 2 < 00. Then , t here exists a sequence v " E H;(.o) , n E N, such that Ilvnlll = 1 and (2.50)
Each v " can be uniquely decompo sed into a rigid bod y motion and a sur plus. To do so, we introduce the projection operator IIrot as follows: (llrot V)l ilk is a rigid body motion on .ok , 1 :S k :S K. If a.ok n To has a non zero measure, we set (llrot v) Iilk = 0, otherwise it sati sfies
J a;
tt.;» da
=
J
V
si;
do;
J
rot(IIrotv) do =
o;
J
rot v da ,
a;
where in 2D rot v := (- Vl ,2 + v 2,d and in 3D rot v := (V3,2 - V 2,3 , Vl,3 V3 ,1, V2,1 - Vl,2 )T . Then , v " = IIrotv n + (v" - IIr otv n ) = : IIrotv n + w" . Moreover by mean s of t he second Korn inequality and the positive definiteness of Hooke's tensor, we find
Thus w" converges st rongly in H; (D) to zero. Using the definition of IIrot , we get IIIIrotvnlll :S Cllvnlll = C . The range of th e projection IIrot is a closed finit e dimensional subspace of H; (D) . Its dimension is bounded by 6 K . Therefore, t here exist s a st rongly convergent subsequence; the indices are
2.3 A Multigrid Method for the Mortar Product Space Formulation
149
denoted by tu , Then the subsequence v n i converges st rongly to an element v E H ~ (fl ), and the limit v restri cted to each subdomain is a rigid bod y motion. Moreover , IIvl11 = 1 and (2.50) yields that v E V H which is a cont radiction to Lemm a 2.30. 0 Although the space V H depends on the triangulati on , th e ellipticity constant can be bounded from below, due to the lower boun d for O'. H , independently of the t ria ngulation. Remark 2.32. Unfortunately, the proof by contradiction gives only the existence of such a cons tant 0 1 , but no inform ati on if 0 1 can be chosen independ ently of th e number of subdom ains. However, the decompos ition in to su bdom ains is given by phy sical param eters, and in m any in teresting applicati ons, e.g. contact problems, th e number of subdom ains is fixed and sm all. It is likely that mor e elaborate techniques yield an ellipticity constant which is indep endent of the numb er of subdomains. For t he scalar elliptic case in 2D, we refer to [Gop99] and for the t hree field approach to [BMOO]. Both techniques are based on du ality arguments and cannot be applied directly to our situation.
2.3.6.2 Numerical Results. Considerin g the algebra ic st ruct ure of t he saddle point problem (2.47) and using the same not ation as in Subsect. 1.3.3 , we can directly apply the previously discussed multigrid method. The following figur es show some num erical results for a linear elastic bod y. In our first example, we consider a L-shap e domain and two different materials; see Fig. 2.29. Material 1 is por celain and its elasticity modulus and its Poisson number are given by E = 7.9· 1010 N 1m2 , v = 0.37, and Material 2 is silver with E = 5.8 . 10 10 N 1m 2 , and v = 0.23. We use simplicial tri an gulations and piecewise linear finit e elements for Material 1 and piecewise bilinear elements on rect angles for Mat erial 2. In both cases, th e trace is piecewise linear at the interfaces. Adapt ive refinement techniques have been used. The refinement is controlled by a residu al ty pe erro r estimator for mortar finite element s; see [Woh99c]. The left picture in Fig. 2.29 illustrates t he problem set ting.
j II ,----_---,
N E 1 ~ 7.9 ' 10" iii'
" N iii'
E2 ~ 5.8 ' 10
v, ~ 0.37, v, ~ 0.23
Materi~ "1 [ul~
I
/ Material 2 Material I
rD
'"
e1'" 'g"
t§ 0.4
g
0.3
'~" 0.2
'" e>
0 .4 0.3
~ 0.2
c: 80.1
00.1
10
2
10
3
10
4
10
5
Numbe r of elements
W(I , I)-cycle
10
2
10
3
10
4
10
5
Number of elements
W(3, 3)-cycle
Fig. 2.29 . P roblem set ti ng and converge nce rates, (Ex am ple 1)
150
2. It er at ive Solvers Based on Dom ain Decompositi on
Two different surface pressur es are applied at t he inhomogeneous Neumann bound ar y par t indi cated by t he arro ws in t he left of Fig. 2.29. The resulting displacements at t hat inh omogeneous Neumann boundar y part are constant and linear , respectively; see Fig. 2.30. In both settings, t he lower left corne r has fixed Dirichlet boundary condit ions and homogeneous Neum ann boundary condit ions are applied elsewhere, an d t he body force is zero . Figur e 2.29 illust rates t he convergence rates for a W (l , I)- cycle with a modified ILU-t ype smoother and for a W(3, 3)-cycle with a symmetric GauBSeidel smoother. In both cases , uniform refinement techniques have been applied. The observed convergence rates are asymptotically constant . We obtain better convergence rat es in the case of t hree pre- and postsmoothing ste ps. However , t he difference is not as distinct as in t he scalar case for Examples 1- 3; see Subsect . 2.3.5. This corres ponds to t he fact that we use a more robust smoother in t he case of only one pre- and postsmoothing ste p .
Fig. 2.30. Displ acem ents and adaptive triangulat ions, (Examp le 1)
In Fig. 2.30, the displacements for t he t wo different sets of boundary conditions scaled by the factor ten and t he final adapt ive triangulat ions are given. The right picture shows a zoom of t he adapt ive t riangulation in t he neighborhood of t he corner singularity. Alt hough, t he triangu lations at t he interfaces are highly non-m at ching, no penetrati on of th e two different mat erials occurs at t he int erface. In our second example, we consider a nut-like geomet ry as depicte d in Fig. 2.31. The non convex domain consists of 13 sub domains, and each of the six inner crosspoints has four adjacent subdo mains. We choose silver as material wit h E = 5.8 · 1010 N 1m2 and v = 0.23. Inh omogeneous Dirichlet bounda ry condit ions corresponding to a rot ati on by an angle of 1r 1500 have been applied on the inner boundary FI, i.e., t he outer norm al on n directs toward t he center of gravity. We work with homogeneous boundar y conditions on Fo := oD \ n. On Fo n ODk , we take Neumann ty pe boun dar y condit ions if a, is a triangle, and Dirichlet ty pe boundary conditions if D k is a square. Figur e 2.31 shows t he initi al nonconform ing triangulat ion, t he displacements scaled by t he facto r 100 on t he final t riangu latio n, and t he multigrid convergence rates of the V- cycl e and W-cycle with t hree pre- and postsmoothing ste ps.
2.3 A Multigrid Method for the Mortar Product Space Formulation 0.9
~
V(3,3) cycle --,-- W(3,3)cycle
0.7
Q)
0
c
151
Q) 0.5 S" Q) c> 0.3
0 0
•__•__ •. _-_-0
-,
0.1
",..
..-
10 10 10 4 10' Numberof elements 2
10'
3
Fig. 2 .31. Initial trian gulation (left) , distorted grid (middle) and convergence rates (right) , (Example 2)
2.3.6.3 A Weaker Interface Condition. In this subsubsection, we consider a weaker coupling condition at the int erface. At the interfaces, arbit rary displacement s in tangential direction are admissible, but the jumps of the displacements in normal direction have to be zero . A suitable coupling condit ion is enforce d in normal dir ection. Thi s setting can be viewed as a simplified lineari zed form of a non linear contact problem. Figur e 2.32 illustrat es t he modified situation at th e int erface. In contrast to the previous subsubs ection, we have no const raints in tan genti al direction of t he int erface. free displacement Material I Material 2 no relative displacement
Fig. 2.32. Free t angential displ acem ent at the interface
We define t he bilinear form bn (' , .) corres ponding to the coupling in normal direction at t he int erface by M
bn(v ,f.l) :=
L ([v · n],f.l)' m'
f.l E
m= l
u, :=
M
IT M hb m)
,
m =l
where n is the outer normal of t he sub domain on the non-m ortar side. Now, we replace the bilinear form b(·,·) in (2.47) by the modified one and obtain the following saddle point problem : Find (u~, AhJ E Xh x Mh such that a ( u~, v)
+ bn(v, A~)
bn(u~ ,f.l)
= f(v) ,
v E Xh
= 0,
f.l E M h
,
(2.51)
At first glance, it has t he same st ruct ure as (2.47). However , there is an essent ial difference. Our new bilinear form bn ( · , · ) is defined on X h x M h ,
152
2. Iterative Solvers Based on Domain Decomposition
where Mh is, in contrast to Mj, , a scalar space. As before, we use a du al Lagran ge multiplier space. Using the same algebraic decomposition, i.e., (ul:) T = ((UR)T , (uN) T), we find for the algebraic represent at ion of (2.51) (2.52) In contrast to D in (1.46) , D n is not a diagonal matrix but a dn h x nh blo ck diagon al matrix, where nh is the dimension of the Lagrang e multiplier space M h . Each blo ck is associate d with an int erior vertex on t he non-mortar side, and the blo ck size is given by d x 1. Thus, we cannot eliminate the Lagrange mul tiplier as easy as in (1.40) . Let P := U~=I {xii 1 ~ i ~ N m } be the set of int erior vertices on the non-mortar sides. Then , we can write D n as D n := diag(dp)PE P , where d p E IRd is defined by
Here, E p is the set of element s, i.e., edges in 2D or faces in 3D, on the nonmortar side sharing t he vertex p, and n , is t he constant out er unit normal on the element e. We assume that d p :j:. O. St arting with b i := dp/ltdpll , we introduce for each vertex pEP an orthonormal basis B := {b I , . . . ,b d } in IRd . The orthogonal tran sform ation which maps B to the canonical basis of IRd is deno ted by Op E IRd x d . An explicit represent ation of Op can be obtain ed , e.g., as Hous eholder transformation. For v E X h , we denote by v p E IRd the degrees offreedom associate d with the vertex p . We set (vp;n, V~ T) T := Opv p and call vp;n and v p;T t he norm al and t ang ential component of v at the vertex p , resp ectively. Then , we define the global orthonormal transformation ON by
Applying the coordinate tran sform ation repr esented by diag(Id, ON, Id) to (2.52) , we find the symmet ric system
The orthonormal tran sform ation ON gives a decomposition of uN into a normal and tangent ial component, i.e., ONllN = ((u~) T , (ur)T) T. Due to t he construction of ON, we have Oy:.ON = Id and Opdp = (1Idptl , O)T. Thus, t he d x 1 blo ck matrices of OND n are given Iidpil el . Observing that D'[;D n is a diagonal matrix, the ent ries of which are tldpl1 2, the Lagr an ge multiplier AI: can be locally eliminated by
2.3 A Multigrid Method for the Mortar Product Space Formulation
153
In our last ste p, we can rearrange the indices. The new index group R includ es now the former index group R plus t he tangential components of t he vectors in the former index group N, i.e., u R'new := ((uR) T, (ur)T)T . The new index group N is a subset of th e former' index group N and contains t he norm al components, i.e., u N;new := u ~ . We observe th at the submatrix of OND n corresponding to the new index group N is diagonal. Thus, we can pro ceed as in Section 1.3, and the proposed multigrid algorit hm on the unconstrained product space can be applied. To obtain t he solution from th e rotated solut ion, we have to carry out one local postprocessing st ep. Finall y, we show some numerical results, illustrating t he difference between the two coupling conditions at the int erfaces. We start with the 2D exa mple of Subsubsect . 2.3.6.2. Compar ed to Subsubsect. 2.3.6.2, we use a weaker coupling condition at the int erfaces. Now, the bodi es are not glued to gether , and free tan gential displacement is permitted . The left picture in Fig. 2.33 shows the modified situation at th e int erface. The coupling condition in normal dir ecti on can be viewed as a kind of lineari zed non penetration conditi on of t he bodies in th e reference configuration. In th e middl e and the right picture of Fig. 2.33, the convergence rates are given for a W(I , I)- cycle with a modified ILU-typ e smoot her and for a W( 3, 3)-cycle with a symm etric GauB-Seidel smoot her. Comp ar ed to Fig . 2.29, we observe t hat the stronger coupling condition at the int erface results in better convergence rates. However , t he difference is not significant and in both cases asymptot ically constant convergence rates are obtained .
J
II
N E 1 = 7.9 ,10" IT?
,--_-----, E2 = 5.8 '10"
~
v, =0.37, v, =0.23
Materi~ --1 [u n]=O
I Material 2 rD
/ Material I
$
~
<1>
'§
0 .4
g0.3
0.4
<1>
g <1>
<1>
0.3
W0.2
~
~ 0.2
c 80.1
80.1
10
2
10
3
4
10
10
5
Number 01elemenls
WeI , I)- cycle
10
2
10
3
10
4
10
5
Number 01elements
W( 3,3)-cycle
Fig. 2 .33. P roblem set t ing and convergence rat es, (weak coupling)
In cont ras t to Fig. 2.30, penetration might be observed at th e int erface, see the right picture in Fig. 2.34. Although , th e proposed algorit hm does not solve a nonlinear contact problem, we can use t he meth od as an inn er iteration scheme within an outer scheme used to det ect th e act ual zone of contact. Once t he act ua l cont act boundar y is known , our algorit hm solves the contact problem, and no penet ra tion occurs. The dr awback of this method is th at in each out er iteration ste p a mass matrix has to be assembled.
154
2. Iterative Solvers Based on Domain Decomposition
Fig. 2 .34. Displacem ent s and ada pt ive triangulations, (weak coupling)
A model problem in 3D is shown in Fig. 2.35. The displ acement of t he solut ion and the coarse trian gulation are given for t he saddle point problems (2.47) and (2.51) . On the left , the Lagran ge multiplier space has t hree degrees of freedom per node, one degree in each dir ection. The mor t ar finit e element solution sat isfies a weak cont inuity condit ion in t an gential and normal dir ection. In t he second situation on t he right of Fig. 2.35, there is no cont inuity condit ion for the tangent ial displ acement. Thus, we replace the bilinear form b(·, ·) in t he saddle point formul ation by bnC .), and work with t he modifications proposed in thi s sub sub section.
Fig. 2.35 . Coupling in both directions (left) and in normal dir ection (right) in 3D
Fi gur e 2.35 shows t he st ructural difference between t he two weak coupling condit ions at t he int erface. On t he left, t here is no relat ive displacement of t he two bodi es in t an genti al dir ection, whereas in the sit uation on the righ t , a sliding between the two bodies is permi tted . A relative displacement of the left body with respect t o t he right body can be observed . We remark t hat the nonconforming constrai ned space V h is a subspace of V "h. Here, V"h is th e kernel of t he op erator (B"h )T : X h --+ M j, associate d with the bilinear form bn ( · , .) . In t he genera l sit uat ion t hat 8fh n rD is empty for some subdom ain ind ices, the ellipti city of a(·,· ) on V"h x V "h is lost , and rigid body motions in tangent ial directi on are contained in V"h . We obtain unique solvabilit y in our example by impo sing Dirichl et boundar y condit ions on one face of each subdomain .
2.4 A Diri chlet - Neumann Typ e Method
155
2.4 A Dirichlet-Neumann Type Method In t his sect ion, we use t he unsymmetric mortar formul ation based on du al Lagran ge multipliers given in Subsect . 1.3.1 as starting point for t he const ruction of th e iterative solver. As shown, t he corre sponding unsymm etric var iationa l formulati on (1.42) is closely related to a Dirichlet- Neum ann coupling. One possibility to const ruct efficient ite ra tive solvers for t he related algebra ic system (1.47) would be to apply a Dirichlet-Neum ann or Neumann- Neumann type pr econditioner. These techniques are well known for t he standard conforming case [BGLV89, BW84, BW86 , DL91, Dry88, DW95, LDV91, Le 94, QV99, SBG9 6, Wid88], and have been successfully adapted to non-m atching t riangulations and mor tar finit e elements in [Dry99, DryOO, LSV94]. For simplicity, we restrict ours elves to th e case of two subdomains and homogeneous Dirichlet boun dary conditions on an, and refer to [Dry99, DryOO] for t he ana lysis and a genera lization to many subdomains. Here, we formulate t he algorit hm as a block GauB-Seidel method. 2.4.1 The Algorithm Using t he not ation of Subsect . 1.3.1, t he mortar finite element solution Uh solves t he following block syste m
-
AUh: =
(AI (U l) (iI) STA~ A-S) Jv U2 12 f=
(2.53)
=:
see (1.39), where t he submatrices and th e modified right hand side are defined by := ((j})T , O)T, := h + S T h and
11
12
f-2 =
(Jf +j]
M f f,
) ,
Ab and AJv are given by (1.35) and (1.36) , respectively. The scaled mass matrix M is sparse due to the use of a du al Lagran ge multiplier space . We note t hat here M T is a mapping from t he mortar side ont o t he interior of t he non-m ortar side, m pq := Jan 1 ntW2 Aq¢l;or do) Jafhnan 2 Aq ¢~on do . Here, ¢;or and ¢~on denot e t he nodal basis functi ons on t he mortar and the nonmort ar side, respect ively. Due to the different ordering of t he unknowns and t he definiti on of t he bilinear form b(·, ') , we have used a different not ati on in Sect . 2.3. The M T of Sect . 2.3 has a minu s sign and is extended by zero to t he int erior unknowns. A damp ed block GauB-S eidel method applied to (2.53) yields for n ~ 0 n +l _
uh
n
- uh +
(
wDId 0 0 Id )
(
A 1D -S ) 0 AJv
- 1 (
Id 0 ) - - n 0 wNId (j - AUh) ,
(2.54)
156
2. It er ative Solvers Based on Dom ain Decomposition
where a < W D, W N ::; 1 are suitable damping parameters and u~ is t he start ite rate. Here, we have used the upp er block t riangle matrix to define t he iterati on scheme . We remark t hat t he matrices A}..r and A1-,r correspond to Neum ann boundar y values on t he interface, and t he matrix Ab is t he st iffness matrix of a Diri chlet problem on [h . The idea of (2.54) is illust rat ed in Fig. 2.36 . Neumann proble~ Dirichlet problem
~ O2
mortar
nd . cond. - residual 01
--------S
trace
non-mortar bnd. condo
-
Fig. 2.36 . Informat ion tran sfer at the interface
Based on t his observation, we can rewrite (2.54), and obtain the following Neumann- Dirichlet algorit hm:
Choose damping parameters: a < WD,WN ::; 1. Initialize: pI = (1- wN) (A1-,ru~ - h ) + WNST (II - A}..ru~) , gO = Abu~ For n = 1, . .. do Solve Neum ann problem on [22 (mortar):
11.
Transfer of the "Dirichlet boundary values" and damping:
Solve Dirichlet problem on Abuf =
[21
(non -mor tar) :
11 + gn
,
Transfer of the "Neumann boundary values" and damping:
We note t hat gn an d v" live on [21 and [22 , respectively. However , t he start iterat es can be chosen in such a way t hat t he compo nents of s" and v" associated with t he interior nodes are zero. Possible choices are u~ = (A b)- 111 or u~ = (A b )- l II an d u~ = (A1-,r )- 112 or u~ = (A1-,r) -l [z- In t hat case s" and pn can be interpreted as bounda ry values.
2.4 A Diri chlet -Neumann Typ e Method
157
Lemma 2.33. The N eumann-Dirichlet algorithm and the block Gauj1-Seidel method (2.54) are equivalent. A st raightforward calculation shows the assumption. In our algorit hm , t he solution of a Neumann pr oblem on il2 is followed by t he solution of a Diri chlet pr oblem on ill ' Set tin g WD = 1 gives t he Neumann- Dirichlet algor it hm pr esent ed in [QV99] for the conforming setting. Moreover in that case, t he it erat es satisfy t he mat ching condit ions at the int erface, and thus are contained in t he constrained space Vh. Replacing the upp er block triangle matrix in (2.54) by t he lower blo ck triangle matrix yields it er at es u h which are in general not contained in t he constrained space Vh , i.e., we start with t he Dirichlet step and not with t he Neum ann step. The converge nce analysis can be carr ied out wit hin the Schwarz fram ework , and we refer to [Dry99] for t he mor t ar setting. In the two subdomain case wher e the measure of aili n ail is non zero for i = 1,2, no coarse problem has to be solved to obtain a convergence rate in the energy norm which depends logarithmically on t he ratio H / h; see [Dry99, Dr yDD]. Here, H stands for t he maximal diam et er of t he subdomains and h for the minimal meshsize. Moreover, the method can serve as a pr econditioner for a Schur complement system on the int erface, and a conjugate gradient method can be used as acceleration. Eliminating s" and pn in our algorit hm yields an it erative scheme for t he unknowns Ul and U2. In t he first half step, a Neum ann pr oblem on il2 has to be solved and in t he second half ste p, we solve a Dirichlet probl em on ill : u~ = (1 - WN)U~- l Ui' = (1 - WD)U~- l
+ wN(A'Jy)-1(j2 -
ST A~U~- l) + wD(Ab) -l(jl + Su~ )
n> l .
(2.55)
An int eresting alte rnative formulation can be obtained by eliminating t he int erior variables. Introducing a sca led Lagran ge mul tiplier '\h defined on t he non-m or t ar side, we can work on t he interface. We recall that the mortar side is defined on il2 and t he non-mort ar side on ill . For simplicity, we restrict ourse lves to t he special case WD = 1. Now, we define our pr econdi t ioned Richardson method for '\h as follows: (2.56) wher e the Schur complement S, := A~r - A~I(A}I )- 1A }r , i = 1,2, is t he discret e St eklov-Poincar e operato r on ili , and q := 5 11ql - M T 5:;1 q2 with qi := f} - A~I(A}I)- l f} . We not e that the discr et e St eklov-Poincare operato rs are symmet ric and positive definit e. In the conforming set t ing, the convergence rate of the Richard son it eration (2.56) depend s only on the cont inuity and coerciveness constants of 51 and 52; see [QV97]. Here, the norm of M which do es not dep end on the meshsize also enters. Und er the assumpt ion t hat t he damping param eter WN is small enough, t he convergence rate of t he preco ndit ione d Richardson it eration (2.56) is boun ded ind ep end ently of t he meshsize, i.e.,
158
2. Iterative Solvers Based on Domain Decomposition
II'\h- '\hll :::; pnll'\~ - '\hll with 0:::; p < 1 independent ofthe meshsize and '\ h := (51 1 +M T S 21 M) -l q. The following lemm a shows th e relation between th e iterates u 2 and '\h'
ur,
Lemma 2.34. Un der the assumpti ons th at A~ = WN( O Id)(1I - A }yu~) and A rvu g = Jz , th e N eumann-Dirichlet algorithm with WD = 1 and the precond ition ed R ichardson ite rati on (2.56) are equivalent. Th e fini te elem ent it erat es can be obtain ed from '\h by
U2 =
(A rv)-I( Jz
ur = (Ab)- I(Jl
+ S T(O (,\~ -I) T) T) + Su 2)
n >l
Moreover, th e scaled Lagrange multiplier satisfie s
'\h =
(1 - WN ),\~-1
= WN(O Id)
n
+ WN(O Id)(1I
I: (1 -
- A }yul)
WN )n-I (II - A }yuD '
(2.57)
1= 0
The assertio ns can be obtained by indu ction and are based on (2.55) , (2.56) and t he definition of t he Schur complement s. Compa ring (2.57) with (1.40) , we find that '\h is the scaled Lagran ge multiplier of the mortar set ting, i.e., '\h = D Ah where D is a diagonal matrix, th e ent ries of which are proportional to the local meshsize. The assumption Arvug = Jz in Lemm a 2.34 can be weakened . For t he equivalence, it s is sufficient if there exists a Yl such t ha t A rvug = h + S T us. A three term recur sion for '\h can be obtained for more general damping par ameters. Remark 2.35. Th is type of algorith m can be us ed to construct an efficient itera tive solver fo r nonlinear problem s. It can be applied to solve multi body cont act problems. Th en in each ite rati on step, a lin ear N eumann problem and a S igno rini problem have to be solved. Th e N eumann data on the mortar side are obtain ed in terms of ST an d the Lagrang e multiplier on th e non-mortar side. Applying S to th e Dirichlet values on th e mo rta r side gives th e onesided obstacl e fo r th e Signorin i problem on the non-morta r side. W e refer to [KWOOc] for th e n onlinear Dirichlet-Neumann algorithm an d numerical resu lts illus tra ti ng th e deformation of the bodies and th e bounda ry stresses at th e contact zone . An extension of th e algorithm to 3D an d Coulomb fr iction can be f ound in [KW01]. Th e nonlinear Signorini problem can be solved effic iently by m on otone m ultigrid methods; see [Kor91, KK99, KKOO, Kratll], 2.4.2 Numerical R esults
In t his subs ection , we pr esent num erical results for the pr oposed NeumannDirichlet algorit hm. We consider t he ext remely simple case of two unit squares Di , i = 1, 2, and -L\u = f on D, D := D 1 U D 2 . The right hand side f is
2.4 A Dirichlet-Neumann Type Method
159
chosen to be const ant on each subdomain, fi n! = -1 and fl n2 = 1. Inhomogeneous Dirichlet boundary condit ions ar e imposed on the sides of the unit squar es parallel to the int erface, and homogeneous Neumann boundary condition ar e applied elsewhere . We work with non-matching triangulations at the int erface and use uniform refinement t echniques. The coarse triangulation has 4 element s on the non-mortar sub domain D 1 and 9 element s on the mor tar subdomain D2 . In the Neumann-Dirichlet algorit hm, the start it erate u~ is given by a random vector on each level, the valu es of the component s are cont ained in [-5,5] . Furthermore, we set pI = ST(J1 - AJvu~) which corresponds t o the choice ug = (A'7v)-l(h + ST(J1 - AJvu~)). Replacing t he upper block triangl e matrix in (2.54) by the lower block triangle matrix yields a Diri chlet -Neum ann algorit hm . In t hat case, we have to initialize gl. We define l = Sug , where ug is a random vector. This choice corresponds t o u~ = (Ab)-1(j1 + Sug). In the Dirichlet -Neum ann algorit hm, we start with a Dirichlet problem on t he non-mortar subdomain D1 followed by a Neum ann problem on the mortar subdomain D 2 . We compare the proposed Neum ann-Dirichlet algorit hm, WN E {0.7,0.75,0.8,0.85,0.9} and WD = 1, with the Dirichlet-Neum ann algorit hm , W D E {O. 7,0.75,0.8,0.85, 0.9} and W N = 1. In Tabl es 2.5 and 2.6, the number of required it er ation st eps to obtain an error redu ction of 10- 6 is given for different damping par am et ers. The numbers in parenthesis show the numeric al results for a different coarse trian gulation. In that case , we st art with 25 element s on D1 and 4 element s on D 2 . Table 2.5. Numbe r of it eration ste ps, (t race norm) level 1 2 3 4 5 6 7
Dirichlet-Neumann WD 0.85 0.75 0.8 7 (9 6 7 6 6 10 11 7 9 5 8 6 6 10 7 9 5 7 6 6 10 7 9 6 7 6 6 11 7 9 5 8 6 6 10) 7 9 6 8 6 6 10 7 9 6 7 6 6
0.7 9 9 9 9 9 9 9
0.9 8 6 8 7 8 7 8 7 8 7 8 7 8 7
Neumann-Dirichlet W N 0.7 0.75 0.8 0.85 9 10) 7 8 6 7 6 5 9 9 7 (8 5 7 6 6 7 8 6 7 5 6 9 9 9 9 7 8 6 7 5 6 9 10 ) 7 8 6 7 5 6 9 (9 7 8 6 7 6 5 9 9 7 8 6 7 5 6)
0.9 7 7) 8 7) 7 7 7 7 7 7 7 7) 7 7)
In Tabl e 2.5, t he error is measur ed in a weighted L 2-norm for the trace. The weighting factor is given by 1/h2 where h2 is the local meshsiz e on the mortar side, i.e., (2.58) Her e, Sh2 is the set of edges at the interface on the mor t ar side , and U 2 is t he exac t discrete solution on the mortar side. We rem ark that this weighted
160
2. It erative Solvers Based on Domain Decomposition
L 2 -nor m is equivalent to the Euclidean vector norm of t he interface nod es on t he mor tar side.
U2 -
u~ restricted to
Table 2.6. Number of it eration steps, (Lagr ange multiplier norm)
level 1 2 3 4 5
6 7
0.7 9 11 9 11 8 10 8 11 8 11 8 (11 8 11
Dirichlet-Neumann WD 0.8 0.75 0.85 7 9 5 8 6 6 7 9 5 8 6 7 7 9 5 7 6 6 7 9 5 8 6 6 7 (9 5 8 6 6 7 (9) 5 (8) 6 (6) 7 (9 5 8 6 (6
0.9 8 6 8 6 8 6 8 6 8 6 8 6 8 6
Neumann-Dirichlet WN 0.7 0.75 0.8 0.85 9 10 7 9 5 7 6 6 9 10 7 9 5 8 6 6 9 11 7 9 6 8 6 6 9 10 7 9 5 8 6 6 9 11 7 9 5 8 6 6 9 (10) 7 (9) 5 (8) 6 (6) 9 (11 7 8 5 8 6 (6)
0.9 8 7 8 7 8 7 8 7 8 7 8 6 8 7
In Table 2.6, th e err or in the Lagrange multiplier defines th e stopping crite ria for t he iterat ive solver. It is measur ed in a weight ed L 2 -nor m . The weighting factor is given by the local meshsize on t he non-mortar side. We note t hat the Lagrange multiplier is an approximation of th e flux. It can be obtained from t he residuum of a Neum ann problem on the non-mortar side by a diagonal scaling, i.e., Ah = D-I(OId)(h - Atvud, where UI is the exac t discret e soluti on on th e non-mort ar sub domain D I. Moreover if we set A~ = D-I(OId)(h - Atvul) , we have the following norm equivalence C
2
II(Atv(UI - ur ))r I1
::;
L
hellAh - A~II~; e ::; C II (Atv(UI - ur))rI1
2
e ESh ,
Here, Sh, is the set of edges at the int erface on th e non-mortar side . This equivalence motivates t he stopping crite ria (2.59) We note th at th e choice of our st ar t vectors gua ra ntee t hat Atv(UI - ur) = 0 for all interior nod es on DI , i.e., II(Atv(UI - ur))rll = IIAtv(UI - ur)ll · The number of requir ed iteration st eps is independent of the refinement level for all damping parameters . But it depends highly on th e damping pa ra mete r, see also Figs. 2.37 and 2.38. Both algorit hms, Dirichlet-Neum ann and Neum ann-Dirichlet , require approximate ly the same number of iteration ste ps to obtain the requir ed error reducti on. The choice of th e norm for th e error plays only a minor role, and almost the same results are obtained for th e stopping crite rias (2.58) and (2.59). Figur es 2.37 and 2.38 show the computed errors and t he best least-square fit for different damping par ameters on level 5 and level 7. The compute d err ors ar e marked by different symbols whereas the corresponding best fit is given by a solid line. In Fig . 2.37, the error reductio n is measure d in the
2.4 A Dirichlet -Neumann Typ e Method
161
• damp ing: 0.7
• damping:0.75 e damping: 0.8
D
o
da mping: 0.85 da m in : 0.9 2
4
6
8
Number of iterations, level 7
10
Fig. 2.37. Error reduct ion (trace norm) on level 5 (left) and level 7 (righ t) for the Diri chlet -Neumann algorit hm
sp ecified £2- norm for t he trace on the mortar side . The results for the err or in t he Lagrange multiplier on the non-mortar side are given in Fig. 2.38.
Fig. 2.38. Error reduct ion (Lagrange multiplier norm) on level 5 (left) and level 7 (right) for t he Diri chlet-Neumann algorit hm
Finally, Tabl es 2.7 and 2.8 give t he asympt ot ic convergence rates. The lines of the two tables corres pond t o the levels 1-7. To define the convergence rate q, we use a first order logari thmic polynomial least- squ ar e fit based on 10 it er ation ste ps. Then , q := lOa where 10
10
2)ek-ak-b) 2= min 2:)ek- ak- (J) 2 , k =l
a,!3 EIRk = l
and ek is the logarithm of the it er at ion err or in one of the sp ecified norms. Tabl e 2.7 shows the convergence rates for different damping factors in the t race norm on the mortar side, and Tabl e 2.8 gives the corresponding results for t he Lagrange multiplier norm on the non-mortar side . The convergence rates dep end highly on t he choice of t he damping par am et er. In pr actic e, t he choice of a good damping par am et er is a delicat e point. T he influence of the damping param et er can be considerably weakened by using the block GauBSeidel method as a pr econditioner for a Krylov subspace method. Another
162
2. Iterative Solvers Based on Dom ain Decomp osit ion
possibility might be to use ada ptive st rategies for the damping par am et er. We not e t hat in general Ul and U2 are unknown. Then , u 2 - u~- l an d uf - U ?-l can be used t o define a stopping criteria. T able 2.7. Asym pt ot ic convergence rat es q , (trace) 0.7 0.151 0.142 0.145 0.147 0.146 0.149 0.148
Dir ichlet-Neumann WD 0.75 0.8 0.85 0.094 0.038 0.052 0.084 0.034 0.048 0.088 0.036 0.051 0.091 0.037 0.053 0.089 0.036 0.053 0.092 0.037 0.051 0.091 0.037 0.052
0.9 0.109 0.108 0.109 0.113 0.113 0.109 0.111
0.7 0.160 0.150 0.159 0.160 0.160 0.158 0.160
Neumann- Dirichlet WN 0.75 0.8 0.85 0.100 0.040 0.046 0.089 0.028 0.050 0.099 0.039 0.042 0.100 0.040 0.044 0.100 0.040 0.040 0.099 0.039 0.047 0.100 0.040 0.042
0.9 0.101 0.107 0.087 0.090 0.084 0.100 0.088
Table 2.8. Asymptot ic convergence rat es q, (Lagra nge multiplier norm) 0.7 0.139 0.140 0.139 0.138 0.138 0.138 0.138
Dirichlet- Neumann WD 0.8 0.85 0.75 0.081 0.035 0.049 0.079 0.026 0.046 0.078 0.027 0.050 0.078 0.028 0.051 0.078 0.028 0.053 0.078 0.027 0.051 0.078 0.028 0.052
0.9 0.109 0.106 0.110 0.111 0.113 0.111 0.112
0.7 0.146 0.142 0.154 0.141 0.145 0.140 0.142
Neumann-Dirichlet WN 0.75 0.8 0.85 0.088 0.036 0.045 0.083 0.027 0.050 0.095 0.037 0.050 0.085 0.035 0.051 0.089 0.036 0.053 0.081 0.032 0.050 0.085 0.034 0.052
0.9 0.105 0.110 0.106 0.112 0.113 0.111 0.111
R emark 2.36. We rema rk that the application of the operators S an d S T
is extrem ely cheap. Th is is due to the use of dual basis fun ctions for the Lagrange m ultiplier.
2.5 A Multigrid Method for the Mortar Saddle Point Formulation In Sect . 2.3 and in Sect. 2.4, two different it erati ve solvers for mortar discret ization t echniques have been discussed . We have considere d the case of du al Lagrange multipliers. T he t heoret ical results also hold if standa rd Lagrange mu ltipliers are used . But in t hat case , the computational cost for one it erati on step is considerably higher. This is due to t he fact , t hat t he scaling of t he mass matrix on t he mor t ar side involves the inverse of a mass matrix on t he non-mortar side. In this section, we analyze a multigrid method based on t he mortar saddle point formulation wit h standard Lagran ge mul t iplier
2.5 A Mult igrid Method for the Mortar Saddle Point Formulati on
163
spaces . In cont ras t to t he two previous secti ons, th e scaled mass matrix does not ente r in t he computation. There are different approaches to th e efficient solution of t he indefinite saddle point problem (1.9). One possibility is to use a good preconditioner for the exac t Schur complement as ana lyzed in [Kuz95a, Kuz95b] and further exploite d in [EHI+98 , EHI+OO, HIK+98]. A different technique is based on an idea of Br aess and Sar azin which is present ed for the Stokes problem in [BS97] . It has been successfully adapte d to mortar situations in [BD98, BDW99, WW98]. In t ha t approac h, a modified Schur complement syste m has to be solved exac t ly in each smoot hing step . More recently, a simplified version of this idea has been studied in [WW99]. The modification is based on ideas introduced in [ZulOl]' and a more genera l approximation property ana lyzed in [WohOOb] . Here, we st udy a multigrid meth od for t he saddle point problem (1.9) . T wo different ty pes of smoot hers are discussed ; a block diagonal one and one reflecting the sad dle point st ruc t ure . There is no need to solve a Schur complement system exact ly. The first smoot her works on the squa red syste m. In t he second case , t he exac t solution of the modified Schur complement syste m is replaced by an iteration, resulting in an inner and outer iterati on cycle. This multi grid method is given for th e standa rd mortar formulation as pr esent ed in Sect. 1.1. In cont rast to [BD98, BDW99 , WW98], we are not workin g in the subspace on which t he saddle point problem is positive definite; th e iterates do not have to satisfy the const ra ints at the int erfaces, and we do not have to solve a Schur complement system exactly . To obtain convergence results, it is therefore necessar y to establish an appro priate approximation property for both vari ables Uh and A h . This issue was originally addressed in [WohOOb] ; a weaker approximation property has also been studied in [BD98, BDW 99]. For bot h smoot hers, we obtain level independent convergence rates for the W- cycle provided t hat the numb er of smoothing ste ps is large enough. We assume full H 2 -regularity. In this section, the multigrid analysis is carr ied out in t he operator setting. We work with t he standa rd Lagrange multiplier spaces M l , 0 ~ l ~ i . as defined by (1.2) , and t he unconstrained product spaces Xl which are asso ciated with a nested famil y of quasi-uniform trian gulations Ti, hi = 2h1H . The t riangulations on the non-mort ar sides are called S m ;l ' We then find for the finit e element spaces t hat
The spaces X; x M; are Hilbert spaces equipped with the mesh dependent bilinear forms
((v, f.1 ), (w, V)) hl ;S? XS := (v,w)o + (f.1 , v) _ ~ , hi
;S
(v, f.1) , (w, v) E X;
X
M, ,
where t he mesh dependent bilinear form (' , ·)h- 3/ 2.S on M, x M, is given by I
'
164
2. Iter ative Solvers Based on Domain Decomposit ion
The corres ponding norm s are denoted by 11 ·llhl;!lXS and 11 ·lIh-s /2.s. 1 , Before focusing on t he approx imation property, we consider suit able t ransfer operators. Let ILl : X l- l --t X, and JL l : M l- l --t M, be th e natural injecti ons. We t hen define If-l and Jf- l by Wl- l E Xl- I , Vl- l E M l- l
Since th e spaces are nest ed , we have Wl- l = IL l Wl - l, Wl-l E X l- I , and Vl-l = JL lvl- l , Vl-l E Mi.:«. A st ra ightforward calculat ion now shows that if (713[ , Vl ) E X, X M, satisfies the saddle point probl em a(u5[ ,Vl)
+ b(Vl' Vl) =
= (~ ,J-Ll)
b(u5[ ,J-Ll)
and if (Wl- l ,Vl - l ) E X l- l a(uh- l,Vl - d
X
a(Wl' vd b(un, J-LI)
_.3.
h i 2 ;S
,
J-LlEMl ,
M l- l satisfies
+ b(Vl- l ,Vl-l)
b({ih- l ,J-Ll-l)
Vl EXl ,
(d; ,Vl )O,
= (If- ld; ,Vl- l) O, 1 = (Jf- 8l ,J-Ll-l) _ .3. , hl_; ;S
+ b(vl ' Vl ) =
_ .3.
hi 2 ;S
J-Ll-l E Ml- l ,
VI E XI ,
(dl ,VI)O,
= (01 , J-Ll)
Vl-l E Xl- I ,
,
J-LlEMI ,
with ((dl , Ol) , (VI- I , J-Ll-d)hl;!l XS = 0, for all (VI- I , J-Ll-d E X l- l x M I - l . After t hese prelimin ar y remarks, we will establish a genera l approximat ion property and introduce two different kind s of smoot hing operators. 2.5.1 An Approximation Property
The following lemm a can be found in an abst rac t form , as well as in th e special case of mixed finit e element approximations of the St oke's equation, in [Ver84 , Lemm a 4.2]. Here, we ada pt it to mortar finite elements . It shows t hat t he coarse grid correction in the multigrid framework yields a good approximation of the solut ion of the defect equation on t he fine level. Lemma 2.37. (Approximation property) Let (d l , 01) E X, x M I be orthogo nal to X l - l x M I -
l ,
i .e.,
2.5 A Multigrid Method for the Mortar Saddle Point Formulation
165
and let (WI , VI ) E XI x MI be the solution of: Find (WI , VI) E XI x M I such that
(2.60) Then, there exists a constant C satisf ying
Proof. The qu asi-uniform ity of t he triangulation s yields 3
c (1lwtllo + h? Ilvtll o;s)
~
lI(wI , vI)llh /;S2 XS
~
3
C (1lwtllo + h? IIvtllo;s)
The estimate for the Lagran ge multipli er term will be based on the discrete inf-sup condition (1.10) whereas the bound for Ilwtllo is based on du ality techniqu es. We start with an estimate for the upp er bound of h; /21Ivtllo ;s . The continuity of a(·, '), th e discrete inf-sup condit ion (1.10) , and the orthogonality of dl on XI-I , i.e., (dl , VI- d o = 0, VI- l E X I- I , yield an upp er bound for Ilvtll h;-1/2;S in terms of Ilwllh and IIdtllo: (dl ' VI -
VI- l)OIlvtlll
a(wI ,VI )
(2.61) The second equation in (2.60) yields b(wI ,J.LI-d = 0, JLI-l E M I- 1 , and thus WI-l E Y . Recallin g t ha t the bilinear form a(· , ·) is ellipt ic on Y x Y , see (1.5) , and using (2.61) , we find for WI-l E XI-l C
Ilwtlli
WI-l )O- (151, VI)h
~ a(wI ,WI) = (dl ,WI )O- b(wI , VI) = (dl ,WI -
~
3
Chi
7
(lIdtilo + h? IIJtllo;s) Ilwtlh + Chl 1ldtllollJtllo;s . 2
To obtain an upp er bound for IIwtlh in terms of sufficient to consider t he quadrati c polynomial g(8) :=
8
2
-
a(al
+ a2)8 -
h;/21IJtllo;s
aa la2,
8
and
_3 I
'1·s '
Ildtl lo, it
is
E IR ,
where al := h~/21I Jtllo;s 2: 0, a2 := htlldtllo 2: 0, and a is a positive constant . In the case that al = a2 = 0, t he uniqu e solvability of (2.60) gives Ilwtlh = O. For al + a2 > 0, an easy calculatio n shows that g(8) > 0 for 82: (1 + a )(a l + a2)' Thus, t he following upp er bound for Ilwtlll holds in t erms of IIJtllo;s and
Ildlllo:
166
2. Iterative Solvers Based on Dom ain Decomposition
(2.62) Combining (2.62) with the upp er bound (2.61) for
Ilvtllh-l/2.S' we find I
'
(2.63) In our next step, we focus on an estimate for Ilwtllo. To obtain an upp er bound for Ilwtllo, we use Aubin - Nit sche typ e arguments. Let w E H6(D) be the solution of the cont inuous vari ational problem: Find w E HJ (D) such t ha t a(w , v) = (WI , v)o, v E HJ(D) . Taking into account t ha t, in genera l, WI E XI is not contained in get IIwtlI6= a(w ,WI ) + b(WI , v) ,
HJ (D) , we
where v .- a'Vw . n is t he flux of w across the interfaces. Using t ha t a(wI' VI-I ) + b(VI_l , VI) = 0 , VI- l E XI- I , b(WI , /-LI-l) = 0 , /-LI-l E MI-l , and observing t ha t b(w , VI) = 0, VI E M I , we find for VI- l E X I- l and /-LI-l E M I - l th at
IIwtlI6 = a(w - VI- I, WI) + b(WI , v- /-LI-d + b(w - VI-I,VI) :::; C(11w - vl- l lll Ilwtll l + II[wdllo;s Ilv - /-LI- lllo;s + It[w- vl- dllo;s Ilvtllo;s) . Here, we have used t ha t th e spaces are nested . We choose VI- l E X I -
local quasi-projection of w such that
111V- vi-l ih :::; Chl- l llwl12 , II[w- vI-l]llo;s :::; Ch?_11IwI12 111-1 E M I -
l
l
as a
(2.65)
3
see, e.g., [SZ90] , and
(2.64)
satisfies 1
1
llv - /-LI-lllo;s :::; Ch?_lllvil k;s :::; Ch?-111w112
(2.66)
Remark 1.13 shows t hat M I - l is also a suit able Lagrange multipli er space for X I. Thus, we can apply Lemm a 1.7, for ~ := { v E X I , b(v ,/-L) = 0, /-L E MI-d , and obtain 1
II[wdllo;s :::; Ch?llwtlll . In a last ste p , we use t he H2-regularity, IIwl12:::; Cllwtll, and a t race theorem. Combing t he upp er bounds (2.64)-(2.66) yields
IIwtlI6:::; Chi (1lwtlh + Ilvtllh;k;s) Ilwtllo , which, t ogether with (2.62) and (2.63) , proves th e assertion.
0
2.5 A Multigrid Method for the Mortar Saddle Po int Formulat ion
167
Before introducing our smoot hing op erators, we consider t he operator K, associated with the saddle point problem (2.60) on levell: Let Al : X, ---7 Xi , Bi : M, ---7 Xl , B] : X, ---7 M, be t he operators defined by (A1Vl ,Wl)O := a(vl ,w l) , (Bl/..Ll ,Wl)O:= b(Wl ,f.Ll) , (Bt wl ,f.Ll) _ ~ h/
;S
b(Wl ,f.Ll )
:=
Then , th e self-adjoint non-singular operator K, : X, X M, associate d with t he saddle point problem (2.60) , is given by Kl(Vl ,f.Ll) := (A1 VI
+ Blf.Ll ,Btvl) ,
(Vl ,f.Ll) E Xi
X
---7
Mi
X,
X
Ml ,
(2.67)
The solution (WI , VI) of t he saddle point problem (2.60) satisfies K; (WI , VI) = (dl ,81) ,
and t hus IIK1- 1(dl ,81)llh/ ;!l XS ::; Ch rll(dl ,81)llh,;!l XS for those (dl ,81) E X, X M, which are orthogonal onto X l - 1 x M l - 1 with respect to (., ·)h/;!l xS. Equivalently, we find (2.68) for (WI , VI) E X, x M, satisfying (Kl(Wl ,Vl),(Vl- l ,f.Ll-l))h/ ;!l XS = 0, for all (Vl - l ,f.Ll-d E X l- 1 x M l- 1 . 2.5 .2 Smoothing and Stability Properties
The second basic tool to establish convergence within t he multi grid framework is the smoothing property. In t his subsection, we introduce two typ es of smoot hers. We show that t hey are st able and sati sfy a suitable smoothing property. Before we give the definition of the smoo th ers, we consider th e opera to r Bl in mor e det ail. The st ability of the block diagonal smoother can be easily established by mean s of the properties of th e operator Bi , The following lemm a shows that th e condition numb er of Bi is uniformly bounded. Lemma 2.38. There exist constants such that
(2.69) P roof. The upper bound is obtained by using an inverse estimate and t he definiti on of Bi :
IIBlf.Llllo = sup
W IE X , w /;cO
168
2. It er ative Solvers Based on Domain Decomposit ion
To establish the lower bound, we observe t hat
Each WI E X, is uniquely defined by it s valu es at the vertices of the t riangulat ion . We define Wl (P) := J-ll(P) for an int erior vert ex P of a non-m ort ar side "(m , 1 :::; m :::; M . For all other vertices q of Ti , we set Wl (q) := O. This special choice yields b(Wl, J-ll) ~ cll[wdllo;sIIJ-ltllo;s; we refer to [Woh99c] for det ails. The lower bound in (2.69) now follows from II[wdllo;s ~ ch~ I/2 1 Iwtl lo . 0 Remark 2.39 . Using a sim ilar construction as in the proof of Lemma 2.38, we fin d by a straightforward computati on and by m eans of (2.69)
WI EXI .
inf
Vt EXl
(2.70)
Bi Vl = B jw l
Using t he definition (2.67) of th e operat or Ki , we can rewrite the saddle poin t problem (1.9) on X, x M, as an operator equation: Find zt := (Ul' AI) E X, X M, such t hat Kl zt
= fl
,
where li > (1;,0) E X, X M, is defined by (j;,Vl )O:= (f,Vl )O, VI E Xl . To establish appropriate smoot hing pr operties, we will use suitable operator norms . We recall t hat if 5 is a linear cont inuous operat or 5 : H I --+ H 2 , where H I and H 2 are Hilb ert spaces with norm s II . 111ft and II .IIII2' respect ively. T hen the standa rd operator norm is given by
11511:=
sup
xE H l
x#O
115xllII2 II x II n, .
In the next two subsubsections, we establish smoot hing and st ability pr operties for two different ty pes of it erations. In par ticular , we do not requ ire that t he iterates satisfy th e weak cont inuity conditions at th e int erfaces exactly. As a consequence, we neither need a good pr econdi tioner for the exact Schur complement nor an exact solver for a modified Schur comp lement. 2.5.2.1 A Block Diagonal Smoother. Following the ideas of [Ver84], we introduce a smoother for t he squa red positiv e definite syste m. The operato r Kl : X, x M, --+ X, X M, is defined by means of the symmet ric posit ive definit e bilin ear forms a (·, ·) and d(·, ·) on X, x X, and M, x Mi , respectively,
(Kl (Vl, J-lt) , (WI , Vl ))h/;S!XS := a(vI' WI ) + d(VI' J-lI) , It has a block diagonal st ructure
K l(vt,J-lI ) = (AIVl, DI J-lI) ,
(WI , vt) E X,
X
M, .
(2.71)
2.5 A Multigrid Method for the Mortar Saddle Point Formulati on
169
where the op erators A l : X I --+ XI , and D I : M I --+ M I are asso ciated with the bilin ear forms a( ·, ·) and d(·, '), resp ectively. One smoothing it er at ion on level l is given by m
m -l
ZI := zi
K IZIm - l ) , + K~I-IKI K~-l(d I 1-
(2.72)
where dl represents the right hand side of t he syst em Ki z, = dl , which has to be solved, ZI is t he exact solution, zl' denotes the it er at e in the mthst ep , and zp is the initial guess. The blo ck diagonal smoother works on the squared syst em which is positive definit e. Each smoot hing ste p can be easily performed provid ed that the applicat ions of Al l and DI I are cheap. The following lemma gives t he smoot hing rat e and can be found in [WohOOb].
Lemma 2.40. (Smoothing property) Let KI be defin ed as in (2. 71), where A I, IIAdl ::; Clh;, an d D I , IIDdl ::; are self- adj oint positive definite operators. If th ere exists for each WI E XI an aWl' 0 < aWl < 1 such that
cn«,
I
(BID I BtwI , wI)o ::; (1- a wl)(Alwl , wl)o ,
(AIWL,WI)O ::; awl (Alw l,wl )o,
then th e follo wing sm oothin g property holds for the it eration (2.72) : IIKl el'II''1 ;!7 Xs ::;
h;~lle?llhl ;!7 XS ' m ~ 1
.
(2.73)
Here, ei := zl' - ZI, m ~ 0, is the iterati on error in the mth-smoothing step, and th e const an t C does not depend on the aWl' Proof. The it er ation err or ei sat isfies m
el
= (Id -
K ~-IK K ~-IK) m 0 I
I
I
l
ei '
Since KI is a self-adjoint positive definit e oper ator and KI is self-adjoint , t here exists a complete set of orthogonal eigenfunctions zj satisfying
.
.
z;, we find for the eigenvalues Si =j:. 0, that w; =j:. 0, .
- _ 1.
Setting (Wi, pi) := K I and t hat
2
T hen , t he ass umpt ions on
Al
and DI yield
170
2. Iterative Solvers Base d on Dom ain Decomp ositi on
and t he norm of K1 ei is bounded by m
IIKl el Ilh/;!2XS
- ! - -! - -! -- -! - -! 2 m- ! 0 :S II K I tc, KIKI (Id - ttc, K IKI )) «, el llh,;!2xS
<
sup I
I
Is(1 -
s2)m I IIKtll ll e?llh,;!2 xs
s Eu (K,- ' K/K,- ' )
We obtain (2.73) by using t hat SUPt E[O;l j (t( l - t 2 )m) :S C/ "fiii and II Kti I :S C / hf. 0 Combining t he approximation pr operty (2.68) and t he smoothing property (2.73) , we obtain level ind ependent convergence rates for t he two-grid algorit hm pr ovided t hat t he number of smoot hing steps is lar ge enough. The analysis of th e full multigrid cycle is based on t he two-grid case, a perturbat ion argument and t he stability of t he smoot hing it eration (2.72); see [Hac85]. Lemma 2.41. (Stability estimate) Under the assumptions of Lemma 2.40 and if furth erm ore IIAlIIi :S C h f, then th ere exists a cons tant C in dependent of m such th at the follow ing stability estim ate holds (2.74) Proof To obtain th e stability est imate (2.74), we use t he same ty pe of arguments as in t he pr oof of Lemm a 2.40. The assumption on D , and (2.70) yield an upp er bound for II D;-I/211
The last inequality, together wit h t he assumption on hi , and t hus
All,
gives IIK I-
I 211 /
:S
o Under t he assumptions of Lemma 2.41, th e convergence rates of t he Wcycle in t he II . IIh, ;!2 xs-nor m are ind epend ent of t he num ber of refinement levels provided t hat t he num ber of smoothing ste ps is large enough; see, e.g., [Hac85, Ver84].
2.5 A Multigrid Method for the Mortar Saddle Point Formulation
171
Remark 2.4 2. A suitable sm oother, in the algebraic formulation of the m ethod, is given by the diagonal m atrix
~
K, := a h
d-2
(Id 0 ) 0 hfId
'
for some constant a > O. Here, nodal basis functions are used for both the finite eleme nts and the Lagrange multiplier, and d stands for the space dim ension, [l C IRd . On e iterat ion ste p using (2.72) re2.u ires the application of Kl- l twice. This is closely related t o t he fact that K, is positive definit e whereas K, is ind efinit e. In the followin g subsubsection, we discuss a second typ e of smoothing ope rato r ori ginally analyzed in [ZuIOl] for an abst ract saddle point problem. It has been applied t o t he mortar set t ing in [WW99]. 2.5. 2.2 An Indefinit e Smoother. The symmetric but ind efinit e op erator Ki can be decomposed as follows
«, =
(~t I~)
(Ar
-Bi~llBJ (~l~)
,
where All has t o be replaced by a suitable pseud o-inverse if Al is singular. This decomposition motivat es t he const ruction of our second smoot hing ope rator Kl . We not e t ha t a smoother wit h the sa me algebraic st ruc t ure as K, was introduced and ana lyzed in [BS97] for t he St okes problem . In t he definit ion of t he smoother, t he operator Al is repl aced by a suitable AI. To app ly one smo ot hing step, one has to solve a modified Schur compleme nt system exactly, where t he Schur complement is defined by 51 := B ] Al l Bi . This approach has been applied successfully t o t he mort ar sit uation in [BD98, BDW99, BraOl , WW98]. A disad vant age of this approach is t ha t t he exact solution of t he modified Schur complement system can be rather expensive. Even t he use of du al Lagran ge multipliers does not , in general, improve t he complexity. A simplified ap proac h has been pr op osed in [ZulOl] (2.75) Then, t he smoothing it eration is defined in terms of m Zl
:= Zlm - l
+ K- I-l(dI -
K
Kl
m -l) l ZI ,
(2.76)
where dl st ands for t he right hand side of the syste m K l z l = d l which has to be solved , Zl is t he exact solution , z i t he iter ate in t he m th-st ep , and z? the initi al guess. Each smoo t hing ste p can be perform ed eas ily pr ovid ed t hat t he applications of All and 51- 1 are cheap. The following lemma has been established in [ZulO l]' and guarantees t he smoothing pr op erty of t he it eration (2.76) un der some ass umptions on 51and AI .
172
2. Iterative Solvers Based on Domain Decompositi on
Lemma 2.43. (Smoothing property) Let K I be defin ed as in (2. 75), with Al and 51 positive definit e self-adj oint operat ors. Und er th e assu mpti ons (AIWI,WI)O ::; (AIWI , wdo, WI E XI , and
we obtain th e follow ing smoothing prop erty fo r th e iteration (2. 76)
Here 1](m) ----+ 0 for m ----+
00.
We refer to [Zula1] for a pro of, and not e that th e assumption (2.77) can be weakened if a damping strategy is used. A central point in th e proof is that th e operat or KI - K I is positive semi-definite. In th e previous subsubsection, we found it easy to const ruct a scaled J acobi-type operator KI satisfying the assumptions of Lemmas 2.40 and 2.41. Here, th e choice of 51 is a delicate matter. To find an adequate 51 satisfying (2.77), we follow an approach proposed in [WW99]. The opera tor 51 is constructed in terms of a positive definite self-adjoint operator 51 satisfying SI < 25 1. Then , th e spect ral radiu s of Id - ~-l SI is bounded by one, i.e., q := p(Id - 51- l SI) < 1. With a functi on k(£) defined by ._ log e k( £ ) . , £ > 0 , logq we find t hat p((Id - 5- 1 S)k) < £ for any int eger k prelimina ry remarks, we define
> k(£). Based on t hese
A straight forward computation shows t ha t 51 (k, a ) is selfadjoint and positive definite for any integer k > 0 and a > O. The idea is now to find an integer k and a value of a such t ha t 51(k, a ) satisfies (2.77). It is easy to see that 1- £ 1+ £ - - SI(k , a) ::; SI ::; - - SI(k, a) a a for k
> k(£). Setting a
:=
(1 - e) yields
1+ £ SI(k, l - e) ::; SI ::; 1 _ £ SI(k ,l - s)
= (1 +
2£ 1- £ ) SI(k , 1 - £)
The last inequality shows th at t he assumpt ions of Lemm a 2.43 are satisfied for 7£ < 1. We define 51 := 51(kc , 1 - s ) for a fixed £ < 1/7, k, > k(£), and refer to Sub sect. 2.5.3 for det ails on the implement ati on of the matrix vecto r mult iplication of 5 1- 1 .
2.5 A Multigrid Method for the Mortar Saddle Point Formulati on
173
To obtain optimal convergence rat es, not only for the two-grid algorit hm but also for t he full multigrid method, a stability estimate for t he smoothing iteration (2.76) is required. Lemma 2.44. (Stability estimate) Let th e assumpti ons of Lemma 2.43 be sat isfied. Furth ermore, if chrllwdlo ~ IIA11welio ~ Chrllwdlo, there exists a constant independent of m such that the follow ing estima te holds
lI elllhl;oxs ~ C lI e?llhl ;OXS ,
m
2: 1 .
Proof. The assumptions on Ae and 5e yield th at semi-definite opera tor . Thus, we find for e1
ic, -
K, is a positive
el = (Id - K e- 1Kc)me?= K e- 1(KI - Ke) (K1- 1(Ke - Kc)) m- l
e? ?
= K1- 1(K e - K 1) 1/2 ((Ke - K 1) 1/2K1- 1(Ke - KI)1/2) m-\K 1 - KI? /2e
Und er t he assumpt ions of Lemm a 2.43 , t he spect ral radius of (KI - K 1)1 /2 K e- 1(Ke - K e)1 /2 is bounded by one; see [ZulOl]' and we obtain
In the second inequalit y, we have used t hat und er the assumptions of Lemm a 2.44, th e norm of K e- 1 is bounded from above by Chr and that of KI - K, by Ch 12 . 0 Remark 2.45. In Lemma 2.44, it was assu m ed that the condition number of Ae is bounded in depen dently of the m eshsize he . This is satisfied f or Jacobityp e smo oih ers, but not for those of IL U-type. N everth eless, level in depen dent convergence rat es can be obtain ed by replacing the m esh dependent norm for the Lagrang e multiplier by a n orm inv olving the S chur complement.
We can now formul ate the cent ra l result of t his sect ion which shows t hat both classes of smoot hers give rise to optimal multigrid methods. Theorem 2.46. Let th e sm oothing iteration (2.6) in the m ultigrid cycle be defined by (2. 72) or (2.76) . Th en under th e assumptions of Lemma 2.41 or Lemma 2.44, th e convergence rates of th e W- cycle are in dependent of th e nu m ber of refin em ent levels, provided that the number of smoothing steps is large enough. Proof. T he abst ract multi grid theory; see, e.g., [BS94, Hac85, Ver84], shows that the approximatio n property given in Lemma 2.37, t he smoot hing properti es given in Lemm as 2.40 and 2.43 , and t he st ability estimates of Lemm as 2.41 and 2.44 gua ra ntee level independent convergence rates of the W- cycle provided t hat t he numb er of smoot hing st eps is lar ge enough. 0
174
2. Iterative Solvers Based on Domain Decompositi on
2.5.3 Numerical Results We show only some numerical results which were originally pr esent ed in [WW99] . In contrast to [BD98, BDW99, Br aOl, WW98J, where an exac t modified Schur complement was solved in each smoot hing st ep , we do not solve any modifi ed Schur com plement syste ms exac tl y. As pr eviou sly shown , t he exact solut ion can be replaced by an it eration . As a consequence the iterates do not belon g t o the subspace for which the saddle point problem is positive definite. In spit e of this , the resulting multigrid convergence rat es are ind ependent of the refinement levels provided that the number of smoot hing ste ps is lar ge enough. We start with a discussion of a good choice of SI in (2.75). We have shown t hat SI := SI(ke , 1 - c) sat isfies the assumptions of Lemma 2.43 provided t hat e < 1/7, SI < 251, and k; > k(c) . We select e := 0.1 in our numeric al examples, and we obtain t he solut ion of SIYI = tl by k e it er ation ste ps and one sca ling ste p. Formally, we can rewrite StYl = tl as YI = 1
~ c SI- l / 2 (Id -
l 2
l
(Id - Sl / 5 1-1 Sl / 2)ke ) SI- / 2tl
The implementation of SI- ltl is based on t he identity
which is established by a st raight forward computation. The right hand side can be int erpret ed as the error propagation of the following it er at ion scheme : n ._
n- l
YI .- YI
+ S~-l(t I I
-
S
n- l )
IYI
,
n;::: l .
(2.78)
Set t ing YP := 0, we find Yl - SI- ltl = -(Id - 5 1- l SI)ns l- l t l ' The choice n = k; yields YI = (1- c) - l y~e . The delicat e point is t he choice of k e . In our approach, we use a stopping crite ria for the it er ation (2.78) to define ke . The application of KI- l in each smoot hing ste p can be easily impl emented in te rms of the inn er it er ation (2.78) . Each smoot hing ste p (2.6) requires the solution of
Using t he decomposition (2.75) , we find t hat t he inverse of KI can be written as a product of an upper and a lower blo ck t ridiagonal matrix
Then , t he solut ion can be obt ain ed by a forward and backward substitution , . (x I' T YIT )T -- UI ( slT ' tT)T h (sl' T tT)T -- L I (fT I , were I I ' glT)T . Th e app l'lcai.e., 1 tion of KI- is carr ied out in t he following way:
2.5 A Multigrid Method for the Mortar Saddle Point Formulation
Al 11 , tl := Bi sl - gl , Y?: = 0 , for n = 1, 2,3, ... do n n-l + S~-l(t B * A--1B n-l) YI := YI I I I I IYI , SI :=
175
- -1
until lltl- Bi All BI yFllhI-3/2.S :S 1
1
C
(2.79)
Iltdlh-3/2.S ' I
I
n
YI := -1- YI ,
- c
XI := SI -
-
1
Al B IYI
The linear it er ation (2.79) can be accelerate d by a conjugate gradient method. In our numerical result s only a few number of inn er iter ati on ste ps were required . In our first exa mple, we consider a problem with highly discontinuous coefficient s similar t o Ex ample 3 in Subs ect. 1.5.1. The domain is decomposed int o four squar es and the coefficient a is 1 or 106 in t he subdomains. For details, we refer t o [WW99] where all the examples of t his subsection were originally discussed . This probl em is a classical test example in multilevel theory; see, e.g., [Den8 2]. Our num erical resul ts are base d on the smoot her given by (2.79). Table 2.9 . Asymptotic convergence rates for highly discontinuous coefficients
level 4 5 6 7 8
number of elements 1024 4096 16384 65536 262144
W(2,2)-cycle damped Jacobi 0.084 0.128 0.137 0.144 0.146
V( 1, 1)-cycle symmetric Caufl-Seidel 0.080 0.091 0.095 0.098 0.102
T wo choices of Al are considered . The first one is a J acobi method with dam pin g fact or 0.7 and t he second one is a symmetric GauB-Seidel smoot her. In Tabl e 2.9, we pr esent t he asy mpt ot ic convergence rat es for a W- cycle with two pre- and postsmoothing st eps and a V- cycle with one pr e- and post smoothing ste p. We find that t he number of inn er it eration steps required in (2.79) is bounded ind epend entl y of the refinement level, and by 4 for t he J acobi-typ e smoother and by 8 for t he Gau B- Seidel smoother. This reflects t he fact t hat the cond it ion number of t he approximated Schur complement is worse for t he Gaufi-Seidel smoother. Wi thin one smoothing step (2.79) , we have to app ly All (n + 2)-times, where n is t he number of inn er it erations . At first glance , this makes t he Gau B-Seidel smoother considerably more expensive t han t he J acobi smoother. However , t he applicat ion of Al lBI can be ext remely simp lified by taking t he st ructure of B I into acco unt.
176
2. It er at ive Solvers Based on Dom ain Decom posit ion
In t he rest of t his subsection, we have also applied t he multigrid method for t he examples given in t he int roduction of Sect . 1.5. We use t he V-cycle as a precondit ioner for a Krylov space method . Since we are not working on t he subspace on which t he operat or Ki is positi ve definite, we use a bicgst ab method . An ILU-type smoot her is chosen in t he example of t he time-dependent pr oblem illustrated in Fig. 1.25. We recall th at in t he case of a nested iterati on , t he redu cti on facto r is much better at the beginning. Only a few ite rations ste ps are necessary to obtain an iteration error of th e same order as t he discreti zati on error. An error reduction of 10- 10 is obtained with 3 iteration ste ps on each level. On the finest mesh, we have 327680 elements . The multigrid meth od is mor e sensitive to t he choice of t he smoot her in t he case when mixed and conforming finit e elements are coupled, as ana lyzed in Sect . 1.4.1 and impl emented for the example illust rated by Fig. 1.26 in Sect. 1.5. Numerical test s with a J acobi-typ e smoother show th at t he damping fact or has to be decreased and that t he number of smoothing steps has to be increased to obtain a robu st method . Stable convergence rates are obtained by using a pr econditioned Krylov space meth od . We use a V (2, 2)-cycle with an ILU smoother and three inner iterations in (2.79) as preconditioner. Table 2.10 shows t he perform an ce of t he preconditioned method. Table 2 .10. Convergence rat e of the precondit ioned Kr ylov space method
I level I eleme nts I conv 1 2 3 4
4028 16112 64448 257792
rat e 0.05 0.08 0.10 0.11
I
In t he case of t he linear elasticity problem discussed in Sect . 1.5; see also Fig . 1.27, t he multi grid method is very sensitive to lar ge aspect ratios of the subdomains and to t he material par ameters. The average convergence rate of t he precondi ti oned bicgstab is 0.5 in our num erical experiments . A V(3,3)cycle with a symmet ric Gaufs-Seidel smoot her is used as preconditioner. On t he finest mesh , we have 360448 elements .
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List of Figures
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37
Decomposition of a global problem int o homog eneous subproblems . . . . . 1 Geometrically conforming (left) and non conforming (right) situation . . . 4 Decomposition and non-matching triangulations in 2D and 3D 4 Different non-matching triangulations at an int erface in 2D . . . . . . . . . . . 5 Structure of the support of a nodal bas is fun ction in Vh, (standard) . . . . 8 Nodal basis fun ction on a mortar and non -mortar side , (standard) . . . . . 9 St ability properti es of (1.8) and (1.16) :. .. . . 15 Tri angulations on mortar and non -mortar side of "[m: in 3D 19 Structure of the support of a nodal basis funct ion in Vh, (dual) . . . . . . .. 28 Different types of basis functions for Lagr ange multiplier spaces. . . . . . . 30 Nodal basis fun ction on a mortar and non-mortar side, (dual) . . . . . . . . . 32 Tr ace of a hexah edral (left) and a simplicial (right) triangulation. . . . . . 33 Indices in the cas e of a rectangle and a triangl e . . . . . . . . . . . . . . . . . . . . .. 33 Isolin es of piecewis e bilin ear and piecewise constant dual basis fun ctions 35 Different types of piecewise bilin ear du al basis functions . . . . . . . . . . . . .. 35 Numbering of the nodal basis fun ctions of WO ;h m (Jm) . . . . . . . . . . . . . . . . 36 Du al basis fun ctions, (nk = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 Decomposition into two sub dom ains 40 Support of the extended fun ction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Decomposition of t he nodes into sets in 2D 47 Wireb asket (left) and detail (right) 48 Coupling between du al and primal finit e element s 51 Equivalence between primal dual coupling and mortar coupling . . . . . . . . 57 Ori entation of the normal vector n e . . . . . . . . . . . . . • • . . . . . . . . . . . . . . • .. 58 The flux of the solution on a rotating geometry. . . . . . . . . . . . . . . . . . . . . . 62 Initial mesh for the channel dom ain (left) and flux (right) 63 Deformation of the composite (left) and zoom of a thin layer (right) 63 Decomposition into 9 sub dom ains and initial triangulation (left) and isolin es of the solution (right), (Ex ample 1) . . . . . . . . . . . . . . . . . . . . . . . .. 65 Decomposition into 4 sub dom ains and initial triangulation (left) and isolines of the solution (right) , (Ex ample 2) . . . . . . . . . . . . . . . . . . . . . . . . . 66 Decomposition into 4 subdomains and initial triangulation (left) and isolines of the solution (right) , (Example 3) . . . . . . . . . . . . . . . . . . . . . . . .. 68 Decomposition into 2 subdomains and initial triangulation (left) and isolines of the solution (right) , (Example 4) . . . . . . . . . . . . . . . . . . . . . . . . . 69 Discretization err ors in the energy norm versus number of elem ents . . . . 70 Discretization errors in the L 2 -nor m versus number of elements . . . . . . . . 71 Vertex X i and midpoint mi do not coincide 72 Sp ecial t riangulat ions on the int erface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Triangulation (left) , solution for Sit . I (middle) and Sit . II (right) . . . . .. 75 Isolines of the solu tion (left) , decomposition into 5 sub domains (left middle) , level difference 2 (righ t middle) and level differenc e 3 (right) of t he initial triangulation, (Ex ample 5) 76
188
List of Figures
1.38 Situati on I (left) and Situati on II (right) , (Example 5) 1.39 Error in the energy (left) and in the £ 2-norm (right) , (Ex ample 5) . . . .. 1.40 P erforman ce ada ptive (left) and un iform refinement , level difference 2 (middle) and level difference 3 (right) , (Example 5) . . . . . . . . . . . . . . . . .. 1.41 Isolines of the solution (left) and non -m atching initial t riangulation (right) , (Example 6) 1.42 Ad ap tive refinem ent for Situations I (left) and II (right) , (Ex ample 6) .. 1.43 Ad aptive refinem ent on Level 4 and Level 5, (Ex ample 7) . . . . . . . . . . . .. 1.44 Ad ap tive refinement on Level 6 for a2 == 0.1 (left) and a2 == 0.01 (right) 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38
Conv ergence rates and number of it erati ons . . . . . . . . . . . . . . . . . . . . . . . . . Converge nce rat es of pr econditioned cg-method . . . . . . . . . . . . . . . . . . . . .. Decomposition used by Schwarz in his original work Coloring of sub dom ain s into four classes . . . . . . . . . . . . . . . . . . . . . . . . . . .. Macro- trian gul ation and fine trian gulation on n ..................... Decomposition of V into t hree typ es of subspaces . . . . . . . . . . . . . . . . . . . . Hierarchical decomposition of t he nodes of V . . . . . . . . . . . . . . . . . . . . . . . . Effect of one sym metric GauE-Seidel smoot hing step. . . . . . . . . . . . . . . .. Local degrees of freedom of a lowest order R aviar t-Thom as vecto r field . Decompositi on of Vh int o t hree ty pes of subspaces Decomposit ion of F Neighborhood of t he wirebasket Constructi on of t he sequence f; in 2D Different refinem ent t echniques on FE F I- 1 . . . . . . . . • . . . . . . . . . . . • • . . contains a divergence free element .. Ad aptive refinem ent such t hat Conditi on number (ast erisk) and least- squ are second ord er logarithmic polyn omi al (solid line) Condition number (ast erisk) and least- squ ar e second order logarithmic polynomial (solid line) Structure of t he sup port of P I Vl J acobi an d sym metric Gaufi-Seidel smoo ther , (Ex ample 1) Non-sy mme t ric Gaufl-Seidel smoother , (Ex ample 1) J acobi and sym metric Gaufi-Seidel smoother, (Ex ample 2) Non-sym me t ric Gaufl-Seidel smoot her , (Example 2) J acobi and sym met ric Gaufi-Seidel smoot her , (Ex ample 3) Non-symmetric Gaufl-Seidel smoo t her , (Example 3) Tri an gul ati on and isolines for Ex ample 4 (left) and Ex ample 5 (right ) .. Jacobi an d symme tric Ga ufi- Seidel smoot her , (Ex ample 4) J acobi and sy mmetric Gaufl-Seidel smo ot her, (Example 5) . . . . . . . . . . . . Test fun ction PH for standa rd (left) and du al (right ) Lagrange multiplier Problem set t ing and converge nce rat es, (Ex ample 1) Displ acem ent s and adaptive t riangulations , (Ex ample 1) In iti al trian gulation (left ), dist ort ed grid (middle) and convergence rates (right) , (Example 2) Free t angenti al displ acement at t he int erface . . . . . . . . . . . . . . . . . . . . . . .. P roblem set t ing and convergence rat es, (weak coupling ) Displacem ent s and ada pt ive trian gulati ons , (weak coupling) Coupling in both direct ions (left) and in normal dir ection (right) in 3D. Informati on tran sfer at t he int erface Error redu cti on (trace norm) on level 5 (left) and level 7 (right) for t he Diric hle t-Neumann algorit hm Error redu ct ion (Lagran ge mul t iplier norm) on level 5 (left) and level 7 (right ) for t he Diri chlet- Neumann algorit hm
Yt;T
77 80 80 81 82 83 84 85 86 87 91 91 92 94 98 102 104 105 111 116 119 120 122 124 128 138 138 139 139 140 140 141 141 142 147 149 150 151 151 153 154 154 156 161 161
List of Tables
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14
Discretization errors in the £2-norm, (Example 1) Discretization errors in the energy norm, (Example 1) . . . . . . . . . . . . . . . . Discretization errors in the £2- nor m, (Example 2) Discretization errors in the energy norm, (Example 2) . . . . . . . . . . . . . . . . Discretization errors in the £2-nor m, (Example 3) Discretization errors in the energy norm , (Example 3) . . . . . . . . . . . . . . . . Discretization errors in the £2 -norm , (Example 4) Discretization errors in the energy norm, (Example 4) . . . . . . . . . . . . . . . . Discretization errors in the £ 2-norm for Examples 1-4 . Discretization errors in the energy norm for Examples 1-4 .. .. .. Effectivity index ( and performance X, (Example 5) . . . . . . . . . . . . . . . . . . Discretization errors in the case of a level difference 3, (Example 5) . . . . Discretization errors in the case of a level difference 2, (Example 5) . . .. Error in the energy and £2-norm for Situations I and II in the case of uniform refinement , (Example 6) 1.15 Effectivity index on [lPl ' [lCR and [l (a2 = 1), (Example 7)
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10
Condition number and number of cg-iterations, (in parentheses) Condition number and number of cg-iterations, (in parentheses) Condition number and number of cg-iterations, (in parentheses) Condition number and number of cg-iterations, (in parentheses) Number of iteration steps, (trace norm) Number of iteration steps, (Lagrange multiplier norm) Asymptotic convergence rates q, (trace) Asymptotic convergence rates q, (Lagrange multiplier norm) Asymptotic convergence rates for highly discontinuous coefficient s Convergence rate of the preconditioned Krylov sp ace method
65 66 67 67 68 68 69 70 73 74 78 79 79 82 84
. 122 123 . 124 . 125 . 159 . 160 . 162 . 162 . 175 . 176
Notations
Basis functions:
basis function of Mh m bm), 1::; i ::; n; basis function of WO ;h m bm), 1 ::; i ::; N m and Whmbm) , 1::; i::; V m , V m > n; basis function of Wh m (rm) , 1 ::; i ::; N m standard nodal basis function
Domains and related notations: D
IDI
r
1m
a
ilk [lii(m) [In(m)
S
Sm
Lipschitz domain ar ea of the domain D interface or boundary interface between two subdomains bounded polygonal domain polygonal subdomain of [l subdomain associated with the mortar side of the interface "[m. subdomain associated with the non-mortar side of the interface union of all interfaces "[m. strip of width h-« on the mortar side of "[rri
Elasticity: €
E E, v
>',J-l
a tr( .)
u, f , p
infinitesimal strain tensor Hooke's tensor, (components E ijkl, 1::; i, j , k, l ::; d) Young's modulus, Poisson ratio Lame constants stress tensor, (components ou, 1 ::; i ,j ::; d) trace of a tensor displacement field, volume force, surface traction
Finite element spaces: Bhl Bh(aT) CRhl ' CRh
M hm (rm) Mh NChl NCg;hl NV(il ; T) NV(T)
space of cubic bubble functions on [ll space of bubbles on aT Crouzeix-Raviart finite element spaces on [ll, [l Lagrange multiplier space on the interface Lagrange multiplier space enriched nonconforming finite element space on [ll subset of NCh l satisfying inhomogeneous boundary conditions Nedelec finite elements in 3D associated with the triangulation T local Nedelec finite elements in 3D
192
Notations
QU) RTh l ;n l
RT(il ;7) RT(T) S h(aT)
Vh Vh
VH Vi Vi
Vi
VF Vr
W h l ;nl
w., (')'m )
(')'m ) (')'m ) O;h m (')'m )
W O;h m W hm W
local sp ace of bilin ear fun ctions on f R aviart-Thomas finite element space of order n l on ill R aviar t - Thomas finit e elements in 3D associate d with T local lowest order R aviart-Thomas finit e elements in 3D sp ace of conforming bilin ear finit e elements on aT const rained finit e element space (Ch apter 1) finit e element space on t he fine triangulation (Ch apter 2) coarse finit e element sp ace associated with the macro-trian gulation finit e element space on level 1 hierarchical basis finit e element space on level 1; C Vi finit e element sp ace associate d with two adjacent substruct ures finit e element space associated with one sub structure finit e element space of piecewise polynomials of order n l on ill trace space of X h n (m ) ;n n (m) on t he int erface "[m trace sp ace with zero valu es on the boundary of "[m: subspace of Wh m (')'m ) subspace of WO ;h m (')'m ) conforming Pn k -finite elements on ilk un constrain ed product sp ace
X h k ;n k X h
Finite element and weak solutions: flux of t he weak solut ion discret e flux on ill flux in normal dir ection discret e Lagran ge multiplier weak solut ion of the model pr oblem finit e element solut ion
Hilbert spaces: C(D) Co(D) Pn(D) H S(D) , L 2(D) HJ(D)
continuous fun ct ions cont inuous fun cti ons with zero values on t he boundary of D pol yn omi als of degree :::; n st andard Hilbert spaces subspace of H I (D ) with zero t race on the boundary of D
Hto(')'m)
interpolation space between L 2 (')'m) and H J (')'m )
1
1
(Hcih(')'m)) H( div ;D)
I
Ho(div ; D ) H(curl ; D) y
1
du al sp ace of Hgo (')'m) vect or valu ed Hilb ert space ; divq E L 2(D) subspace of H(div ; D) ; q . n = 0 on aD vect or valu ed Hilb ert space; curlq E (L 2(D) ? subspace of H l(il k ) ; da = 0, 1 :::; m :::; M
nc,
I-yJv]
Miscellaneous: Kr onecker symbo l local contribut ion of an error estimato r local contribut ion of a simplified erro r est imator number of sub dom ain s condition number eigenvalue in Subsect . 2.3.3 number of int erfaces
Not ations dimension of the Lagr ange multiplier space Mhm (-ym) dim ension of the trace space Wh m (-ym) sp ectral radius spectrum of an operator performan ce effect ivity ind ex
p CT
X
<;
Norms and semi norms: Euclidean vector norm II -II L 2-norm on 52 II -110 br oken HI -norm on 52 II -Ih broken HI-semi norm on I· h energy norm [I -III 1-!s;D
11 -lls;D 1-ls;oD 11 -lls;oD II . II H! h'm) II · [I H ! (I'm ) oo -II ( H 1-2 h'm » 1 oo -llh:" ;l'm -Ilh:" ;5 · IIM -11M' '11M -IIMI m m
52
H S-semi norm on D H i-uotu: on D (scaled or uns ealed) H S-semi norm on aD HS-norm on aD (scaled or uns ealed) H I/ 2_n orm on "[m.
/ H 00 -norm on 1m I 2
H~{/ -dual norm on 1m mesh depend ent L 2 -norm mesh depend ent L 2 -norm Lagran ge mul tipli er norm du al norm of II. IIMm Lagran ge multiplier norm du al norm of II -11M
on "[m (local fact or h;S) on S (local factor h; S) on 1m on S
Operators:
t,
JIm
Pm
Qm p n , PI
Lagrange int erpolation operator on 52k mortar pro ject ion pr ojection onto Wh m (-Ym) du al projection onto Mhm (-Ym ) inte rpolat ion operat or for Raviart-Thomas vect or fields
Triangulations and related notations: hm o r hnon he
H/ h
[h, [ H
E ,e F h, F H
F,f
s ;«: S m ;1
ti«, T" , TH t;
T ,t
meshsize on the mort ar side meshsize on the non-m ort ar side lengt h of t he edge e (2D), diam et er of the element e (3D) ratio between substruct ure diam et er and element diam et er sets of edges edges of the triangulation sets of faces faces of the triangulation (in 3D) t riangulat ion on the int erface "[rn (non-mortar side) famil y of triangulati ons on t he int erface , m trian gulat ion on 52 k with meshsize hk globa l trian gulations family of shap e regular triangulat ions elements of t he trian gulat ion
193
Index
a post eriori error est imat or, 60, 75 a pri ori est imates
- L 2 -n or m
- - abstract, 23 - - standard, 6 - energy norm - - abstract, 23 - - standard, 6 - Lagran ge multiplier - - abst ract, 26 - - st andard, 10 - primal-dual coupling, 54 adaptiv refinem ent , 150 adaptive refineme nt , 76, 80, 154 approximation pr op erty, 29, 129, 165 - abstract constra ined space, 15, 18 Lagran ge mul tiplier space (Sb) , 12, 34,37 - modifi ed t race space (Sc) , 12 - saddle p oint pr oblem , 164 asy m ptotic rates, 66, 79 Aubin-Nitsche, 23, 166 biorthogonality, 27- 29, 31, 34, 38 block diagon al smoot her , 168 block struct ure , 50, 134, 155 bri cks and mort ar , 63 bubble funct ions, 6, 55, 109 Cauchy- Schwarz , 90, 116 coarse space, 92, 93, 103 coercivity , 7, 53, 145 coloring, 17, 90, 104, 114, 120, 122 condit ion number , 85, 92-94, 114, 121 132 ' consistency err or , 7, 19, 22, 51, 71, 82 const rai ned space, 5, 71, 145, 146, 154 - abstrac t , 13 - standa rd , 5 const ra ints, 7, 27, 56, 128, 133, 138, 139, 163
convergence rat e, 85- 87, 95, 135, 138, 173
Crouzeix-Raviar t , 55, 59, 83 decomposition, 87, 88, 90-92 , 94, 128 - horizont al, 115 - verti cal, 114, 118 dir ect sum, 88, 104, 115 Dirichlet condit ions, 51 Dirichlet pr oblem , 40, 55 - algebraic formul ati on, 41 Diri chlet-Neumann 158 160 Diri chlet-Neumann' cou~ling, 40, 52 - algebra ic formulati on , 43 disconti nuous coefficients , 83 discreti zati on error , 5, 23, 38, 53, 54 57, 64-70 ' diver gence free, 119 du al basis, 11, 27- 29, 31, 33, 34, 36 37 125, 126 ' , du al norm, 9, 24 du al space, 11, 137 du al tec hniques , 62 du ality pairing, 9, 51 effectivity index, 78, 84 eigenvalue , 89, 90, 99, 122, 124, 131, 132, 134 element space, 103 ellipticity, 6, 59, 127, 148, 154 energy norm , 7, 100 err or pr opagati on , 95 exte nsion, 24, 42, 107, 127 - by zero, 18 - discret e harmonic, 17, 113 - divergence free, 108 face space, 104 fine triangulation, 91, 105 flow pr oblem , 62 C aufl- Seidel, 95, 135, 139, 155 generic constants, 3
196
Index
geometrically conforming, 4 Gr een 's formula , 100, 105 hier ar chical basis, 93, 114, 121 Hooke's law, 143 ind efinite smoother , 171 inf-sup condition, 10, 26, 53, 165 int erpolation, 105, 115 it eration error , 131, 133, 135, 169, 170, 172,173 it erative substructuring method, 92, 99, 121 J acobi-method, 85, 91, 173 jum p, 5, 8, 17, 19, 22, 74, 75, 77, 83, 114 Korn 's inequ ality, 145 Lagran ge multiplier , 44, 57, 60 Lagrange multiplier space, 6, 64 - abst rac t, 13 - exa m ples, 27 - linear case in 2D , 29 - linear case in 3D , 33 - piecewise constant , 59 - qu adrat ic case in 2D , 36 - st andard, 6 Lam e constant s, 63, 143 locality of the support , 27, 28 - Lagran ge multiplier space (Sa) , 11 logari thmic bound, 107 mass matrix, 43, 44, 49, 54, 131 mesh dependent norm , 10, 127 mix ed finit e eleme nt s, 62 model problem , 3 mortar formulations - algebraic, 47 - non conforming - - abstract , 13 - - standa rd, 6 - product sp ace - - algebraic, 50 - - vari ational, 45 - sa ddle point pr obl em abs tract , 14 - - primal-dual, 56 - - standa rd, 9 mortar proj ection, 32 - modified , 15 - origina l, 8 mortar side, 74
multigrid method, 48, 97, 125, 137 - W- cycle, 173 - V- cycl e, 137 - corr ection, 98 - prolongation, 98 - restricti on , 129 - saddle point, 162 - smoothing, 98, 132, 168, 171 Nedelec, 118, 123 n atural boundary condit ions, 52 nest ed sequence , 114 nest ed spaces, 126 Neumann conditio ns, 51 Neumann problem , 40 - algebraic formulation , 41 Neumann-Dirichlet , 156-158, 160 Neumann-Neumann algorithm, 125 Nits che, 38 non-matching, 1, 5, 38, 42, 67, 74, 80, 150 non-mortar side, 5, 6, 8, 11, 18, 28, 35, 44, 47, 48, 50, 54, 74, 131 non-optimal method, 64, 71 non-symmetric, 136 non conforming formulation - abst ract , 13 - standa rd, 6 non conformity, 22, 75 norm equivalence, 102 op erator form , 167, 168 op erator norm, 168 penalty method , 38 performance, 77 Po isson' s ratio, 143 polylogari thmic , 114, 123 pos tprocessing , 39, 57, 60, 140 primal and dual eleme nt s, 51, 57, 62 product space, 7, 51, 136 proj ection , 15, 20, 29, 45, 55, 94, 105, 117 prolongation, 127, 140 qu asi-op timal bounds, 113 qu asi-projection , 88, 94, 118 R aviart-Thomas , 52, 100, 101, 114, 123 - divergence free, 118 regularity, 66, 166 remeshing, 62 restriction , 98, 127, 140 Richardson method , 157
Index rigid b od y moti on , 148, 154 rotating geom etry, 62 sa ddle po int probl em , 52, 53, 60 sca led norm, 100 Schur compleme nt , 125, 157, 171 Schwar z H.A., 87 Schwarz method, 87 - additi ve, 89, 114, 118 - condit ion number , 89 - multigrid , 98 - multiplicative, 95 - overlapping, 91 - subst ruct ur ing, 103 - sym metric, 97 sha pe regular , 114 sha rp int erface, 77, 80 smoother, 98, 132 - block dia gon al, 168 - indefinite, 171 smoothing iteration, 132, 136, 171 smoot hing property, 98, 133, 169, 172 spec tral equivalence (Sd) , 12 spectral radius, 94, 115, 134, 172, 173 stability, 16, 135 - H J -st abl e, 16 H~£ 2 -st able, 8, 17 L -st abl e, 15, 112, 123
-
197
stability est imate , 105, 170, 173 St eklov-Poincar e, 157 substructur ing , 93, 104 support of a nodal basis fun ct ion - du al basis , 28 - standard, 8 surplus, 119 trace space - mod ified , 12 - st an dard, 7 t race theorem, 101, 166 t ransfer opera tor, 128, 140, 164 two-grid algorit hm , 170 unc onst rain ed product space, 7, 39, 45, 48, 54, 58, 126, 131 uniform refinem ent , 78, 82 uniformly elliptic , 7, 59 vect or valu ed space, 114 vect or valu ed space , 100, 118 vert ical decomp ositi on , 118, 119 wirebasket , 93, 106, 111 Young's modulus, 143
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