Discontinuous Galerkin methods for viscous incompressible flow

Guido Kanschat Discontinuous Galerkin Methods for Viscous Incompressible Flow

TEUBNER RESEARCH Advances in Numerical Mathematics Herausgeber/Editors: Prof. Dr. Wolfgang Hackbusch, Max-Planck-Institut, Leipzig Prof. Dr. rer. nat. Hans Georg Bock, Universität Heidelberg Prof. Mitchell Luskin, University of Minnesota, USA Prof. Dr. Rolf Rannacher, Universität Heidelberg

Guido Kanschat

Discontinuous Galerkin Methods for Viscous Incompressible Flow

TEUBNER RESEARCH

Bibliographic information published by Die Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

Habilitationsschrift Universität Heidelberg, 2004

1st Edition November 2007 All rights reserved © Deutscher Universitäts-Verlag | GWV Fachverlage GmbH, Wiesbaden 2007 Readers: Ute Wrasmann / Anita Wilke Deutscher Universitäts-Verlag and Teubner Verlag are companies of Springer Science+Business Media. www.duv.de www.teubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: Regine Zimmer, Dipl.-Designerin, Frankfurt/Main Printed on acid-free paper Printed in Germany ISBN 978-3-8350-4001-4

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PREFACE

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Contents

1 Basics

15

1.1

Meshes and shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.2

Spaces and approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19



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2

3

4

Linear Diffusion I

33



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Linear Diffusion II

67



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Stokes Equations

89



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CONTENTS

8 5 Flow Problems

105

5.1

Advection-Diffusion-Reaction Equation . . . . . . . . . . . . . . . . . . . . . 107

5.2

Oseen Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3

Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Linear Solvers

131

6.1

Krylov space methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2

Interior Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3

Local multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.4

LDG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.5

Advection Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.6

Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.7

Oseen equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A Example problems

167

A.1 Meshes and domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.3 Advection-diffusion-reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.4 Stokes and Oseen equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

List of Figures 1.1

The set  and   (if /  /). . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.2

The cel % (dark grey) and the neighbor set %! (grey). Note that a common corner does not result in neighborhood. . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.1

Smallest eigenvalue of - 54 5 on a mesh of squares . . . . . . . . . . . . . . .

38

2.2

Eigenfunction to the smallest eigenvalue of - 54 5 for ?  5 (left) and ?  5 (right) on a mesh of squares . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.3

Accuracy of the interior penalty method depending on the parameter ?. . . . .

44

2.4

Pointwise errors on irregular meshes . . . . . . . . . . . . . . . . . . . . . . .

55

2.5

Adapted mesh (level
 % ) for the point value computation . . . . . . . . . .

60

2.6


 -error over number of degrees of freedom for CG- , DG- and DG- elements (left) and (bi-)quadratic elements (right) . . . . . . . . . . . . . . . .

62

2.7 2.8

3.1 3.2




-error over number of matrix entries for CG- , DG- and DG- elements (left) and (bi-)quadratic elements (right) . . . . . . . . . . . . . . . . . . . . .

63




-error versus degrees of freedom (left) and matrix entries (right) for polynomials of degree 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy of the standard LDG method on Cartesian grids depending on M  . . . Accuracy of the standard LDG method on distorted grids depending on M  . . .

3.3

Naming of the boundary components of a rectangular grid cell % . . . . . . . .

3.4

Error L  • •L with inhomogeneous Dirichlet boundary values . . . . . . . .



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64 76 77 78 83

3.5

Error L  •

with homogeneous Dirichlet boundary values . . . . . . . . .

84

3.6

Performance of the different DG methods . . . . . . . . . . . . . . . . . . . .

87

4.1

Discretization error depending on the two parameters. Errors  (top) and  (bottom) with homogeneous  elements . . . . . . . . . . . . . . . . . . . .

99

LIST OF FIGURES

10 

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5.1

Comparison of DGFEM and CGFEM (both  for a reaction dominated diffu  VLRQ SUREOHP ZLWK D   (left) and D   ULJKW               



 &*)(0 OHIW DQG '*)(0 ULJKW VROXWLRQ RI D UHDFWLRQGRPLQDWHG D   ) diffusion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110



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6.3

Comparison of point and block smoothers for different shape function spaces . 144



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 -norms of Kovasznay solutions L # and  GHSHQGLQJ RQ WKH 5H\QROGV QXPEHU 

List of Tables The lowest stable ? depending on the boundary stabilization on an equidistant square grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Convergence of the interior penalty method with tensor product polynomials on Cartesian and distorted meshes . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Convergence of the interior penalty method with polynomials " on Cartesian and distorted meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Convergence of the interior penalty method with  - and  -elements on a circle with bilinear mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.5


 -errors and their scaled differences for discontinuous  elements. . . . . .

54

2.6

Point error - and -  -  ; #  and the corresponding a posteriori estimators ; #  and ; #  (
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2.7

Point error - and a posteriori estimator ; #                 





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71

3.2

Convergence of the standard LDG method on Cartesian and distorted meshes .

74

3.3

Convergence results for the superconvergent LDG method in two and three space dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Convergence results for the superconvergent LDG method with homogeneous Dirichlet boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.5

Convergence results for the superconvergent LDG method with " elements . .

85

3.6

Convergence results for the superconvergent LDG method with 6-stabilization DW WKH ERXQGDU\  • •,1                        



3.7

Convergence results for the superconvergent LDG method on non-Cartesian grids ( • •,1                              



2.1 2.2 2.3 2.4

3.4

LIST OF TABLES

12 4.1

Convergence of LDG discretization for Stokes equations . . . . . . . . . . . .

98

4.2

Convergence of LDG discretization for Stokes equations in 3D . . . . . . . . . 100



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5.2

Errors and orders of convergence for D  5. . . . . . . . . . . . . . . . . . . 128

5.3

5.4

Errors and orders of convergence for D  5 in the jump seminorm ••' •• ) 3, 5 7.  ••# ' • •&&••  . . . . . . . . . . . . . . . . . . . . . . . . 128 
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5.5

Number of iterations for convergence of the non-linear iteration. . . . . . . . . 128



(UURUV IRU .RYDV]QD\ ÀRZ D  ) and pairs RT" /" . . . . . . . . . . . . . . . 130



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6.3

!

ZKHQ *DX‰6HLGHO VPRRWKHU DQG ELTXDGUDWLF Condition numbers of 
and ( ) and bicubic ( ) shape functions are used. 1  • • 4  5 . . . . . . . . . 141

6.4

Condition numbers and contraction numbers for L-shaped and slit domains usLQJ ELOLQHDU VKDSH IXQFWLRQV DQG *DX‰6HLGHO VPRRWKLQJ             



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5HGXFWLRQ DQG FRQYHUJHQFH UDWHV IRU WKH YDULDEOH 9F\FOH ZLWK EORFN*DX‰ Seidel smoother (" shape functions) . . . . . . . . . . . . . . . . . . . . . . . 143

6.7

Contraction and convergence rates on a three-dimensional cube . . . . . . . . . 144



5HGXFWLRQ DQG FRQYHUJHQFH UDWHV IRU WKH YDULDEOH 9F\FOH ZLWK EORFN*DX‰ Seidel smoother on non-Cartesian grids ( shape functions) . . . . . . . . . . 145



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 -errors and orders of convergence in the velocity and
 -norm of the divergence of the post-processed solution # for D  5. . . . . . . . . . . . . . . 128

!  &RQMXJDWH JUDGLHQW FRQYHUJHQFH UDWHV IRU VWDQGDUG /'* ZLWK SUHFRQGLWLRQHU .

153

6.11 Values of stabilization parameters yielding optimal convergence . . . . . . . . 153 6.12 Iteration counts  for superconvergent LDG . . . . . . . . . . . . . . . . . . 155  3HUIRUPDQFH RI FJ PHWKRG IRU PRGL¿HG VXSHUFRQYHUJHQW /'*          

LIST OF TABLES

13

6.14 GMRES performance for the preconditioned LDG Schur complement and the preconditioned system ( -elements) . . . . . . . . . . . . . . . . . . . . . . 157 6.15 Convergence rates for GMRES with multilevel preconditioner using downwind block-Gauß-Seidel smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.16 Performance of GMRES with multilevel preconditioner using upwind blockGauß-Seidel smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.17 Convergence rates for GMRES with multilevel preconditioner using downwind block-Gauß-Seidel smoothing in a vortex . . . . . . . . . . . . . . . . . . . . 160 6.18 GMRES performance for the Stokes system . . . . . . . . . . . . . . . . . . . 162 6.19 Convergence of exact Kay/Loghin preconditioner for two dimensional PoisVHXLOOH ÀRZ ZLWK  elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.20 Convergence of Kay/Loghin preconditioner for two dimensional Poisseuille ÀRZ ZLWK RQH DQG WZR 9F\FOHV  elements) . . . . . . . . . . . . . . . . . . 165 6.21 Iteration steps  .D\/RJKLQ SUHFRQGLWLRQHU IRU OLQHDU GULYHQ FDYLW\ ÀRZ ZLWK RQH YDULDEOH 9F\FOHV %LFJVWDE  elements) . . . . . . . . . . . . . . . . . . 165

Chapter 1 Basics 7KLV FKDSWHU VXPPDUL]HV GH¿QLWLRQV DQG UHVXOWV IURP JHQHUDO ¿QLWH HOHPHQW WKHRU\ IRU IXU WKHU UHIHUHQFH ,W VHWV RXW IURP GH¿QLWLRQ RI PHVKHV DQG ¿QLWH HOHPHQW IXQFWLRQ VSDFHV 7KHQ LW VXPPDUL]HV UHVXOWV RQ 6REROHY IXQFWLRQV DQG WKHLU DSSUR[LPDWLRQ E\ SRO\QRPL DOV RQ JULGV 6HFWLRQ  SUHVHQWV DQ DEVWUDFW ¿QLWH HOHPHQW DQDO\VLV VR WKDW SURRIV LQ ODWHU FKDSWHUV FDQ UHIHU WR WKH VSHFL¿F VWHSV ZLWKRXW UHSHDWLQJ WKHP VHYHUDO WLPHV )LQDOO\ WKH GLVFRQWLQXRXV *DOHUNLQ PHWKRG IRU SXUH DGYHFWLRQ SUREOHPV LV SUHVHQWHG

1.1 Meshes and shape functions 1.1.1 Let 1 be a domain in     4  :H GHQRWH WKH ERXQGDU\ RI 1 E\ /  =1 ,Q HDFK SRLQW 0 • •=1

0 denotes the outward unit normal vector )RU HOOLSWLF SUREOHPV ZH GLVWLQJXLVK EHWZHHQ WKH 'LULFKOHW ERXQGDU\ / DQG WKH 1HXPDQQ ERXQGDU\ /  7KH 1HXPDQQ ERXQGDU\ LV DQ RSHQ VXEVHW RI =1 LQ WKH WRSRORJ\ LQKHULWHG E\ =1 from  and =1  / • •/  1.1.2 /HW WKH GRPDLQ 1 be subdivided into a triangulation  FRQVLVWLQJ RI FORVHG WULDQJXODU RU TXDGULODWHUDO DQG WHWUDKHGUDO RU KH[DKHGUDO JULG FHOOV % LQ WZR DQG WKUHH VSDFH GLPHQVLRQV UHVSHFWLYHO\ 4XDGULODWHUDOV DQG KH[DKHGUD ZLOO EH UHIHUUHG WR DV tensor product cells DQG ZLOO DOZD\V EH DVVXPHG FRQYH[ 7KLV DVVXPSWLRQ LV QR UHVWULFWLRQ VLQFH LW LV QHFHVVDU\ LQ RUGHU WR KDYH D XQLIRUPO\ SRVLWLYH GHWHUPLQDQW RI WKH PDSSLQJ IURP WKH 7ULDQJOHV DQG WHWUDKHGUD DUH refered to as simplicial cells  'H¿QLWLRQ The reference cell % LV HLWKHU WKH XQLW VLPSOH[ VSDQQHG E\ WKH RULJLQ DQG WKH  FRRUGLQDWH YHFWRUV VLPSOLFLDO FHOOV  RU WKH XQLW K\SHUFXEH #• • 4 &  Grid cells % of a triangulation  DUH LPDJHV RI WKH UHIHUHQFH FHOO VXEMHFW WR D  PDSSLQJ 2  7KLV PDSSLQJ ZLOO EH UHTXLUHG WR EH FRQIRUPLQJ LQ WKH VHQVH WKDW WKH IDFHV RI QHLJKERULQJ FHOOV DUH RQ WRS RI HFK RWKHU DQG GR QHLWKHU SURGXFH RYHUODS QRU JDSV  )ROORZLQJ FRPPRQ SUDFWLFH ZH ZLOO FRPPLW WKH HW\PRORJLFDO FULPH WR XVH WKH WHUP WULDQJXODWLRQ V\QRQ\ PRXVO\ IRU VXEGLYLVLRQ HYHQ LI WKH PHVK FHOOV DUH QRW WULDQJOHV  7KHVH QDPHV DUH WR EH XQGHUVWRRG LQ D WRSRORJLFDO VHQVH VLQFH ZH ZLOO DOORZ IRU FXUYHG ERXQGDULHV RI WKH FHOOV

CHAPTER 1. BASICS

16

1.1.4 Assumption: 9 For the analysis of the methods presented here, we will consider only   % . Furthermore, we will assume that each edge or face of a mesh cell domains, where 1 which is subset of the boundary =1 is either a subset of / or of / . This assumption is purely technical and proofs can be extended to the general case by generic arguments, provided that the mapping 2 UHSURGXFHV WKH ERXQGDU\ VXI¿FLHQWO\ DFFXUDWHO\ 1.1.5 We will consider sequences of triangulations generated by successive (possibly local) UH¿QHPHQW RI WKH coarse grid  ,  • •5 5 5 • •< • •5 5 5 • •
4 ZKHUH ZH GHQRWH SRVVLEO\ E\ WKH UH¿QHPHQW RSHUDWRU “• ´•  *OREDO UH¿QHPHQW LH UH¿QHPHQW RI DOO PHVK FHOOV LV GHQRWHG E\ ³
”. +HUH UH¿QHPHQW RI D FHOO PHDQV GLYLGLQJ WKH UHIHUHQFH FHOO LQWR  VLPLODU FKLOGUHQ DQG PDSSLQJ WKRVH ZLWK VXLWDEOH PDSSLQJV 2 HDFK 1.1.6 :LWK HDFK WULDQJXODWLRQ <  ZH DVVRFLDWH WKH PHVK VL]H IXQFWLRQ  • •
 1 VXFK WKDW 0     %  if 0 • •% ZLWK % • •< . We will denote by < WKH PD[LPXP RI 0 for a triangulation <  ,I ZH ZDQW WR VWUHVV WKH PHVK VL]H GHSHQGHQFH RI WKH WULDQJXODWLRQ ZH ZULWH  instead of < . We write  instead of  if no confusion between grid levels can arise. A grid cell % LQ WKLV KLHUDUFK\ EHORQJV WR H[DFWO\ RQH OHYHO O ZKLFK LV WKH QXPEHU RI UH¿QHPHQW VWHSV QHFHVVDU\ WR FUHDWH LW IURP RQH RI WKH FRDUVH JULG FHOOV  'H¿QLWLRQ By  (< ,  ZH GHQRWH WKH VHW RI FORVHG faces (edges and surfaces)  of WKH JULG FHOOV LQ WZR DQG WKUHH GLPHQVLRQV D IDFH LV WKH  • •GLPHQVLRQDO LQWHUVHFWLRQ RI WZR PHVK FHOOV :H GR QRW UHTXLUH WKDW WKH HGJHV RI FHOOV PDFK H[DFWO\ VXFK WKDW D IDFH PD\ EH WKH D VXEVHW RI VXFK DQ HGJH 7KH VHW  is subdivided into the subsets of edges on the Dirichlet and 1HXPDQQ ERXQGDULHV DQG WKH VXEVHW RI LQWHULRU HGJHV  7 3    • •  • •/     • •  • •   7 3

   • •  • •/  7 3

 5 

   • •  • •/

,Q WKH FDVH RI ORFDO UH¿QHPHQW DQ HGJH  of a cell % PD\ QRW EH LQ WKH VHW   7KHQ  is the union of   elements of   EHLQJ WKH HGJHV RI WKH FKLOGUHQ RI WKH UH¿QHG QHLJKERU VHH )LJXUH   7KH XVH RI WKH ZRUG HGJH KHUH DQG EHORZ ZLOO DOZD\V LPSO\ WKH ZRUG VXUIDFH LQ WKUHH VSDFH GLPHQVLRQV 1.1.8 2Q WKH UHIHUHQFH FHOO ZH GH¿QH WKH SRO\QRPLDO VSDFHV 3 7 "  0 " 

3 

7  0  4

ZKHUH  and  DUH SRO\QRPLDOV RI GHJUHH DW PRVW (. We will refer to " as the complete polynomial space of degree ( and to " as the tensor product polynomials of degree (. If WKH GLPHQVLRQ RI WKH GRPDLQ LV LPSRUWDQW ZH ZLOO ZULWH  " and  " instead of " and " , respectively.

1.1. MESHES AND SHAPE FUNCTIONS

17

  

Figure 1.1: The set  and   (if /  /). 1.1.9 We choose a convenient basis for " and " , respectively. This may either be Legendre polynomials for both spaces or Lagrange polynomials for a grid of support points (usually equidistant) for " . Although Legendre polynomials seem to be more convenient due to their orthogonality properties, this advantage may be lost on general quadrilaterals due to the mapSLQJ GH¿QHG EHORZ 7KH EDVLV IXQFWLRQV DUH FDOOHG <  ZKHUH  4 5 5 5 4 " and " is the dimension of " and " , respectively. 1.1.10 The mapping 2 ZLOO EH HLWKHU LQ "  or in "  for simplicial and tensor product cells, respectively. Remark that the surfaces of hexahedral grid cells are not necessarily plane quadrilaterals; they are bilinear mappings of those.  'H¿QLWLRQ The shape functions on the grid cell % are the basis functions on the reference cell % GH¿QHG LQ 6HFWLRQ  VXEMHFW WR WKH PDSSLQJ 2 , that is, the th shape function of mesh cell % is <, 0  < 2  05

(1.1)

The shape function space of the cell % is the space spanned by all functions • •<  • •--- DQG ZH denote it by " %  and " % , respectively. These are also called mapped polynomial spaces, but the functions are not polynomials due to the mapping 2  . With   ZH GHQRWH WKH QXPEHU of shape functions on cell % . We remark that the spaces " %  have optimal approximation properties only if 2 LV DQ DI¿QH mapping on all cells % (see [ABF02]). 1.1.12 Finite element spaces on  DUH JHQHUDWHG E\ MX[WDSRVLWLRQ RI WKH VKDSH IXQFWLRQV RI DOO mesh cells: $   " %  ,"  

 ," 

$

" % 

 

 By choosing the double subscript    for basis functions, the application to -methods becomes evident, since numbering on different cells is completely independent

CHAPTER 1. BASICS

18

The generic symbol ," will be used for both of these spaces. Whenever we need a basis for these spaces, we will use the basis functions in (1.1) with double indices %4 ; this notation is very convenient for discontinuous Galerkin methods. 1.1.13 Assumption: We will often assume that the triangulation  is shape regular, that is, for each cell % , there is a mapping 2 , such that the minimal and maximal eigenvalues of • •2 admit the estimate •  0 ( A  • 2   4 
( A 
• •2 0

(1.2)

with a constant  independent of the grid cell % . In the same way, a family of triangulation • •< • •is shape regular if is independent of O, too. 1.1.14 Assumption: Usually, we will assume that the mesh size variation of the triangulation  is bounded, that is, there is a constant  9  such that for any two cells % and %  holds: " &       %4 %   5 

(1.3)

This implies in particular that for two adjacent cells    5 

(1.4)

1.1.15 Remark: $ FRPPRQ W\SH RI ORFDO PHVK UH¿QHPHQW DOORZV WKDW WKH UH¿QHPHQW OHYHO RI two neighboring cells may differ by one (one-irregular mesh). If such a mesh is generated from a single parallelogram cell, this constant is    , where  is the dimension of the domain. If the coarse mesh consists of more than one cell,     , where  is the corresponding value for the coarse mesh and $ LV WKH PD[LPXP GLIIHUHQFH RI UH¿QHPHQW OHYHOV DFKLHYDEOH DW a single point of the coarse grid. Similar relations can be derived for cells with general tensor product shape.  'H¿QLWLRQ By %! , we denote the union of all mesh cells sharing at least part of an edge, that is  %!  !5 (1.5)  


This union explicitly includes % itself. In Figure 1.2, all the set %! of the dark grey cell % is the whole grey area. 1.1.17 Lemma: Let assumptions 1.1.13 and 1.1.14 hold for a mesh  . Then, there is a constant 8 independent of the grid cell such that the number of neighbors of a cell % does not exceed 8, that is, • •% ••% • •%! • • 8

•% • • • 5

(1.6)

1.2. SPACES AND APPROXIMATION

19

Figure 1.2: The cel % (dark grey) and the neighbor set %! (grey). Note that a common corner does not result in neighborhood.

1.2 Spaces and approximation 1.2.1 For a set ! • •  ,
 !, denotes the space of functions which are -integrable on the set !. Its norm will be denoted by • •5• •
  . If !  1, we abbreviate • •5• •  • •5• •
  and • •5• • • •5• • . Furthermore, the spaces 3 ",! are the usual Sobolev spaces of functions in
 ! with weak derivatives of order up to ( in
 ! (cf. e. g. [Ada75]). Their norms will be indexed by the space. We sometimes write " " ! instead of 3 ",!. The space ", ", 3 ",   6   is the dual space of 3 !. The norm on 3 ! is denoted  ! with 6 by 5+    5", . )XUWKHUPRUH ZH GH¿QH WKH VSDFH RI IXQFWLRQV ZLWK GLYHUJHQFH LQ
 1, namely (cf. for instance [Mon03]), " div 1 ) • •' • •
 1*   ) • •••' • •
 1• •5

(1.7)

1.2.2 Theorem (Sobolev embedding): A continuous embedding of 3 " , 1 into 3 " , 1 exists for bounded 1 • •  , if ( • •   ( • •  5 In this case, we have for any # • •3 " , 1:     #  #" ,  5 " , 

(1.8)

(1.9)

Let ! be a smooth manifold of dimension  in the closure of 1. Then, a trace of functions # • •3 " , 1 exists in 3 " , ! if ( • •   ( • •  and (1.9) holds accordingly with ! replacing 1 on the left. Proof. This result can be found for instance in [Ada75].

(1.10)

CHAPTER 1. BASICS

20

1.2.3 Lemma (Friedrichs inequality): For a function # • • 3 , 1 on Lipschitz bounded domains 1, the following two estimates hold with constants  depending on 1:       ! #   • •#  # (1.11) 2   ! !     #   • # (1.12) •  # 2

1.2.4 On a domain ! • • , we abbreviate The
 !-scalar product  " & #4 '  ) #0'0 05 

For a surface  in   ZH GH¿QH DQDORJRXVO\ 

 #4 '  ) #0'0 5 

On the sets  and

 (

DQG VR RQ DQDORJRXVO\  ZH DEEUHYLDWH DJDLQ " & " & #4 '  #4 '  )



#4 '






 

)

  #4 '  5




 'H¿QLWLRQ The broken Sobolev space 3 ",  on the triangulation  and its norm DUH GH¿QHG DV  3 7 3 ",   ' • •
 1• % • • • ) '•• • •3 ", %      • •5• •+    • •5• •+    

 'H¿QLWLRQ 6LQFH IXQFWLRQV LQ EURNHQ VSDFHV GR QRW KDYH ZHOOGH¿QHG YDOXHV RQ WKH HGJHV RI D WULDQJXODWLRQ QXPHULFDO ÀX[ RSHUDWRUV ZLOO EH GH¿QHG WKHUH 7KHLU SXUSRVH LV H[ WUDFWLQJ D XQLTXH YDOXH IURP D GRXEOH YDOXHG IXQFWLRQ 7KHVH ÀX[HV ZLOO EH GH¿QHG IRU WKH SDUWLFXODU PHWKRGV EHORZ LQ WHUPV RI PHDQ YDOXH DQG MXPS RSHUDWRUV 2Q DQ LQWHULRU HGJH EH WZHHQ WKH JULG FHOOV / and /  , let # and # be the traces of the function # from the cells / and /   UHVSHFWLYHO\ 7KHQ ZH GH¿QH IRU HDFK SRLQW 0 on an interior edge  the mean value operator •• •#•• • •0 • )

# 0  # 0 5 

Let
and
 EH WKH RXWZDUG XQLW QRUPDO YHFWRUV RI WKH WZR FHOOV 7KHQ IRU DQ\ PXOWLSOLFDWLYH operation • • ZH GH¿QH WKH jump operator #
• •#&& )
• •# 
 • •# 5

1.2. SPACES AND APPROXIMATION

21

For instance, for vector valued functions # we have, for the scalar product, the vector product and the Kronecker product, respectively, #
••#&& 
••# 
 ••# #
• •#&& 
• •# 
 • •# #
• •#&& 
• •# 
 • •# 5 $GGLWLRQDOO\ ZH GH¿QH # #&&  # • •# 5 This jump will only occur squared, such that the product does not depend on the orientation of the face normal.  'H¿QLWLRQ Given a vector 2 in the point 0 • • for any edge  not parallel to 2, we GH¿QH WKH XSZLQG ÀX[ # # 0    #0 • •K25

(1.13)

:

The GRZQZLQG ÀX[ # is the limit from the opposite side, i.e., # 0    #0  K25 :

If  is parallel to 2, we let #  #   7KLV GH¿QLWLRQ ZLOO EH FRQVLVWHQW VLQFH WKHVH ÀX[HV DUH always multiplied with a term of the form 2 ••
 . )RU D YHFWRU ¿HOG 2 on 1 WKLV GH¿QLWLRQ KDV WR EH DSSOLHG SRLQWZLVH  /HPPD 7KH PHDQ YDOXH DQG MXPS RSHUDWRUV GH¿QHG DERYH DGPLW WKH IROORZLQJ UXOHV # #'&&  # #&&•• •'•• • •• •• •#•• • •# '& • &4 •• •#'•• • • • •• •#•• • •• •'•• • ••• • •# #&•&# '&&5

(1.14) (1.15)

•• •#•• • • • •• •#••• •5 •

(1.16)

Furthermore,

Proof. elementary computation.   'H¿QLWLRQ %URNHQ "  QRUP  on the space 3  4   ZH GH¿QH WKH QRUP 5 E\           # ) #  ? # 4 ? # #&&    + 




(1.17)


 

which is the discontinuous Galerkin equivalent of the 3 ,VHPLQRUP ,Q RUGHU WR REWDLQ RSWL mal estimates, the parameter ? is chosen positive and of the order of (  6 . For details on its choice, confer Lemma 2.1.5.

CHAPTER 1. BASICS

22

1.2.10 Remark: Friedrichs’ inequality holds for this norm as well by applying it cellwise, such that we have     #   # 5 (1.18)
 

Furthermore, we have the Sobolev inequality     #   # 5
 

(1.19)

Both of these inequalities hold because their cellwise versions hold. 1.2.11 The
 -projection , )
 1 • •," of a function # LV GH¿QHG E\ WKH RUWKRJRQDOLW\ relation " & # • •, #4 '   • •' • •," 5 We will denote the projection error by ;  # • •, #.

1.2.12 Lemma: Let 2 )  • •  be the scaling of a vector by the factor  9 , that is 20  0 and % a bounded domain. Then, for any function # • •3 ",%      # • •2   ." #," 5 (1.20) ,"! Proof. For any multi-index B with ••B•• ( holds    5   " 5 "  & 5 & = # • •2  = 2 0 = #0 •• 20••0   " = 5 # 5 ! 

1.2.13 Lemma: Assume that ' • •3 " ,%  and that Assumption 1.1.13 holds. Then, the
 -projection onto the space " %  on an arbitrary cell % admits the projection error estimate 



••' • •, '••),  ••% ••   ")  ••' ••" , 5

(1.21)

Proof. see [Cia78, Theorem 3.1.5] 1.2.14 Lemma: For any function # • •3 , %  with  7  7 • •, the trace estimates        # #  •   • # 2   



  

     • •#   #    

hold with 6  6  . In particular, for # •    #    •  2

•3 , , we have    •#    #

(1.22) (1.23)

(1.24)

1.2. SPACES AND APPROXIMATION

23

Proof. On the reference cell % , the estimate        #  #   4  (#   2 

holds (cf. [BS02, Theorem 1.6.6]. Since the terms on both sides scale with the same order of , estimate (1.22) holds. The second estimate follows by applying Young inequality -2  - 6  2 6 and balancing the scaling appropriately. 1.2.15 Corollary: Assume that '  3 " ,  and Assumption 1.1.13 hold. Then, the
 projection , admits the estimate   ")    '  '  , '    5 ),2 " ,

(1.25)

Proof. This result is an immediate consequence of estimates (1.21) and (1.22).  1.2.16 Lemma: For any function #  ," , the inverse estimate

    (#     #  4   

holds.

(1.26)

Proof. The estimate holds on the reference cell by norm equivalence in discrete spaces. The power of  follows again by a scaling argument.  1.2.17 Corollary: For any function #  ," , the trace estimate

holds for  7  7 .

    = #    (# 5 2

(1.27)

Proof. This corollary is an immediate consequence of lemmas 1.2.14 and 1.2.16. 1.2.18 Lemma: Let # be a polynomial either in " or in " , such that  (# : 0   :   4

(1.28)



where  $ ". If furthermore #   on an open subset of at least one of the edges of % , then #  .

CHAPTER 1. BASICS

24

Proof. First, if #  " , then for any directional derivative = #  " . Therefore, (1.28) implies that # is constant on % and #  . If #  " and   , the directional derivative = # is in "    " . Therefore, using test as functions polynomials in 0, , and separately in (1.28) implies that = #  B," 0  ," 0 , where ," is the (th Legendre polynomial on #4 & and B an arbitrary number. Integrating yields #0 5 5 5 0   B0 ," 0  ," 0   5 Analogous results are obtained for the remaining components, such that # is constant if (   or, if (    #0 5 5 5 0   B 0  5

Finally, if this polynomial is zero on an open subset of the boundary of % , it is zero everywhere on % .

 *HQHULF ¿QLWH HOHPHQW DQDO\VLV 1.3.1 &RQYHUJHQFH DQDO\VLV RI ¿QLWH HOHPHQW GLVFUHWL]DWLRQV XVXDOO\ IROORZV D ZHOOHVWDEOLVKHG path. Since most of the proofs in later chapters follow this path, we present these techniques here in an abstract way. Later we will only refer to the steps of this outline. 1.3.2 Let -54 5 be a bounded bilinear form on the space . and  a bounded linear functional on . . Then, the generic differential equation subject to discretization will be written in the weak form -#4 '   ' '  .5

(1.29)

We assume that this equation is uniquely solvable and we will call # the continuous solution or solution to the continuous problem.  5HPDUN Since we will use this analysis in the context of discontinuous methods, we assume that boundary conditions are imposed weakly in the forms -54 5 and  ; we will not treat them separately in the following paragraphs. 1.3.4 On a sequence of meshes   ZH LQWURGXFH ¿QLWH GLPHQVLRQDO VXEVSDFHV . and the EL OLQHDU IRUPV - 54 5 and  . These forms are assumed to be consistent in the following sense: if # is the solution to (1.29), then for any : - #4 '   ' '  .  .5

(1.30)

Furthermore, the discrete solution is the function #  . , such that - # 4 '   '

'  . 5

(1.31)

1.3. GENERIC FINITE ELEMENT ANALYSIS

25

 1.3.5 Assumption: There exists a norm 5 on . called energy norm, such that the stability estimate   '   - '4 ' '  . 4 (1.32) 

holds with a constant  independent . If - 54 5 LV D V\PPHWULF SRVLWLYH GH¿QLWH ELOLQHDU   of4 form, then a typical choice is '   - '4 ', such that (1.32) holds trivially with +  ' and   . 1.3.6 Lemma (Lax-Milgram): Let -54 5 be a bilinear form on a Hilbert space . such that for a constant  and #4 '  . holds     -#4 '  # '  &

  -#4 '   #

  &    ' !& &   -#4 '   '  4

  &  & #

&

&

then, for any linear functional  on . there is a unique solution # to -#4 '   ' and     #      5 & &

This version of the Lax-Milgram lemma can for instance be found in [BF91]. 1.3.7 Assumption: There is a projection , ) .  . admitting the projection error estimate     '  , '    '   4 (1.33)  + 

,

for any '  3  . The projection error of # will be abbreviated by ; ) #  , #. Typically, with second order elliptic problems, and discretization with polynomials of degree ( this estimate holds for   (  

1.3.8 Assumption: If #  3 ,   is the solution of (1.29), then the projection error ;  #  , # and the bilinear form - 54 5 admit the estimate     - ; 4 '   #+   '  5 '  . (1.34)

1.3.9 Theorem: Let #  3 ,   be a solution to (1.30) and #  . be a solution to (1.31) and   #  #  )XUWKHUPRUH OHW DVVXPSWLRQV  WR  EH IXO¿OOHG WKHQ         #  5 (1.35) + 

CHAPTER 1. BASICS

26 Proof. First, we apply triangle inequality,       #  #   #  , #  , #  #  4   

(1.36)

DQG HVWLPDWH WKH ¿UVW WHUP E\   )RU WKH VHFRQG ZH VXEWUDFW HTXDWLRQV  DQG   yielding the Galerkin orthogonality relation - 4 '  

'  . 5

(1.37)

    /HW QRZ +  . with +   , ##  EH D IXQFWLRQ VXFK WKDW WKH VWDELOLW\ HVWLPDWH  holds for '  , #  #  7KHQ *DOHUNLQ RUWKRJRQDOLW\ DQG $VVXPSWLRQ  \LHOG   , #  #   - , #  # 4 +



 - ; 4 +      , #  #   #+  5

(QWHULQJ WKLV LQ  FRQFOXGHV WKH SURRI RI WKH WKHRUHP

1.3.10 (VWLPDWHV RI WKH
 QRUP DQG RI ZHDNHU QRUPV RI WKH HUURU DUH FRQGXFWHG XVLQJ WKH GXDOLW\ DUJXPHQW LQWURGXFHG E\ $XELQ DQG 1LWVFKH 7KLV DUJXPHQW UHTXLUHV DGGLWLRQDO DVVXPS WLRQV 1.3.11 Assumption: Let  
 and let for arbitrary   3 , 1 the solution of the dual problem

admit the estimate

" & -<4   4 < <  .5

(1.39)

      
     5 +  + 



In the case   , this assumption is often referred to as HOOLSWLF UHJXODULW\. In the case of Poisson’s equation (2.1) D VXI¿FLHQW FRQGLWLRQ IRU  is for instance that -  3 , 1, /  and every corner of the domain is convex. More detailed analysis can be found in [Gri85]. 1.3.12 Assumption: If  3  ,   is the solution of (1.29) and '  . , then the bilinear form - 54 5 admits the estimate     (1.41) - '4 ;   '     
 5 

+



1.3.13 Assumption: In extension of Assumption 1.3.7, the estimate     - #  , #4  ,    #    # 
 +

for functions #  3 ,   and  3  ,  .



+



4

(1.42)

1.4. ADVECTION PROBLEMS

27

1.3.14 Theorem: Let Theorem 1.3.9 hold and  
 . If furthermore Assumptions 1.3.11    1, admits the estimate to 1.3.13 hold, the 3 , 1-norm of the error, where 1        
    #  5 (1.43) +  + 

Proof. Let . be the space of functions <  3 , 1 with <0   almost everywhere on  %\ GH¿QLWLRQ RI WKH QHJDWLYH QRUPV IRU  -  we have to estimate 1  1 & "    4 <  
      5 (1.44) +    1& < +   

  instead of the supremum. Here G1  is the characteristic In the case    ZH WDNH <  G1  function of the set 1: " &      1       4 G  (1.45)
     5


8VLQJ WKH GXDO SUREOHP  DQG FRQVLVWHQF\   ZH ZULWH " &  4 <  -  4   -  4 ;   - , #  # 4 ;   - ; 4 ; 5 'XH WR HVWLPDWH   ZH KDYH

    - , #  # 4 ;   , #  #    + 
  4

ZKLFK WRJHWKHU ZLWK  DQG  \LHOGV   " &  4 <   #

+  

      
 5 + 

,Q YLUWXH RI WKH GXDO UHJXODULW\ HVWLPDWH  ZH FRQWLQXH    " &   4 <    #+   <+   5

(QWHULQJ WKLV UHVXOW LQWR  DQG   UHVSHFWLYHO\ \LHOGV WKH UHVXOW RI WKH WKHRUHP 1.3.15 Remark: The powers of  involved in this section were adapted to second order probOHPV ZLWK IXOO UHJXODULW\ 7KH VDPH PHFKDQLVP LV DSSOLFDEOH ZLWK GLIIHUHQW -dependence. If the powers of  LQ HVWLPDWHV  DQG  DQG WKH JDLQ RI UHJXODULW\ LQ  DUH GLIIHUHQW LW LV REYLRXV KRZ WR FKDQJH WKH UHVXOWV RI WKH WKHRUHPV WR DFFRXQW IRU WKH QHZ VLWXDWLRQ $Q example is Section 1.4 on advection problems below.

1.4 Advection Problems 1.4.1 7KH RULJLQ RI GLVFRQWLQXRXV *DOHUNLQ PHWKRGV OLHV LQ WKH GLVFUHWL]DWLRQ RI 1HXWURQ WUDQV SRUW SUREOHPV LQYHVWLJDWHG E\ /H6DLQW DQG 5DYLDUW >/5@ :H ZLOO GLVFXVV WKLV VFKHPH LQ D VOLJKWO\ JHQHUDOL]HG IRUP LQ WKH IROORZLQJ SDUDJUDSKV

CHAPTER 1. BASICS

28

1.4.2 We consider the following stationary linear advection problem as a prototype for hyperbolic problems: Let 2 
 1*   EH D YHFWRU ¿HOG ZLWK ( 2 
 1 and ( 20   DOPRVW HYHU\ZKHUH 7KHQ ¿QG # such that 2 (#  # 

in 1 on / 4

(1.46)

where the LQÀRZ ERXQGDU\ / is given by  3 7 /  0  =1 
0 2 7  5

(1.47)

1.4.3 The discontinuous Galerkin formulation of (1.46) is constructed by multiplying with a test function ' and integrating by parts on each cell % of a triangulation  , yielding " & " &

  #4 ( 2'   #4 2
' 2  4 '  5 (1.48) 7KLV IRUP LV ZHOOGH¿QHG LQ % , if # and ' belong to the space  6  .  ' 
 1%   ) 2 (' 
 %  4 1.4.4 6LPSOLI\LQJ WKH QRWDWLRQ ZH GH¿QH WKH IRUP



 #4 ' * ) #4 ' 2
  4

(1.49)

for any edge  of the triangulation and on sets of edges  and so on, accordingly. Here,
LV DQ\ RI WKH WZR QRUPDO YHFWRUV VLQFH FKDQJH RI RULHQWDWLRQ GRHV QRW FKDQJH WKH GH¿QLWLRQ Considering ( 2#  #( 2  2 (#, equation (1.48) becomes " & " &



 " &  #4 2 ('   ( 2#4 '   #4 ' *2
 #4 ' *2
 4 '  5 1.4.5 The traces of the function # DUH QRW ZHOOGH¿QHG RQ LQWHULRU HGJHV VLQFH WKH YDOXHV IURP both adjacent cells are not required to coincide. Therefore, the trace of # is replaced by the QXPHULFDO ÀX[ # 7KH ÀX[ FKRVHQ KHUH LV WKH XSZLQG ÀX[ #  # RI 'H¿QLWLRQ  7KXV WKH GLVFRQWLQXRXV IRUPXODWLRQ LV LQWHJUDWLQJ E\ SDUWV DJDLQ  ¿QG #  . such that   " & " &

E #4 '  2 (#4 '   #  # 4 '  *
 #4 '  *





 " &  4 '  # 4 '  *



where        & / and / is the LQÀRZ ERXQGDU\  3 7 /  0  =12
0 7  5

'  .4 (1.50)

" & 1.4.6 Remark: Heuristically, the form E 54 5 is constructed in such a way that the value of # DW WKH LQÀRZ ERXQGDU\ RI HDFK FHOO LV ZHDNO\ SUHVFULEHG WR EH HTXDO WR WKH YDOXH LQ WKH XSVWUHDP cell.

1.4. ADVECTION PROBLEMS

29

" & 1.4.7 Lemma: The form E 54 5 LV SRVLWLYH VHPLGH¿QLWH RQ 3 ,  . Proof. Following [LR74], we integrate by parts to obtain

yielding

"

2 (#4 #

&





 " & " &   #4 2 (#   ( 2 #4 #   #4 # *2
 #4 # *2
4

" & " &



  2 (#4 #    ( 2 #4 #   #4 # *2
 #4 # *2
5

(1.51)

Summing up over all cells % , we get

" &   " &

 2 (#4 #    ( 2 #4 #   # 4 # *


 # 4 # *


5











By entering this result into (1.50), we obtain

  " & " &

E #4 #   ( 2 #4 #   # 4 # *


 # 4 # *










   #  # 4 '  *
  # 4 '  *

 

" & 

  ( 2 #4 #   # 4 # *

 # 4 # *

 # #&&4 # #&& *
  (1.52)







" &  'H¿QLWLRQ The energy norm for the form E 54 5 is

8 "   & # ) E #4 #  .. 2 (#. 5 

8

(1.53)

7KLV WHUP LV D VHPLQRUP GXH WR WKH SUHFHGLQJ OHPPD DQG EHFRPHV GH¿QLWH RQ WKH VSDFH . by the additional term. Due to (1.52), we have  

   #   # 4 #
  # 4 #
  # #&&4 # #&&  .. 2 (#. 

*


*


*


8   

1.4.9 7KH GLVFRQWLQXRXV *DOHUNLQ GLVFUHWL]DWLRQ RI  LV DFKLHYHG ¿QDOO\ E\ UHSODFLQJ WKH spaces . E\ D ¿QLWH HOHPHQW VXEVSDFH .  ," .

1.4.10 Remark: The following stability result is a special case of a result in [GK03c] which will be generalized to advection-diffusion-reaction problems in Lemma 5.1.5 on page 108. A more general result including
 -norms (   7 ) was obtained earlier in [JP86] by a technique using a global exponential function decaying in direction 2.

CHAPTER 1. BASICS

30

1.4.11 Lemma: ,I WKH YHFWRU ¿HOG E is constant on each mesh cell, there exists constants  and M independent of the mesh size such that for any #  . the following stability estimate holds:   " & #   E # 4 #  M2 (# 5 (1.54) 8 Proof. We omit the index  of # for simplicity and start with   " & " & E #4 #  M2 (#  E #4 #  M . 2 (#

 

 M #  # 4  2 (# *
 M # 4  2 (# *

(1.55)



Applying Cauchy-Schwarz and Young inequalities to the boundary terms, and the trace esti  mate (1.27) to 2 (#  , we obtain     " & " & E #4 #  M2 (#  E #4 #  M . 2 (#  JM . 2 (#





 ! M  # #&&4 # #&& *
 #4 # *

5 



J

We can now set J  6 and use (1.52) to choose M to obtain (1.54) with  independent of . 1.4.12 Remark: 7KLV OHPPD FDQ EH H[WHQGHG WR VPRRWK YHFWRU ¿HOGV E by using a suitable SURMHFWLRQ LQ WKH GH¿QLWLRQ RI WKH WHVW IXQFWLRQ LQ   VWLOO WKH QDWXUH RI WKLV H[WHQVLRQ LV QRW completely clear, since it must break down in the case that E has closed integral curves (see the remark at the end of this chapter). 1.4.13 Theorem: Let #  3 ,1 be the solution to (1.46) with 6    (   and #  .  ," EH WKH ¿QLWH HOHPHQW VROXWLRQ WR (1.50). Then, the error   #  # admits the estimate          # 5 (1.56) , 8

Proof. The proofs follows the generic Theorem 1.3.9. Stability Assumption 1.3.5 was proven in Lemma 1.4.11. Therefore, we begin by proving approximation Assumption 1.3.7, in this case with the approximation order . . From the projection estimate (1.21), we infer for the projection error ;  #  , # that       ;   (;    # 5 ,

Applying trace inequality (1.24), we deduce     ;     # 5 , 8

Reverting to the error  , we use triangle inequality to obtain          ;   ,   5 8 8 8

(1.57)

1.4. ADVECTION PROBLEMS

31

7KH ¿UVW WHUP LV DOUHDG\ FRYHUHG E\ WKH SUHYLRXV DSSUR[LPDWLRQ UHVXOW 6LQFH ,   . , we XVH /HPPD  DQG *DOHUNLQ RUWKRJRQDOLW\ \LHOGLQJ   " & ,    E ,  4 ,   M2 (,   8 " &  E ; 4 ,   M2 (,   5

:H HVWLPDWH WKLV WHUP XVLQJ WKH IROORZLQJ DUJXPHQW XVLQJ LQWHJUDWLRQ E\ SDUWV \LHOGV " & " & " & E '4 +  '4 2 (+   ( 2 '4 + 





  ' 4 +  +  *
 ' 4 + *

5 (1.58)



7KHUHIRUH DEEUHYLDWLQJ @  ,  ,   " & " &       E ; 4 @  M2 (@    .2.   ;    @   ; 
# @&&4 # @&& *






          ; 

@ 4 @ *

 M   2 (;    2 (@ 



  



   M # #&&4 # #&& *
 2 (@ 
 M # #&&4 # #&& *

 2 (@ 







         ;     ;   ;  8








!  @  5 8

)LQDOO\ WKH UHVXOW IROORZV E\ DSSO\LQJ WUDFH HVWLPDWH     LQYHUVH HVWLPDWH DQG DSSUR[LPDWLRQ /HPPD  WR WKH ¿UVW WHUP DQG GLYLGLQJ E\ @ 8 . 1.4.14 Corollary: Under the assumptions of Theorem 1.4.13, the
 -norm of the error  admits the estimate          # 5 (1.59) ,

1.4.15 Remark: )RU VROXWLRQV #  3 " , 1 HVWLPDWH  LV RSWLPDO ZLWK UHVSHFW WR WKH DSSUR[LPDWLRQ UHVXOW   ZKLOH  LV VXERSWLPDO E\ .  )RU &DUWHVLDQ JULGV ZH KDYH WKH IROORZLQJ VXSHUFRQYHUJHQFH UHVXOW GXH WR /H6DLQW DQG 5DYLDUW >/5@ 1.4.16 Theorem: Let the triangulation  consist of rectangles only and let 2 be constant in  1. Additionally, .  ," and the continuous solution #  3 " ,1 ! 3 " ,1. Then, the error  admits the estimate        "  # 
  +  1.4.17 $ VROXWLRQ WR  FDQ EH FRQVWUXFWHG E\ FRQVLGHULQJ WKH VHW RI characteristic curves, ZKLFK DUH WKH LQWHJUDO FXUYHV RI WKH YHFWRU ¿HOG 2 /HW  EH VXFK D FXUYH 7KHQ  #5  /HW  EH WKH ORZHU OLPLW RI DOO  such that   1 and  WKH OHDVW XSSHU ERXQG RI WKRVH . Assume that  LV /LSVFKLW] FRQWLQXRXV RQ # 4  & and # 4  & ! 1 LV VLPSO\ FRQQHFWHG 7KHQ # FDQ EH REWDLQHG DV WKH VROXWLRQ RI WKH RUGLQDU\ GLIIHUHQWLDO HTXDWLRQ 2 (# 

#   4 ZLWK VWDUW YDOXH #   #  .



CHAPTER 1. BASICS

32

1.4.18 Remark: ,I WKH VHW RI LQWHJUDO FXUYHV RI WKH YHFWRU ¿HOG 2 FRQWDLQV FORVHG ORRSV WKHQ  KDV D VROXWLRQ LI DQG RQO\ LI IRU DQ\ VXFK ORRS  0:  :  5 7KLV IROORZV LPPHGLDWHO\ E\ VROYLQJ  RQ VXFK D ORRS ZLWK    %  7KHUHIRUH D VWDELOLW\ UHVXOW FDQQRW EH REWDLQHG IRU WKLV FDVH 1.4.19 Remark: ,I WKH YHFWRU ¿HOG 2 does not have closed integral curves and the mesh is VXI¿FLHQWO\ ¿QH WKH JULG FHOOV FDQ EH RUGHUHG LQ VXFK D ZD\ WKDW WKH GLVFUHWH OLQHDU V\VWHP FDQ EH VROYHG FHOO E\ FHOO IURP LQÀRZ WR RXWÀRZ ERXQGDU\ FI >-3@ 

Chapter 2 Linear Diffusion I This chapter treats Poisson equation in its primal form. It sets out from Nitsche’s method for ZHDNO\ LPSRVHG 'LULFKOHW ERXQGDU\ FRQGLWLRQV VHH 1LWVFKH >1LW@ WR GH¿QH WKH LQWHULRU penalty method (see Arnold [Arn82]). The standard energy norm and
 error analysis of these methods is quoted from the cited articles. We add a new study of the behavior of the discretization depending on the stabilization parameter. Then, the
 analysis by Kanschat and Rannacher (cf. [KR02]) is presented, followed by a posteriori error estimates from the same source. Finally, we compare continuous and discontinuous methods.

2.0.1 In this chapter, we will investigate discontinuous Galerkin discretizations of Poisson’s equation " & ( D(#   in 1 (2.1) #  #

on /



on /

= #  L 

(2.2)

(2.3)

where D 
1 with D0  D 9  almost everywhere in 1. / and / are the Dirichlet  " /  . and Neumann parts of /  =1, respectively. We have / ! /  and /  / 2.0.2 The presentation will focus on the generic equation +#  4

(2.4)

under boundary conditions (2.2) and (2.3). Results for the more general problem (2.1) are UHFRYHUHG E\ VWDQGDUG DQDO\WLFDO WHFKQLTXHV XVXDOO\ UHTXLULQJ WKDW WKH FRHI¿FLHQW IXO¿OOV D  D 9  DQG LV VXI¿FLHQWO\ VPRRWK 2.0.3 Theorem: Equation (2.4) has a unique solution #  3 , 1 with trace  on / , provided   3 , 1 (cf.[GT98]). ,I WKH ERXQGDU\ RI WKH GRPDLQ LV VXI¿FLHQWO\ UHJXODU DQG   3 5, 1 with B 9 , then #  3  5, 1 and (see [Gri85] for details)     # 
    
 5 (2.5) +  + 

CHAPTER 2. LINEAR DIFFUSION I

34

In particular, if all corners and edges of the domain are convex and  
 1, then #  3 , 1 and (2.6) holds with B  , namely   #

+  

    
  5

(2.6)

We will refer to this last estimate as elliptic regularity.

2.1 Weak boundary conditions 2.1.1 In [Nit71], Nitsche proposed a fully conforming method of treating Dirichlet boundary values in weak form . The weak formulation for Nitsche’s method is " & " &





  ? #4 ' 2  = #4 ' 2  #4 = ' 2 -

 #4 '  (#4 ('



   '  4 '  ? # 4 ' 2  # 4 = ' 2 4 (2.7)

where ? 9  is a function constant on each edge     achieving stability of the form as well as penalizing violation of the boundary condition #  # . Suitable values for ? will be  determined below. The term  = #4 ' 2 achieves conformity of the method since it eliminates

 the natural boundary condition.  #4 = ' 2 was introduced to symmetrize the operator and

 has to be matched by  # 4 = ' 2 on the right hand side to maintain conformity. Since this method exhibits the essential properties of the interior penalty method below, we will summarize the most important analytical results. 2.1.2 :H GH¿QH WKH HQHUJ\ QRUP IRU  DV 8   

 # ) #  ? #4 # 2 5 ,

(2.8)

7KLV IRUP LV REYLRXVO\ QRQQHJDWLYH 7KH GH¿QLWHQHVV LV HVWDEOLVKHG E\ WKH IROORZLQJ OHPPD 2.1.3 Lemma: Let ? 
 =1 and ?  ? 9  almost  everywhere on =1. Then, for a function #  3 , 1, #   implies #  . Therefore, 5 is in fact a norm on 3 , 1.

  Proof. First, (#   implies that #    in 1 (cf. e. g. [GT98]). Since ? 9 , 

? #4 # 2   LV SRVLWLYH GH¿QLWH RQ WKH VSDFH RI IXQFWLRQV LQ
 =1. Since this space includes the traces of functions in 3 , 1, the trace of # must be zero and consequently # itself.

2.1.4 Discretization of (2.7) is achieved by choosing a mesh  on 1 and restricting test and  trial functions to the space .  ," !  1. 

For simplicity of the presentation, we will assume that 


2.1. WEAK BOUNDARY CONDITIONS 2.1.5 Lemma: Assume that ? in (2.7) is chosen such that ?

4 ? 9  4 ?  

35

(2.9)

where ? is independent of  and  is the length of the cell % orthogonal to the face .  is the constant in the trace estimate (cf. corollary 1.2.17)      =      ( 4 2 2

for all mesh cells % adjacent to the boundary =1 and for all polynomials   " % . Then, the stability estimate   " & #  - #4 # 4 (2.10) 

holds for all #  . with a positive constant  independent of the mesh size and of the actual value of ? .

Proof. Application of Young inequality with  7 B   6? 7  and the trace estimate (1.23) yields

 " &       ? #4 # -

  = #4 # 2  #4 #  (# 2 0  + B        = #  #2 2  #  2 2   B   

   #          #  B (#  B ? # 2      B # 5

2.1.6 Theorem: Assume that the solution # of (2.1) is in 3 , 1 with   . Let .  ," with (    . If ? is chosen according to (2.9), then the error   #  # admits the estimate         #  5 (2.11)

+  Proof. ,Q YLHZ RI 7KHRUHP  LW LV VXI¿FLHQW WR HVWLPDWH WKH HUURU & #  # . By the stability estimate (2.10), we have   " & & #  #   -

 ; 4 & #  #  #    #   (; (& #  #    ? ; 2  ? & #  # 2          = ; 2 & #  # 2  ; 2 = & #  # 2      !   #   & #  #    (;    ? ; 2  ?  = ; 2 5  Applying approximation Lemma 1.2.13 (and ?  6) yields     & #  #    #  5

+ 

CHAPTER 2. LINEAR DIFFUSION I

36

2.1.7 Remark: Obviously, the analysis above extends to the situation where Dirichlet boundary conditions are imposed on an open subset / & =1 only.

2.2 The Interior Penalty Method 2.2.1 Applying Nitsche’s method of weakly imposed boundary conditions to each grid cell and and averaging the test functions over neighboring cells yields the interior penalty method (IP):

 " & " &

 -IP  #4 '  (#4 ('   ? # #&&4 # '&&
 ? #4 '








 

(#

4 # '
&&
 # #
&&4

('


 = #4 '
  #4 = '






  " &  4 '  ? # 4 '
  # 4 = '
 5 (2.12)



" & 2.2.2 Remark: The form -IP  54 5 GH¿QHG LQ  GLIIHUV IURP WKH GH¿QLWLRQ IRXQG LQ WKH majority of publications (e.g. [Arn82]) in the stabilization parameter on the boundary. The analysis below shows, that the stabilization in our version is indeed more equilibrated (see also [HL02]). See paragraph 2.2.9 on page 40 for detailed results.  'H¿QLWLRQ The energy norm for (2.12) —analogously to (2.8)— is      #   # # ) #   ? # #&&
  ? #
 5 , 



(2.13)



7KH GH¿QLWHQHVV RI WKLV QRUP IROORZV ZLWK )ULHGULFKV LQHTXDOLW\ LQ /HPPD  E\ WKH VDPH arguments as in the proof of Lemma 2.1.3. 2.2.4 We discretize the differential equation by choosing the space .  ," with polynomials in " RQ DI¿QH FHOOV RU " on each cell and (  . 2.2.5 Lemma: If in (2.12) for each edge between cells % and %  ? 

 ?   ? 4 

?  9

 4 

(2.14)

with positive B 7  and  is the constant from the trace estimate (1.27) in corollary 1.2.17 on page 23 for the adjacent cells % and %  , respectively. Then, the stability estimate   " & '   -IP '4 ' 4  

(2.15)

holds for all '  . with a constant  independent of the mesh size and of ?. Proof. We start with " &  



   -IP  '4 '  '   

('

4 # '
&&
  = '4 '
 5



2.2. THE INTERIOR PENALTY METHOD

37

7KH LQGH¿QLWH WHUP LV HVWLPDWHG E\ FRQVLGHULQJ LW FHOO E\ FHOO



 

('

4 # '
&&

   = '4 '




0       + B       = '   # '&&  B = '    '     2 2 2 2    B  B  

 +  #    #  0  ('    ?##'&&   ?'  B  B'  5 2 2    

7KHUHIRUH WKH OHPPD LV SURYHQ ZLWK     B.

2.2.6 $GGLWLRQDOO\ ZH LQWURGXFH WKH IROORZLQJ ³H[WHQGHG´ HQHUJ\ QRUP   '  ) 

) %+       0 . ('   ? # '&&   

= '

 4  2 2  ?  

(2.16)



ZKLFK LV HTXLYDOHQW WR WKH HQHUJ\ QRUP RQ . E\  DQG WKH SUHYLRXV OHPPD

2.2.7 Lemma: For functions #  3 ,  with   (  , the interior penalty norm admits the projection error estimate     ;    # 5 (2.17)  ,

Proof. 7KLV OHPPD LV DQ LPPHGLDWH FRQVHTXHQFH RI DSSUR[LPDWLRQ UHVXOW  DQG WUDFH HVWL PDWH  " & 2.2.8 If ? LV FKRVHQ WRR VPDOO WKH GH¿QLWHQHVV RI WKH IRUP -IP  54 5 LV ORVW 7KLV LV UHÀHFWHG LQ the fact that the smallest solution Amin RI WKH HLJHQYDOXH SUREOHP & " & " -IP '  . 4 (2.18)  # 4 '  A #4 '

EHFRPHV GHSHQGHQW RI ? DQG QHJDWLYH ,Q )LJXUH  ZH VKRZ WKH GHSHQGHQF\ RI Amin on ? for WKH &DUWHVLDQ PHVKHV RI H[DPSOH $ RQ SDJH  7KH YDOXHV DUH WKH PLQLPDO HLJHQYDOXHV RI WKH WULGLDJRQDO /DQF]RV PDWUL[ JHQHUDWHG E\ WKH FRQMXJDWH JUDGLHQW PHWKRG 7KH PHVK ZLGWK IRU HDFK SRO\QRPLDO GHJUHH ZDV FKRVHQ VXFK WKDW WKH .U\ORY VSDFH KDG D GLPHQVLRQ RI DW OHDVW  WR HQVXUH VXI¿FLHQW DSSUR[LPDWLRQ RI WKH GLVFUHWH HLJHQYDOXHV 7KH JUDSKV VKRZ WKDW ZKHQHYHU ? EHFRPHV ORZHU WKDQ D FHUWDLQ WKUHVKROG GHSHQGLQJ RQ WKH SRO\QRPLDO GHJUHH VWDELOLW\ LV ORVW DOPRVW LQVWDQWO\ WKH VWHS VL]H LQ ? is   :KHQHYHU ? LV ODUJHU WKDQ WKLV WKUHVKROG WKH smallest eigenvalue of - 54 5 LV FRQVWDQW +HUH WKH SRO\QRPLDO VSDFH  LV DQ H[FHSWLRQ WKH VPDOOHVW HLJHQYDOXH LQFUHDVHV VORZO\ VLQFH WKH GHJUHHV RI IUHHGRP GR QRW DOORZ WKDW WKH MXPSV RI WKH FRUUHVSRQGLQJ HLJHQIXQFWLRQ WHQG WR ]HUR )LJXUH  VKRZV WKH FRUUHVSRQGLQJ HLJHQIXQFWLRQ IRU ? MXVW EHORZ WKH VWDELOLW\ UHTXLUHPHQW RQ WKH OHIW DQG IRU WKH VWDEOH IRUP RQ WKH ULJKW :KLOH WKH VWDEOH YHUVLRQ VKRZV DSSUR[LPDWLRQ WR WKH continuous eigenfunction  I0  I WKH XQVWDEOH RQH VKRZV RVFLOODWLRQV ZLWK WKH KLJKHVW IUHTXHQF\ XVXDOO\ REVHUYHG IRU WKH ODUJHVW HLJHQYDOXH 7KH LQGH¿QLWH MXPS WHUPV VHUYH KHUH WR RXWZHLJK WKH ODUJH JUDGLHQWV LQVLGH WKH FHOOV

CHAPTER 2. LINEAR DIFFUSION I

38

21

P1 P2 P3 P4 P5

A 


20.5

20

19.5

19 1

3

6

10

15

20

15

20

?

21

A 


20.5

Q1 Q2 Q3 Q4 Q5

20

19.5

19 1

3

6

10 ?

Figure 2.1: Smallest eigenvalue of - 54 5 on a mesh of squares

2.2. THE INTERIOR PENALTY METHOD

39

1 0.5 -1

-0.5

0 0

0.5

-0.5 1 -1

1 0.5 -1

-0.5

0 0

0.5

-0.5 1 -1

Figure 2.2: Eigenfunction to the smallest eigenvalue of - 54 5 for ?  5 (left) and ?  5 (right) on a mesh of squares

CHAPTER 2. LINEAR DIFFUSION I

40 H 1.0 1.5 2.0 4.0

 1.34 1.08 1.01 1.00

 4.43 3.30 3.00 2.98

     9.46 16.41 25.39 1.35 4.44 6.76 11.47 17.50 1.08 3.30 6.00 9.97 14.96 1.01 3.01 5.97 9.85 14.78 1.00 2.99

   9.47 16.49 25.50 6.77 11.51 17.55 6.01 10.00 15.00 5.98 9.86 14.83

Table 2.1: The lowest stable ? depending on the boundary stabilization on an equidistant square grid

2.2.9 We already remarked in 2.2.2 that there is some inconsistency of handling stabilization at the boundary in the literature. We investigate this fact by introducing the additional parameter H and the generalized form of equation (2.12)



 " & (#4 ('   ? # #&&4 # '&&
 ? # #&&4 # '&&










 

(#

4 # '
&&
 # #
&&4

('


 = #4 '
  #4 = '






" &



  4 '  H? # 4 '
  # 4 =
'
 5 (2.19)



Table 2.1 shows, that the minimal value of ? yielding a stable bilinear form depends on H, as long as H 7 . For H   this threshold value remains nearly constant. We conclude, that for H 7 , the grid cells causing the instability are located at the boundary. Therefore, all remaining examples in this chapter will be computed with the form (2.12), where H  .

2.2.10 Theorem: Let ? be chosen according to (2.14). If the solution # of (2.4) is in 3 ,  with  7   (  , then the error   #  # between # and the solution # of (2.12) admits the estimate         # 5 (2.20)  ,

Proof. Again, we follow the outline of Theorem 1.3.9 and estimate the error remaining after projection. The arguments are similar to the proof of Theorem 2.1.6, but we can use the cellwise
 -projection , . We abbreviate @  , #   and estimate   " & " & IP @   -IP  @ 4 @  - ; 4 @ "  &

  (; 4 (@     ? # ; & 4 # @ &
   ? ; 4 @
 





 

 

(;

4 # @ &
   (; 4 @
 





  # ; & 4

(@


   ; 4 (@
 5



%\ +ŽROGHU LQHTXDOLW\ DQG DSSUR[LPDWLRQ UHVXOWV  DQG   ZH REWDLQ IRU WKH ¿UVW WHUPV         & "  (; 4 (@    (;  @    #, @ 



       #

  @   ? # ; & 4 # @ &    ? ;  @    #







,



Here and in the remaining estimates, the boundary terms are treated like the interior terms. We have         



(;

4 # @ &
  ? 

(;


? # @ & 4 # @ &
  #, @  4







2.2. THE INTERIOR PENALTY METHOD

41

again by approximation result (1.25) and since ?   . Finally,

      #  

= @ 4 = @
  #, @  4  # ; & 4

(@


   ? # ; & 




?

by estimate (1.25) and inverse estimate (1.26). Summing up and dividing by  approximation  @  yields the result of the theorem. 

2.2.11 Remark: In the last theorem we assumed #  3 ,   to keep the presentation simple. In fact, in [HN01] a technique using weighted norms is presented which allows for solutions with less regularity. This technique applies to the energy estimate above as well as to weaker and pointwise estimates below. Especially in the last case, a combination of weights balancing the singularities of the solution and of the Green function would render the presentation completely unreadable.

2.2.12 Theorem: Assume that #  3 ,   and (2.6) holds for any solution of (2.4) with arbitrary right hand side  
 1. Then, we have the
 -error estimate         #  5 (2.21) + 

Proof. The proof follows the lines of 1.3.14 with the concrete elliptic regularity estimate (2.6) entering in place of (1.40). 2.2.13 :H SUHVHQW H[SHULPHQWDO FRQ¿UPDWLRQ RI WKH UHVXOWV DERYH IRU " shape functions in Table 2.2. We solve example A.2.2 on page 170 on the Cartesian (see example A.1.1 on page 167) and distorted meshes (see example A.1.3), respectively. The table shows the
 -norm and the 3 ,  -norm of the error   #  # , as well as the experimental order of convergence determined by the formula ord
   

err
5 err


On both types of meshes, the results are in good correspondence with the theory presented above. 2.2.14 We perform the same experiments with shape function spaces " in Table 2.3. On Cartesian grids, we obtain the same orders of convergence as in Table 2.2, even if the
 -errors of the  -element are considerably larger than those of the  -element. 6LQFH WKH GLVWRUWHG PHVKHV FRQVLVW RI QRQDI¿QH JULG FHOOV WKH " -elements suffer from a lack of approximation (cf. [ABF02]). This is clearly visible in the reduced approximation orders. Therefore, " -elements should be avoided on grids not consisting of parallelogram cells. 2.2.15 The situation of the distorted grid in the previous paragraph occurs if the computational JULG LV WKH UHVXOW RI D JULG JHQHUDWRU ,I WKH JULG LV JHQHUDWHG E\ FRQVHFXWLYH UH¿QHPHQW RI D coarse mesh, grid cells approximate parallelograms on coarser meshes. We investigate behavior in this case discretizing on the circle 1    (for meshes see example A.1.6 on page 170). The solution is again the exponential function. Table 2.4 shows that on such a sequence of grids the  -element converges again with the same order as the  -element.

CHAPTER 2. LINEAR DIFFUSION I

42    




err

2 3 4 5 6 7

1.64e-2 4.38e-3 1.15e-3 2.96e-4 7.53e-5 1.90e-5

2 3 4 5 6 7

1.87e-02 5.26e-03 1.43e-03 3.79e-04 9.89e-05 2.55e-05

        ,

ord err ord err  , ?   Cartesian mesh 1.85 3.56e-1 0.99 1.98e-05 1.90 1.73e-1 1.04 1.39e-06 1.93 8.45e-2 1.04 9.16e-08 1.96 4.16e-2 1.02 5.88e-09 1.97 2.06e-2 1.01 3.86e-10 1.99 1.02e-2 1.01 8.17e-11 distorted mesh  , ?  $ 1.74 3.23e-01 0.97 4.31e-04 1.83 1.66e-01 0.97 6.12e-05 1.88 8.46e-02 0.97 8.45e-06 1.91 4.30e-02 0.98 1.14e-06 1.94 2.17e-02 0.98 1.51e-07 1.96 1.10e-02 0.99 1.96e-08

    ,

ord err ord  , ?  $ 2.99 3.00 3.00 3.00 3.00 3.00

1.15e-2 2.88e-3 7.20e-4 1.80e-4 4.50e-5 1.13e-5

1.99 2.00 2.00 2.00 2.00 2.00

 , ?  ' 2.74 1.22e-02 2.82 3.30e-03 2.86 8.75e-04 2.89 2.29e-04 2.92 5.90e-05 2.94 1.51e-05

1.87 1.89 1.91 1.94 1.95 1.97

Table 2.2: Convergence of the interior penalty method with tensor product polynomials on Cartesian and distorted meshes




2 3 4 5 6 7

2 3 4 5 6 7

                ,

,

err ord err ord err ord err ord  , ?    , ?  $ Cartesian mesh 2.24e-2 1.94 4.77e-1 0.98 8.08e-4 3.00 2.80e-2 1.97 5.83e-3 1.94 2.37e-1 1.01 9.99e-5 3.02 7.04e-3 1.99 1.50e-3 1.96 1.17e-1 1.01 1.24e-5 3.01 1.76e-3 2.00 3.83e-4 1.97 5.82e-2 1.01 1.55e-6 3.00 4.41e-4 2.00 9.70e-5 1.98 2.90e-2 1.01 1.93e-7 3.00 1.10e-4 2.00 2.44e-5 1.99 1.44e-2 1.00 2.42e-8 3.00 2.76e-5 2.00 distorted mesh  , ?  $  , ?  ' 4.71e-2 1.08 6.66e-1 0.76 3.18e-3 2.86 7.20e-2 1.77 1.73e-2 1.44 3.54e-1 0.91 3.85e-4 3.04 1.95e-2 1.88 5.27e-3 1.72 1.91e-1 0.89 5.15e-5 2.90 5.54e-3 1.82 1.53e-3 1.78 1.12e-1 0.77 8.90e-6 2.53 1.79e-3 1.63 4.81e-4 1.67 7.93e-2 0.49 1.91e-6 2.22 7.04e-4 1.35 1.74e-4 1.46 6.74e-2 0.24 4.52e-7 2.08 3.17e-4 1.15

Table 2.3: Convergence of the interior penalty method with polynomials " on Cartesian and distorted meshes

2.3. LOCAL ERROR ESTIMATES    




err

2 3 4 5 6 7

1.67e-2 4.46e-3 1.16e-3 2.97e-4 7.54e-5 1.90e-5

43

   

   

,

ord err  , ?  1.80 3.20e-1 1.90 1.62e-1 1.94 8.08e-2 1.96 4.02e-2 1.98 2.00e-2 1.99 9.99e-3

ord

err

0.89 0.98 1.00 1.01 1.01 1.00

4.24e-2 1.33e-2 3.82e-3 1.05e-3 2.76e-4 7.12e-5

   

,

ord err  , ?  $ 1.36 6.32e-1 1.67 3.27e-1 1.80 1.66e-1 1.87 8.31e-2 1.92 4.14e-2 1.95 2.06e-2

ord 0.88 0.95 0.98 1.00 1.01 1.01

Table 2.4: Convergence of the interior penalty method with  - and  -elements on a circle with bilinear mapping 2.2.16 Lemma: Let the space .! be the largest continuous subspace of . and .! the subspace such that for all '!  .! holds

 #  '!4 +    +  .! 5 (2.22) 

Then, if ?  , the solution #3 of (2.12) converges to the solution #!  .! of the variational problem " & " & (#4 ('  4 ' '  .! 4 where .! is the subspace of .! with zero traces on / . Proof. By Theorem 2.2.10, we have that   , #  #3 

 3

   
 #  '  !&

 3

4

with  independent of ? 1RZ ZH ¿UVW DVVXPH #   on / and choose '  .! ; then, the right hand side does not depend on ?, yielding   # #  #3&    5 ?

,QKRPRJHQHRXV ERXQGDU\ FRQGLWLRQV DUH LQFOXGHG E\ SURMHFWLQJ WKHP LQWR WKH ¿QLWH HOHPHQW VSDFH ¿UVW 2.2.17 7KH UHVXOW RI WKLV OHPPD LV YHUL¿HG LQ )LJXUH  ,W VKRZV WKDW D YHU\ VPDOO HUURU LV achieved with ? only slightly (by  ) larger than the stability threshold. If  HOHPHQWV DUH XVHG WKH HUURU FRQYHUJHV WR D ¿[HG YDOXH YHU\ IDVW :LWK  -elements, the error starts to increase slowly with growing ?. In fact, if ?  , the  VROXWLRQ PXVW FRQYHUJH WR D JOREDOO\ OLQHDU IXQFWLRQ WKHUHIRUH QRW DSSUR[LPDWLQJ WKH VROXWLRQ RI WKH GLIIHUHQWLDO HTXDWLRQ DW DOO

2.3 Local error estimates 2.3.1 7KH IROORZLQJ SDUDJUDSKV HVWDEOLVK HVWLPDWHV RI WKH HUURU ZHLJKWHG ZLWK VPRRWK DSSUR[ LPDWLRQV WR 'LUDF IXQFWLRQDOV ,Q WKH FDVH RI TXDVLXQLIRUP JULGV ZH REWDLQ RSWLPDO
 -error

CHAPTER 2. LINEAR DIFFUSION I

44

   

 

0.001 0

2

4

6

8

10

12

14

16

18

20

? Figure 2.3: Accuracy of the interior penalty method depending on the parameter ?.

estimates. The results are taken from [KR02], following the outlines in [FR76]. Therefore, in this whole section, we restrict ourselves to the two-dimensional case    and to bilinear shape functions  . In order to keep the presentation of the rather lengthy computations simpler, we will use the convention # # '&&  '   '4

'

   throughout this whole section. This way, the boundary terms in (2.12) assume the same form as the interior terms. Furthermore, we assume for simplicity that /  . The analysis follows essentially the framework for weak norms in Section 1.3. In order to obtain optimal estimates, this has to be performed in weighted norms. Therefore, we will begin this section by collecting estimates for weight functions and Green functions. The main result of this section can be found in Theorem 2.3.7 on page 48.  'H¿QLWLRQ For some point -  1 and a parameter K 9 , let %: & 1 be a set, such that -  %: 4

 %:   K4

K  %:   K 5

(2.23)

Typically, we consider a ball of radius K around - or a patch of mesh cells containing -. Further, let J%: 
 )%:  be a weighting function (regularized Dirac function) with support in %: and    (2.24) J%: 00  4 J%: 
    K 5 



2.3. LOCAL ERROR ESTIMATES

45

We remark, that the function being constant 6%:  inside %: DQG ]HUR RWKHUZLVH IXO¿OOV WKHVH conditions. Still, the function J%: is not required to be constant on the set %: . In particular, projections of the Dirac functional into the discrete space . will be allowed. )LQDOO\ ZH GH¿QH WKH HYDOXDWLRQ IXQFWLRQDO *%: # 





#0J%: 004

(2.25)

ZKLFK DSSUR[LPDWHV WKH SRLQW HYDOXDWLRQ #- if # LV VXI¿FLHQWO\ UHJXODU DQG K  .  'H¿QLWLRQ We introduce N  N:% )

4

0  -  K 4

(2.26)

the regularized distance function with respect to a point -. With %% , we denote a mesh cell containing - and %    )XUWKHUPRUH ZH GH¿QH N as the
 -projection of N into the space , of cell-wise constant functions and denote by N, its value on the cell % .  /HPPD Let the shape regularity Assumption 1.1.13 and the assumption on mesh size variation 1.1.14 hold. Then, there exist constants  independent of K,  and % , such that as soon as % 7 K, the following estimates hold with OK      K.   N,  N   
N   4    N    OK5

%   4 %   4

(2.27) (2.28) (2.29)

Proof. :H EHJLQ SURYLQJ   /HW % be a cell, such that  -4 %   K. Then, 
 N  K and by (1.4)  N  K     K  "     K 5 

Now, if  -4 %   K holds, then   %   %% 4 %  by Assumption 1.1.14. Since  %% 4 %    -4 %  ZH KDYH IRU 0  % N0   -4 %      K    -4 %  4  N0   -4 %   K   -4 %  5  7KHUHIRUH  LV SURYHQ DQG  IROORZV VLQFH WKH PHDQ YDOXHV RI N on a cell is between the minimum and the maximum. ,Q RUGHU WR SURYH   ZH H[WHQG WKH QRUP WR D EDOO  - in  around - and containing 1, yielding       N   N        I    K      I   K    K 5 Under the reasonable assumptions that K *  1 and K 7 , estimate (2.29) follows.

CHAPTER 2. LINEAR DIFFUSION I

46

 'H¿QLWLRQ The regularized Green function   %:  3 , 1 associated with the regularized Dirac function J%: is the solution of the problem +%:  J%: %:  

in 14 on / 5

Its Ritz projection   . is the solution to the discrete problem " & : -

 '4   J% 4 ' '  . 5

(2.30)

(2.31)

  /HPPD Let    and .  , . Let furthermore the elliptic regularity estimate (2.6) KROG 7KHQ WKH UHJXODUL]HG DQG GLVFUHWH *UHHQ IXQFWLRQV GH¿QHG DERYH DGPLW WKH HVWLPDWHV 4     (   N (     OK4 (2.32)          (      OK4 (2.33)   

 N   ,   N (  ,    OK5 (2.34) 

Proof. :H EHJLQ ZLWK SURYLQJ   %\ GH¿QLWLRQ WKHUH KROGV   (   4 J%:     5  

(2.35)

1RZ OHW 04  be the true Green function of the Laplacian on the domain 1 IRU ZKLFK ZH have the following bound (see [FR76]):   04    0    5

7KHUHIRUH

0  04 4 J    This implies (    OK. .

 %: 





04    OK5

1H[W ZH HVWLPDWH XVLQJ WKH HOOLSWLF UHJXODULW\ HVWLPDWH            0 (    H K (   N (    N(     

        ( 0   (   H K (    

         +0   (   H K +   (    

       0 +   (   H K +   

     N+   (  5

2.3. LOCAL ERROR ESTIMATES

47

&RQVHTXHQWO\ E\ WKH GH¿QLWLRQ RI ,       N (    NJ :   (   OK5 %

ZKLFK LPSOLHV WKH DVVHUWHG ERXQG   8VLQJ WKH HVWLPDWHV  DQG  DQG WKH 6REROHY LQHTXDOLW\     ('   ( '   4 '  . ! 3 , 14 


ZH FRQFOXGH IRU '  . ! 3 , 1 and '  . ! 3 , 1 UHVSHFWLYHO\ WKH JOREDO DSSUR[LPDWLRQ estimate      ( '  '  , '      
  5  ( '   
 )RU '4 +  .  .  WKHUH KROGV  +        ('  (+   ? # '&& # +&& - '4 +    2 2  

DQG FRQVHTXHQWO\

        0  

= '

2 # +&&2  # '&&2 

= +

2 4

    - '4 +  '  +  5



%\ WKH FRHUFLYLW\ HVWLPDWH  DQG *DOHUNLQ RUWKRJRQDOLW\ ZH FRQFOXGH IRU WKH GXDO HUURU  )    that   ,    - ,  4 ,    -  4 ,    -   , 4 ,     -   , 4 ,  5

&RQVHTXHQWO\ E\  DQG WKH HTXLYDOHQFH RI WKH HQHUJ\ QRUPV            ,      ,      ,   5    

+HQFH DSSUR[LPDWLRQ HVWLPDWH  \LHOGV        (    5
 

1H[W ZH HPSOR\ D GXDOLW\ DUJXPHQW /HW  . be the solution of   +     in14   VDWLVI\LQJ WKH D SULRUL HVWLPDWH      %\  DQG HOOLSWLF UHJXODULW\ ZH KDYH       ,   (   5 

8VLQJ *DOHUNLQ RUWKRJRQDOLW\ WKHUH KROGV      -  4   -  4  , 5









CHAPTER 2. LINEAR DIFFUSION I

48 Consequently,

                 ,      (    5
   

It remains to bound the
 norm on the right. There holds       (     N N(  


(2.43)

(2.44)

which, in view of (2.29) and (2.32) yields the asserted
 error estimate (2.33). Finally, we derive the interpolation estimate (2.34). In virtue of the local interpolation estimates (1.21), we have      N   ,   N,   ,    

 



 

and analogously,

    N, (     N (   4

    N (  ,    N (   5 

Combining this with the a priori bound (2.32) completes the proof. 2.3.7 Theorem: Let K 9  and for some point -  1, let    - be a ball around - with radius  9 K. Then, if  7   and % 7 K, there holds     *%: #  *%: #    OK #
    #, 5 (2.45)

Proof. Using Galerkin orthogonality, for  and  , we conclude  4 J%:   -  4   -  4    - ; 4   + ?  (; 4 (   ##; & 4 #  & 2  

= ;

4 #  & 2      

0   ##; & 4

= 

2 4 

and abbreviate

.

.

 4 J%:   - - 4 where, with some parameter M  4 &,  +  N (;   ? 

N

# ; & 4 # ; & 2 -    M  

0  

N



= ;

4

= ;

2 4  ?  +  N (   ? 

N

## & 4 #  & 2 -      

0  M 

N

= 

4

= 

2 5

2.3. LOCAL ERROR ESTIMATES

49

The two terms - and - will be estimated separately. First, we estimate - . For %! GH¿QHG LQ   ZH FRQFOXGH IURP WKH LQWHUSRODWLRQ HVWLPDWHV  DQG   REVHUYLQJ DVVXPS WLRQV  DQG  WKDW         (;   . # ; &   . 

= ;

   ( #  4    2 2  1RZ ZH VSOLW WKH VXPPDWLRQ RYHU %   DV IROORZV   3 7 555  -    ,

  , 

3

7 555 5

2Q FHOOV % -& %  ZH KDYH N    and therefore estimate

      ( #    ( #   N M   ,    ,               ( # 5   ( #
  N  M M

- 

)URP WKLV ZH REWDLQ



 M

- 

        OK( #
    ( # 5 M M



Next, we estimate -  2EVHUYLQJ WKDW N is constant on each cell %  ZH KDYH + ? - N  4    (N  4 (   ##N  & 4 #  & 2    

0    ##N  & 4

= 

2  

= N  

4 #  & 2   + ?  N ( 4 (   

N

## & 4 #  & 2    

 ? ##N &



4 #  & 2  ##N &



4

= 

2    0    

N

## & 4

= 

2  ##N & # =  & 4 #  & 2 5 ' 

7KLV OHDGV XV WR

-  - N  4    - 4



where - 

+  M 

N

= 

4

= 

2  ##N &



4

= 

2   

  

N

## & 4

= 

2  ##N & # =  & 4 #  & 2 ' 0 ? ##N &



4 #  & 2 5   

:H SURFHHG ZLWK WKH ¿UVW WHUP RQ WKH ULJKW LQ   8VLQJ *DOHUNLQ RUWKRJRQDOLW\ RI  , we obtain - N  4    - N   , N  4  5



CHAPTER 2. LINEAR DIFFUSION I

50

Since N is constant on % and the projection , is local, we have , N    N ,  . Hence, it follows that + ? (N ; 4 (   ##N ; & 4 #  & 2 - N  4       

0    ##N ; & 4

= 

2  

= N ; 

4 #  & 2 5  

The four terms on the right hand side are now estimated separately using the trace inequalities (1.24), and the interpolation estimates (1.21), (1.25). Furthermore, we employ the a priori bound (2.33) and the error estimate (2.33) for the Green function  and the dual error ; . Note that in order to estimate the average values

= ;

2  LW VXI¿FHV WR HVWLPDWH HDFK RI WKH WHUPV = ; 2 separately. In virtue of Assumption 1.1.14, the weights N can be estimated by ##N & 2   N,   

N

2 4

(2.49)

and we use that (   . By (2.28), N  N. We will use a free parameter M  4 &. )RU WKH ¿UVW WHUP ZH ¿QG    +   0      N(    M N (  (N ; 4 (      M  

 

   OK  M- 4 M and, analogously, for the second term,  ?    ##N ; & 4 #  & 2      

+ ?     0 

N

 # N ; &   M? 

N

## &   2 2 M   

+ 0      M?  

N

## &   N(     2 M    

  OK  M- 5 M The third term is estimated by     ##N ; & 4

= 

2    

  +" & 0    ##N &

N

;

 # N & 

;



N

 # ; & 4

= 

2   



    N ;  N =   2 2

 





 

    " .   & . .  N(    N (    N(  

+    0   M N (    N(   M  

  M-  OK 5 M



2.3. LOCAL ERROR ESTIMATES

51

)LQDOO\ IRU WKH IRXUWK WHUP ZH ¿QG         

N



= N ; 

 

N

## &  

= N ; 

4 #  & 2    2 2  

 

 +  "    &   0 N (;    N ( ;   M ? 

N

## &   2   M   

+ 0      M?  

N

## &   N(     2 M    

  OK  M- 5 M

&RPELQLQJ WKHVH HVWLPDWHV \LHOGV - N  4    M- 

 OK 5 M

(2.50)

1H[W ZH HVWLPDWH WKH ¿YH WHUPV LQ - separately again by using the local trace estimates (1.24), DQG WKH D SULRUL ERXQG DQG HUURU HVWLPDWH  DQG   )RU WKH ¿UVW WHUP LQ - , we obtain    +    0   N (    N (   M 

N

= 

4

= 

2   M  

 

 

 M- 



 OK 4 M

DQG DQDORJRXVO\ IRU WKH VHFRQG WHUP      ##N &



4

= 

2     

 +     0  



2  M 

N

= 

2  M  

 +  "    &    &0 "     (   M N (    N (        M  

  M-  OK 5 M 7KH WKLUG WHUP LQ - LV WKH PRVW FULWLFDO RQH VLQFH LW GRHV QRQ FRQWDLQ D IDFWRU  . Therefore, ZH KDYH WR DEVRUE WKLV WHUP LQWR WKH RWKHU GH¿QLWH WHUPV LQ - using the stabilization parameter ?. We recall that for '  . , 

= '

4 # ' & 2  

     ('   ? # ' &  4 2  ? 

(2.51)

with %! GH¿QHG LQ   6SOLWWLQJ WKH GXDO HUURU OLNH   ;  ,  with ;    , , we have 

N

## & 4

= 

2  

N

## & 4

= ;

2  

N

## & 4

= , 

2 7KH ¿UVW WHUP RQ WKH ULJKW LV WUHDWHG DQDORJRXVO\ DV WKH RWKHU WHUPV EHIRUH OHDGLQJ WR WKH HVWLPDWH      (2.52) 

N

## & 4

= ;

2   M-   OK5  M  

CHAPTER 2. LINEAR DIFFUSION I

52 The second term is treated as follows:     

N

## & 4

= , 

2    



0 + ?  

N

= , 

4

= , 

2 5 

N

## & 4 #  & 2   ?  

Using relation (2.51), we conclude by a lengthy but standard calculation:   

N

= , 

4

= , 

2   N,   N, 

= , 

 2

     N,   N, (,      0  H !  + N (,      (,   4        H

with an arbitrary constant H  4 &. Consequently,   

N

= , 

4

= , 

2 ?  

  8 +   0  H !  N (,      (,   5      ? H  

From

!       (,      H (    (;  4    H

follows

  

N

= , 

4

= , 

2 ?  



  8      HN (  ?  

+       0  (   N (;    (;  5   

Hence, observing the results of Lemma 2.3.6, we obtain    8  

N

= , 

4

= , 

2     H-  OK 5 ? ? H  

)LQDOO\ WKH IRXUWK DQG WKH ¿IWK WHUP DUH HVWLPDWHG LQ D VLPLODU ZD\ DV EHIRUH E\     ##N & # =  & 4 #  & 2    



 +

 

M

    0  



2  M 

N

= 

2

 +  "     &   &0 "     (   M N (    N (       M  

  M-  OK 4 M



(2.53)

2.3. LOCAL ERROR ESTIMATES

53

and by   ?   ##N &



4 #  & 2      

 ? +    &0 "  # N &  M 

N

## &    



  2 2   M  

+ ?  0     M 

N

## & 4 #  & 2      (     

  M-  OK 5 M Collecting these results, we obtain !  " !  8  OK 4 -  M     H -  ? M H

(2.54)

!  " !  8  OK 5 -  M     H -  ? M H

(2.55)

and consequently, in virtue of (2.47) and (2.50),

1RZ ZH ¿[ ? according to condition (2.14) and then choose H and M VXI¿FLHQWO\ VPDOO VXFK that ! "  (2.56) M     H  8 7 5 B ? :LWK WKLV FKRLFH RI WKH SDUDPHWHUV ZH REWDLQ WKDW

-  H4 M OK5

(2.57)

ZKLFK WRJHWKHU ZLWK  FRPSOHWHV WKH SURRI 2.3.8 Corollary: $VVXPH WKDW WKHUH LV QR ORFDO UH¿QHPHQW DURXQG WKH SRLQW -, i. e., that there is a mesh-independent constant  such that   %  5

(2.58)

Then, Theorem 2.3.7 implies the point-wise error estimate      -   O #
    #, 5

(2.59)

Proof. Let K  % and %:  %% LQ   7KHUH H[LVWV D J%:   %% , such that for any    %%  there holds  -  (2.60) 0J%: 005 

CHAPTER 2. LINEAR DIFFUSION I

54

0 1 2 3 4 5 6 7

 . . 
 

 5.0e-01 – 2.0e-01 -2.83e-01 6.3e-02 -2.27e-01 2.0e-02 -2.50e-01 5.7e-03 -2.04e-01 1.6e-03 -2.11e-01 4.6e-04 -2.15e-01 1.3e-04 -2.20e-01

Table 2.5:
 -errors and their scaled differences for discontinuous  elements.

Choosing   , we see that 



J%: 00  4

 J%:   %%  4

(2.61)



as is required for the application of Theorem 2.3.7. Using this construction, we derive the estimate     (2.62)  J%: 05 -    #  , #   



Hence, by the approximation properties of  elements and the result of Theorem 2.3.7, we obtain       (2.63) -   ( #
    O( #
    ( #5 which completes the proof.

2.3.9 Remark: It is well known that the logarithmic term in corollary 2.3.8 is observed only on irregular grids in two dimensions with  or  ¿QLWH HOHPHQWV :H FKHFN WKLV UHVXOW E\ solving example A.2.1 on page 170 on the irregular grid in example A.1.4 on page 167. As can be seen in Figure 2.4, the maxima of the error are located at the two irregular points of the mesh. Since the right graph is scaled up by a factor of 16 (i. e.  ), the peaks are indeed growing by the logarithmic factor. 2.3.10 The scaled error  )  .#  # . admits the asymptotic estimate          #-  # -   #-  # -

 5 

7KHUHIRUH WKLV GLIIHUHQFH VKRXOG UHPDLQ FRQVWDQW XQGHU UH¿QHPHQW LI  LV VKDUS ,Q 7D ble 2.5, we display these values together with the
 -norm of the errors. This table clearly supports our theoretical result.

2.3. LOCAL ERROR ESTIMATES

55

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 1 0.5 -1

-0.5

0 0

0.5

-0.5 1 -1

0.001 0.0008 0.0006 0.0004 0.0002 0 1 0.5 -1

-0.5

0 0

0.5

-0.5 1 -1

Figure 2.4: Pointwise errors on irregular meshes

CHAPTER 2. LINEAR DIFFUSION I

56

2.4 A posteriori error analysis 2.4.1 We will now derive a posteriori estimates for the interior penalty scheme (2.12). A similar analysis based on duality arguments has been developed by Becker et al. in [BHL03] and [BHS03], but there the emphasis is on
 -norm error bounds and on non-matching meshes in the context of domain decomposition. Let *  be an arbitrary linear functional on . with respect to which the error   #  # is to be estimated. Examples are local averages as considered above, contour integrals or integrals over subdomains:    #00* #004 *#  #04 *#  *#  :  





for more examples see [BR01]. With the functional * , we associate a dual solution  . as the solution of the auxiliary problem (<4 (   *<

<  .5

(2.64)

If the functional *  has an
 representation $, then the dual problem can be written in strong form as +  $

in 14

2



 4 = 2



 5

(2.65)

Recalling that *<  - <4 4

<  .  . 4

we have by Galerkin orthogonality *   -  4   -  4  C 4 for arbitrary C  . . Integrating cell-wise by parts and reordering terms, we conclude by elementary but tedious calculation that, for ; )  C , *   -  4  F  +  + 4 ;   =  4 ; 2  

= 

4 # ; & 2    

0  ?  ## & 4

= ;

2  ## & 4 # ; & 2   +     +# 4 ;   ##= # & 4

;

2   0 ?  ### & 4 # ; & 2 5  ### & 4

= ;

2    This is rewritten in the form *  

 



; 

+  # 4 ;   =# 4

;

   0 ?  # ; & 2 4  #4

= ;

  

(2.66)

2.4. A POSTERIORI ERROR ANALYSIS

57

with the following notation of cell and edge residuals: #  )   +# 4  
 # # & 4 if / & =%  =14  #  # 4 if / & =1 4 #  ) 4

 4 if / & =1 4  
 # = # & 4 if / & =%  =14  4 if / & =1 4 5 =#  )



 #  = # 4 if / & =1 4

From the error representation (2.66), we obtain the following error estimate: 2.4.2 Theorem: For the error   #  # in the interior penalty scheme, there holds the a posteriori error estimate 0  +       *   (2.67) N >  N >  N > 4  

with the cell residuals

 N

and weight factors > EHLQJ GH¿QHG E\ 





N  .# . 4

>  .  C . 4

 N

 >



.

N   

.=# .2 4

.  .#.2 4



.

>   .  C .2 4 

.  .

= 

 C 

 ?  #  C & .2 4

for arbitrary C  . . 2.4.3 Corollary: The mean quadratic error admits the asymptotically optimal a posteriori estimate 0. +       . .     N   N   N 5 (2.68)  

Proof. Theorem 2.4.2 provides a posteriori estimates for arbitrary functionals of the error. This also includes the
 -error estimates. To see this, we take the special functional *< )

 4 < 5 . .

The corresponding dual solution  . VDWLV¿HV #  "  1 and the a priori bound .( .   4

(2.69)

where the stability constant  only depends on the domain 1. From (2.67) and the interpolation estimates (1.21), (1.25), we infer that 0. +        N   N   N 5 (2.70) . .     

CHAPTER 2. LINEAR DIFFUSION I

58

2.4.4 Next, we state an a posteriori error estimate for the locally averaged error as considered in our a priori error analysis. In this case the dual solution is just the regularized Green function,  %: introduced in the proof of Theorem 2.3.7. We have the estimate 0  +       *:%    ; #  ) N >  N >  N > 4 (2.71)  





where the residual terms N DUH DV GH¿QHG DERYH DQG WKH ZHLJKWV > can be estimated as follows: 

>  .%:  C . 4 

.

>   .%:  C .2 4  >

.

  .

= %:  C 



? : #   C & .2 5  %

2.4.5 :H WHVW RXU HUURU HVWLPDWH ZLWK WKH VDPH FRQ¿JXUDWLRQ DV LQ WKH SUHYLRXV VHFWLRQ WKDW LV the solution of example A.2.1 on page 170 on the irregular grid of example A.1.4. The error is evaluated at the point -  5"4 5" , which is the point with maximum error in Figure 2.4 (the value is taken from the solution in the upper right cell adjacent to this point) . We compare WKUHH W\SHV RI ³HUURU HVWLPDWRUV´ 7KH ¿UVW RQH ; #   LV REWDLQHG GLUHFWO\ IURP WKH HUURU representation (2.66), avoiding the use of triangle and H¨older inequalities:   ; 5 (2.72) *  ; #  ) 



7KH VHFRQG HVWLPDWRU XVHV WKH ORFDO UH¿QHPHQW LQGLFDWRUV ; : ; #  )





; )

 





; 5

The third one, ; #   LV JLYHQ E\ 7KHRUHP  DV GHVFULEHG DERYH   3       7 N >  N >  N > 5 ; #  ) ; ) 



)RU SUDFWLFDO HYDOXDWLRQ RI WKH HUURU HVWLPDWRUV ZH VROYH WKH GXDO SUREOHP  RQ WKH FXUUHQW PHVK ZLWK ELTXDGUDWLF SRO\QRPLDOV REWDLQLQJ   . . We decided for exact computation in WKH KLJKHU RUGHU VSDFH WR DYRLG DGGLWLRQDO HUURU FRQWULEXWLRQV )RU D PRUH HI¿FLHQW FRPSXWDWLRQ  FDQ EH REWDLQHG E\ SRVWSURFHVVLQJ VHH >%5@ IRU VXFK VWUDWHJLHV DQG WKHLU LQÀXHQFH RQ WKH estimator. The quality of the resulting approximate error estimators ; # 4 the “effectivity index”:  ; #  5 &" ) : *%  



 4 4   LV PHDVXUHG E\

0HVK DGDSWDWLRQ LV EDVHG RQ ³HUURU LQGLFDWRUV´ ; and ;  UHVSHFWLYHO\ )RU PHVK UH¿QHPHQW D ¿[HG IUDFWLRQ KHUH  RI WKH JULG FHOOV ZLWK ODUJHVW LQGLFDWRU DUH UH¿QHG

2.4. A POSTERIORI ERROR ANALYSIS
3 4 5 6 7 8 9 10 11

- 4.306e-3 1.655e-3 6.540e-4 2.920e-4 1.402e-4 6.756e-5 3.376e-5 1.743e-5 9.069e-6

; #  4.288e-3 1.652e-3 6.528e-4 2.917e-4 1.401e-4 6.751e-5 3.373e-5 1.742e-5 9.068e-6

59 &eff 0.996 0.998 0.998 0.999 0.999 0.999 0.999 1.000 1.000

- 1.82e-5 2.75e-6 1.27e-6 2.66e-7 8.60e-8 5.47e-8 2.47e-8 3.10e-9 1.74e-9

; #  6.127e-3 2.296e-3 9.431e-4 4.182e-4 2.000e-4 9.678e-5 4.875e-5 2.522e-5 1.324e-5

&eff 1.42 1.39 1.44 1.43 1.43 1.43 1.44 1.45 1.46

Table 2.6: Point error - and -  -  ; #  and the corresponding a posteriori estimators ; #  and ; #  (
LV PD[LPXP UH¿QHPHQW OHYHO 
3 4 5 6 7 8 9 10 11

- 3.877e-03 1.459e-03 6.508e-04 3.146e-04 1.608e-04 8.205e-05 4.203e-05 8.205e-05 4.203e-05

; #  2.021e-02 7.310e-03 3.050e-03 1.430e-03 6.985e-04 3.574e-04 1.834e-04 3.574e-04 1.834e-04

&eff 5.211 5.009 4.686 4.544 4.344 4.355 4.365 4.355 4.365

Table 2.7: Point error - and a posteriori estimator ; # 

2.4.6 Table 2.6 presents results for estimators ; and ; REWDLQHG E\ DGDSWLYH UH¿QHPHQW EDVHG  on indicators ; . Estimator ; #  is asymptotically optimal while estimator ; #  appears to be off by a factor of about 6 7KH HUURU UHSUHVHQWDWLRQ  VXJJHVWV WR FRQVLGHU *#  ) *#   ; #  DV QHZ DSSUR[LPDWLRQ 7KLV SRVWSURFHVVLQJ VWHS FDQ LPSURYH DFFXUDF\ GUDPDWLFDOO\ DV VKRZQ in Table 2.6. The mesh obtained in the eighth step of this iteration is shown in Figure 2.5. 2.4.7 ,Q 7DEOH  ZH VKRZ UHVXOWV IRU HVWLPDWRU ; #  6LQFH WKLV HVWLPDWRU LQYROYHV +ŽROGHU DQG WULDQJOH LQHTXDOLWLHV RQ HDFK FHOO WKHUH LV QR FDQFHOODWLRQ EHWZHHQ WKH GLIIHUHQW WHUPV RI WKH HVWLPDWRU 7KHUHIRUH WKH HUURU LV RYHUHVWLPDWHG E\ D IDFWRU EHWZHHQ  DQG  7KH HUURUV UHVXOWLQJ IURP DGDSWLYH UH¿QHPHQW ²EDVHG RQ WKH HVWLPDWRU ; #  KHUH² DUH FRPSDUDEOH WR WKRVH RI 7DEOH  7KHUHIRUH ZH FRQFOXGH WKDW ERWK HVWLPDWRUV DUH VXLWHG DV UH¿QHPHQW FULWHULD 2.4.8 )LQDOO\ LQ 7DEOH  ZH FRPSDUH WKH ³HUURUPHVKUDWLR´ - for uniform and adapWLYH UH¿QHPHQW 2Q XQLIRUPO\ UH¿QHG PHVKHV GXH WR WKH DV\PSWRWLF EHKDYLRU -   O DV GHPRQVWUDWHG LQ 7DEOH  ZH H[SHFW - to grow like O6  ZKLOH WKH PHVK UH¿QH ment at the point - VKRXOG VXSSUHVV WKLV GHIHFW 7KLV LV GHPRQVWUDWHG E\ )LJXUH  ZKHUH WKH UDWLR HYHQ LPSURYHV XQGHU DGDSWLYH UH¿QHPHQW

CHAPTER 2. LINEAR DIFFUSION I

60

Figure 2.5: Adapted mesh (level
 % ) for the point value computation


1 2 3 4 5 6 7 8 9

XQLIRUP UH¿QHPHQW - - 3.811e-02 1.985e+00 1.126e-02 2.348e+00 3.212e-03 2.677e+00 8.956e-04 2.983e+00 2.464e-04 3.281e+00 6.716e-05 3.577e+00 1.818e-05 3.872e+00 – – – –

DGDSWLYH UH¿QHPHQW - - 3.811e-02 1.985e+00 1.448e-02 1.433e+00 4.306e-03 8.404e-01 1.655e-03 6.261e-01 6.540e-04 4.804e-01 2.920e-04 4.238e-01 1.402e-04 3.945e-01 6.756e-05 3.664e-01 3.376e-05 3.507e-01

7DEOH  (I¿FLHQF\ RI FRPSXWDWLRQ RI SRLQW YDOXH #- RQ XQLIRUPO\ DQG DGDSWLYHO\ UH¿QHG meshes (
UH¿QHPHQW OHYHO number of cells).

2.5. COMPARISON WITH CONTINUOUS FINITE ELEMENTS

61

2.4.9 Remark: 7KH HUURU UHSUHVHQWDWLRQ  FDQ DOVR EH DSSOLHG WR FRQWLQXRXV ¿QLWH HO ement spaces. In that case, #    on interior edges and we recover the representation in [BR01]. Therefore, (2.66) can be viewed as a straightforward extension of the continuous representation to discontinuous elements. Here, an advantage of weak boundary conditions becomes clear, since the approximation error DW WKH ERXQGDU\ QDWXUDOO\ HQWHUV WKH HVWLPDWH 8VLQJ FRQWLQXRXV ¿QLWH HOHPHQWV ZLWK VWURQJ boundary conditions, this error must be assessed in a different way.

 &RPSDULVRQ ZLWK FRQWLQXRXV ¿QLWH (OHPHQWV 2.5.1 ,Q WKLV VHFWLRQ ZH GR D VKRUW H[SHULPHQWDO FRPSDULVRQ ZLWK D FRQWLQXRXV ¿QLWH HOHPHQW method (CG). Since we do not have an optimized solver for this method at hand, we only compare discretization errors with respect to degrees of freedom. 2.5.2 Figure 2.6 shows
 HUURUV IRU H[DPSOH $ RQ &DUWHVLDQ JULGV GHSHQGLQJ RQ WKH QXP EHU RI GHJUHHV RI IUHHGRP IRU ELOLQHDU FRQWLQXRXV DQG GLVFRQWLQXRXV DV ZHOO DV OLQHDU GLVFRQ WLQXRXV SRO\QRPLDO VSDFHV )RXU ZDV FKRVHQ DV WKH ORJDULWKPLF EDVH EHFDXVH WKLV LV WKH DS SUR[LPDWH LQFUHPHQW IDFWRU DIWHU D VLQJOH JOREDO UH¿QHPHQW VWHS )LUVW ZH REVHUYH WKDW WKH discontinuous  HOHPHQW QHHGV DERXW IRXU WLPHV DV PDQ\ GHJUHHV RI IUHHGRP WKDQ LWV FRQ tinuous counterpart. This is exactly the ration of degrees of freedom per cell. Therefore, we conclude that the approximation of continuous and discontinuous elements per grid cell is about the same. For biquadratics, this ratio is slightly better than the expected 9/4. The spaces  and  KDYH OHVV GHJUHHV RI IUHHGRP SHU FHOO WKDQ WKH GLVFRQWLQXRXV WHQVRU SURG uct spaces, while admitting the same asymptotic convergence order. Still, their approximation in relation to the number of degrees of freedom is worse, due to larger interpolation constants. Therefore, the additional numerical effort due to more degrees of freedom due to " HOHPHQWV pays off.

2.5.3 Considering the number of matrix entries as basis of our comparison, we expect the continuous method to be even more superior, since its stencil is smaller. This can be observed in Figure 2.7 at least compared to the discontinuous tensor product elements. Here, the smaller number of degrees of freedom per cell of the " spaces (which enters quadratically into the QXPEHU RI PDWUL[ HQWULHV UHQGHUV WKHP PRUH HI¿FLHQW WKDQ GLVFRQWLQXRXV " , but still less HI¿FLHQW WKDQ FRQWLQXRXV HOHPHQWV 2.5.4 The advantage in degrees of freedom per cell of the continuous element reduces with increasing polynomial degree and the " element eventually has less degrees of freedom per FHOO )RU SRO\QRPLDOV RI GHJUHH  LQ WZR GLPHQVLRQV WKHVH QXPEHUV DUH   DQG  IRU &*   '* DQG '*  1HYHUWKHOHVV )LJXUH  VKRZV WKDW WKH HI¿FLHQFLHV JHW FORVHU EXW WKDW the missing basis functions of  UHGXFH DSSUR[LPDWLRQ WRR PXFK WR PDNH LW PRUH HI¿FLHQW WKDQ the other elements.

CHAPTER 2. LINEAR DIFFUSION I

62 4-1

CGQ DGQ DGP

4-2 -3

L2-error

4

4-4 4-6 4-6 4-7 -8

4

4-10

41

42

42

44 45 45 degrees of freedom

46

48

49

-3

4

CGQ DGQ DGP

4-4 4-6 4-6 L2-error

4-7 4-8 4-10 4-11 4-12 4-12 4-13 4-14

42

42

44

45 45 46 degrees of freedom

48

49

Figure 2.6:
 -error over number of degrees of freedom for CG- , DG- and DG- elements (left) and (bi-)quadratic elements (right)

2.5. COMPARISON WITH CONTINUOUS FINITE ELEMENTS 4-1

63

CGQ DGQ DGP

-2

4

L2-error

4-3 4-4 4-6 4-6 4-7 4-8 4-10

42

42

44

45

45 46 48 matrix entries

4-3

49

410

411

CGQ DGQ DGP

4-4 4-6 -6

4 L2-error

4-7 -8

4 4

-10

4-11 4-12 4-12 4-13 4-14

42

44

45

45 46 48 matrix entries

49

410

411

Figure 2.7:
 -error over number of matrix entries for CG- , DG- and DG- elements (left) and (bi-)quadratic elements (right)

CHAPTER 2. LINEAR DIFFUSION I

64 4-6

CGQ DGQ DGP

-8

4

-11

L2-error

4

4-12 4-14 4-16 4-19 4-21

42

42

44 45 45 degrees of freedom

4-6

46

48

CGQ DGQ DGP

-8

4

L2-error

4-11 4-12 -14

4

4-16 4-19 4-21

45

45

46

48 49 matrix entries

410

411

Figure 2.8:
 -error versus degrees of freedom (left) and matrix entries (right) for polynomials of degree 4

2.5. COMPARISON WITH CONTINUOUS FINITE ELEMENTS

65

2.5.5 Concluding this comparison, we may say that for Poisson’s equation we have to expect a penalty for using discontinuous Galerkin. On the other hand, this penalty is a factor of four in the worst case and decreasing for higher order polynomials. Therefore, we may say that the interior penalty method is not optimal for Poisson equation, but within a reasonable range. Still, we expect it to develop its full potential in singularly perturbed problems, especially with a dominant nonlinear hyperbolic part.

Chapter 3 Linear Diffusion II This chapter treats discretization of Poisson problems in mixed form by the local discontinuous Galerkin method introduced by Cockburn and Shu in [CS98]. We present the analytical results on the so called “standard” method by Castillo et al. from [CCPS00] together ZLWK RZQ H[SHULPHQWDO UHVXOWV DVVHVVLQJ WKH LQÀXHQFH RI WKH VWDELOL]DWLRQ SDUDPHWHU 7KHQ ZH IRFXV RQ WKH ³VXSHUFRQYHUJHQW´ YHUVLRQ RI WKLV PHWKRG VHH &RFNEXUQ .DQVFKDW HW DO >&.36@  +HUH ZH VKRZ QHZ QXPHULFDO HYLGHQFH IRU DSSOLFDWLRQV RI WKH VFKHPH EH yond the limits of the presented analysis. Since this method approximates the “stresses” of WKH PL[HG IRUPXODWLRQ EHWWHU WKDQ WKH VWDQGDUG VFKHPH LW LV RI VSHFLDO XVH ZKHUH DFFXUDWH YDOXHV RI WKH GHULYDWLYHV DUH QHHGHG )LQDOO\ D FRPSDULVRQ RI WKH PHWKRGV ZLWK UHVSHFW WR FRPSXWDWLRQ WLPH XVLQJ WKH HI¿FLHQW VROYHUV IURP &KDSWHU  LV FRQGXFWHG

3.0.1 ,Q WKLV FKDSWHU ZH FRQVLGHU WKH PL[HG IRUPXODWLRQ RI 3RLVVRQ HTXDWLRQ L  (#   ( L   #  # L
 L

in 1 on / on / 5



 'H¿QLWLRQ The dual mixed formulation RI  LV WKH IROORZLQJ YDULDWLRQDO SUREOHP given the spaces . 
 1 and  7 3 0  " div 1 ) '  " div 1'
  on / 4     :  div ) ( :   5 

¿QG L4 #  0  .








such that

" & " & " L4 : &  #4 ( :    ( L4 '  4



holds for any :  0 and '  . ,Q WKLV IRUPXODWLRQ #   on / is the natural boundary FRQGLWLRQ ZKLOH L
  is explicitly built into the space 0 IRU VLPSOLFLW\ ZH RQO\ FRQVLGHU KRPRJHQHRXV ERXQGDU\ FRQGLWLRQV 

CHAPTER 3. LINEAR DIFFUSION II

68

 'H¿QLWLRQ The primal mixed formulation of (3.1) is the following variational problem: given the spaces .   3, 1 and 0 
 1*   ¿QG L4 #  0  .  such that " & " & " L4 : &  (#4 :   (3.3) L4 ('  4 KROGV IRU DQ\ :  0 and '  .   +HUH WKH QDWXUDO ERXQGDU\ FRQGLWLRQ LV L
  on / and WKH 'LULFKOHW ERXQGDU\ FRQGLWLRQ LV REH\HG E\ WKH VSDFH .  .

 5HPDUN %RWK YDULDWLRQDO IRUPXODWLRQV RI WKH PL[HG /DSODFLDQ KDYH D XQLTXH VROX tion #4 L 7KH SULPDO PL[HG IRUPXODWLRQ KDV SURSHUWLHV VLPLODU WR WKH SXUHO\ SULPDO IRUPXOD WLRQ   7KH GXDO PL[HG IRUPXODWLRQ DOORZV IRU DFFXUDWH VROXWLRQV LI WKHUH LV D URXJK GLIIXVLRQ FRHI¿FLHQW D. In this case, L  D(# PD\ EH D VPRRWK IXQFWLRQ HYHQ LI # is not in 3 , 1. 3.0.5 'LVFUHWL]DWLRQ RI HTXDWLRQV  DQG  E\ ¿QLWH HOHPHQWV UHTXLUHV DQ DSSURSULDWH choice of subspaces 0 & 0 . and . & . .  UHVSHFWLYHO\ VXFK WKDW WKH %DEXßVND%UH]]L condition " & ( L4 '    M 9 4 (3.4)


    !& /  :  '    & KROGV LQGHSHQGHQW RI WKH GLVFUHWL]DWLRQ SDUDPHWHU  FI >%)@  )RU WKH GXDO IRUPXODWLRQ WKH VSDFHV LQYROYHG DUH TXLWH LQWULFDWH VHH >%')0 %'0 57@  FRQVLVWLQJ RI YHFWRU YDOXHG VKDSH IXQFWLRQV ZLWK VSHFLDO FRQWLQXLW\ UHTXLUHPHQWV DW WKH ERXQGDU\

3.0.6 7KH GLVFUHWL]DWLRQ ZH LQWURGXFH QRZ ZLOO QRW VHW RXW IURP D ZHOOSRVHG YDULDWLRQDO SURE OHP OLNH  RU   EXW XVHV VWDELOL]DWLRQ RI WKH ELOLQHDU IRUP LQVWHDG 7KLV LV DFKLHYHG E\ DGGLQJ D SRVLWLYH VHPLGH¿QLWH ELOLQHDU IRUP  #4 ' WR WKH VHFRQG HTXDWLRQ LQ   7KLV VWD ELOL]DWLRQ DOORZV XV WR FRQGXFW D RSWLPDO FRQYHUJHQFH DQDO\VLV LQ WKH SDLU RI 6REROHY VSDFHV 0  3 ,  *   and .  3 ,  . 7KH QHFHVVDU\ FRQGLWLRQ  IRU XQLTXH VROYDELOLW\ FDQ EH UHOD[HG WR WKH FRQGLWLRQ VHH>)6@ that there are two positive numbers M and M such that " &   ( :4 '  

  M ' &  M  '4 '4  :  / 



KROGV IRU DQ\ '  . .

3.1 The LDG Method 3.1.1 7HVWLQJ  RQ D VLQJOH JULG FHOO %   with functions : and ' DQG LQWHJUDWLQJ E\ SDUWV \LHOGV " &

 " & " L4 : &  #4 ( :   #4 :
2  "  & (3.6)  L
4 ' 2  4 '  5 L4 (' 

7KHVH HTXDWLRQV DUH ZHOOGH¿QHG IRU IXQFWLRQV L4 :  0  3 , *   and #4 '  .  3 ,  .

3.1. THE LDG METHOD

69

"  &  3.1.2 Assumption: The discrete subspaces of 0 and . are chosen as ," and ," ,   respectively. Here, (   and (;  (  4 ( . In the case of parallelogram cells, we also  allow the spaces ," . 3.1.3 Using these discrete spaces and replacing the traces on the cell boundaries in (3.6) by QXPHULFDO ÀX[HV L  and # WR EH VSHFL¿HG EHORZ WKH JHQHULF /'* PHWKRG UHDGV DV IROORZV ¿QG L 4 #   0  . such that for all : 4 '   0  .

 " & & " " L 4 : &  # 4 ( :   # 4 :
2  "  & (3.7)  L 
4 ' 2  4 '  5 L 4 ('  3.1.4 :LWK WKH ÀX[ RSHUDWRUV RI 'H¿QLWLRQ  ZH GH¿QH WKH QXPHULFDO ÀX[HV JHQHUDOO\ L  and # as   

L

 M" # #
&&  M&; # L
&& for    4

(3.8) L   L  M #
 #
for    4 



L
for   

4   

#

 M; # #
&&  M;; # L && for    4

#  # (3.9) for    4  " &

#  M;; L
 L

for   

 5

The scalar parameters M and M;;  DV ZHOO DV WKH YHFWRU YDOXHG SDUDPHWHUV M; and M; ZLOO EH  ¿[HG EHORZ 5LJKW DZD\ ZH FKRRVH M;  M; WR PDLQWDLQ V\PPHWU\ RI WKH V\VWHP 1XPHULFDO ÀX[HV GH¿QHG WKLV ZD\ DUH FRQVLVWHQW LQ WKH VHQVH WKDW D ³FRQWLQXRXV´ VROXWLRQ L4 #  3 , 1*    IXO¿OOV L 
 L
and #  #. 3.1.5 )RU WKH DQDO\VLV RI WKH VFKHPH   ZH UHDUUDQJH WKH WHUPV VR WKDW ZH REWDLQ :  0 '  . 5

- L 4 :   2 # 4 :    :  2 '4 L    # 4 '   '

7KH ELOLQHDU DQG OLQHDU IRUPV DUH " &



 - L4 :   L4 :   M;; # L
&&4 # :
&&
 M;; L
4 :

 4







 " & 2 #4 :   #4 ( :  

#

 M; # #
&&4 # :
&&
 #4 :

 4







  #4 '  M # #
&&4 # '
&&
 M #4 '
 4

(3.10)

(3.11)  (3.13)



and



  :   # 4 :

  M;; L 4 :





 " &  '  4 '   M # 4 '
  L 4 '
 5

'H¿QLQJ



(3.14) (3.15)



" &
 L4 #* :4 ' ) - L4 :   2 #4 :   2 '4 L   #4 '  : * ' )  :    '4

ZH DUULYH DW WKH ZHDN IRUPXODWLRQ ¿QG L 4 #   0  . such that " &
 L 4 # * :4 '   : * ' :4 '  0  . 5

(3.16)

CHAPTER 3. LINEAR DIFFUSION II

70

3.1.6 Lemma: Let / - and let the spaces 0 and . be chosen according to Assumption 3.1.2. Furthermore, let M 9  and M;;   on  . Then, the discrete system (3.10) with ÀX[HV FKRVHQ DFFRUGLQJ WR (3.8) and (3.9) is uniquely solvable in 0  . . Proof. (from [CCPS00]) We show that the only solution for homogeneous data is zero. Testing (3.16) with L and # yields - L 4 L    # 4 #   5 Under the assumptions on the parameters, we conclude .L .  - L 4 L 4 and  # 4 #   5  From the Therefore, L   in 1 and # # &   on  . Consequently, # is continuous in 1. ¿UVW HTXDWLRQ RI   ZH GHGXFH 2# 4 :    :  0 5 Integrating by parts in (3.12) yields



 " & 2 # 4 :    (# 4 :  

:

 M; # :
&&4 # #
&&
 # 4 :

 4

and therefore we obtain



" & 2 # 4 :   (# 4 :   

:  0 5

(3.17)



(3.18)

Now, applying Lemma 1.2.18 to all grid cells starting at the Dirichlet boundary yields #  . 3.1.7 Remark: By choosing M;;  , the form - 54 5 does not contain any boundary terms between elements and is just the
 -scalar product on 0 . Therefore, the scheme is called local discontinuous Galerkin (LDG) method. Consequently, the resulting matrix is a block diagonal matrix where the size of each block is the number of degrees of freedom on a single cell. ,QGHSHQGHQW RI WKH FKRLFH RI WKH SRO\QRPLDO EDVLV WKLV PDWUL[ FDQ EH LQYHUWHG HI¿FLHQWO\ LQ D preprocess before starting an iterative linear solver. 3.1.8 The remaining parameters are used for tuning the scheme. In Table 3.1 on the next page, we give an overview over the convergence orders achieved with different parameter values, summarizing the results of the following sections.

3.2 The standard LDG method 3.2.1 Choice of the parameter M; OHDGV WR GLIIHUHQW ÀDYRUV RI WKH /'* PHWKRG )LUVW LQ WKH standard LDG method, we set M;   and choose M as  6     M for 0  % ! %  4   

(3.19) M  M   

 for 0  % ! =14 

where M  9  is independent of the mesh size, but may depend on the shape of the mesh cells. In our experiments, it will always be constant.

3.2. THE STANDARD LDG METHOD M 6 6  

71

M; .L  L . .#  # . 0 ( ( upwind ( ( 0 ( (   upwind (   (

7DEOH  2UGHUV RI FRQYHUJHQFH RI WKH /'* PHWKRG IRU GLIIHUHQW FKRLFHV RI WKH ÀX[ SDUDPHWHUV " & 3.2.2 With the form
 54 5* 54 5 , we associate the energy seminorm or LDG seminorm 8 "   & L4 #
 L4 #* L4 # 5 (3.20) ) LDG In case of the standard LDG method, this seminorm is         !       L4 #  L   M # #&&
 M #
 LDG



3.2.3 Theorem: Let   (   where ( is the polynomial degree of the space . according to Assumption 3.1.2 and let L4 #  3 , *    3 ,  be a solution to the mixed Poisson problem (3.1). Let L 4 #   0  . be a solution to the LDG formulation (3.10) with parameters M;; 4 M;   and M according to (3.19). Furthermore, let assumptions 1.1.13 and 1.1.14 on the triangulation hold. Then, the error admits the estimate      & " ; 4   (3.21)    L ,   #, 5 LDG



Proof. The proof of Theorem 3.2.3 will follow the generic proof of Theorem 1.3.9. Therefore, we must prove concretizations of assumptions 1.3.5 to 1.3.8. We observe that the stability estimate (1.32) in Assumption 1.3.5 holds with a constant    and +  '  : 4 '  due to WKH GH¿QLWLRQ RI WKH /'*VHPLQRUP LQ   7KH IROORZLQJ OHPPDV SURYLGH $VVXPSWLRQ  (in its more general form Assumption 1.3.12) and Assumption 1.3.8. 3.2.4 Lemma: Let L  3 ,1*   and :  3  , 1*  , furthermore #  3 ,1 and '  3  , 1 with   (   and   (  , where ( is the polynomial degree of the space . and 0 is chosen according to Assumption 3.1.2. Then, the
 -projection errors ;; , ;/ , ; and ;! admit the estimates  "    & " & 
 ;; 4 ; * ;/ 4 ;!     L    #,  ,

   " &    :  4    '   ,

 ,

and

   & " & "  
 ;; 4 ; * ;; 4 ;    L ,   #,



)XUWKHUPRUH LI IRU WKH GLIIXVLRQ FRHI¿FLHQW KROGV D  3  , then  "    & 
 ;; 4 ; * ;; 4 ;    # , ,



CHAPTER 3. LINEAR DIFFUSION II

72

Proof. For functions L4 :  0  0 and #4 '  .  . , we write " &
 L4 #* :4 '  - L4 :   2 #4 :   2 '4 L   #4 '4 and estimate each term separately. First,

     - L4 :    7  L : 5

(3.22)

Using form (3.17) of 2 54 5, we obtain by trace estimate (1.23)



 " & 2 #4 :    (#4 :  

:

 M; # :
&&4 # #
&&
 #4 :





           ! # :     #   :   ,





2



2

 

     &"    &!   "  #  :  :    # #  :      ,  ,  ,  

 & &"   " 

  #,   # : ,  :  5



Due to symmetry, we obtain the same for the third term. Finally,          #   '   #4 '   

2



2

 

 &"   & " 

  #,   # ' ,   ' 



Entering ;; for L, ; for # DQG VR RQ DQG DSSUR[LPDWLRQ UHVXOW  WKXV \LHOG WKH ¿UVW HVWLPDWH of the lemma. The second follows by entering ;; for : and ; for '. The third estimate follows by observing that L  D(#  3 , 1*  . 3.2.5 Lemma: Assume that M;  is independent of  and M chosen according to (3.19). Let :  0 , '  . . Furthermore, let L  3 ,  and #  3 ,  with   (   and ;; 4 ;  the
 -projection error. Then, the following estimate holds for   : !  "        & 
 :4 '* ;; 4 ;   :4 '  L    #, 5 (3.23) LDG ,



Proof. We set out with  " & 
 :4 '* L4 #   - :4 ;;   2 '4 ;;   2 ; 4 :    ; 4 '4

and estimate each term separately.     - :4 ;;   D   .- :4 : .  ;; 4      !          '4 ;    M # '&&  M '   M ; 









(3.24) 5

(3.25)

Since we have to avoid derivatives on # and L, we use 254 5 in the forms (3.12) and (3.17), respectively. Using the orthogonality of the
 -projection, the contributions of the cells vanish and we obtain     &    "   2 '4 ;;    M # '&&
 M ' 
 M ;; 







      :4 'LDG M ;; 

 4 (3.26)



3.2. THE STANDARD LDG METHOD

73

and, using inverse estimate, 2 ; 4 : 

   &     &  "    "      # ; & 
   ; 
    # :
&&
   :

 



      :4 'LDG   ; 

 5 (3.27)

Summing up yields    " & 
 :4 '* L4 #   :4 '



      &  ! "    ;;   M ;; 

      M ; 
4



 !                :4 ' LDG  L ,   # , LDG





3.2.6 Remark: Reviewing the proof of Theorem 3.2.3, we can extend the result to a more general choice of the parameters. First, if M; is any quantity obeying the estimate M;   independent of the mesh size , all estimates remain valid and so does the theorem. M    is the optimal value in view of # estimate (3.27). Using M   instead will cause the the energy estimate to be sub-optimal by  with respect to the approximation of L due to estimate (3.26) . 3.2.7 Theorem: Let the assumptions of Theorem 3.2.3 hold and additionally estimate (2.5) with B    , where     (  . Then, the error  is bounded by the negative norm estimate       (3.28)    #, 5 , Proof. ,Q YLHZ RI WKH JHQHULF 7KHRUHP  LW ZLOO EH VXI¿FLHQW WR YHULI\ $VVXPSWLRQV  and 1.3.13. Let   3  , 1 be the solution of the dual problem (2.4) with right hand side <  3 , 1 and ;  D(   3  , 1. Then,          ;  
  D  
 # 
  <  5 +  +  +  +  Therefore, Lemma 3.2.4 and Lemma 3.2.5 (with     ) yield the required estimates.

3.2.8 Corollary: If the domain is convex in the vicinity of every corner of its boundary and the solution # in 3 " ,  , then        "  # " , 3.2.9 We give numerical evidence of the sharpness of these estimates by solving example A.2.2 on page 170 on Cartesian (see example A.1.1 on page 167) and distorted meshes (see example A.1.3), respectively. Table 3.2 shows indeed the predicted convergence orders for L and # with " and " elements .  We have not considered    in detail since experiments showed that the gain in accuracy with equal order elements always outweighed the additional degrees of freedom.  The loss of order in the last row for (bi-)quadratic elements is due to the solver accuracy of 



CHAPTER 3. LINEAR DIFFUSION II

74




  ;  err ord

      ;  err ord err ord Cartesian mesh

1 2 3 4 5 6 7

2.01e-1 1.50e-1 9.39e-2 5.21e-2 2.74e-2 1.41e-2 7.12e-3

1 2 3 4 5 6 7

2.01e-1 1.49e-1 9.39e-2 5.21e-2 2.74e-2 1.41e-2 7.12e-3

 — 5.13e-2 0.42 1.55e-2 0.68 4.30e-3 0.85 1.14e-3 0.93 2.95e-4 0.96 7.52e-5 0.98 1.90e-5  — 8.31e-2 0.43 2.21e-2 0.67 5.80e-3 0.85 1.50e-3 0.93 3.83e-4 0.96 9.69e-5 0.98 2.44e-5

1 2 3 4 5 6 7

2.00e-1 1.55e-1 9.80e-2 5.45e-2 2.88e-2 1.48e-2 7.57e-3

 — 5.31e-2 0.37 1.69e-2 0.66 4.92e-3 0.84 1.36e-3 0.92 3.65e-4 0.95 9.62e-5 0.97 2.50e-5

— 1.72 1.85 1.92 1.95 1.97 1.98

5.04e-3 1.22e-3 3.06e-4 7.70e-5 1.93e-5 4.82e-6 1.20e-6

— 5.09e-3 1.91 1.22e-3 1.93 3.06e-4 1.95 7.70e-5 1.97 1.93e-5 1.98 4.83e-6 1.99 1.20e-6 distorted mesh — 1.65 1.78 1.86 1.90 1.92 1.95

7.28e-3 1.81e-3 5.11e-4 1.32e-4 3.38e-5 8.76e-6 2.26e-6

    err ord

 — 1.82e-3 2.04 2.29e-4 2.00 2.86e-5 1.99 3.58e-6 2.00 4.48e-7 2.00 5.65e-8 2.00 1.02e-8  — 1.82e-3 2.06 2.29e-4 2.00 2.86e-5 1.99 3.58e-6 2.00 4.48e-7 2.00 5.65e-8 2.00 1.02e-8  — 2.14e-3 2.00 3.07e-4 1.83 4.28e-5 1.95 5.82e-6 1.97 7.80e-7 1.95 1.03e-7 1.95 1.34e-8

— 2.99 3.00 3.00 3.00 2.99 2.47 — 2.99 3.00 3.00 3.00 2.99 2.47

— 2.81 2.84 2.88 2.90 2.93 2.94

Table 3.2: Convergence of the standard LDG method on Cartesian and distorted meshes

3.2. THE STANDARD LDG METHOD

75

3.2.10 Since the size of M  is not relevant for stability, as long as it is positive, it is interesting WR H[DPLQH LWV LQÀXHQFH RQ DFFXUDF\ )LJXUH  VKRZV WKDW WKH DFFXUDF\ RI WKH VFKHPH IRU GLIIHUHQW SRO\QRPLDO GHJUHHV YDOXHV KDYH EHHQ VFDOHG E\ VXLWDEOH SRZHUV RI  WR ¿W LQWR WKH VDPH ¿JXUH IXUWKHUPRUH WKH PHVK VL]HV KDYH EHHQ FKRVHQ IRU HDFK SRO\QRPLDO GHJUHH VXFK WKDW WKH HUURU ZDV LQ WKH DV\PSWRWLF UHJLPH DQG ZHOO DERYH PDFKLQH DFFXUDF\  $ TXDOLWDWLYH GLIIHUHQFH EHWZHHQ RGG DQG HYHQ RUGHU SRO\QRPLDOV FDQ EH REVHUYHG WKH QRUP RI the error remains constant at its minimum if M  WHQGV WR LQ¿QLW\ DQG WKH SRO\QRPLDO GHJUHH LV RGG LQ WKLV FDVH WKH HUURU JURZV LI M  EHFRPHV VPDOO DQ HIIHFW UHGXFLQJ IDVW IRU KLJKHU RUGHU polynomials.

7KH ¿JXUH RQ WKH ULJKW VKRZV WKDW WKH HUURU JURZV ZLWK M   ZKLOH LW UHPDLQV FRQVWDQW LI M  LV VPDOO DW OHDVW LQ WKH UDQJH LQYHVWLJDWHG KHUH  7KLV GLIIHUHQW EHKDYLRU RII RGG DQG HYHQ SRO\QRPLDOV VKRXOG EH H[SODLQHG E\ LPSURYHG DQDO\VLV ODWHU :H FRQFOXGH WKDW WKH /'* PHWKRG LV PXFK PRUH UREXVW ZLWK UHVSHFW WR WKH VWDELOL]DWLRQ SDUDP HWHU WKDQ WKH LQWHULRU SHQDOW\ PHWKRG )LJXUH  VKRZV WKDW WKLV LV QRW UHVWULFWHG WR &DUWHVLDQ JULGV EXW KROGV RQ GLVWRUWHG JULGV WRR 3.2.11 Since the bilinear form - 54 5 RI WKH /'* PHWKRG GRHV QRW LQWURGXFH FRXSOLQJ EHWZHHQ PHVK FHOOV VHH 5HPDUN   WKH YDULDEOH L can be eliminated from the system on a cell by FHOO EDVLV YHU\ PXFK OLNH VWDWLF FRQGHQVDWLRQ IRU VWDQGDUG KLJKHU RUGHU HOHPHQWV $V UHVXOW RI WKLV FRQGHQVDWLRQ ZH REWDLQ D ELOLQHDU IRUP  54 5 on .  .  WKH 6FKXU FRPSOHPHQW IRUP RI the LDG method. 7KLV ELOLQHDU IRUP DPRQJ RWKHUV LV DQDO\]HG LQ >$%&0@ +HUH ZH ZLOO TXRWH WKH PRVW LPSRUWDQW UHVXOWV IRU RXU ZRUN IURP WKLV DUWLFOH 3.2.12 Lemma: For any positive parameter  , the form  54 5 is a bounded, coercive bilin  M ear form on . equipped with the norm 5 . Namely, there are constants  and independent of  such that for #4 '  .      #4 '  # '        #4 #    # 5  

" & 3.2.13 Corollary: The bilinear forms -IP  54 5 and  54 5 are spectrally equivalent in the sense that there are constants  and independent of  such that for #  . " & " & IP -IP  #4 #   #4 #  - #4 # 5



3.2.14 Remark: 'XH WR WKH FRQGHQVDWLRQ SURFHVV WKH 6FKXU FRPSOHPHQW IRUP  54 5 FRQ WDLQV FRXSOLQJV QRW RQO\ EHWZHHQ GHJUHHV RI IUHHGRP RI D FHOO DQG LWV QHLJKERXUV EXW DOVR WR WKHLU QHLJKERUV 7KHUHIRUH LQ SUDFWLFDO FRPSXWDWLRQV WKH IXOO V\VWHP ZLOO XVXDOO\ EH SUHIHUUHG

CHAPTER 3. LINEAR DIFFUSION II

76 2-9

Q1 213 Q 12 3 2 Q5

||eu||

2-10

2-11

2-12

2-13 -6 2

2-4

2-2

20 Guu

22

||eu||

2-17

24

26

Q 9 2 25 Q4 2 Q6

2-18

2-19 2-6

2-4

2-2

20 Guu

22

24

26

Figure 3.1: Accuracy of the standard LDG method on Cartesian grids depending on M  .

3.3. THE “SUPERCONVERGENT” LDG METHOD

77

2-9

Q1 27 Q2

||eu||

2-10

2-11

2-12 -6 2

2-4

2-2

20 Guu

22

24

26

Figure 3.2: Accuracy of the standard LDG method on distorted grids depending on M  .

3.3 The “superconvergent” LDG method 3.3.1 The parameter M; can improve stability of the LDG method if it is chosen as an upwind ÀX[ IRU WKH JUDGLHQW RSHUDWRU ,Q WKLV FDVH M VKRXOG EH SRVLWLYH LQGHSHQGHQW RI . 3.3.2 Remark: The method presented here was introduced in [CKPS01] on Cartesian grids. A PRGL¿FDWLRQ DOORZLQJ WR XVH WKH PHWKRG RQ JHQHUDO JULGV DQG LPSURYLQJ WKH FRQGLWLRQ QXPEHU RI WKH PDWUL[ ZLOO EH VXJJHVWHG ODWHU ZLWKRXW PDWKHPDWLFDO SURRI RI FRQYHUJHQFH RUGHUV  'H¿QLWLRQ 6XSHUFRQYHUJHQW ÀX[HV  Let   4 5 5 5 4     7KHQ ZH FKRRVH  M  M  on  and M; on  such that M;




 5 

(3.32)

2EYLRXVO\ WKLV GH¿QLWLRQ GRHV QRW GHSHQG RQ WKH FKRLFH RI
and both values are independent of . On each edge  OHW # and # be the upwind and downwind values of # with respect to the YHFWRU ¿HOG  and let
(
 ) be the vector normal to  with
   (
 7   7KHQ WKH bilinear form 2 54 5 of the superconvergent LDG method assumes the form " &



 #4 ( :   #
4 :  :
 #
4 :
 4 (3.33) 2 #4 :  



"



 &      (#4 :   #  # 4 :

 #4 :

 5 (3.34)





CHAPTER 3. LINEAR DIFFUSION II

78

/ 

/ 

T

/ 

/ 

Figure 3.3: Naming of the boundary components of a rectangular grid cell %

3.3.4 Remark: Unique solvability of equation (3.10) with the form 2 54 5 in (3.33) follows already from Lemma 3.1.6. Furthermore, the proof of Theorem 3.2.3 applies to this method as well, yielding non-optimal convergence estimates in the energy norm. It remains to show improved error estimates with an extra half power of .  'H¿QLWLRQ In order to facilitate the analysis below, we split the Dirichlet and Neumann

boundary into the components 

and 

with
 9  and the remaining parts   and     . 

 1

  The boundary of a grid cell %  #-  4 - & is split into the components /  00  - according to Figure 3.3.  7KHRUHP /HW WKH PHVK FRQVLVW RI UHFWDQJOHV DQG OHW WKH ¿QLWH HOHPHQW VSDFHV EH 0    ,"  and .  ," . Let L 4 #  be the solution to (3.10) with the form 2 54 5 and M FKRVHQ DFFRUGLQJ WR 'H¿QLWLRQ  DQG WKH VROXWLRQ L4 # of (3.1) be in 3 " , *     3 " , . Let furthermore #  3 " , 

 . Then, the error ; 4   admits the estimates    ; 4    "  LDG      "  5

(3.35) (3.36)

In particular, this implies

  ;   "  5

(3.37)

Proof. Again, the proof follows the general framework of Theorem 1.3.9. It relies on the introduction of special projection operators and intermediary results in lemmas 3.3.9 and 3.3.10.

3.3. THE “SUPERCONVERGENT” LDG METHOD

79

 'H¿QLWLRQ 8SZLQG SURMHFWLRQV  For an interval &  #- 4 - & &  ZH GH¿QH WKH one-dimensional upwind/downwind projection I  )  &  " & E\ WKH FRQGLWLRQV " & '  I  '4 +   +  "& (3.38) '-   I  '-   5 1

On a rectangular axiparallel cell %  #-  4 - & ZH GH¿QH DQ upwinding projection from the  space of continuous functions %  onto " %  by ,  )

(

I 5

(3.39)



+HUH WKH VXEVFULSWV LQGLFDWH WKH DSSOLFDWLRQ RI WKH RQH GLPHQVLRQDO RSHUDWRU I  LQ WKH FRU UHVSRQGLQJ FRRUGLQDWH GLUHFWLRQ :H ZLOO DSSO\ WKLV RSHUDWRU WR IXQFWLRQV LQ 3 , %  and 3 , %  7KHVH VSDFHV DUH FRQWLQXRXVO\ HPEHGGHG LQWR  %  for   4  VXFK WKDW WKH SURMHFWLRQ RSHUDWRU LV ZHOOGH¿QHG ,  )XUWKHUPRUH ZH GH¿QH SURMHFWLRQV ,  " %  E\ WKH FRQGLWLRQV ; ) 3 " & :  ,  '  " %  ; :4 ('    

  '  " / :  ,; :4 '
     5 

(3.40) (3.41)



On a face /  SHUSHQGLFXODU WR WKH 0 D[LV ZH GH¿QH WKH IDFH SURMHFWRU ( , I 5  

(3.42)



 /HPPD Let % be a rectangular mesh cell with edges parallel to the coordinate axes and diameter  . Then, the projection errors for functions '  3 ,%  and :  3 ,% *   with   (   and (  (  (; admit the estimates     '  , '  '  (3.43)  )   , ),      )  '  , '  (3.44)    ' ,     ),  )   :  ,; :  : 

 4 (3.45)  ), ,      )   :  , :   4 (3.46) ; : ),   , where  is any of the edges(faces) of % . Furthermore, if '  3 , , then       '  ,   '    ' , 5

Finally, for the edges /  holds     +  , +      +  5   , 

(3.47)

(3.48)



 Proof. 6LPSOH FDOFXODWLRQ VKRZV WKDW WKH RSHUDWRUV ,  and ,; preserve polynomials in " % . 7KHUHIRUH WKH HVWLPDWHV IROORZ E\ WKH %UDPEOH+LOEHUW OHPPD DQG E\ WUDFH HVWLPDWHV

CHAPTER 3. LINEAR DIFFUSION II

80

3.3.9 Lemma: Let L  3 , 1*   and #  3 ,1 with   (  , where (  (  (;

in Assumption 3.1.2. Then, the projection errors ;;  L  , ; L and ;  #  , # admit the estimate  & & " "      
 ;; 4 ; * ;; 4 ;     L ,    #,

Furthermore, if #  3

 ,



,

  and D  3  , then  "   &   
 ; 4 ; * ; 4 ;     #  ;  ; ,



Proof. The proof is analogous to Lemma 3.2.4, with the difference that M   and that powers of  in the form 2 54 5 are equilibrated in a different way. 3.3.10 Lemma: Let :  0 and '  . , furthermore, L  3 ,  *   and #  3 ,  with   (  . Then, " & 
 :4 '* ;; 4 ; 

   " &           L ,    #,    #,

:4 'LDG 4 (3.49)





with a constant  independent of the mesh size.

Proof. (from [CKPS01]) For simplicity, we omit the superscripts on ;  and writing " & 
 :4 '* ;; 4 ;   - :4 ;;   2 ; 4 :   2 '4 ;;    '4 ; 4

ZH HVWLPDWH WKH WHUPV VHSDUDWHO\ )RU WKH ¿UVW DQG WKH ODVW WHUP ZH UHIHU WR HTXDWLRQV  and (3.25), observing M   now. The second is used in the form



 2 '4 ;;   ('4 ;;   # '
&&4 ;;
 '4 ;;

 5



,Q YLUWXH RI WKH GH¿QLWLRQ RI WKH SURMHFWLRQ WKH ¿UVW WZR WHUPV RI WKLV VXP DQG WKH WKLUG UHVWULFWHG WR WKH ³LQÀRZ´ ERXQGDU\ YDQLVK &RQVHTXHQWO\

 2 '4 ;;    '4 ;;




        M ' 

M ;; 





      :4 ' M ;;  
LDG




The estimate of the third term is more involved and uses the techniques of the superconvergence estimate for advection operators found in [LR74]. We begin with " &



 2 ; 4 :   ; 4 ( :   ;
4 :  :
 ;
4 :
 5



Therefore, the contribution of an interior cell is



 " &  :4 #  ; 4 ( :   ;
4 : 2
  ;
4 : 2
 4



3.3. THE “SUPERCONVERGENT” LDG METHOD

81

where we used that # refers to # of the cell % on =  % . Since #  3 , 1 and due to the GH¿QLWLRQ RI , we have ; 
 &  ,  #
and obtain

" &  :4 #  ; 4 ( :  



  

 !



#  ,  #4 :

 #  , #4 :

5

(3.50)





6XPPLQJ XS DQG DGGLQJ WKH PLVVLQJ WHUPV DW WKH ERXQGDU\ \LHOGV 2 :4 # 



 

  :4 # 

     #  ,  #  ,  #4 :
 5  #4 :
  









Here, denotes the coordinate direction orthogonal to . Since ,   , on an edge  



 "   ZH FDQ HVWLPDWH ZLWK  IRU DQ\    "  :

  

      :  4  #  ,  #4 :
    # ,  



with % the cell having  RQ LWV ³GRZQZLQG´ VLGH

 On 

  ZH HVWLPDWH E\ LQYHUVH LQHTXDOLW\    



    #  ,  #4 :
  #  , # ,, :
,              #  ,  # , :
          #, :
 5 

&RQVHTXHQWO\ 2 :4 # 



 

  :4 # !              #,   #,

:4 'LDG 4



1RZ LW UHPDLQV WR HVWLPDWH WKH WHUPV  :4 # ZLWK WKH VDPH SRZHUV RI  )LUVW ZH VSOLW  :4 # 



  : 4 #5



,Q WZR GLPHQVLRQV ZH KDYH & "



  : 4 #  #  ,  #4 = :   #  , #4 : 



and   with 9  DQDORJRXVO\ GXH WR WKH RUWKRJRQDOLW\ RI WKH HGJHV WR WKH FRRUGLQDWH SODQHV :H FODLP WKDW IRU DQ\ :  " %    : 4 #  

 #  " % 5

(3.51)

CHAPTER 3. LINEAR DIFFUSION II

82

Since the cell % is just a scaling of the reference cell %  #4 & and the orthogonality (3.51) is preserved by scaling, we will assume %  % . Furthermore, we will show the proof for   : 4 # only, the other directions following the same pattern.

First, we remark that projections ,  and , preserve polynomials. Thus (3.51) holds for #  " . It remains to show that (3.51) holds for the two polynomials #0  0"  and  #0  0"   . 

The function #0  0"  is constant on /   and / , thus #  , #  . In the interior, we

conclude from = :  " that #  , #  , therefore   : 4 0"     .

For #0  0"   , we integrate by parts and obtain & " & "

#  ,  #4 = :    = #  , #4 : 

,  #





 #  ,  #4 : 
 #  , #4 : 
5 



/  ,

Since # is independent of 0 and , #  on this sum is zero. We conclude that the linear functional   : 4 5 vanishes on " %  for any :  " %  and by the Bramble-Hilbert lemma we obtain      :4 #   #, :  4 for   (  . In higher dimensions, the additional polynomials to be tested are #0  0"  ,  9 . These can be treated the same way as 0"   .

3.3.11 First, we solve example A.2.2 on the Cartesian grids in example A.1.1. The results in 7DEOH  FRQ¿UP WKH VKDUSQHVV RI RXU HVWLPDWH . 3.3.12 Figure 3.4 shows that the error of the stress L is concentrated at the boundary. Therefore, we compute approximations for example A.2.1 with homogeneous Dirichlet boundary values for comparison. Table 3.4 shows that the convergence orders of L are indeed better by one half and the error distribution in Figure 3.5 is equilibrated over the domain. Similar results are obtained with inhomogeneous boundary values lying in the approximation space. We conclude that the projection error of # into the space . at the boundary is a major source of error in L. 3.3.13 In Table 3.5, we present some results not covered by our analysis. These results are for the polynomial spaces  and  and inhomogeneous Dirichlet boundary conditions (example A.2.2). They show, that tensor product polynomials are not required to achieve the convergence rates predicted by the theory of the superconvergent # method. In particular, piecewise constant functions for # and L still yield convergence with  for L in
 . 3.3.14 Remark: 7KH OLQHDU V\VWHP RI WKH XSZLQG /'* PHWKRG LV QRW VROYDEOH HI¿FLHQWO\ VLQFH D VSHFWUDOO\ HTXLYDOHQW SUHFRQGLWLRQHU LV GLI¿FXOW WR REWDLQ 6XFK D SUHFRQGLWLRQHU FDQ EH IRXQG LI WKH ÀX[ DW WKH 'LULFKOHW ERXQGDU\ LV UHSODFHG E\ WKH VWDQGDUG ÀX[   7KLV FDVH is not covered by the analysis either, since the negative power of  in front of the jump term FDQQRW EH FRPSHQVDWHG 1HYHUWKHOHVV 7DEOH  VKRZV WKDW WKLV PRGL¿FDWLRQ GRHV QRW DIIHFW WKH convergence rates —compared to Table 3.4— in two and three space dimensions. 

The last row in the  -column is probably spoiled by the accuracy of the linear solver

3.3. THE “SUPERCONVERGENT” LDG METHOD




  ;  err ord



1 2 3 4 5 6 7

2.04e-1 7.25e-2 2.44e-2 8.05e-3 2.68e-3 9.10e-4 3.14e-4

— 1.49 1.57 1.60 1.59 1.56 1.53

1 2 3 4 5 6

2.36e-01 1.07e-01 4.32e-02 1.63e-02 5.98e-03 2.15e-03

— 1.14 1.31 1.40 1.45 1.48

    err ord

83

  ;  err ord

    err ord

 unit square in two dimensions 4.99e-2 — 9.64e-3 — 1.99e-3 1.32e-2 1.92 1.67e-3 2.53 2.73e-4 3.62e-3 1.86 2.83e-4 2.56 3.75e-5 9.76e-4 1.89 4.78e-5 2.57 4.99e-6 2.55e-4 1.93 8.16e-6 2.55 6.47e-7 6.54e-5 1.96 1.42e-6 2.53 8.27e-8 1.66e-5 1.98 2.69e-7 2.40 1.28e-8 unit cube in three dimensions 9.67e-02 — 5.44e-3 — 9.53e-4 2.88e-02 1.75 1.15e-3 2.24 1.79e-4 8.15e-03 1.82 2.25e-4 2.36 2.72e-5 2.19e-03 1.90 4.19e-5 2.43 3.74e-6 5.67e-04 1.95 7.58e-6 2.46 4.90e-7 1.44e-04 1.97 1.36e-6 2.48 6.30e-8

— 2.87 2.87 2.91 2.95 2.97 2.70 — 2.41 2.72 2.86 2.93 2.96

Table 3.3: Convergence results for the superconvergent LDG method in two and three space dimensions

0.1 0.05 0 -0.05 -0.1

0

0.2

0.4

0.6



0.8

1 0

0.2

0.4

0.6

0.8

1

Figure 3.4: Error L   L with inhomogeneous Dirichlet boundary values

CHAPTER 3. LINEAR DIFFUSION II

84   ;  err ord






1 2 3 4 5 6 7

4.87e-01 1.62e-01 4.88e-02 1.37e-02 3.66e-03 9.47e-04 2.41e-04

— 1.58 1.73 1.83 1.91 1.95 1.97

1 2 3 4 5 6

3.80e-01 1.00e-01 2.52e-02 6.33e-03 1.59e-03 3.98e-04

— 1.92 1.99 1.99 1.99 2.00

    err ord

  ;  err ord

    err ord

 unit square in two dimensions 1.93e-01 — 7.25e-02 — 2.07e-02 4.65e-02 2.06 1.27e-02 2.52 2.67e-03 1.18e-02 1.97 1.83e-03 2.79 3.59e-04 3.07e-03 1.94 2.40e-04 2.93 4.80e-05 7.94e-04 1.95 3.03e-05 2.99 6.25e-06 2.03e-04 1.97 3.78e-06 3.00 7.99e-07 5.12e-05 1.98 4.73e-07 3.00 1.06e-07 unit cube in three dimensions 1.99e-01 — 2.98e-02 — 1.12e-02 5.46e-02 1.87 5.14e-03 2.54 1.64e-03 1.36e-02 2.00 6.90e-04 2.90 2.61e-04 3.37e-03 2.02 8.65e-05 3.00 3.64e-05 8.36e-04 2.01 1.07e-05 3.01 4.78e-06 2.08e-04 2.01 1.33e-06 3.01 6.11e-07

— 2.95 2.89 2.90 2.94 2.97 2.92 — 2.77 2.65 2.84 2.93 2.97

Table 3.4: Convergence results for the superconvergent LDG method with homogeneous Dirichlet boundary values

0.025 0 -0.025 1 0.5 -1

-0.5

0 0

0.5



-0.5 1 -1

Figure 3.5: Error L   L with homogeneous Dirichlet boundary values

3.3. THE “SUPERCONVERGENT” LDG METHOD




  ;  err ord



1 2 3 4 5 6 7

1.01e+0 6.01e-1 4.21e-1 3.01e-1 2.13e-1 1.50e-1 1.06e-1

— 0.75 0.51 0.48 0.50 0.50 0.50

1 2 3 4 5 6

1.92e+0 1.06e+0 6.60e-1 4.60e-1 3.28e-1 2.33e-1

— 0.86 0.68 0.52 0.49 0.49

    err ord

  ;  err ord

85     err ord

 unit square in two dimensions 6.84e-1 — 1.99e-1 — 8.14e-2 4.21e-1 0.70 6.73e-2 1.56 2.28e-2 2.36e-1 0.84 2.37e-2 1.51 5.91e-3 1.24e-1 0.93 8.49e-3 1.48 1.49e-3 6.29e-2 0.97 3.05e-3 1.48 3.73e-4 3.17e-2 0.99 1.09e-3 1.49 9.34e-5 1.59e-2 1.00 3.87e-4 1.49 2.34e-5 unit cube in three dimensions 1.44e+0 — 4.43e-1 — 2.27e-1 8.22e-1 0.81 1.51e-1 1.55 6.19e-2 4.57e-1 0.85 5.38e-2 1.49 1.59e-2 2.42e-1 0.92 1.96e-2 1.46 3.99e-3 1.24e-1 0.97 7.08e-3 1.47 9.98e-4 6.24e-2 0.99 2.54e-3 1.48 2.49e-4

— 1.83 1.95 1.99 2.00 2.00 2.00 — 1.87 1.96 1.99 2.00 2.00

Table 3.5: Convergence results for the superconvergent LDG method with " elements




1 2 3 4 5 6 7 1 2 3 4 5

        ;    ;    err ord err ord err ord err ord #   #  (   unit square in two dimensions 3.74e-1 — 1.97e-1 — 1.44e-1 — 4.70e-2 — 1.03e-1 1.86 5.19e-2 1.93 5.80e-2 1.31 1.41e-2 1.74 2.58e-2 2.00 1.29e-2 2.01 2.16e-2 1.43 3.93e-3 1.84 6.44e-3 2.00 3.24e-3 1.99 7.77e-3 1.47 1.03e-3 1.93 1.61e-3 2.00 8.19e-4 1.99 2.76e-3 1.49 2.64e-4 1.97 4.01e-4 2.00 2.06e-4 1.99 9.80e-4 1.50 6.66e-5 1.99 1.00e-4 2.00 5.17e-5 2.00 3.47e-4 1.50 1.67e-5 1.99 unit cube in three dimensions 4.06e-1 — 2.23e-1 — 2.56e-1 — 9.68e-2 — 1.17e-1 1.79 6.34e-2 1.81 1.19e-1 1.11 2.91e-2 1.73 2.95e-2 1.99 1.58e-2 2.00 4.75e-2 1.33 8.33e-3 1.81 7.38e-3 2.00 3.98e-3 1.99 1.77e-2 1.42 2.23e-3 1.90 1.85e-3 2.00 1.00e-3 1.99 6.39e-3 1.47 5.74e-4 1.96

Table 3.6: Convergence results for the superconvergent LDG method with 6-stabilization at the boundary (  ,1)

CHAPTER 3. LINEAR DIFFUSION II

86


1 2 3 4 5 6 7

    ;    err ord err ord irregular 1.31e-1 — 3.60e-2 — 4.53e-2 1.53 9.78e-3 1.88 1.56e-2 1.54 2.56e-3 1.93 5.41e-3 1.53 6.56e-4 1.97 1.89e-3 1.52 1.66e-4 1.98 6.65e-4 1.51 4.18e-5 1.99 2.34e-4 1.50 1.05e-5 2.00

    ;    err ord err ord distorted 1.44e-1 — 4.70e-2 — 5.80e-2 1.31 1.41e-2 1.74 2.16e-2 1.43 3.93e-3 1.84 7.77e-3 1.47 1.03e-3 1.93 2.76e-3 1.49 2.64e-4 1.97 9.80e-4 1.50 6.66e-5 1.99 3.47e-4 1.50 1.67e-5 1.99

Table 3.7: Convergence results for the superconvergent LDG method on non-Cartesian grids (  ,1) 3.3.15 Table 3.7, we show experiments indicating, that the Cartesian meshes required for the DQDO\VLV DUH QRW QHFHVVDU\ HLWKHU LQ H[SHULPHQWV :H VKRZ UHVXOWV ZLWK WKH PRGL¿HG VFKHPH VHH SUHYLRXV UHPDUN RQ WKH XQLW VTXDUH H[DPSOH $ DJDLQ  RQFH ZLWK WKH LUUHJXODU FRDUVH PHVK LQ $ DQG RQFH ZLWK WKH GLVWRUWHG PHVKHV LQ $ 7KH ¿UVW PHVK FRQWDLQV LUUHJXODU SRLQWV VSRLOLQJ SRLQWZLVH VXSHUFRQYHUJHQFH VHH   WKH RWKHU PHVKHV GR QRW FRQYHUJH WR SDUDOOHOR gram cells. Even more, the additional half order of convergence observed with homogeneous boundary conditions is still obtained for these examples. A theoretical explanation for these results is still missing. 3.3.16 Remark: LDG methods are based on a mixed system of equations and thus require a larger number of degrees of freedom than the interior penalty method. Furthermore, the VROXWLRQ RI WKH UHVXOWLQJ OLQHDU V\VWHPV UHTXLUHV PRUH LWHUDWLRQ VWHSV VHH &KDSWHU   7KHUHIRUH a comparison of these methods should consider approximation properties and VROYHU HI¿FLHQF\ The following discussion should give a hint when to choose which of the methods. For the results in Figure 3.6, the linear system was solved down to a residual norm of  . Then, for each method, solution time is plotted over the
 -norm of the error of the solution # itself (left) and of its derivative L ULJKW  UHVSHFWLYHO\ +HUH WKH GHULYDWLYHV RI WKH LQWHULRU SHQDOW\ VROXWLRQ DUH VLPSO\ FRPSXWHG LQVLGH WKH FHOOV 7KH ¿JXUH VKRZV WKDW WKH LQWHULRU SHQDOW\ PHWKRG LV IDVWHU than the standard LDG by a factor of two, if only the solution # is considered. In this case, the superconvergent LDG is slower by far. On the other hand, if derivatives are important, the picture is just reversed. Superconvergent LDG is fastest due to its half order gain in accuracy, followed by standard LDG. Therefore, a decision between these three methods can only be made according to the quantities that have to be simulated accurately.

3.3. THE “SUPERCONVERGENT” LDG METHOD

87

10000

IP standard LDG upwind LDG

Solution time

1000

100

10

1 1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

L2-error of u 10000

IP standard LDG upwind LDG

Solution time

1000

100

10

1 1e-08

1e-07

1e-06

1e-05

0.0001

L2-error of S

Figure 3.6: Performance of the different DG methods

0.001

Chapter 4 Stokes Equations Several discretizations of Stokes equations by discontinuous functions have been proposed. First, there is the method by Baker et al. (cf. [BJK90, KJ98]). It uses piecewise solenoidal trial functions for the velocity and a “pressure” penalizing the discontinuities of the velocity at boundaries between cells. A fully discontinuous scheme was devised by Hansbo and Larson in [HL02]. Their method is based on the interior penalty method of Chapter 2 and has the special merit of providing a smooth transition between compressible and incompressible elasticity. The scheme presented here was introduced and analyzed by Cockburn, Kanschat, Sch¨otzau and Schwab in [CKSS02] and is based on the LDG method in Chapter 3. It is similar to the one by Hansbo and Larson, but is based on the mixed form of the Laplacian, offering improved approximation of the stress tensor. Numerical evidence for the analytical converJHQFH UDWHV LV JLYHQ DQG WKH LQÀXHQFH RI WKH VWDELOL]DWLRQ SDUDPHWHUV LV VWXGLHG ,Q WKH HQG ZH GLVFXVV VWDEOH ¿QLWH HOHPHQW SDLUV ZLWKRXW VWDELOL]DWLRQ

4.0.1 :H H[WHQG WKH /'* PHWKRG WR 6WRNHV ÀRZ SUREOHPV PRGHOHG E\ Stokes equations D+#  (   ( # 

in 1 &  4

(4.1)

ZLWK 'LULFKOHW DQG RXWÀRZ ERXQGDU\ FRQGLWLRQV #  # = #    

on / on / 4

(4.2) (4.3)

respectively. If / is empty, it is well known that the pressure is only determined up to a constant. Therefore, we require the additional condition " &  4   5 (4.4) and the compatibility condition



2

#
  5

(4.5)

CHAPTER 4. STOKES EQUATIONS

90

4.0.2 7KH FODVVLFDO YDULDWLRQDO SUREOHP DVVRFLDWHG ZLWK  VHH >%)@ LV GH¿QHG DV IROORZV Let  3 7 .  '  3 , 1*  '   on / (4.6)   3 7    if /  5  
16   
1 4     (4.7)
 1 else,  ZKHUH ZH XQGHUVWDQG WKH ERXQGDU\ FRQGLWLRQ LQ WKH VHQVH RI WUDFHV 7KHQ ¿QG #4   .   such that " & " & '  . 4 "D(#4 ('&  ( '4     ( #4     5  exists a velocity function 4.0.3 Lemma For any pressure function     (inf-sup  condition):  '  . with '    , satisfying ,

"

  & ( '4   M   4

(4.8)

where the constant J depends on 1 only.

4.0.4 ,Q YLHZ RI WKH PL[HG IRUP RI WKH /DSODFLDQ  XQGHUO\LQJ WKH /'* PHWKRG ZH WUDQV IRUP  LQWR WKH V\VWHP L  D(# ( L  (   ( #  

in 14

(4.9)

ZLWK ERXQGDU\ FRQGLWLRQV  DQG   :H ZLOO DVVXPH WKDW L4 #4   0  .   and 0  3 ,  *   4 .  3 ,  *  4    3 ,   !  4.0.5 Assumption: For estimates of the velocity error in
 and weaker norms, we have to assume that the solution   4   of the Stokes problem ( D(    (    (    4 admits the stability estimate          & "   D  #  D   5,   D    5,  D   5, 5 5 ,

(4.10)

(4.11)

4.0.6 Remark: 7KLV DVVXPSWLRQ KROGV IRU LQVWDQFH LI WKH GRPDLQ 1 is bounded and for its  boundary holds =1   5 , ZLWK 'LULFKOHW ERXQGDU\ YDOXHV #  3 5   =1 and /  (see [T´em79, Proposition 2.2]).

4.1. LDG DISCRETIZATION

91

4.1 LDG discretization 4.1.1 Assumption: Following Section 3.1, we derive the LDG discretization of 4.9 by choosLQJ ¿QLWH HOHPHQW VXEVSDFHV 0 , . and  of 0, . and , respectively: 0  ,"   4

.  ,"  4

  ," 4

(4.12)

where (   and (; , ( are either equal to ( or to (  , independent of each other. Again,  we limit spaces ," to the case where the mesh consists of parallelograms only. 4.1.2 Multiplication with a test function on each cell % and integration by parts yield "

 " & &   "#4 ( : &   #4 :
 2  "  & " 7 L4 : & L4 ('   L4 ' 
 2  "4 ( '&   4 '
 2  4 '   #4 (   #4 
 2  

(4.13)

Now, the traces of L, # and  on =% PXVW EH UHSODFHG E\ VLQJOH YDOXHG QXPHULFDO ÀX[HV WR UHQGHU WKLV HTXDWLRQ ZHOOGH¿QHG 4.1.3 The trace of # DSSHDUV LQ WKH ¿UVW DQG WKLUG HTXDWLRQ RI  DQG ZH ZLOO GHQRWH WKH FRUUHVSRQGLQJ ÀX[HV E\ #; and #  UHVSHFWLYHO\ 7KHQ ZH GH¿QH D JHQHUDO VHW RI FRQVLVWHQW ÀX[HV RQ LQWHULRU HGJHV E\ L  

L

 # L
&&  M;  M # # 
&&4   M; # # 
&&4 #; 

#

# 

#

 M # #
&&  M # 
&&4  



 M # 
&&5

(4.14)

Remark that  does not contain the jump of #, since the corresponding term is already found in WKH GH¿QLWLRQ RI L  )XUWKHUPRUH ZH KDYH FKRVHQ WKH ÀX[HV WR UHQGHU WKH SUREOHP V\PPHWULF The parameters M( will be chosen according to the analysis later in this chapter. 4.1.4 2Q WKH 'LULFKOHW ERXQGDU\ WKHVH ÀX[HV DUH WDNHQ DV L   L  M #  #  
4 #;  # 4 #  # 4   4

(4.15)

where traces are taken from the interior only.  'H¿QLWLRQ 7KH /'* IRUPXODWLRQ RI WKH PL[HG 6WRNHV SUREOHP  UHDGV DV IROORZV ¿QG IXQFWLRQV L 4 # 4    0  .   such that  ; :  - L 4 :   2 # 4 :   2 '4 L    # 4 '   '4     '   # 4     4    

:  0 '  .    4

(4.16)

CHAPTER 4. STOKES EQUATIONS

92

ZKHUH WKH ELOLQHDU IRUPV DUH GH¿QHG DV " & - L4 :   7 L4 :  4

 " & 2 '4 :   '4 ( :  

#

 # # 
&&M; 4 # :
&&
4





  #4 '  M # # 
&&4 # ' 
&&
 M # 
4 ' 

 4



" &   '4    4 ( '  



 M # 
&&4 # '
&&
 4 '

 4





  4   M # 
&&4 # 
&&
5

(4.17) (4.18) (4.19) (4.20) (4.21)



The linear forms on the right hand side of (4.16) are

 ; :   # 4 :

 4

 " &  '  4 '   M # 
4 ' 

 4



     4 #

 5

(4.22) (4.23) (4.24)



4.1.6 Remark: By integration by parts, the forms 2 54 5 and  54 5 admit the alternative representation 



" & (4.25) 2 '4 :    ('4 :   # ' 
&&4

:

 # :
&&  M;
 '4 :





" &

 (4.26)  '4   '4 (  

'

 M # '
&&4 # 
&&




4.1.7 :H GH¿QH WKH ELOLQHDU IRUP DVVRFLDWHG ZLWK WKH 6WRNHV SUREOHP DV " &  L4 #4 * :4 '4  ) - L4 :   2 #4 :   2 '4 L   #4 '   '4    #4    4 4 DQG WKH DVVRFLDWHG HQHUJ\ VHPLQRUP 4   L4 #4  ) - L4 L   #4 #   4 5 S

(4.27)

4.1.8 Lemma: Let M and M be positive and the discrete spaces chosen according to Assumption 4.1.1. Let the Dirichlet boundary / be nonempty. Then, the discrete equation " &  L 4 # 4  * :4 '4   ; :    '   

:4 '4   0  .   4 (4.28)

admits a unique solution in 0  .   . Proof. :H VKRZ WKDW WKH KRPRJHQHRXV V\VWHP DGPLWV WKH RQO\ VROXWLRQ  ,QGHHG WDNLQJ :  L , '  # and    in (4.28) yields   L 4 # 4    5 S

4.1. LDG DISCRETIZATION

93

Therefore, L  . Furthermore, # # 
&&   and # 
&&   on  and #   on  . &RQVHTXHQWO\ LQ YLHZ RI WKH ¿UVW HTXDWLRQ LQ  & " (# 4 :    :  0 4

yielding #   E\ WKH VDPH DUJXPHQW DV LQ WKH SURRI RI /HPPD  )LQDOO\ LQ YLUWXH RI WKH VHFRQG HTXDWLRQ LQ   & " '4 (   5

7KHUHIRUH WKH SUHVVXUH  is constant on 1 LQ WKH FDVH RI WHQVRU SURGXFW SRO\QRPLDOV E\ /HPPD   1RZ HLWKHU    on / RU  \LHOGV   .  'H¿QLWLRQ 7KH VWDQGDUG /'* PHWKRG IRU 6WRNHV HTXDWLRQV LQ >&.66@ XVHV WKH IRO ORZLQJ FKRLFH RI SDUDPHWHUV OHW  be an edge and % , %  WKH FHOOV DGMDFHQW WR  RQO\ % if  LV D ERXQGDU\ HGJH  7KHQ  

4 
if      ) 
if    4 and M  M 

D M  4    M   D

 

5

+HUH WKH SDUDPHWHUV M   and M  PD\ GHSHQG RQ WKH VKDSH RI WKH JULG FHOOV EXW QRW RQ . Furthermore, we choose M;  4

M  5

and

 7KHRUHP HQHUJ\ HUURU  Assume that the mesh is shape regular and the mesh size variation is bounded according to assumptions 1.1.13 and 1.1.14. Let L4 #4  be a solution to equation  and let L 4 # 4   be a solution to  , where the discrete spaces are FKRVHQ DFFRUGLQJ WR $VVXPSWLRQ  DQG WKH FRHI¿FLHQWV DFFRUGLQJ WR 'H¿QLWLRQ  ,I #  3 ,  *   and   3 ,   and D  3 , with   (  , then the energy seminorm of the error admits the estimate        ; 4  4    D   #  D    - 5 S ,





Proof. $JDLQ ZH KDYH VWDELOLW\ E\ GH¿QLWLRQ RI WKH HQHUJ\ VHPLQRUP 7KHUHIRUH LW LV VXI¿FLHQW WR YHULI\ DVVXPSWLRQV  DQG  RI WKH JHQHULF WKHRU\ 7KLV LV DFKLHYHG LQ OHPPDV  DQG  EHORZ ZKHUH ZH HQWHU L  D(#  3 , *   .

CHAPTER 4. STOKES EQUATIONS

94

4.1.11 Lemma: Assume that the mesh is shape regular and the mesh size variation is bounded according to assumptions 1.1.13 and 1.1.14. Let ,; , , and , be the
 -projections onto the spaces 0 , . and  according to Assumption 4.1.1, respectively. Let L  3 ,  *   , #  3 , *   and   3 ,  and furthermore let :  3  ,  *   , '  3  ,  *   and   3  ,  . If M and M DUH FKRVHQ DFFRUGLQJ WR 'H¿QLWLRQ  then the
 -projection errors admit the estimate " &  ;; 4 ; 4 ; * ;/ 4 ;! 4 ;        "  &   D    L ,  D   #,  D    ,



      "      & D   :   D    '  ,  D       , 5 (4.32)  ,





In particular,

" &  ;; 4 ; 4 ; * ;; 4 ; 4 ; "       D   L 

,

      &  D   #,  D    , 5 (4.33)



" & Proof. We split the form  54 54 5* 54 54 5 into its components and estimate the single forms separately. Since the forms - 54 5, 2 54 5 and  54 5 are the same as in Lemma 3.2.4, only in a YHFWRUYDOXHG YHUVLRQ LW LV VXI¿FLHQW WR SURYH WKH HVWLPDWH IRU  54 5 and  54 5. For 4   3 ,  , we obtain from trace estimate (1.24)  4   

          D     D     



 

"    &"    &       D     D   , D      D    , 5



Inserting now ; for  and ; for  and applying approximation result 1.2.13 yields        ; 4 ;   D    , D       , 5



The estimate of the form  54 5 uses the same trace estimates as the one for 2 54 5. Finally, the second estimate follows immediately by letting :  L, '  # and   . 4.1.12 Lemma: Let the functions L, # and  as well as the projection operators be as in Lemma 4.1.11. Then, for any :4 '4   0  .   , the estimate

" &  ;; 4 ; 4 ; * :4 '4    "         &  :4 '4 S D    L ,  D   #,  D    , 4 (4.34)





holds with a constant c independent of the mesh size.

Proof. Again, we split the Stokes bilinear form into its components and estimate each of them separately. For the forms - 54 5, 2 54 5 and  54 5, the estimates proven in Lemma 3.2.5

4.1. LDG DISCRETIZATION

95

KROG VLQFH WKH /'* VHPLQRUP LV ERXQGHG E\ WKH 6WRNHV VHPLQRUP LW LV VXI¿FLHQW WR SURYH FRUUHVSRQGLQJ HVWLPDWHV IRU  54 5 and  54 5 here. %\ &DXFK\6FKZDU] LQHTXDOLW\ DQG DSSUR[LPDWLRQ UHVXOW  ZH REWDLQ LPPHGLDWHO\ M  )         4   ; 4   D     ; 
 4   D    , :4 '4 S 5



6LPLODU WR WKH HVWLPDWH RI 2 54 5 ZH XVH  54 5 LQ ERWK IRUPV  DQG  WR H[SORLW RUWKRJRQDOLW\ RI WKH
 SURMHFWLRQ     #      ; 4   

;

4 # 
&&
    ; 
 ##&&




4     D   #,  4 4 



and 

     '4 ;    

;

4 # '
&&
 ; 4 '

 





4     D    ,  '4 '5 

4.1.13 Lemma (inf-sup condition): Assume that the parameters of the LDG method are choVHQ DFFRUGLQJ WR 'H¿QLWLRQ  DQG WKH GLVFUHWH VSDFHV DFFRUGLQJ WR $VVXPSWLRQ  7KHQ there exist positive constants  and  independent of the mesh size  such that the following UHOD[HG  LQIVXS FRQGLWLRQ   holds: for all :4 '4   0  .   , there is a function +  . with 4 +4 S    such that      " &  :4 '4 * 4 +4    D       :4 '4 S 5



Proof. Let :4 '4  EH JLYHQ %\ WKH FRQWLQXRXV LQIVXS FRQGLWLRQ   WKHUH H[LVWV D IXQFWLRQ #  3 , 1 ZLWK "

   & ( #4    D    

     D  #   5 ,



1RZ OHW +  , # be the
 SURMHFWLRQ RI # into .  7KHQ ZH KDYH " &  :4 '4 * 4 +4   2 +4 :    '4 +   +4 5

:H FRQWLQXH E\ HVWLPDWLQJ WKHVH WHUPV VHSDUDWHO\ XVLQJ WKH VDPH HVWLPDWHV DV LQ WKH SURRI RI /HPPD  )LUVW 2 +4 :   2 ; 4 :   2 #4 :      "          &   #, :    (#4 :   D  #, D   :    :4 '4 S 5

CHAPTER 4. STOKES EQUATIONS

96 Then,

4        '4 +   '4 ;   D  #,  '4 '   :4 '4 S 5

Finally, we split

 +4    #4    ; 4 5 Since # is in 3 , 1*   WKH ERXQGDU\ WHUPV LQ WKH ¿UVW SDUW YDQLVK DQG WKH WHUP LV HVWLPDWHG by (4.38). The remainder is estimated by 4        ; 4   D  #,  4    :4 '4 S 4

as soon as   . Combining these estimates, we obtain

" &  :4 '4 * 4 +4    #4    ; 4   2 +4 :    '4 +         D       :4 '4 S          :4 '4 S 4   D     

by Young inequality.

4.1.14 Theorem: Under the assumptions of Theorem 4.1.10, the
 -norm of the error in the pressure admits the estimate  !          D #   , 5 ,

(4.39)



Proof. The key ingredient to the proof of this theorem is the relaxed inf-sup condition of Lemma 4.1.13, which we apply to   ,  . It allows us to estimate      " &  D   ,     ,; ; 4 ,  4 ,  * 4 +4   ;; 4 ; 4 ; S   " &   ;; 4 ; 4 ; * 4 +4   ;; 4 ; 4 ;  5 S

1RZ ZH HVWLPDWH WKH ¿UVW WHUP E\ /HPPD  DQG UHPDUN WKDW DQ HVWLPDWH IRU WKH VHFRQG LV an intermediate result of Theorem 4.1.10. Therefore, !             D   ,    4 +4 S D   #,  D    ,



!         D   #,  D    ,



!          4  ,     D   #,  D    ,





E\
4.1. LDG DISCRETIZATION

97

4.1.15 Theorem (weak estimates): Let additionally to the assumptions of Theorem 4.1.10 the GRPDLQ DQG FRHI¿FLHQWV RI WKH SUREOHP EH VXFK WKDW IRU D QRQQHJDWLYH LQWHJHU   (   the regularity estimate (2.5) holds with B    . Then, the error admits the estimate !            D     D     (4.40)  D    #,  D     , 5  

In particular, for   , we obtain



        # 5 ,

(4.41)



Proof. Like in the generic Theorem 1.3.14, let < be either and arbitrary function in 3 , 1*   if  9  or the error  if   . Furthermore, let <  3  , 1 arbitrary. Then, the theorem is proven if we can show that " & " & !           4<   4< D    # D          5    , ,  < 



 <  , ,

Let   4   be the solution of the dual problem (4.10) with right hand side < 4 <  and ;  D(  . Then, by consistency of the LDG method with the dual problem, we have for any function :4 '4   0  .  : & " & " & "  :4 '4 * ; 4  4   < 4 '  < 4  5 Entering :4 '4   ; 4  4   and applying Galerkin orthogonality yields & " & " < 4   < 4  & "   ; 4  4  * ; 4 ; 4 ; " & " &   ;; 4 ; 4 ; * ; 4 ; 4 ;   ,; ; 4 ,  4 ,  * ; 4 ; 4 ; 5

7KH ¿UVW WHUP LV HVWLPDWHG E\ /HPPD  DQG WKH UHJXODULW\ HVWLPDWH   " &  ;; 4 ; 4 ; * ; 4 ; 4 ;     "  &   D    #,  D     , 



   & "    D    ,  D     ,



    & "    D    #,  D     , 



   "   &  D  < ,  D   <  , 5



" & Since the bilinear form  54 54 5* 54 54 5 is symmetric, we can apply Lemma 4.1.12 to estimate the second term " &  ,; ; 4,  4 ,  * ; 4 ; 4 ;     "    &  ,; ; 4 ,  4 ,  S D      ,  D       ,



    & "    D    #,  D     , 



   & "    D    ,  D     ,



      & "       D   #,  D   , 



   & "    D  < ,  D   <  , 5



CHAPTER 4. STOKES EQUATIONS

98


  ; 

1 2 3 4 5 6 7

8.2e-1 5.1e-1 3.5e-1 2.2e-1 1.2e-1 6.2e-2 3.2e-2

1 2 3 4 5 6 7

8.5e-2 2.2e-2 5.5e-3 1.4e-3 3.5e-4 8.9e-5 2.2e-5

1 2 3 4 5 6

5.4e-3 1.2e-3 2.3e-4 3.4e-5 4.4e-6 5.6e-7

    order   order   order  , M    , M   5$ — 3.6e-1 — 5.2e-1 — 0.68 1.0e-1 1.80 5.6e-1 -0.11 0.52 2.5e-2 2.04 4.3e-1 0.39 0.70 5.6e-3 2.16 2.5e-1 0.75 0.86 1.3e-3 2.08 1.4e-1 0.91 0.94 3.3e-4 2.01 7.0e-2 0.96 0.97 8.3e-5 1.99 3.5e-2 0.98  , M    , M   5 — 1.4e-2 — 8.0e-2 — 1.96 1.5e-3 3.18 1.8e-2 2.16 1.99 1.8e-4 3.05 4.9e-3 1.87 1.98 2.3e-5 3.01 1.3e-3 1.91 1.99 2.9e-6 3.00 3.3e-4 1.96 1.99 3.6e-7 3.00 8.5e-5 1.98 2.00 4.5e-8 3.00 2.1e-5 1.99  , M    , M   5" — 2.7e-3 — 2.9e-3 — 2.14 1.9e-4 3.86 9.7e-4 1.59 2.41 1.3e-5 3.81 2.2e-4 2.13 2.79 9.2e-7 3.85 3.3e-5 2.74 2.94 6.1e-8 3.92 4.4e-6 2.93 2.98 3.9e-9 3.96 5.6e-7 2.98

Table 4.1: Convergence of LDG discretization for Stokes equations

4.1.16 We verify our results approximating the solution in example A.4.2 on page 172 on Cartesian grids (see example A.1.1. We use D   and refer to Section 5.2 on Oseen equations for different viscosity parameters. Table 4.1 shows that the convergence rates predicted by the analysis are indeed achieved and optimal. 4.1.17 )LJXUH  VKRZV WKDW WKH LQÀXHQFH RI WKH VWDELOL]DWLRQ SDUDPHWHU M  on the discretization accuracy is similar to the results for Poisson equation shown earlier in Figure 3.1 on page 76). Moreover, the parameter M  GRHV QRW LQÀXHQFH WKH HUURU DW DOO RYHU D ZLGH UDQJH We will exploit this fact later and use this value to adjust the condition number of the Schur FRPSOHPHQW 3RLVVHXLOOH ÀRZ H[DPSOH $ RQ &DUWHVLDQ    grid,  elements). 4.1.18 We demonstrate convergence of the discretization in three dimensions computing PoisVHXLOOH ÀRZ LQ D F\OLQGULFDO SLSH VHH H[DPSOH $ 7KH FXUYHG ERXQGDU\ LV DSSUR[LPDWHG E\ a bilinear mapping (yielding isoparametric  elements).) The solution on the grid after one UH¿QHPHQW VWHS LV VKRZQ LQ )LJXUH  7KH FRQYHUJHQFH UHVXOWV DUH VKRZQ LQ 7DEOH  7KH\ exhibit the same expected convergence rates for L and # as the results in two dimensions. The pressure exhibits superconvergence here, an effect also observed with stabilized conforming methods (cf. e.g. [Ber01]). 4.1.19 Remark: Like in the case of the LDG method for the Poisson equation, we can eliminate the variable L by cellwise static condensation. Obviously, our results still hold for the

4.1. LDG DISCRETIZATION

-4

2

2

-3

99

-2

2

2

-1

2

0

2

1

2

2

2

3

Guu

-4

2

2

-3

-2

2

2

-1

2

0

Guu

2

1

2

2

2

3

24 3 2 22 21 20 -1 Gpp 2 2-2 -3 2 2-4 2-5 2-61.2e-04 4 2 1.0e-04 8.0e-05 7.0e-05 6.0e-05

24 23 22 21 20 Gpp 2-1 2-2 -3 2 2-4 2-5 2-6 4 2 6.0e-03 5.0e-03 4.0e-03 3.0e-03

Figure 4.1: Discretization error depending on the two parameters. Errors  (top) and  (bottom) with homogeneous  elements

CHAPTER 4. STOKES EQUATIONS

100

)LJXUH  'LVFUHWL]DWLRQ RI WKUHH GLPHQVLRQDO 3RLVVHXLOOH ÀRZ JULG DIWHU ¿UVW UH¿QHPHQW DQG solution      
;  order   order   order   M     M   5" 1 1.3e+0 — 1.3e-1 — 2.3e+0 — 2 5.3e-1 1.28 3.8e-2 1.79 6.3e-1 1.85 3 2.5e-1 1.08 1.0e-2 1.88 1.7e-1 1.86 4 1.2e-1 1.01 2.8e-3 1.92 4.7e-2 1.88 Table 4.2: Convergence of LDG discretization for Stokes equations in 3D UHVXOWLQJ V\VWHP VLQFH LW LV DOJHEUDLFDOO\ HTXLYDOHQW 8VLQJ WKH HTXLYDOHQFH LQ &RUROODU\  LW LV FOHDU WKDW WKH UHVXOWV RI WKLV FKDSWHU DSSO\ WR D 6WRNHV GLVFUHWL]DWLRQ ZLWK WKH LQWHULRU SHQDOW\ PHWKRG DV ZHOO ,Q SDUWLFXODU H[LVWHQFH DQG FRQYHUJHQFH UHVXOWV IRU # and  H[WHQG WR WKH system " &  -IP  '  .  # 4 '   ' 4     '  (4.42)  # 4      4           4 " & ZKHUH WKH IRUP -IP  54 5 is understood to be applied component by component.

4.2 Stable Finite Element Pairs 4.2.1 8S WR QRZ ZH FRQVLGHUHG RQO\ GLVFUHWL]DWLRQV ZLWK D SUHVVXUH VWDELOL]DWLRQ 6LQFH VWDEOH ¿QLWH HOHPHQW SDLUV ZLOO EH RI LPSRUWDQFH LQ &KDSWHU  ZH ZLOO GLVFXVV WKHVH SDLUV KHUH 7KH\ DUH DQDO\]HG LQ WKH FRQWH[W RI PL[HG ¿QLWH HOHPHQWV VHH IRU LQVWDQFH >%)@  6LQFH PL[HG HOHPHQWV DUH EH\RQG WKH VFRSH RI WKLV ERRN ZH ZLOO FLWH WKH PRVW LPSRUWDQW UHVXOWV ZLWKRXW proof. Stable pairs are characterized by 4.2.2 Assumption (inf-sup condition): There exists a constant M 9  independent of the mesh-size such that for any    holds

 !&

   '4     M  5 '  ,

(4.43)

4.2. STABLE FINITE ELEMENT PAIRS

101

Equivalently, we may write   

'  .

  with M ' &  ..



  and 2'4     5

(4.44)

4.2.3 Lemma: If Assumption 4.2.2 holds, the system (4.16) admits a unique solution even if M  . Theorems 4.1.10 and 4.1.14 hold analogously.

Proof. This result is an immediate consequence of the well-posedness of the LDG method in the previous chapter and Theorem 1.1 in [BF91, Chapter II].

4.2.4 Proofs of Assumption 4.2.2 for special DG schemes can be found for instance in [HL02, 7RV 667@ $OO WKHVH SURRIV UHO\ RQ SURYLQJ WKH DVVXPSWLRQ ¿UVW IRU WKH VSDFH .div  . ! " div 15

(4.45)

Actually, if we can prove (4.44) for .div as a subspace of . , it is automatically proven for . as well. Therefore, we introduce " div 1FRQIRUPLQJ ¿QLWH HOHPHQW VSDFHV DQG LQYHVWLJDWH WKHLU SURS erties.  'H¿QLWLRQ 5DYLDUW7KRPDV HOHPHQWV  7KH ¿QLWH HOHPHQW VSDFH %"   of vector valued functions consists of piecewise polynomials with each element  satisfying the following conditions:

1. On the reference cell, we have #" )   0" 4 % "

that is, it consists of tensor product polynomials of 1degree ( plus in each component ' the additional set of polynomials of the form 0"    ' 0  with ' in " .

2. The restriction ' to a mesh cell % with transformation 2 VHH 'H¿QLWLRQ  LV D #" mapped by the Piola transformation polynomial '  % '2 0 

 (2 0 ' 0   (2 0

(4.46)

3. The normal components of ' at interfaces between cells are continuous, that is, on an edge  between cells % and % holds
' 
' 5

CHAPTER 4. STOKES EQUATIONS

102

4.2.6 Remark: )URP WKH GH¿QLWLRQ ZH VHH HDVLO\ .div  %"   & " div 1 and  ( % '  " 5

(4.47)

)XUWKHUPRUH %"   is a subset of "    EXW QRW RI "  1 7KHUHIRUH GH¿QLQJ MXPS WHUPV RQ HGJHV LV VWLOO UHDVRQDEOH DQG D GLVFUHWL]DWLRQ RI WKH YHORFLW\ ¿HOG LQ 6WRNHV HTXDWLRQV E\ %" LV VWLOO D '* PHWKRG 6LQFH WKH QRUPDO FRPSRQHQWV RI IXQFWLRQV LQ .div DUH FRQWLQXRXV WKHLU MXPSV LQ WKH IRUP  54 5 YDQLVK 7KHUHIRUH WKH ZHDN LQFRPSUHVVLELOLW\ FRQGLWLRQ  #4      

(4.48)

VLPSOL¿HV WR "

& ( #4       5

(4.49)

4.2.7 Lemma: The pair .  % and    with norms   and 5
  and 5  UHVSHFWLYHO\ IXO¿OOV $VVXPSWLRQ 

4.2.8 Lemma  of Raviart-Thomas elements): The pair .  %" and   "  (Stability with norms 5, and 5
  UHVSHFWLYHO\ IXO¿OOV $VVXPSWLRQ  IRU DQ\ ( with a constant M independent of .

4.2.9 $Q DOWHUQDWLYH WR 5DYLDUW7KRPDV HOHPHQWV DUH WKH %UH]]L'RXJODV0DULQL %'0 HO HPHQWV LQ >%'0@ :KLOH WKH PDWFKLQJ SUHVVXUH VSDFH IRU %" is "  WKH YHORFLW\ VSDFH " LV PDWFKHG E\ " .  'H¿QLWLRQ %'0 HOHPHQW  7KH VSDFH RI %'0 SRO\QRPLDOV RI GHJUHH ( RQ WKH UHIHU ence cell #4 & is "  "  (  0"    (  0 " 5 7KHVH VKDSH IXQFWLRQV DUH PDSSHG WR DQ DFWXDO PHVK FHOO E\ WKH 3LROD WUDQVIRUPDWLRQ   4.2.11 Remark: 7KLV VSDFH LV FKRVHQ VXFK WKDW ( " & " DQG IRU HYHU\ #  "  #
 "   on each face  .  'H¿QLWLRQ %'0 SURMHFWLRQ  :H GH¿QH IRU D VXI¿FLHQWO\ VPRRWK IXQFWLRQ # WKH FHOO ZLVH LQWHUSRODWLRQ RSHUDWRU & LQWR WKH VSDFH " E\ WKH LQWHUSRODWLRQ FRQGLWLRQV

  & #
4    #
4      "    " & & "  '  " %   & #4 '   #4 ' 

4.2. STABLE FINITE ELEMENT PAIRS

103

4.2.13 Remark: 7KH LQWHUSRODWLRQ RSHUDWRU LQ WKH SUHYLRXV DERYH GH¿QHV D SRO\QRPLDO & #  " XQLTXHO\ )XUWKHUPRUH LI DSSOLHG RQ WKH ZKROH PHVK LWV UHVXOW LV LQ " div 1 and the FRPPXWDWRU SURSHUW\ ( &  #  , ( # KROGV )RU GHWDLOV ZH UHIHU WKH UHDGHU WR >%)@ 4.2.14 Remark: 2Q WULDQJOHV WKHVH PRPHQWV DUH QRW VXI¿FLHQW WR GH¿QH WKH LQWHUSRODWLRQ XQLTXHO\ 7KHUHIRUH WKH PRPHQWV " & & " <  ." % 4 & #4 <   #4 < 

DUH DGGHG ZKHUH

." %   . 
 %  ) 2 < ' 2  ." %/ 5

+HUH 2 ) %/  % GHQRWHV WKH HOHPHQWDO PDSSLQJ DQG 2 LWV -DFRELDQ 2Q WKH UHIHUHQFH triangle %/  / 0 4 0 /  ) 0 / 9 4 0 /  0 / 7   WKH VSDFH 2" %/ LV GH¿QHG E\ 2" %/  2  " %/ ) ( 2   in %/4 2    on = %/ 5

4.2.15 Remark: Both vector spaces " and %" FRQWDLQ WKH FRPSOHWH SRO\QRPLDO VSDFHV  "  7KHUHIRUH WKH LQWHUSRODWLRQ HVWLPDWH  KROGV IRU WKHVH VSDFHV DV ZHOO

Chapter 5 Flow Problems In the following sections, we study the effect of the addition of lower order terms to the Laplacian or to the Stokes operator. The convergence results of the preceding chapters are used in combination with the stable advection discretization of Section 1.4. This way, results for advection-diffusion-reaction and Oseen equations are easily obtained. Advection-diffusion problems are a straight-forward extension of the methods developed for Poisson equation (see e. g. Cockburn and Dawson [CD00] and Gopalakrishnan and Kanschat [GK03c]). We present an example with a dominant reaction term, showing that the DG method yields considerably better approximation on coarse grids. 'LVFRQWLQXRXV *DOHUNLQ PHWKRGV IRU 2VHHQ HTXDWLRQV ZHUH ¿UVW VWXGLHG E\ &RFNEXUQ .DQ schat and Sch¨otzau in [CKS03b]. We present an alternative way of analyzing the scheme presented there in Section 5.2. We close this chapter presenting results on incompressible Navier-Stokes equations. A formulation using discontinuous piecewise solenoidal velocities and continuous pressures ZDV SURSRVHG E\ .DUDNDVKLDQ DQG -XUHLGLQL LQ >.-@ 7KH ¿UVW JHQXLQHO\ GLVFRQWLQXRXV Galerkin scheme was presented by Girault, Rivi`ere and Wheeler in [GRW02]. It uses a nonsymmetric version of the interior penalty method for the Stokes part of the equation and a stabilized trilinear form for the advection term. We combine their trilinear form with the LDG discretization of Stokes equations and show computational results indicating the feasibility of the method. $OWHUQDWLYHO\ WKH XVH RI GLYHUJHQFH FRQIRUPLQJ ¿QLWH HOHPHQW VSDFHV IRU WKH YHORFLWLHV KDV been proposed in [CKS05, CKS07]; this approach yields pointwise a. e. divergence free discrete solutions in  div
, such that the stability of the linearized problems is guaranteed, but it avoids the construction of a (piecewise) solenoidal basis. For the latter reason, the pressure of this method still approximates the physical quantity. This method is presented LQ WZR GLIIHUHQW ÀDYRUV DW WKH HQG RI WKH FKDSWHU

5.0.1 :H FRQVLGHU LQFRPSUHVVLEOH ÀRZ PRGHOHG E\ WKH VWDWLRQDU\ LQFRPSUHVVLEOH NavierStokes equations D+#  # (#  (   ( # 

in 1 & 

(5.1)

with boundary conditions (4.2) and (4.3) as for the Stokes problem; we restrict the analysis to the pure Dirichlet case, since the question of unique solutions in the presence of natural

CHAPTER 5. FLOW PROBLEMS

106

boundary conditions is still open. The following analytical results are quoted from [QV94, Chapter 10]. For the question of existence and uniqueness of solutions to this equation as well as for the numerical solution a suitable sequence of linear problems must be solved. This can be achieved by  'H¿QLWLRQ 3LFDUG LWHUDWLRQ  Let #  3 , 1 be given, Picard iteration GH¿QHV D sequence #" 4 (  4 5 5 5 through the recurrence relation D+#"  #" (#"  ("   ( #" 

(5.2)

 'H¿QLWLRQ The resulting linear system is usally refered to as Oseen equations. Their  (compare (4.7)) such that weak form is: let .  3, 1*   DQG ¿QG #4   .   " & " & " & " D(#4 (' &  + (#4 '  ( '4   4 ' ( #4   

'  .    

(5.3)

" &  /HPPD If the domain 1 is two or three dimensional, the trilinear form + (#4 ' admits the estimate " & + (#4 '  ++  #+  '+  5

(5.4)

Proof. Using H¨older inequality twice, we obtain       " & + (#4 '  ++  #' 
  ++  #
 ' 
 5

Applying the embedding Theorem 1.2.2, the estimate (5.4) follows with being the square of the constant in (1.9).  /HPPD Assume that +  3, 1*   and "

& ( +4 '  

' 
 15

(5.5)

 and Then the Oseen problem (5.3) has a unique solution #4   .     #    . .
  4 + D

(5.6)

5.1. ADVECTION-DIFFUSION-REACTION EQUATION

107

Proof. By partial integration and condition (5.5), we have for any +4 '  . " & " & " & " & + ('4 '   '4 + ('  ( +'4 '   '4 + (' 4

" & and thus + ('4 '  . Therefore, for any '  . " & " & D('4 ('  + ('4 '  D',

(5.7)

independent of +. Now, let

 3 7 .  '  . ( '   5

Then, the reduced problem " & " & " & D(#4 ('  + (#4 '  4 ' '  .

(5.8)

admits a unique solution #  . admitting estimate (5.6) due to (5.7) and the Lax-Milgram Lemma 1.3.6. 5.0.6 Theorem: If the right hand side  is divergence free and admits the estimate . . 

D 4

(5.9)

where is the diameter of 1 and the constant in (5.4), then the Navier-Stokes equa tions (5.1) in weak form have a unique solution #4   .  . Proof. By contraction property of Picard iteration; see e. g. [QV94, Ch. 10]. 5.0.7 We will approach the discretization of the Navier-Stokes equations (5.1) by combining the techniques of the previous chapters step by step. Therefore, we start by adding linear advection to the Laplacian and Stokes operators in order to end up using the discretization obtained in Picard iteration for the complete nonlinear problem.

5.1 Advection-Diffusion-Reaction Equation 5.1.1 The simplest model for diffusive transport problems is the linear advection-diffusionreaction equation D+#  2 (#  N#  4

(5.10)

with non-negative parameters D and N 7KH YHFWRU ¿HOG 2 
 1*   IXO¿OV WKH VWDELOLW\ condition ( 2  N   a.e. in 15

(5.11)

CHAPTER 5. FLOW PROBLEMS

108

5.1.2 We derive a discontinuous Galerkin discretization of equation (5.10) by simply adding the stable advection discretization (1.50), the interior penalty discretization of the Laplacian LQ  DQG DQ DGGLWLRQDO PDVV PDWUL[ \LHOGLQJ WKH ZHDN IRUPXODWLRQ ¿QG #  . " & " & " & '  . 5 (5.12) D-IP  # 4 '  E # 4 '  N# 4 '   4 '



Here, the space . is chosen as in Section 2.2. Accordingly, the energy norm is chosen as         # (5.13)  D #  #8  N# 5 ADR

5.1.3 Remark: The stability and convergence analysis of this discretization are conducted by combining the results of sections 1.4 and 2.2. If the test functions used for the stability analysis in both sections would be the same, we could just add the results and yield the combined estimates. As it is, the test functions for the advection equation had to be augmented by the term M2 (# in Lemma 1.4.11 on page 30. Therefore, we will have to show that this term does not spoil stability and boundedness of the other operators involved. 5.1.4 Lemma: For any function '  3 , , the energy norm admits the
 -projection estimate     '  , '   B' , 4 (5.14) ADR

#



with B   D

4

4

.

2

# 4 N .

Proof. By summing up the projection estimates for each term of the energy norm separately, namely (1.57), (2.17) and (1.21). 5.1.5 Lemma: There exist constants  and M such that the following stability estimate holds for any '  . (abbreviating '  M2 ('):    " & " & " & ! '  (5.15)   E '4 '  '  N'4 '  '  5   D-IP  '4 '  ' ADR

Proof. First, we apply Lemma 1.4.11 to the advection term and obtain constants % and M% such that   & " '  M% 2 ('   % E '4 '  M% 2 (' 5 8 We observe that this estimate holds with a different constant  for any positive M  M%

The diffusion and reaction terms are both handled using inverse estimate, e. g. " & " & N'4 M 2 ('  M N'4 ' 4 " & and choose M such that M  6. The terms in D-IP  '4 M 2 (' are estimated accordingly. Finally, we chose M as the minimum of M% , M and M to obtain the stability estimate. 

An -analysis of a formulation with a nonsymmetric interior penalty discretization can be found in [HSS02]

5.1. ADVECTION-DIFFUSION-REACTION EQUATION 100

100

DG CG

-1

10 error

error

DG CG

-1

10

10-2 -3

10-2 -3

10

10-4 101

109

10

102

103 104 105 matrix entries

106

107

10-4 101

102

103 104 105 matrix entries

106

107

Figure 5.1: Comparison of DGFEM and CGFEM (both  for a reaction dominated diffusion   problem with D   (left) and D   (right) 5.1.6 Corollary: By the same inverse estimates as used in the last proof, we obtain a constant  independent of  and the parameters such that for any function '  .     '  M2 ('  (5.16)  ' ADR 5 ADR 5.1.7 Theorem: Let #  3 ,  with 6    (   and #  . be solutions of equations (5.10) and (5.12), respectively. Then, the error   #  # admits the estimate       (5.17)  B#, 4 ADR

#



where B   D

4 # 4 2. 4 N .

Proof. The proof follows the generic proof of Theorem 1.3.9. Assumptions 1.3.5 to 1.3.8 hold true as a consequence of the preceding lemmas and corollaries. 5.1.8 In Figure 5.1, we compare the accuracy of solutions to the discontinuous Galerkin VFKHPH  DQG FRQWLQXRXV ¿QLWH HOHPHQWV IRU GLVFUHWL]DWLRQ RI H[DPSOH $ :H REVHUYH that the discontinuous method is better on coarse meshes, as long as the boundary layer is not UHVROYHG SURSHUO\ $Q H[SODQDWLRQ IRU WKLV EHKDYLRU LV WKDW WKH GLVFRQWLQXRXV PHWKRG GRHV QRW REH\ WKH ERXQGDU\ YDOXHV DV ORQJ DV WKH OD\HU LV QRW UHVROYHG WKHUHIRUH EHLQJ PXFK FORVHU WR the limit solution. On the other hand, the continuous method is known to produce oscillations VHH )LJXUH  XQOHVV VSHFLDO PHDVXUHV OLNH PDVV OXPSLQJ RU VWDELOL]DWLRQ DUH WDNHQ 7KHVH on the other hand must be implemented carefully. Mass lumping for instance spoils Galerkin orthogonality of the error. The discontinuous scheme shows that the additional freedom introGXFHG WKURXJK WKH ÀX[ IXQFWLRQV FDQ EH XVHG WR DGDSW WKH DFFXUDF\ RI ERXQGDU\ YDOXHV WR WKH GLVFUHWL]DWLRQ DFFXUDF\ For mesh parameters  RI WKH VL]H RI WKH ERXQGDU\ OD\HU DQG VPDOOHU ZH REVHUYH WKDW WKH DV\PS WRWLF EHKDYLRU RI WKH PHWKRGV LV VLPLODU EXW WKH FRQWLQXRXV PHWKRG QHHGV OHVV PDWUL[ HQWULHV WR  The error was integrated using a  point Gauß quadrature rule. This way, we asserted that the error was PHDVXUHG FRUUHFWO\ HYHQ RQ FRDUVH PHVKHV

CHAPTER 5. FLOW PROBLEMS

110 1.4 1.2 1 0.8 0.6 0.4 0.2 0

1.4 1.2 1 0.8 0.6 0.4 0.2 0

Figure 5.2: CGFEM (left) and DGFEM (right) solution of a reaction-dominated (D  diffusion problem

 ) 

achieve the same accuracy as the DG scheme. A quantitative investigation reveils that the error RQ ¿QH PHVKHV IRU WKHVH VFKHPHV LV DERXW WKH VDPH LI FHOO VL]HV DUH HTXDO %XW LQ WKLV FDVH WKH '* VFKHPH KDV PRUH GHJUHHV RI IUHHGRP DQG D GHQVHU PDWUL[ VWHQFLO VXFK WKDW LWV QXPHULFDO effort is larger.

5.2 Oseen Equations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mented by an advection term and an additional reaction term to account for instance for time stepping schemes reads -DL  (#   ( L  2 (#  M#  (   ( #  

in 15

(5.18)

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5.2. OSEEN EQUATIONS

111

5.2.4 We apply the discretizations of the previous chapter and the previous section to the weak form of (5.18) to obtain the discontinuous Galerkin bilinear form - L 4 :   2 # 4 : "  ; :  &  2 '4 L    # 4 '  E # 4 '   '4     '   # 4     4    

:  0 4 '  . 4    4

(5.20)

ZKHUH WKH SDUWLFXODU ELOLQHDU IRUPV DUH GH¿QHG LQ HTXDWLRQV  WR   LQ SDUWLFXODU 2 54 :  is the gradient on the velocity space tested with stress functions : and "  54 &  is the divergence of the velocity tested with a pressure test function. E # 4 ' GH¿QHG LQ  LV   " The & form " & , applied to each component of # VHSDUDWHO\ L H E #4 '  E #4 ' . The bilinear form of the complete Oseen problem is then " & " &  L4 #4 * :4 '4  ) - L4 :   2 #4 :   2 '4 L   #4 '  E #4 '   '4    #4     4  /LNH LQ WKH SUHYLRXV VHFWLRQ ZH GH¿QH WKH HQHUJ\ VHPLQRUP IRU WKLV SUREOHP E\ DGGLQJ WKH HQHUJ\ VHPLQRUP RI WKH 6WRNHV SUREOHP DQG RI WKH DGYHFWLRQ SUREOHP \LHOGLQJ        #  5 L4 #4   L4 #4   8 O S 

6LQFH ERWK IRUPV DUH VHPL GH¿QLWH VR LV WKHLU VXP (YHQ PRUH ZH KDYH WKH IROORZLQJ VWDELOLW\ result. 5.2.5 Lemma: There exist constants  and M VXFK WKDW WKH 2VHHQ ELOLQHDU IRUP GH¿QHG in (5.20) admits the stability estimates " &    L4 #4 * L4 #4    " & L4 #4    L4 #4 * L4 #  M2 (#4  5 O

(5.21) (5.22)

Proof. This result is obtained by combining the proofs of stability of the& Stokes discretization " LQ &KDSWHU  DQG /HPPD  6LQFH WKH WHUP  4 #4 * 4 M2 (#4  can be controlled by LQYHUVH HVWLPDWHV H[DFWO\ OLNH LQ /HPPD  ZH OHDYH WKH GHWDLOV WR WKH UHDGHU 5.2.6 Theorem: Under the assumptions of Theorem 4.1.10 and those made in paragraph 5.2.4, the error ; 4  4   between the discrete solution L 4 # 4   of (5.20) and the continuous solution L4 #4  of (5.18) admits the estimates !       ; 4  4     B#   , 4 O ,

# where B   D 4 2. .



(5.23)

CHAPTER 5. FLOW PROBLEMS

112

Proof. Replacing the Stokes bilinear form by the Oseen bilinear form, estimates (4.33) and (4.34) are transformed into !          L4 #4    D   L   B#,  D   , 4 , O





and

" &  ;; 4 ; 4 ; * :4 '4 

!            :4 '4 O D   L ,  B#,  D   , 5





Combined with stability estimate (5.22) and Galerkin orthogonality, these two estimates allow us to obtain an energy norm estimate following the generic proof of Theorem 1.3.9. Estimate (5.23) follows by letting L  D(#.

5.2.7 Remark: Again, like in Remark 4.1.19, the stability result holds if the additional stress tensor L is eliminated from the system by static condensation. Then, combining the stability analysis for the Stokes problem with the analysis for the advection-reaction-diffusion problem in (5.12), we obtain stability and convergence results in the norm consisting of the
 -norm of WKH SUHVVXUH DQG WKH QDWXUDO QRUP IRU WKH DGYHFWLRQ SUREHOP GH¿QHG LQ        #  #ADR   5 O;IP 5.2.8 :H VKRZ UHVXOWV IRU .RYDV]QD\ ÀRZ H[DPSOH $ RQ SDJH  ZLWK GLIIHUHQW 5H\QROGV numbers   6D in Figure 5.3, where we used  elements for . and  . We observe  convergence for # and  for L over the whole range of meshes and viscosities. Again, this is the superconvergence effect of the advection discretization observed earlier in this chapter. For WKH SUHVVXUH ZH REVHUYH WKH SUHGLFWHG FRQYHUJHQFH UDWHV RQ ¿QH JULGV 2Q FRDUVHU JULGV WKH SUHVVXUH DSSUR[LPDWLRQ VHHPV WR VXIIHU UHVXOWLQJ LQ D KLJKHU FRQYHUJHQFH UDWH 7KH ¿JXUH DOVR VKRZV WKDW WKH PHWKRG LV UREXVW ZLWK UHVSHFW WR WKH 5H\QROGV QXPEHU DV WKH HUURUV GR QRW LQFUHDVH 7KH VFDOLQJ RI WKH HUURUV LQ WKH ¿JXUH LV DFFRUGLQJ WR WKH WKHRUHWLFDO results. The errors in L and  are in fact decreasing, which is due to the fact that the functions themselves are of order D VHH )LJXUH $ RQ SDJH   &RQFOXGLQJ ZH REVHUYH WKDW WKH UHODWLYH errors in #,  and L are nearly independent of the viscosity.

5.3 Navier-Stokes Equations 5.3.1 After an accurate and stable discretization for the Oseen problem has been found, we would like to use this discretization in the Picard iteration (5.2). We cannot do this in a straight forward fashion, since the discrete velocities are only weakly divergence free, when tested with the pressure space (see [CKS05]). We will shortly discuss the problems arising by considering the continuous form of the equations (5.1).

5.3. NAVIER-STOKES EQUATIONS

113

1e+02 1e+01

N

-1/2

||eS||L2

1e+00 1e-01 1e-02 Re = 1 Re = 10 Re = 20 Re = 50 Re = 100 Re = 200 Re = 500

1e-03 1e-04 1e-05 2

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7

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7

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Refinement level 1e+01 1e+00

1e-02 1e-03

N

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1e-01

1e-04 Re = 1 Re = 10 Re = 20 Re = 50 Re = 100 Re = 200 Re = 500

1e-05 1e-06 1e-07 2

3

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Refinement level 1e+02 1e+01 1e+00

1e-02 1e-03

N

-1/2

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1e-01

1e-04

Re = 1 Re = 10 Re = 20 Re = 50 Re = 100 Re = 200 Re = 500

1e-05 1e-06 1e-07 2

3

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Refinement level

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2 3 4 5 6

  ;  7.1e-1 4.3e-1 2.5e-1 1.4e-1 7.3e-2

order — 0.73 0.75 0.86 0.94

    7.2e-1 2.7e-1 6.8e-2 1.9e-2 5.1e-3

order — 1.43 1.97 1.88 1.88

    order 1.3e+0 — 4.8e-1 1.44 2.1e-1 1.18 9.7e-2 1.12 4.7e-2 1.04

steps 38 35 25 19 18

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5.3.10 The DG discretization of " & the stationary Navier-Stokes equations is obtained by replacing the advection form E # 4 ' in (5.20) by  # * # 4 ' (and here replacing the LDG method E\ LQWHULRU SHQDOW\  \LHOGLQJ WKH QRQOLQHDU V\VWHP RI HTXDWLRQV " & -IP  # * # 4 '   '4     ' '  . 4  # 4 '    # 4     4        5

(5.28)

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(5.29)

and let #  4     #4 

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5.3.13 The remainder of this chapter is devoted to schemes using the original form (5.1) of the Navier-Stokes equations. They are based on enforcing exact incompressibility in the space " div 1 GH¿QHG LQ   ,W LV FOHDU IURP WKH UHDVRQLQJ LQ 3DUDJUDSK  WKDW WKLV \LHOGV VWDEOH 2VHHQ SUREOHPV :H FRQVLGHU WZR YHUVLRQV RI WKLV PHWKRG WKH ¿UVW UHOLHV RQ D VWDEOH '* VFKHPH ZLWK PL[HG RUGHU " 6" discretization for . and  ZLWK D SRVWSURFHVVLQJ RI WKH YHORFLW\ LQWR D GLYHUJHQFH FRQIRUPLQJ VSDFH /DWHU RQ ZH SUHVHQW D '* VFKHPH GLUHFWO\ formulated in a subspace of " div 1.

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+HUH ZH KDYH XVHG LQWHJUDWLRQ E\ SDUWV WKH SURSHUWLHV RI  DQG   7KXV ZH KDYH ( #   in 1.

CHAPTER 5. FLOW PROBLEMS

118

5.3.17 Lemma: Let '  . . Then we have     '    '  4  

(5.31)

with a stability constant  9  that is independent of the mesh-size. The proof is rather technical and lengthy, so we refer the reader to [CKS05].  'H¿QLWLRQ Using the postprocessing operator  ZH FDQ GH¿QH WKH ZHDN IRUP RI RXU GLVFUHWL]DWLRQ RI WKH 1DYLHU6WRNHV HTXDWLRQV DV " &  -IP  # 4 '   # * # 4 '   '4     ' '  . 4 (5.32)    4        4  # 4  ZKHUH WKH IRUPV DUH WKH VDPH DV LQ  DQG  +* #4 ' LV VLPSO\ WKH DGYHFWLRQ IRUP  with 2 replaced by +. 7KH 3LFDUG LWHUDWLRQ IRU WKLV SUREOHP LV DJDLQ REWDLQHG E\ UHSODFLQJ WKH ¿UVW DUJXPHQW WR  by WKH VROXWLRQ RI WKH SUHYLRXV VWHS 5.3.19 Lemma: Let #  .  . and + 4 + 4 '  . . Then there is a Lipschitz constant  independent of the mesh-size, such that        + * #4 '   + * #4 '   +  +  # '  5 " & Proof. $FFRUGLQJ WR WKH GH¿QLWLRQ RI E 54 5 LQ   ZH PXVW HVWLPDWH WKH IRUP " &  + * #4 '   + * #4 '  +  +  (#4 ' 



  +
#  #  4 ' 
 +
#  #  4 ' 






  +
#4 '

  +
#4 '

 4



ZKHUH WKH LQGLFHV LQGLFDWH XSVWUHDP DQG GRZQVWUHDP GLUHFWLRQV IRU + and + , respectively. By WKH GLVFUHWH 6REROHY HPEHGGLQJ  DQG /HPPD  WKH ¿UVW WHUP FDQ LPPHGLDWHO\ EH estimated through       & " +  +  (#4 '   +  + 
 # ' 


       +  +  # '  5 )RU WKH VHFRQG WHUP ZH REVHUYH WKDW IRU DQ\ YHFWRU ¿HOG 2  

2
#  # 4 ' 
  2
#  # 4 '
4



where the adjacent cells % and % of an edge are chosen arbitrarily. Therefore,  

 +
#  #  4 ' 
 +
#  #  4 ' 






  +  + 
#  # 4 '


    +  +   # # 
&&  








  '   5


5.3. NAVIER-STOKES EQUATIONS

119

    8VLQJ LQYHUVH HVWLPDWH DQG WKH IDFW WKDW E\ GH¿QLWLRQ # #&&
 # yields



 +
#  #  4 '

 






 +
#  #  4 '

 






       +  +  # '  5

)LQDOO\ LQ RUGHU WR HVWLPDWH WKH ERXQGDU\ WHUPV ZH REVHUYH WKDW IRU DQ\ YHFWRU ¿HOG 2 ZH REYLRXVO\ KDYH



 2
#4 '

 2
#4 '
 5



7KHUHIRUH DSSO\LQJ WULDQJOH LQHTXDOLW\ WR +
  +
 ZH REWDLQ



  +
#4 '

  +
#4 '

 





  +  + 





 

    #  '  



DQG ZH FRQWLQXH OLNH RQ LQWHULRU HGJHV 6XPPLQJ XS WKH UHVXOWV SURYHV WKH HVWLPDWH 5.3.20 Theorem: Assume that  

    8 ) 7 4 D  B

(5.33)

where B is the constant of the stability estimate for the interior penalty method (2.15),  the Lipschitz constant of Lemma 5.3.19 and  the constant from Friedrichs inequality (1.18). Then the DG method (5.32) GH¿QHV D XQLTXH VROXWLRQ # 4    .    ,W VDWLV¿HV WKH ERXQGV    

  #    4 (5.34)  D% B )        M    % B    4 (5.35)   B " & where % is the constant in the boundedness of -IP  54 5 . Finally,  

   5 #
 ### 
&&   #
#    DB  

 









(5.36)

Proof. )LUVW ZH HOLPLQDWH WKH SUHVVXUH IURP WKH SUREOHP E\ UHVWULFWLQJ RXUVHOYHV WR WKH ZHDNO\ GLYHUJHQFHIUHH VXEVSDFH RI . ,   '  . )  '4   

   5

7KH DSSUR[LPDWH YHORFLW\ LV WKXV FKDUDFWHUL]HG DV WKH RQO\ IXQFWLRQ #   such that " & " & -IP '   5  # 4 '   # * # 4 '  4 '

(5.37)

(5.38)

CHAPTER 5. FLOW PROBLEMS

120

Let us introduce the solution operator ! of the Oseen problem. For #   , #  !# denotes WKH VROXWLRQ RI WKH SUREOHP ¿QG #   such that " & " & '   5 -IP  #4 '   #* #4 '  4 ' Note that since #   we have, by Lemma 5.3.16, that ( #  . As a "consequence, this & problem is uniquely solvable. Furthermore, by the coercivity of the form -IP  54 5 in (2.15) and  in Lemma 5.2.5,   " &    " &    DB#  -IP  #4 #   #* #4 #  4 #    #  5 By Friedrichs inequality (1.18), we obtain      DB#     # 5 Hence, the solution # WR WKH DERYH SUREOHP VDWLV¿HV    

  #   4  DB

(5.39)

and ! maps

  into itself.

     

    '    '    DB



Next, we show that ! is a contraction on  under the smallness condition (5.33). To do so, let # 4 # be in  , and set #  !# , #  !# . Then   " & DB#  #  -IP  #  # 4 #  # 5

Since

" & -IP  #  # 4 '   # * # 4 '   # * # 4 '  4

for any '   , taking '  #  # we get   DB#  #    # * #  # 4 #  #    


  # * # 4 #  #    # * # 4 #  # 5

By the continuity property of  in Lemma 5.3.19, the bound (5.39) and the continuity of the post-processing operator  in Lemma 5.3.17,         DB#  #    #  #  #  #  #      

     #  #  #  #     DB     DB8 #  #  #  #  5 This implies that

  #  # 



   8 #  #  4

(5.40)

5.3. NAVIER-STOKES EQUATIONS

121

and so, if 8 7  WKDW LV LI WKH VPDOOQHVV FRQGLWLRQ  LV VDWLV¿HG WKH PDSSLQJ ! is a contraction. Hence, ! KDV D XQLTXH ¿[HG SRLQW #    ZKLFK LV WKH VROXWLRQ WR SUREOHP   Now that the velocity # KDV EHHQ FRPSXWHG WKH SUHVVXUH LV WKH VROXWLRQ    of " & " &  '4    4 '  -IP  # 4 '   # * # 4 '

'  . 5

(5.41)

'XH WR ERXQGHGQHVV RI WKH LQYROYHG IRUPV DQG WKH )ULHGULFKV LQHTXDOLW\ LQ /HPPD WKH ULJKWKDQG VLGH GH¿QHV D FRQWLQXRXV OLQHDU IXQFWLRQDO RQ . and   7KH LQIVXS FRQGLWLRQ LQ /HPPD  WKHQ JXDUDQWHHV WKH H[LVWHQFH RI D XQLTXH VROXWLRQ  WR WKH DERYH SUREOHP ,W FDQ WKHQ HDVLO\ EH VHHQ WKDW WKH WXSOH # 4   is the unique solution to the DG method in 5.20 with velocity 2  # .   1H[W ZH VKRZ WKH VWDELOLW\ ERXQGV IRU # 4   7KH ERXQG IRU #  in (5.34) follows since #    7R REWDLQ WKH ERXQG IRU WKH XSZLQG WHUP LQ   QRWH WKDW

                      DB#  5 DB #    # * # 4 #      #    DB  " & 6LPLODUO\ WR WKH SUHYLRXV DUJXPHQWV KHUH ZH KDYH XVHG WKH FRHUFLYLW\ RI -IP  54 5  HTXDWLRQ    with test function '  #  DQG WKH )ULHGULFKV LQHTXDOLW\ %ULQJLQJ WKH WHUP  DB#  to the OHIWKDQG VLGH DQG REVHUYLQJ WKH FRHUFLYLW\ RI  JLYH WKH VWDELOLW\ ERXQG LQ  

0RUHRYHU XVLQJ WKH LQIVXS FRQGLWLRQ LQ /HPPD  WKH )ULHGULFKV LQHTXDOLW\ LQ /HPPD  WKH FRQWLQXLW\ RI WKH ELOLQHDU IRUPV /HPPD  DQG WKH VWDELOLW\ RI  LQ /HPPD  ZH have from (5.41)          '4    M           D % #    #  5   ' !&



7DNLQJ LQWR DFFRXQW WKH VWDELOLW\ ERXQG IRU # DQG $VVXPSWLRQ  JLYHV     % )    

           M      %    

    B D  B ) %  

%  M       5 B 7KLV JLYHV WKH GHVLUHG ERXQG  IRU  .

<  5.3.21 Corollary: If #<   4  <,,--- is the sequence produced by the Picard iteration for the postprocessed scheme, then      

   8< #  #<    4   DB   8 ) %     8<   <     M  %  B    4    B   8

for any initial guess # 4    .   .

CHAPTER 5. FLOW PROBLEMS

122

Proof. Since #<   !#<  and ! is a contraction with Lipschitz constant 8, we immediately  JHW \ WKH %DQDFK ¿[HG SRLQW WKHRUHP   #  #<     

 8<  #  #  5     8 

The result now follows from the fact that, by the stability bound (5.39),  ) #  

for 1  .



 

   4  DB

To obtain the estimate for the pressure, we proceed as follows. First, we note that, from the momentum equation, we have " & " <  & <  < IP  '4 <     4 '  - # 4 '   # * # 4 '

'  . 5

This implies that

" & <  < IP  '4   <     - #  # 4 '   # * # 4 '   # * # 4 '  #< * # 4 '   #< * #<   4 ' " & <  <  -IP  #  # 4 '   # * # 4 '   # * # 4 '  # 4 '5  #< * #<  

" & IP :H LQVHUW WKLV H[SUHVVLRQ LQ WKH LQIVXS FRQGLWLRQIRU  , use  stability properties of - 54 5 , 
    <    M  D % 8      8 #  #<    DB    M  8 D %  DB #  #<       then follows from the bound for #  #<  . The desired bound for   <      5.3.22 Remark: If we set B   
4 B 4

  

4  4

  

4  4

(5.42)

WKHQ ERWK WKH VPDOOQHVV DVVXPSWLRQV LQ  DQG  DUH VDWLV¿HG LI ZH KDYH WKDW  

    7 5 D B 

+HQFH ERWK WKH 1DYLHU6WRNHV HTXDWLRQV DQG WKHLU '* DSSUR[LPDWLRQ DUH XQLTXHO\ VROYDEOH Under a smallness condition that is slightly more restrictive, we obtain the following estimates.

5.3. NAVIER-STOKES EQUATIONS

123

5.3.23 Theorem: Assume that

 

      4 D B   and that the exact solution #4  of the Navier-Stokes equations (5.1) VDWLV¿HV #  "   1 4

  "  14

  5

(5.43)

(5.44)

Then    & "  #  #    
", #  D   4        & "  #  #    
", #  D   4        & "         
", D #   4    

where the constants are independent of .

Proof. We modify the approach used in the previous section to get error estimates for the DG method for the Oseen problem in two ways. First, we use the non-conforming approach introduced in [PS02] and later used in [SST03], and consider the expression  #4  ) 

!&

where

 #4 * '   4 '  

" & " &  #4 * ' ) D-IP  #4 '   #* #4 '   '4   4 ' 4

'  . 5

(5.45)

7KH VHFRQG PRGL¿FDWLRQ LV RI FRXUVH GXH WR WKH SUHVHQFH RI WKH FRQYHFWLYH QRQOLQHDULW\   Let us begin with the estimate for the error #  # , for which we are going to prove an estimate in terms of best approximation, namely %       #  #  
 #  '  
 #  '     !&

!& div )    
       #4  5 (5.46)  D D To prove this estimate, we proceed as in the error analysis of standard mixed methods, see HJ >%)@ DQG FRQVLGHU ¿UVW DQ HOHPHQW '   . Using a standard triangle inequality       #  #   #  '   '  #  4    ZH IRFXV RQ WKH VHFRQG WHUP 8VLQJ WKH GH¿QLWLRQ RI  in (5.45), we obtain

  " & DB'  #  -IP  '  # 4 '  # " & -IP  '  #4 '  #   #4 * '  #    '  # 4      # * # 4 '  #    #* #4 '  #     


CHAPTER 5. FLOW PROBLEMS

124

:KLOH WKH ¿UVW WZR WHUPV LQ WKLV VXP FDQ EH HVWLPDWHG LPPHGLDWHO\ E\ HOOLSWLFLW\ DQG WKH GH¿QL tion of  #4  UHVSHFWLYHO\ WKH QRQOLQHDULW\ DQG LQFRPSUHVVLELOLW\ QHHG VRPH PRUH WKRXJKW 6LQFH '  #    ZH KDYH  '  # 4       '  # 4    '  # 4   4 IRU DQ\     )URP WKH ERXQGHGQHVV RI WKH IRUPV IROORZV      '  # 4      '  #     

    5

)LQDOO\

  # * #4 '  #    '* #4 '  #    # * '  # 4 '  #    '* #4 '  #    #* #4 '  #    # * '  #4 '  #  )%  %  %  % 5 1RWH WKDW GXH WR /HPPD  /HPPD  WKH VWDELOLW\ ERXQG IRU # LQ   DQG WKH GH¿QLWLRQV RI WKH SDUDPHWHUV LQ   ZH KDYH    %   # '  #    

 1-    '  #    D   

     '  #    DB     DB  '  #  4  

E\ WKH VPDOOQHVV FRQGLWLRQ  

1H[W E\ /HPPD  %   DQG E\ /HPPD  /HPPD  DQG WKH ERXQG IRU # in 7KHRUHP      

     #  '  '  #  %    DB     

     #  '  '  #     DB            #  '  '  #  4  DB  E\ WKH VPDOOQHVV FRQGLWLRQ  

)LQDOO\ E\ WKH /LSVFKLW] SURSHUW\ RI WKH IRUP  LQ /HPPD  ZH KDYH      %   #  '  # '  #      

    #  '  '  #     D      #  '  '  #  4

5.3. NAVIER-STOKES EQUATIONS

125

by the bound for # in (5.6) and the smallness condition (5.43). Now, take an arbitrary function '' in .div . Since  reproduces functions in .div , we have ' '  '', and so       #  '   #  ' '   ' '  '       '  '   #  ' '    '         #  ''   #  '  4 



by Lemma 5.3.17. This implies that %% ) )      DB      '  #  5 #  ''  #  '  %     



Thus, gathering all the estimates above, we obtain        " & '  #   #  '   #  ' '         #4  4  

and by triangle inequality        " & #  #   #  '   #  ''        #4  4    

(5.47)

for any '   , ''  .div , and    .

It remains to replace '   by an arbitrary function in '  .  7R WKLV HQG ¿[ '  . and consider the problem: Find   . such that  4    #  '4 

   5

The inf-sup condition in Lemma 4.2.2 guarantees that such a solution  H[LVWV )XUWKHUPRUH LW can be easily seen that we have        M  #  '  4  

in view of the inf-sup condition for  and the continuity of the form  . By construction and since  #4    for any    , we further have that   '   . Inserting   ' in (5.47), employing the triangle inequality, and taking into account the above bound for  yield the abstract error estimate (5.46) for the velocity. $V D FRQVHTXHQFH RI WKH DSSUR[LPDWLRQ UHVXOW   ZH REWDLQ WKH IROORZLQJ HVWLPDWH RI WKH error between # DQG LWV JOREDOO\ VROHQRLGDO DSSUR[LPDWLRQ #       #  #      
 #  ' '    #  #  5 (5.48)  !
& div

To see this, note that       #  #   #  ' '   ' '  #       '  #   #  ' '    '         #  ''   #  #  4

where ' ' is any element of .div . Here, we have used the stability bound in Lemma 5.3.17 and the fact that  reproduces polynomials in .div . This shows the inequality (5.48).

CHAPTER 5. FLOW PROBLEMS

126

1RZ OHW XV REWDLQ WKH HVWLPDWH IRU WKH SUHVVXUH :H FODLP WKDW WKH HUURU LQ WKH SUHVVXUH VDWLV¿HV           
 D #  '  
 D #  ' '  (5.49)   !&

!
& div 2   
       #4  5 

To see this, we proceed in a way similar to the one used to deal with the velocity. Thus, we begin by noting that for                             '4               M    ' !&



M



  '4      '4       

  M         4     '  '  !&

!&

where we have use the inf-sup condition in Lemma 4.2.2. Therefore,

             M    '4      M  *      4  '  !&



(5.50)

by the continuity of the form  .

7R ERXQG WKH ¿UVW WHUP RQ WKH ULJKWKDQG VLGH RI   ZH QRWH WKDW " &  '4      -IP  #  # 4 '   #* #4 '   # * # 4 '   #4 * '4 for any '  . , and proceed as in the previous step to obtain " &  '4      -IP  #  # 4 '   # * #  #4 '   # * #4 '   #* #4 '   #4 * '      #  #   D %  D B  2    DB   #  #    #4  '  4     

and since

      #  #      #  ' '   #  #  4 

from (5.40), we get

   '4      %  B   D #  #  % )   B    

D #  ''   #4 5  



  Inserting this inequality in (5.50) and using the bound (5.46) for #  #  from Step 1, we immediately obtain the abstract estimate (5.49) for the pressure.

5.3. NAVIER-STOKES EQUATIONS

127

Under the regularity Assumption (5.44), the following standard approximation property holds      . *  


 D #  '  
        
", D #    5 

!&

Moreover, from the results in [BF91], see also [HL02, Section 3], we have    


 #  ' '     
",#  5 !
& div

Finally, we have that

 . *    #4     
", D #    4

with a constant  independent of the mesh-size.

To see the estimate of the residual, we proceed as follows. For '  . , it is easy to see that  #4 * ' is given by    #4 * ' 

D(#   D(#

) # '  &&  

   

##' && 4 









with  )
 1  0 and  )
 16   denoting the
 -projections onto 0 and  , respectively. The desired estimate follows then by proceeding as the proof of [SST03, Proposition 8.1] and using standard approximation results for  and  . It is now a simple matter to see that the error estimates of Theorem 5.3.23 follow by inserting the approximation estimates obtained in the previous step into the abstract bounds for the velocity, (5.46), for its globally solenoidal post-processing (5.48) and the pressure (5.49). This completes the proof of Theorem 5.3.23. 5.3.24 In Table 5.2 we show the errors and convergence rates in , # and L obtained for D  5. The errors in  and L are measured in the
 -norm while #  # and #  # are evaluated in the broken "  QRUP :H REVHUYH WKH SUHGLFWHG ¿UVW RUGHU FRQYHUJHQFH IRU DOO WKH error components, in full agreement with the results of Theorem 5.3.23. Notice that we have scaled the
 -error in L by D  so that this error can be directly compared to the "  -errors in #  # and #  # . These three errors are all of the same magnitude, with a slight advantage for the post-processed solution. In Table 5.3 we show the seminorm of the errors ## and ## which measures their jumps. Notice that it superconverges with order 6. This means that the relative contribution of this  seminorm to the   norm diminishes as  decreases. An analysis of this phenomenon remains to be carried out.

In Table 5.4, we show the
 -errors in the velocities and their corresponding convergence orGHUV ,Q WKH ¿UVW FROXPQ ZH REVHUYH WKDW WKH YHORFLWLHV FRQYHUJH ZLWK VHFRQG RUGHU ,Q WKH second column, we notice that by post-processing the error is reduced by a factor of roughly 3/2. Therefore, the post-processed solution should be used as the best approximation obtained by our scheme. Furthermore, we show the
 -norms of the divergence of # (evaluated at the points of a 4-by-4 Gauss formula on each cell). These are of the order of the residual of the QRQOLQHDU LWHUDWLRQ FRQ¿UPLQJ WKDW WKH SRVWSURFHVVHG VROXWLRQ LV LQGHHG GLYHUJHQFHIUHH

CHAPTER 5. FLOW PROBLEMS

128


3 4 5 6 7 8

.   . 2.2e+0 — 1.0e+0 1.12 4.8e-1 1.10 2.3e-1 1.04 1.2e-1 1.01 5.8e-2 1.00

  #  # 

1.2e+1 5.4e+0 2.4e+0 1.1e+0 4.7e-1 2.2e-1



  #  # 

— 8.1e+0 1.11 3.2e+0 1.16 1.4e+0 1.18 6.8e-1 1.17 3.4e-1 1.13 1.7e-1



— 1.33 1.18 1.06 1.02 1.01

D  .L  L . 7.0e-0 — 3.4e-0 1.05 1.6e-0 1.07 7.8e-1 1.04 3.9e-1 1.02 1.9e-1 1.02

Table 5.2: Errors and orders of convergence for D  5.


3 4 5 6 7 8

 #  #   9.1e+0 — 4.2e+0 1.11 1.8e+0 1.23 7.2e-1 1.32 2.8e-1 1.39 1.0e-1 1.44

 #  #  4.8e+0 — 1.5e+0 1.72 4.7e-1 1.64 1.6e-1 1.54 5.6e-2 1.52 2.0e-2 1.51

Table 5.3: Errors and orders of convergence for D  5 in the jump seminorm ' ) 3, 7. 5  ##'  &&  . 
 ? 
3 4 5 6 7 8

.#  # . 6.4e-1 — 1.6e-1 2.03 3.3e-2 2.22 7.1e-3 2.24 1.6e-3 2.19 3.5e-4 2.13

.#  # . 4.9e-1 — 1.1e-1 2.22 2.0e-2 2.37 4.2e-3 2.27 9.8e-4 2.12 2.4e-4 2.04

.( # . 1.4e-12 1.4e-12 3.2e-12 1.5e-11 1.8e-12 2.9e-11

Table 5.4:
 -errors and orders of convergence in the velocity and
 -norm of the divergence of the post-processed solution # for D  5.


3 4 5 6 7 8

D   D  5 D  5 14 33 865 10 21 106 8 16 54 6 12 29 5 10 18 6 10 13

Table 5.5: Number of iterations for convergence of the non-linear iteration.

5.3. NAVIER-STOKES EQUATIONS

129

The convergence of the non-linear iteration under consideration is illustrated in Table 5.5. Displayed is the number of steps required to reduce the start residual by a factor of  . The initial guess for the iterations is the vector #  . The linear system in each step is solved by a preconditioned GMRES method up to a relative accuracy of  . Therefore, the error of the linear iterations is small enough to be neglected. Table 5.5 shows that the number of iteration steps is not only bounded independently of the mesh-size, but in fact decreasing. This is in perfect agreement with our theoretical results in Theorem 5.3.20. If we decrease the viscosity, WKH LQFUHDVH RI WKH QXPEHU RI LWHUDWLRQ VWHSV IRU FRQYHUJHQFH LV TXLWH PRGHUDWH RQ ¿QH JULGV 2I course, this only holds as long as there is convergence. With D   , the non-linear iteration does not converge anymore, probably because the stationary solution is not stable in this case. 5.3.25 Remark: For a DG method using the div-conforming space .div in (4.45), it can be UHDGLO\ VHHQ WKDW D ¿HOG #  .div satisfying (4.49) already is exactly incompressible. Hence, for such a DG method, we can take  as the identity; we will study this approach in the following pargraphs. :H ¿UVW QRWLFH WKDW VLQFH WKH VSDFH RI YHORFLWLHV . is included in " div 1, the jump of the normal component of ' vanishes over edges. Therefore, unlike for the case treated before, there LV QR QHHG WR LQWURGXFH D QXPHULFDO ÀX[ DVVRFLDWHG ZLWK  . 5.3.26 Theorem: Under the above assumptions, theorems 5.3.20 and 5.3.23 hold for the divergence conforming method. In particular, we have the optimal error bound         ! #  #        " #  " 4   " 

with a constant independent of the mesh size. Moreover, the approximate velocity # is exactly divergence-free.

5.3.27 To carry out our numerical experiments, we use the symmetric interior penalty method. We consider quadratic mesh cells and the local spaces are, since this method does not use a separate space for L, . %   %   %" %   " % 4 5.3.28 Again, we use example A.4.3. Since this function has boundary values different from zero, special precautions must be taken. As pointed out in Section 2, in order to obtain a divergence-free solution in the elements adjacent to the boundary, the Dirichlet boundary condition for the normal component must be implemented in a strong way, while the tangential components obtain their boundary values weakly through the form (5.20). The strong boundary values are obtained by interpolating the normal component of the prescribed boundary function in the set of Gauss points required to integrate RT" exactly on a face and then initializing the start value of the nonlinear iteration to this value. In all subsequent iteration steps, the residual and update vectors are set to zero in the corresponding components. The tangential component RI WKH YHORFLW\ ¿HOG LV SUHVFULEHG LQ ZHDN IRUP DV ZLWK VWDQGDUG '* VFKHPHV 5.3.29 Absolute errors and convergence rates for several pairs of RT" /" polynomials are listed in Table 5.6. It exhibits clearly the expected convergence orders for error of the velocity in "    and
 1. The pressure errors in this table converge much faster than expected; a fact we cannot explain right now.

CHAPTER 5. FLOW PROBLEMS

130 (

1

2

3

4


4 5 6 7 8 3 4 5 6 7 4 5 6 3 4 5

.( . 9.7e+0 4.9e+0 2.4e+0 1.2e+0 6.1e-1 3.9e+0 9.9e-1 2.5e-1 6.2e-2 1.5e-2 6.5e-2 8.0e-3 9.8e-4 5.3e-2 3.4e-3 2.1e-4

ord. 1.01 1.00 1.00 1.00 1.00 2.22 1.97 2.00 2.00 2.00 2.99 3.03 3.02 4.27 3.97 4.00

. . 2.3e-1 6.2e-2 1.6e-2 4.2e-3 1.1e-3 9.3e-2 1.2e-2 1.5e-3 1.9e-4 2.3e-5 6.8e-4 4.6e-5 3.0e-6 6.8e-4 2.1e-5 6.6e-7

ord. .. ord. .( #. 1.82 3.9e+0 1.63 8.7e-10 1.89 1.2e+0 1.75 2.8e-09 1.93 3.3e-1 1.81 6.7e-09 1.96 9.3e-2 1.84 3.3e-08 1.98 2.6e-2 1.86 2.8e-09 3.12 2.6e+0 2.01 4.8e-10 2.96 4.5e-1 2.53 1.7e-09 3.00 6.5e-2 2.79 4.1e-09 3.00 9.2e-3 2.82 2.0e-08 3.00 1.4e-3 2.77 7.3e-08 3.78 4.2e-2 3.46 1.9e-09 3.87 3.2e-3 3.71 7.9e-09 3.93 2.5e-4 3.72 2.0e-08 5.31 6.3e-2 3.91 7.6e-10 5.00 2.7e-3 4.52 3.6e-09 5.01 9.5e-5 4.85 6.7e-09

7DEOH  (UURUV IRU .RYDV]QD\ ÀRZ D  ) and pairs RT" /" .

Chapter 6 Linear Solvers The solution of the linear systems arising from discontinuous Galerkin discretizations has been neglected for a long time. Time dependent hyperbolic problems are solved mostly by explicit time-stepping schemes, while a Gauß-Seidel method with suitable ordering of WKH GHJUHHV RI IUHHGRP \LHOGV VXI¿FLHQW UHVXOWV LQ WKH VWDWLRQDU\ FDVH ,Q WKH SUHVHQFH RI D VHFRQG RUGHU HOOLSWLF WHUP WKHVH PHWKRGV IDLO WR EH HI¿FLHQW ,PSOLFLW WLPHVWHSSLQJ LV required to avoid unreasonably small space grids due to the Courant-Friedrichs-Lewy condition. Furthermore, the condition number of the linear system in the stationary case or for ODUJH WLPH VWHSV LQFUHDVHV ZLWK PHVK UH¿QHPHQW UHVXOWLQJ LQ YHU\ VORZ LWHUDWLYH VROYHUV RQ ¿QH JULGV Therefore, preconditioners are required to counter the effect of the increasing condition number. Before our work in [GK03c], very few results in this direction were known. ,Q >59:@ 5XVWHQ HW DO KDG XVHG DQ LQFRQVLVWHQW PXOWLOHYHO LQWHULRU SHQDOW\ PHWKRG to precondition the conforming mixed discretization of Poisson equation. Furthermore, (ZLQJ HW DO KDG LQYHVWLJDWHG GRPDLQ GHFRPSRVLWLRQ >(/39@ DQG PXOWLOHYHO >%(36@ PHWKRGV IRU WKH FHOOFHQWHUHG ¿QLWH GLIIHUHQFH PHWKRG ZKLFK FRUUHVSRQGV WR WKH LQWHULRU SHQDOW\ PHWKRG LQ 6HFWLRQ  ZLWK FRQVWDQW VKDSH IXQFWLRQV 5HFHQWO\ D GRPDLQ GHFRP position preconditioner for the interior penalty method was presented by Lasser and Toselli in [LT01]. After summarizing the most important results on Krylov-space solvers, we present the multilevel method of Gopalakrishnan and Kanschat in [GK03c, GK03b], which applies to the interior penalty method as well as to the Schur complement of the matrix produced by the /'* PHWKRG DV VKRZQ LQ >*.D@ :H GHPRQVWUDWH KRZ GDWD VWUXFWXUHV FDQ EH VHW XS IRU D GLVFRQWLQXRXV PXOWLOHYHO PHWKRG VR WKDW ORFDOO\ UH¿QHG JULGV FDQ EH KDQGOHG LQ DQ HI¿FLHQW ZD\ ZLWKRXW DGGLWLRQDO FRGLQJ RI WKH SDUWLFXODU DSSOLFDWLRQ A general framework for preconditioning of saddle point problems was presented by MurSK\ HW DO LQ >0*:@ ,W VXFFHVVIXOO\ DSSOLHG WR YDULRXV GLVFUHWL]DWLRQV RI 6WRNHV DQG 2VHHQ HTXDWLRQV LQ >(6: ./: 6(.:@ +HUH ZH DSSO\ WKLV FRQFHSW VXFFHVVIXOO\ to the LDG saddle point problem, where we obtain an analytical eigenvalue estimate, and to the Stokes system. Computational results show its feasibility in a DG context. The remaining sections of this chapter are devoted to the application of the preconditionLQJ FRQFHSWV WR DGYHFWLRQ GRPLQDWHG SUREOHPV ,Q WKH VHFWLRQ RQ DGYHFWLRQ ZH VKRZ UH sults from [GK03c], indicating that the multilevel scheme combined with a Gauß-Seidel smoother with downwind ordering improves when advection becomes dominant and can therefore be considered robust with respect to the Peclet number. Furthermore, we show QHZ UHVXOWV H[KLELWLQJ WKLV UREXVWQHVV HYHQ IRU ÀRZ ZLWK YRUWLFHV )LQDOO\ ZH VKRZ WKDW D

CHAPTER 6. LINEAR SOLVERS

132

preconditioner following the ideas by Kay et al. in [KLW02] yields good results for Oseen equations.

6.1 Krylov space methods 6.1.1 .U\ORY VSDFH PHWKRGV ²LQ SDUWLFXODU SUHFRQGLWLRQHG YHUVLRQV² DUH WKH PRVW HI¿FLHQW iterative schemes known for solving sparse linear systems of equations   5 They exist in several variants for linear systems that are symmetric, nonsymmetric, positive GH¿QLWH RU QRW Still, their performance usually depends on a good preconditioner for the matrix . It is those preconditioners the following sections deal with after we presented the most important facts on the iterative schemes. 6.1.2 Algorithm (Preconditioned conjugate gradient): With a given start vector  , compute the initial residual  )   and the preconditioned residual  )  . Let  )  .   While "  9 K do for (  4 4 5 5 5 " " "  " ) "  B" " ) "  B"  " ) " 

B" )

"  "  " 

 "  "  " " ) "  E" "

E" ) " 

6.1.3 Algorithm  (GMRES with right preconditioning): Compute the initial residual  )   , E )   and  )  6E.   While "  9 K do for (  4 4 5 5 5 1. Compute  )  " .

2. Compute column ( of the matrix   " ," and orthogonalize  by  4 5 5 5 4 ( " )    )   "  5

6.1. KRYLOV SPACE METHODS

133

3. Let   " ," )   "  ) " ,"   4. Compute " )  
E   .

Finally, let

   



 



6.1.4 Remark: The GMRES algorithm requires an auxiliary vector in each iteration step. Therefore, it is customary to restart the iteration after a certain number of steps.   6.1.5 Lemma: Let 
  . The conjugate gradient method minimizes the error in step ( in the sense that there is a polynomial   " such that "     4

and  minimizes the expression        




(6.1)

in the space " . The GMRES method minimizes the norm of the residual in each step such that       (6.2) is minimal in " .

Proof. see [Saa00].  'H¿QLWLRQ The spectral condition number RI D V\PPHWULF SRVLWLYH GH¿QLWH PDWUL[  is GH¿QHG E\ 
  

A   4 A 


where A   and A 
 are the largest and smallest eigenvalues of , respectively.  &RUROODU\ The error after the th conjugate gradient iteration step admits the estimate # ) %#            #
        #
       5 (6.3)



    
    Proof. 7KH VHFRQG LQHTXDOLW\ IROORZV LPPHGLDWHO\ E\ LWHUDWLRQ 7KH ¿UVW LQHTXDOLW\ IROORZV from (6.1) by using Chebyshev polynomials for the interval #A 
4 A  & (cf. [Saa00]).

CHAPTER 6. LINEAR SOLVERS

134

6.1.8 Corollary: Let the matrix  KDYH UHDO HLJHQYDOXHV DQG EH SRVLWLYH GH¿QLWH EXW QRW necessarily symmetric). Then, the residual of the th step of the GMRES iteration admits the estimate ) %#          #
      5 (6.4) 
    Proof. This estimate follows from (6.2) in the same way as (6.3) from (6.1).

6.2 Interior Penalty 6.2.1 Using the basis of . chosen in paragraph 1.1.12, equation (2.12) is equivalent to the linear system of equations   4

(6.5)

where    LV WKH FRHI¿FLHQW YHFWRU RI # with respect to the basis functions and the entries of  and DUH GH¿QHG E\ ZH FRQWLQXH XVLQJ GRXEOH LQGLFHV IRU WKH EDVLV IXQFWLRQV & & " " ,,  -IP ,  4 <,  5 (6.6)  <, 4 <, 4 6.2.2 Lemma: Assume that the shape regularity Assumption 1.1.13 holds. Then, the spectral condition number 
   of the operator  GH¿QHG E\ " & & " #4 '  . 4 (6.7)  #4 '  -IP  #4 '

behaves asymptotically like


     
 
    ?

  4 ?  5

Here,  
is the diameter of the smallest grid cell. The condition number of the matrix  behaves asymptotically like the condition number of the operator  . Proof. Let #  3 , 1 be an eigenfunction to the smallest eigenvalue A of the continuous Poisson operator. It depends on the domain 1 only. Furthermore, it is known to be a smooth function. Let # be its interior penalty approximation, i. e., "  & "  & IP '  . 5 -IP  # 4 '  - # 4 ' " & Due to the consistence of -IP  54 5 , we have   "  & "  & "  -IP A #   -IP  # 4#  # 4 #   5

Therefore, we conclude that independent of ? 9 ? ,

A 
   A  " 5

(6.8)

6.2. INTERIOR PENALTY

135

If ? 9 ? , a lower bound of the minimal eigenvalue follows from the stability of the method with respect to the energy norm and Friedrichs inequality in Lemma 1.2.3. We estimate the maximal eigenvalue by using the fact that due to symmetry of  for any #  . " & - #4 #  A    #4 # 5

Choosing # such that #   on a cell % such that the diameter of % is minimal and zero everywhere else, we conclude that - #4 # ? A          5 #  


An upper bound for the maximal eigenvalue with the same asymptotic behavior follows from the boundedness of the energy norm and inverse estimates. 6.2.3 Corollary: Solution of the discrete problem (6.5) by the conjugate gradient method requires  
 iteration steps, where  
is the diameter of the smallest grid cell. 6.2.4 The previous lemma and corollary imply that the solution of the discrete linear sysWHP  ZLOO EH YHU\ WLPHFRQVXPLQJ LI SHUIRUPHG RQ ¿QH JULGV ZLWKRXW D VXLWDEOH SUHFRQGL tioner. The remainder of this section is devoted to develop such a preconditioner. We will show, that the suggested preconditioner is uniform, that is, the condition number of the product of preconditioner and matrix is bounded independent of the mesh size. 6.2.5 We recall that we assume a hierarchy of triangulations  , 5 5 5 , < , 5 5 5 , 
5 6LQFH GXH WR $VVXPSWLRQ  WKLV KLHUDUFK\ FRUUHVSRQGV WR D QHVWLQJ RI WKH ¿QLWH HOHPHQW spaces, .  & & .< & & .
5

(6.9)

We remark that the
 -projection ,< )
 1  .< is obviously an operator also from .<  to .< . By its projection properties we have for any #  .< and '  .<  & & " " & " #4 '  #4 ,< '  , < #4 ' 5 Therefore, the adjoint operator , < or prolongation is simply the embedding of .< into .<  .

 'H¿QLWLRQ The matrix < associated with the projection ,< is called restriction, and its transpose < , the matrix representing the embedding operator, prolongation On each grid cell %  ZH GH¿QH WKH cell matrix  by & " < <    -IP  <, 4 <, 5

(6.10)

Here, O is the grid level on which the cell % LV ORFDWHG $QDORJRXV WR   ZH GH¿QH OHYHO matrices <    by using shape functions !4 %  < .

CHAPTER 6. LINEAR SOLVERS

136

 'H¿QLWLRQ The mass matrix < on level O is the matrix with entries " & < ,,  <, 4 <, 5

,Q WKH FRQWH[W RI GLVFRQWLQXRXV ¿QLWH HOHPHQWV WKLV PDWUL[ LV DOZD\V EORFN GLDJRQDO WKDW LV < ,,   if ! - %  )XUWKHUPRUH LI WKH VKDSH IXQFWLRQV DUH /HJHQGUH SRO\QRPLDOV LW LV D GLDJRQDO PDWUL[ /HJHQGUH SRO\QRPLDOV IRU WKH DFWXDO JULG FHOOV ZRXOG \LHOG WKH LGHQWLW\ matrix.  'H¿QLWLRQ A main ingredient of multilevel methods is the smoother < . We consider IRXU VLPSOH VPRRWKHUV KHUH SRLQW DQG EORFN YHUVLRQV RI -DFREL DQG *DX‰6HLGHO PHWKRGV The diagonal and block diagonal of the matrix < are <    < ,,

<       

*LYHQ DQ\ RUGHULQJ ³7´ RI WKH JULG FHOOV ZH GH¿QH DQ RUGHULQJ RI WKH LQGH[ VHW %4  E\ !4  7 %4 $  ! 7 %  !  % % 7 $5 7KHQ WKH lower triangle and lower block triangle of the matrix are the matrices  < ,, if !4  7 %4 $ 
< ,,   else4  <  if ! 7 % 
<    else4 7KH WZR YDULDQWV RI WKH -DFREL VPRRWKHU QDPHO\ point Jacobi and block Jacobi DUH #<  > < 4

# <  > <

with relaxation parameter > DQG WKH SRLQW DQG EORFN *DX‰6HLGHO VPRRWKHUV DUH <  < 
<  4

 <  < 
<  5

! < on a vector <  $OJRULWKP :H GH¿QH WKH DFWLRQ RI WKH variable V-cycle RSHUDWRU UHFXUVLYHO\ :LWK < DQ\ RI WKH VPRRWKHUV LQ 'H¿QLWLRQ  OHW  odd <  <  < even5 /HW WKH QXPEHUV VPRRWKLQJ VWHSV 1< EH SRVLWLYH LQWHJHUV VXFK WKDW E 1<  1<  E 1< 4

(6.11)

with  7 E  E . !    . If O -  VHW 0   DQG FRPSXWH ! < < (assuming that ! < LV DOUHDG\ 7KHQ OHW  GH¿QHG E\ WKH IROORZLQJ VWHSV

6.2. INTERIOR PENALTY

137

1. (Pre-smoothing) Compute ) iteratively by 

    < <  <  4

 4 5 5 5 4 1< 5

2. (Coarse grid correction) Let ! < < <  < ) 5   )  < 3. (Post-smoothing) Compute ) iteratively by ) 

    <

<  <  4

 4 5 5 5 4 1< 5

! < <  ) . 4. Set 6.2.10 Remark: A typical choice is E  E   and 1
 . In this case, the number of smoothing steps on level O is 
< . Then, the asymptotic complexity of the variable V-cycle is the same as that of the W-cycle. Choosing E  E   yields the standard V-cycle. This algorithm is not covered by our analysis, but numerical results later show that it is feasible, too. !
is symmetric and 6.2.11 Theorem: Suppose that regularity Assumption (2.5) holds. Then, SRVLWLYH GH¿QLWH DQG WKHUH LV D FRQVWDQW 3 independent of the level (or of < ) such that ! < <    <  F F  <  5.

  
4

(6.12)

5.

with F  3  1
61
and B from equation (2.5). Consequently, the spectral condition !

VDWLV¿HV 
  !

  F  . number of 6.2.12 Remark: This theorem is the main result of [GK03c]. In virtue of the abstract theory JLYHQ LQ >%3;@ WKH SURRI UHGXFHV WR YHUL¿FDWLRQ RI WZR FRQGLWLRQV 1. There exists >   4  such that for all O  4 5 5 5 4
>    <    <    < < < 4 A<

0   5

2. Let < ) .<  .< EH WKH 5LW] SURMHFWLRQ GH¿QHG E\ " " & & IP #  .< 4 '  .< 5 -IP < < #4 '  -< #4 '

Then, there is ;  4  and  9  such that    < #  IP " & &0   IP " -< #  < #4 #     5 -< #4 # A<

(6.13)

(6.14)

The following lemmas present some intermediary results required in the proof of Theorem 6.2.11, which is deferred to paragraph 6.2.15.

CHAPTER 6. LINEAR SOLVERS

138

6.2.13 Lemma: Let < EH DQ\ RI WKH VPRRWKLQJ RSHUDWRUV LQ 'H¿QLWLRQ  ,I WKH VPRRWKHU is of Jacobi type, we assume further that > LV VXI¿FLHQWO\ VPDOO 7KHQ < is a smoothing operator, that is, there exists an >   4  such that >    <    <    < < < 4 A<

0  

(6.15)

Proof. This result has been proven in a very general context in [BP92] 6.2.14 Lemma: Assume that regularity Assumption 2.5 holds. Then, for all #  .< , with O  4 5 5 5 4
holds     #  < #  ?5< < #

 5 < +  Proof. We start with

where +  3

 5,

      #  < #  #  +   +  < # 4 < < <

(6.16)

1 is the solution of the dual problem ++  < # in 15

%\ FRQVLVWHQF\ RI WKH LQWHULRU SHQDOW\ PHWKRG DQG WKH GH¿QLWLRQ RI < , we have " & " & " & IP '  .< 5 -IP < +4 '  < #4 '  -< #4 '

Therefore, # is the Ritz projection of + in .< , and in virtue of estimate (2.5), we have     #  +   5< +  
 5 < + 

(6.17)

For the estimation of the second term in (6.16), we use the same arguments as above, yielding " & " " & & IP '  .< 5 -IP < +4 '  < #4 '  -< < #4 '

E\ GH¿QLWLRQ RI < . Therefore,     +  < #  5< + + 
  5 <   It remains toshow, that the term +  < #< LV ERXQGHG VXI¿FLHQWO\ E\ WKH FRDUVH OHYHO QRUP  +  < # . Observe that # +  < #&& is zero on each edge   < which is not a subset < of an edge in < .. Therefore,  ?  ?   # +  < #&&   # +  < #&& 4       






and consequently

      +  < #  +  < #  5< + + 
  5 < <

Adding up (6.17) and (6.18) completes the proof.

(6.18)

6.2. INTERIOR PENALTY

139

6.2.15 Proof of Theorem 6.2.11 We will prove (6.14) with ;  B6 and B from estimate (2.5). Lemma 6.2.14 and interpolation of Sobolev norms yield      5  5 #  < #  5 < #  5< < #, < # 5 < <  5, Assuming for the moment that

    < #  #< 4 ,

(6.19)

the proof of estimate (6.14) is completed as follows:   " &     -IP < #  < #4 #  #  < # < # <  5  5 5   < < # #<    < #  " &  -IP 4  < #4 # A<

which is (6.14) with ;  B6. Thus, it remains to prove (6.19). " &   < #4 C < #   

 , C  6+   , " " & & -IP < #4 ;6 < #4 , C       

 5 C  C  6+   6+   , ,

)RU WKH ¿UVW WHUP ZH XVH DSSUR[LPDWLRQ HVWLPDWH  WR REWDLQ " &   < #4 ;6    < < #5 C  , Stability of the projection yields

"

&   < #4 , C    #< 5 C  ,

Finally, since < LV V\PPHWULF SRVLWLYH GH¿QLWH ZH KDYH   & " & " < < #  < A< < #4 #  -IP < #4 # 4 which completes the proof.

6.2.16 Remark: The preceding analysis focused on the case of the multigrid method as a !

 is preconditioner in an outer Krylov space iteration, where the condition number 
  VLJQL¿FDQW IRU HI¿FLHQF\ Alternatively, the multilevel method can be used as a solver in an iteration like !

 
 5     

Here, the contraction number RI WKH LWHUDWLRQ PDWUL[

!

   A  !

 4 N  

(6.20)

!

 are the eigenvalues of   !

. Without covering this case by our analysis, where A   we will show some numerical results below.

CHAPTER 6. LINEAR SOLVERS

140


2 3 4 5 6 7 8

?
 10 22 79 312 1246 4981 19921

1O  
< ! !

 ?

 N  1.36 0.19 1.71 0.26 1.97 0.36 2.08 0.41 2.11 0.42 2.11 0.42 2.12 0.42

1O   !

 N  !

 ? 1.36 0.19 1.72 0.27 2.11 0.42 2.39 0.54 2.56 0.62 2.66 0.66 2.73 0.69

Table 6.1: Condition numbers and contraction numbers, when Gauß-Seidel smoother, and  elements are used. 1  4  , ?  


2 3 4 5 6 7

1O  
< 1O  
<  ! ! ! !

 ?

 N 

 ?

 N  1.62 0.30 1.14 0.10 2.23 0.40 1.36 0.17 2.72 0.54 1.49 0.21 2.95 0.64 1.54 0.22 3.02 0.67 1.56 0.22 3.04 0.68 1.56 0.22

Table 6.2: Condition numbers and contraction numbers when Jacobi smoother and  elements are used. 1  4  , ?  

6.2.17 The following numerical experiments are performed on hierarchies of Cartesian meshes JHQHUDWHG E\ JOREDO UH¿QHPHQW VHH $ RQ SDJH   XQOHVV VWDWHG RWKHUZLVH 7KH ULJKW KDQG side was chosen according to example A.2.2 on page 170, since trigonometric functions are approximated by eigenfunctions of the matrix 
and cause spurious convergence effects. The start vector for all iterations is   . 6.2.18 We start tests of the multilevel method with the standard situation of  elements and the block-Gauß-Seidel smoother  < RI 'H¿QLWLRQ  RQ SDJH  7DEOH  VKRZV UHVXOWV IRU WKLV FDVH ,Q WKH ¿UVW FROXPQ ZH GLVSOD\ WKH FRQGLWLRQ QXPEHU RI WKH matrix 
without preconditioning for comparison; they exhibit the expected growth behavior DQG LQFUHDVH E\ D IDFWRU RI IRXU ZLWK HDFK UH¿QHPHQW VWHS 7KH QH[W FROXPQ VKRZV WKH SUHFRQ ditioner according to the preceding analysis. Clearly, the condition numbers stay bounded by a very moderate number, resulting in a conjugate gradient iteration with contraction number 5(. 7KLV QXPEHU LV PXFK VPDOOHU WKDQ WKH FRQWUDFWLRQ QXPEHU RI WKH PXOWLJULG PHWKRG  LWVHOI shown in the third column. 6.2.19 The block-Jacobi smoother # < is subject of Table 6.2. Here, the relaxation parameter is >  . The left column shows that the condition numbers of the preconditioned system  The condition numbers in the following paragraphs are taken from [GK03c], where the stabilization on the ERXQGDU\ ZDV HTXDO WR WKH RQH LQ WKH LQWHULRU    LQ   7KH UHVXOWV VKRZQ LQ 5HPDUN  UHÀHFW WKH ¿QHU VWDELOLW\ DQDO\VLV DQG DUH FRPSXWHG IRU WKH IRUP LQ  

6.2. INTERIOR PENALTY


2 3 4 5 6 7

141

 elements (?  ') !

 N !

 ?
 ? 23 2.07 0.36 69 2.11 0.39 263 2.14 0.40 1041 2.16 0.41 4154 2.16 0.41 16605 2.16 0.41

 elements (?  ) !

 N !

 ?
 ? 77 2.97 0.53 269 2.88 0.50 1061 2.90 0.52 4235 2.92 0.52 16934 2.92 0.52 67731 2.92 0.52

!

when Gauß-Seidel smoother and biquadratic Table 6.3: Condition numbers of 
and ( ) and bicubic ( ) shape functions are used. 1  4  5

5.5 4.5 4

condition number

Q1 Q2 Q3 Q4 Q5 Q6

5

3.5 3 2.5 2 1.5 1 1

2

3

4

5 level

6

7

8

9

Figure 6.1: Condition numbers for different shape function spaces on a sequence of meshes

as well as the contraction number of the linear multigrid method are considerably higher than for the Gauß-Seidel smoother. Since both methods can be implemented with about the same numerical effort, Gauß-Seidel should be preferred. The right column shows that doubling the number of smoothing steps can improve the performance of the Jacobi smoother a lot. 6.2.20 Convergence for shape function spaces  and  is displayed in Table 6.3. We see that the condition number increases only slightly with polynomial degree, a result also evident from Figure 6.1, where the condition numbers on a sequence of meshes are displayed up to polynomials in  . 6.2.21 In the analysis, we assumed a mild regularity assumption (2.5), that holds for nonconvex polygonal domains. We now investigate the performance of the preconditioner on the domains with reentrant corners of example A.2.3 on page 170. While (2.5) holds with B 7 6 on the L-shaped domain 1 with interior angle of %, it does not hold for the slit domain 1 , where B  6. We conclude from Table 6.4 that the algorithm yields a good preconditioner in both cases.

CHAPTER 6. LINEAR SOLVERS

142


2 3 4 5 6 7

1 1 !

 N  !

 ? !

 N  !

 ? 1.70 0.28 1.70 0.31 1.96 0.38 1.92 0.35 2.08 0.41 2.06 0.40 2.11 0.42 2.10 0.41 2.11 0.42 2.11 0.42 2.12 0.42 2.12 0.42

Table 6.4: Condition numbers and contraction numbers for L-shaped and slit domains using bilinear shape functions and Gauß-Seidel smoothing


1 2 3 4 5 6 7

 6 8 9 9 9 9 9

 , ?   N   0.017 1.763 8 0.053 1.275 11 0.058 1.235 12 0.062 1.210 12 0.062 1.208 12 0.062 1.209 12 0.062 1.211 12

 , ?  N 0.056 0.117 0.139 0.143 0.144 0.143 0.143

 , ?  '   N  1.253 9 0.076 1.116 0.932 12 0.134 0.874 0.858 13 0.162 0.792 0.846 13 0.167 0.777 0.843 13 0.168 0.776 0.843 13 0.168 0.776 0.844 13 0.168 0.776

Table 6.5: Contraction and convergence rates for the variable V-cycle with block-Gauß-Seidel smoother

6.2.22 Remark: Condition numbers are more of theoretical interest and their relation to the number of steps required for a certain accuracy is only indirect. Furthermore, their meaning LQ WKLV FRQWH[W LV OLPLWHG WR V\PPHWULF SRVLWLYH GH¿QLWH PDWULFHV 7R EH DEOH WR FRPSDUH ZLWK preconditioners for other discretizations later, we summarize the results of the preceding paragraphs showing values of immediate practical relevance . First, there is  , the number of steps required to reduce the initial residual by  . Since this value is discrete, we also show the average contraction rate N and the decadic convergence rate      N

(6.21)

according to Varga [Var99]. 6.2.23 The method can be applied to " VKDSH IXQFWLRQ VSDFHV DV VKRZQ LQ 7DEOH  7KH ¿J ures here are very similar to Table 6.6 and we conclude that the multilevel preconditioner 6.2.9 LV DV HI¿FLHQW IRU FRPSOHWH SRO\QRPLDO VSDFHV DV IRU WHQVRU SURGXFW SRO\QRPLDOV 6.2.24 Much like the condition number of the original matrix 
itself, the condition number of the preconditioned system depends on the stabilization parameter ?. Figure 6.2 shows how the value 6  varies with ?. We observe a dependence like   6? for large values of ?, which corresponds to condition numbers depending linearly on ? ,Q SDUWLFXODU WKLV ¿JXUH shows that ? PXVW EH FKRVHQ FDUHIXOO\ WR \LHOG DQ HI¿FLHQW VROYHU :H UHPDUN WKDW WKH VKDSH RI the curve is in good correspondence with the results for incomplete Cholesky preconditioners in [Cas01, Cas02]. 

From now on, values are computed with the improved boundary stabilization

6.2. INTERIOR PENALTY


1 2 3 4 5 6 7

 5 9 9 9 9 9 9

143

 , ?    , ?   , ?  ' N   N   N  0.000 4.382 7 0.024 1.617 8 0.042 1.378 0.054 1.268 11 0.100 0.999 13 0.145 0.838 0.070 1.158 12 0.130 0.885 14 0.172 0.765 0.072 1.142 12 0.135 0.870 14 0.179 0.748 0.071 1.150 12 0.135 0.868 14 0.179 0.747 0.070 1.155 12 0.135 0.871 14 0.179 0.747 0.069 1.162 12 0.133 0.875 14 0.179 0.747

Table 6.6: Reduction and convergence rates for the variable V-cycle with block-Gauß-Seidel smoother (" shape functions)

6 

2

1

1

2

4

8

16

? Figure 6.2: Solver performance depending on stabilization parameter ?.

CHAPTER 6. LINEAR SOLVERS

144

70

block point

Iteration steps

60 50 40 30 20 10 1

2

3 4 Polynomial degree

5

6

Figure 6.3: Comparison of point and block smoothers for different shape function spaces


1 2 3 4 5 6

 7 9 10 11 11 12

 , ?    , ?    , ?  " N   N   N  .025 1.598 6 .017 1.766 8 .040 1.395 .068 1.170 9 .064 1.194 12 .123 0.910 .098 1.008 10 .093 1.032 14 .180 0.744 .113 0.945 11 .111 0.957 15 .211 0.676 .121 0.915 11 .121 0.917 16 .227 0.644 .123 0.911 12 .123 0.909 16 .230 0.638

Table 6.7: Contraction and convergence rates on a three-dimensional cube

6.2.25 Comparison between the point Gauß-Seidel smoother 
and the block smoother  
in Figure 6.3 (from [GK03b]) exhibits the superiority of the block version. While the iteration counts  of the block-preconditioned system increase only moderately with the polynomial degree, the number of steps seems to increase exponentially with point-Gauß-Seidel.

6.2.26 Results for the three-dimensional case are shown in Table 6.7, where we observe the same behavior as in two dimensions. 6.2.27 All experiments above have been performed on equidistant Cartesian grids. In Table 6.8, we show that the method is robust with respect to the shape of the grid cells. First, we VKRZ UHVXOWV IRU WKH LUUHJXODU FRDUVH JULG LQ )LJXUH $ RQ SDJH  UH¿QHG UHJXODUO\ ,Q WKLV case, the inclusion of the spaces .< VWLOO KROGV DQG RXU WKHRU\ LV DSSOLFDEOH 7KH UHVXOWV FRQ¿UP that the reduction factors remain bounded independent of the mesh width.

6.3. LOCAL MULTIGRID


1 2 3 4 5 6

145 Irregular, ?  5" Distorted, ?  $  N   N  17 0.248 0.606 7 0.030 1.517 20 0.307 0.513 16 0.228 0.642 21 0.326 0.487 20 0.301 0.521 22 0.337 0.472 22 0.340 0.469 21 0.331 0.480 23 0.364 0.439 21 0.329 0.483 25 0.387 0.412

Table 6.8: Reduction and convergence rates for the variable V-cycle with block-Gauß-Seidel smoother on non-Cartesian grids ( shape functions) 7KH VHFRQG FROXPQ VKRZV UHVXOWV RQ UH¿QHPHQWV RI D VLQJOH VTXDUH FHOO ZKHUH WKH PHVK LV GLVWRUWHG ZLWK D UDQGRP YHFWRU ¿HOG LQ HDFK UH¿QHPHQW VWHS VHH )LJXUH $ RQ SDJH   ,Q this case, inclusion of the level spaces .< GRHV QRW KROG DQ\PRUH )XUWKHUPRUH WKH VKDSH RI WKH FHOOV GHJHQHUDWHV PRUH DQG PRUH GXULQJ UH¿QHPHQW 7KHUHIRUH ZH KDG WR FKRRVH ?  $ Still, the results show reasonable convergence rates.

6.3 Local multigrid 6.3.1 This section explains the implementation of the multi-level preconditioner presented in 6HFWLRQ  RQ ORFDOO\ UH¿QHG JULGV 7KH JRDOV RI WKLV LPSOHPHQWDWLRQ DUH 1. Optimal complexity: the method should use the same amount of operations per degree of IUHHGRP DV LQ WKH JOREDOO\ UH¿QHG FDVH 7KLV UHTXLUHV local smoothing to avoid repeated VPRRWKLQJ RQ D VLQJOH FHOO 2Q WKH RWKHU KDQG ORFDO VPRRWKLQJ PD\ QRW UHGXFH WKH TXDOLW\ as a preconditioner. 2. The multilevel method should use the same matrices for local smoothing and matrixvector multiplications. Assembling of these matrices should be possible in a cell-by-cell IDVKLRQ VLPLODU WR WKH FDVH RI JOREDO UH¿QHPHQW )ROORZLQJ >%UD@ ZH SHUIRUP D PXOWLOHYHO PHWKRG RQ WKH FRPSOHWH OHYHO VSDFHV .< , but we UHVWULFW WKH VPRRWKHU WR WKH SDUW WKDW LV UHDOO\ UH¿QHG WR OHYHO O.  'H¿QLWLRQ :H FDOO D JULG FHOO LQ D KLHUDUFK\ RI ORFDOO\ UH¿QHG PHVKHV < active, if LW LV QRW UH¿QHG LQ DQ\ RI WKH WULDQJXODWLRQV RI WKH KLHUDUFK\ $FWLYH FHOOV DUH WKH ORFDOO\ ¿QHVW FHOOV RI WKH KLHUDUFK\ ,Q )LJXUH  DFWLYH FHOOV DUH VKDGHG ZKLOH LQDFWLYH RQHV DUH ZKLWH 7KH shape functions of all active cells constitute the discretization space .
RI WKH ¿QHVW OHYHO  'H¿QLWLRQ :H GH¿QH WKH OHYHO O of a grid cell % in the triangulation hierarchy < recursively as follows: if a cell belongs to the coarse mesh, its level is zero. Otherwise, it was REWDLQHG E\ UH¿QHPHQW RI DQRWKHU JULG FHOO % and we set O  O  5 Since there is no coarsening in a single hierarchy <  WKLV OHYHO LV XQLTXHO\ GH¿QHG 

This does not rule out coarsening in the process of obtaining the hierarchy

CHAPTER 6. LINEAR SOLVERS

146

)LJXUH  $ KLHUDUFK\ RI WKUHH PHVKHV ZLWK ORFDO UH¿QHPHQW

)LJXUH  6SOLWWLQJ RI < into < (shaded cells) and 
< (white)

6.3.4 Remark: Figure 6.4 shows that a cell of level O PD\ EHORQJ WR VHYHUDO PHVKHV < 4 <  DQG VR RQ )RU LQVWDQFH WKH VKDGHG FHOOV RQ WKH LQWHUPHGLDWH OHYHO FHOO OHYHO  EHORQJ WR WKH meshes  and  .  'H¿QLWLRQ (DFK WULDQJXODWLRQ < LV SDUWLWLRQHG LQWR WKH VHW RI FHOOV VWULFWO\ RQ OHYHO O  3 7 <  %  < O  O 4 DQG WKH VHW RI FHOOV RQ ORZHU OHYHOV

 3 7 
<  %  < O 7 O 5

7KLV SDUWLWLRQLQJ LV H[SODLQHG LQ )LJXUH  :H UHPDUN WKDW FHOOV LQ 
< PD\ FRQWDLQ JULG FHOOV RI GLIIHUHQW ORZHU OHYHOV 2EYLRXVO\ WKHUH KROGV <  < " 
< 5

6.3. LOCAL MULTIGRID

147

We call the divider between those two sets the UH¿QHPHQW HGJH  $ between levels O and O  , that is  + 0     %  % ! %  <    <  !  




6.3.6 The partitioning of the spaces .< into subspaces follows the splitting of < : we let .<  .<  .<
4

(6.22)

where .< are the functions in .< with support in < and .<
accordingly. Equation (6.22) KROGV IRU GLVFRQWLQXRXV ¿QLWH HOHPHQWV $ EDVLV IRU WKH VXEVSDFHV LV REWDLQHG E\ UHVWULFWLQJ WKH GH¿QLWLRQ RI WKH EDVLV LQ SDUDJUDSK  WR WKH VXEVHWV RI WKH WULDQJXODWLRQ :H ZLOO DVVXPH that the basis is ordered in a way that cells in < are before those in 
< . Then, a function #  .< LV UHSUHVHQWHG E\ D FRHI¿FLHQW YHFWRU < of the form < 4 
< . 6.3.7 Remark: 7KH RUGHULQJ UHTXLUHG LQ WKH SUHYLRXV SDUDJUDSK GRHV QRW LPSRVH DQ\ RUGHULQJ of the basis functions inside .< and .<
 6XFK DQ RUGHULQJ PD\ EH FKRVHQ WR RSWLPL]H WKH VPRRWKHU VHH 5HPDUN   6.3.8 $ V\VWHP RI HTXDWLRQV RQ .< written in coordinates with respect to this basis, < <  < 4

(6.23)

is then split into %


 < <

 
 < <

)%

< 
<

)



% ) < 5 <


(6.24)

6.3.9 Remark: ,I WKH PDWUL[  LQ  ZDV REWDLQHG E\ DSSO\LQJ D ELOLQHDU IRUP -< #4 ' on

.< to the basis functions of .<  WKHQ WKH PDWULFHV  correspond to the restriction of < and < -< 54 5 to the subspaces .< and .<
, respectively.
 6.3.10 Remark: 7KH PDWULFHV 
< and < FRQVLVW RI ERXQGDU\ ÀX[HV RQ WKH UH¿QHPHQW HGJH  < RQO\ 6LQFH ZH H[SHFW WKH FHOOV DGMDFHQW WR WKHVH HGJHV WR EH D VPDOO QXPEHU FRPSDUHG WR WKH ZKROH PHVK WKHVH PDWULFHV KDYH YHU\ IHZ QRQ]HUR HQWULHV DQG FDQ EH VWRUHG YHU\ HI¿FLHQWO\

 'H¿QLWLRQ The restriction operator < ) .<  .< acts as
 SURMHFWLRQ < on .< and as identity on .<
, < <  < <  
< 5

(6.25)

CHAPTER 6. LINEAR SOLVERS

148 

6.3.12 Algorithm: Let < ,  4 5 5 5 be a sequence of smoothers as in algorithm 6.2.9 on page 136, but acting on .< only. Then, the action of the local variable V-cycle operator < on   ): a vector < LV GH¿QHG UHFXUVLYHO\ E\     and ( )

1. (Pre-smoothing) Compute  



   )

and let )  

LWHUDWLYHO\ E\ 



 < <   < 

4

 4 5 5 5 4 1< 4

4  .

2. (Coarse grid correction) Let " ) ) &   )  < < < <   5   
<  
 <  < 

3. (Post-smoothing) Compute ) LWHUDWLYHO\ E\ 



  

) 

 <



<   < 

4

 4 5 5 5 4 1< 5



where <  <  
< 
)

4. Set < <  



4 
.

6.3.13 Theorem: Algorithm 6.3.12 is equivalent to algorithm 6.2.9 if the smoothers there are taken of the form ) %  <   <  5  



Proof. First, due to the structure of the smoother and due to 0   ZH KDYH 0
  during the whole pre-smoothing iteration. Therefore,  %  )   <   <    < 0 < <  < 0      
<  
 < 0 %  )  < <    <    The form of the restricted residual in the coarse grid correction is an immediate consequence  of the splitting of the restriction operator in (6.25). Finally, 
remains constant during postsmoothing and therefore  % )  )    
 <    < )   <   < 
<  <    < 

   
<  
 <   < 
% )   )  < <    <   4  which concludes the proof.

6.3. LOCAL MULTIGRID

149



6.3.14 Remark: For < DQ\ RI WKH VPRRWKLQJ PHWKRGV LQ 'H¿QLWLRQ  DSSOLHG WR  < PD\ EH XVHG +HUH LW LV LPSRUWDQW WKDW WKH PDWUL[  XVHG E\ WKH UHOD[DWLRQ PHWKRG LV WKH < VDPH PDWUL[ XVHG LQ WKH FRPSXWDWLRQ RI UHVLGXDOV 7KHUHIRUH QR DGGLWLRQDO PDWUL[ KDV WR EH DVVHPEOHG RU VWRUHG 6.3.15 Remark: 7KH VSOLWWLQJ RI RSHUDWRUV LQ WKH DOJRULWKP UHQGHUV LW SRVVLEOH WKDW VKDSH IXQFWLRQV RI DQ DFWLYH JULG FHOO RQ D ORZHU OHYHO DUH XVHG only RQ WKLV OHYHO QRW RQ ¿QHU RQHV ,Q IDFW WKH IXQFWLRQ FRUUHVSRQGLQJ WR WKH YHFWRU < RQ DQ DFWLYH FHOO HTXDOV WKH IXQFWLRQ UHSUHVHQWHG by 
RQ WKH ¿QHVW OHYHO SRVVLEO\ PRGL¿HG E\ ERXQGDU\ FRQWULEXWLRQV IURP QHLJKERULQJ UH¿QHG FHOOV )XUWKHUPRUH WKLV PHDQV WKDW PDWULFHV 

< DUH QHLWKHU DVVHPEOHG QRW XVHG 6.3.16 ,Q WKH IROORZLQJ SDUDJUDSKV ZH H[SODLQ KRZ WKLV DOJRULWKP FDQ EH LPSOHPHQWHG XVLQJ RQO\ WKH URXWLQHV XVHG IRU LQWHJUDWLQJ WKH ELOLQHDU DQG OLQHDU IRUPV RQ JULG FHOOV DQG HGJHV 6LQFH WKHVH DUH XVHG IRU DVVHPEOLQJ WKH JOREDO V\VWHP PDWUL[ DV ZHOO ZH DVVXUH WKDW QR DGGLWLRQDO ZRUN LV UHTXLUHG IRU WKH ORFDO PXOWLJULG PHWKRG 6.3.17 :H DVVXPH WKDW WKHUH DUH IXQFWLRQV CellMatrix(T) and EdgeFlux(E) FRPSXW ing the local contributions of the bilinear form - 54 5 E\ VRPH LQWHJUDWLRQ PHWKRG 7KH UHVXOW of the function CellMatrix(T) LV D VLQJOH PDWUL[  of dimension     7KH UHVXOW of EdgeFlux(E) is a set of four matrices  , ,, of dimensions    . Here, % and % DUH WKH WZR FHOOV DGMDFHQW WR  )RU VLPSOLFLW\ ZH DVVXPH WKDW WKLV IXQFWLRQ FDQ KDQGOH ERXQGDU\ ÀX[HV DV ZHOO WKHQ WKH SDUDPHWHU % LV YRLG DQG RQO\  , is returned. 6.3.18 Algorithm: :LWK WKH PDWULFHV GH¿QHG LQ WKH SUHYLRXV SDUDJUDSK WKH V\VWHP PDWUL[  LV DVVHPEOHG LQ D WZRVWHS VXPPDWLRQ SURFHVV VWDUW ZLWK    RPLWWLQJ WKH LQGH[  for REYLRXV UHDVRQV   )RU HDFK DFWLYH FHOO %   let 

,,  ,,  

4 $   5 5 5  5

 )RU HDFK LQWHULRU HGJH    let % and % EH WKH DGMDFHQW FHOOV DQG OHW  ,

 , ,   , ,   

  5 5 5  $   5 5 5 

84 D  4 

2Q HDFK ERXQGDU\ HGJH GR WKH VDPH ZLWK 84 D   RQO\ 6.3.19 Remark: $OJRULWKP  ZKLFK LV WKH XVXDO ZD\ RI EXLOGLQJ PDWULFHV LQ ¿QLWH HO HPHQW FRGHV DSSOLHV WR ORFDOO\ UH¿QHG PHVKHV ZLWK KDQJLQJ QRGHV ,Q WKLV FDVH WKH HGJHV  LQ WKH VHFRQG VWHS PXVW EH WKH HGJHV RI WKH ¿QHU FHOOV VHH )LJXUH   +HUH WKH ODUJHU FHOO DSSHDUV WZLFH LQ WKH VXPPDWLRQ SURFHVV DV %% and %*  6WDELOL]DWLRQ SDUDPHWHUV PXVW EH FKRVHQ SURSHUO\ LQ WKLV FDVH VHH WKH GLVFUHWL]DWLRQ FKDSWHUV IRU GHWDLOV 7KLV DOJRULWKP DOVR DSSOLHV WR DGDSWLYH GLVFUHWL]DWLRQV ZKHUH WKH QXPEHU RI GHJUHHV RI IUHHGRP YDULHV EHWZHHQ JULG FHOOV

CHAPTER 6. LINEAR SOLVERS

150

%%

 % %%

%*

 * %*

)LJXUH  $VVHPEOLQJ RQ ORFDOO\ UH¿QHG PHVKHV 6.3.20 Algorithm: $ VFKHPH VLPLODU WR DOJRULWKP  LV XVHG WR DVVHPEOH WKH PDWULFHV
 
 < , < and <  6WDUW ZLWK DOO PDWULFHV ]HUR DQG  IRU HDFK FHOO % RI WKH WULDQJXODWLRQ KLHUDUFK\ <  OHW O EH WKH OHYHO RI WKH FHOO VHH   /HW 

  < ,,  < ,,  

4 $   5 5 5  5

 )RU HDFK LQWHULRU HGJH  RI WKH KLHUDUFK\ OHW % and % EH WKH DGMDFHQW FHOOV DQG O the OHYHO RI WKH HGJH ,I WKHUH LV QR QHLJKERU FHOO RQ OHYHO O RQ RQH VLGH WKH FHOO RQ WKH FRDUVHU OHYHO LV WDNHQ (a) If O  O  O  OHW  ,

   <  , ,  <  , ,  

  5 5 5  $   5 5 5 

84 D  4 5

7KH VDPH LV GRQH RQ WKH ERXQGDU\ ZKHUH 84 D   only. E 2WKHUZLVH    < and one of the neighbors % and % , say % is on level O  . 7KHQ OHW  ,

   <  , ,  <  , ,  

 ,

   <   , ,  <   , ,  

 ,


  
 <  , ,  <  , ,  


<  , ,




<  , ,

 ,

  

6.3.21 Remark: 7KH DOJRULWKPV IRU DVVHPEOLQJ WKH V\VWHP PDWUL[ DQG IRU DVVHPEOLQJ WKH OHYHO PDWULFHV XVH WKH VDPH FHOO PDWULFHV ZKLFK VLPSOL¿HV WKH FRGLQJ IRU FRPSOH[ V\VWHPV RI HTXDWLRQV FRQVLGHUDEO\  6.3.22 Remark: 6LQFH WKH PDWUL[ 


is never assembled and since the matrices < are not VSOLW LQWR DQ DFWLYH SDUW DQG DQ RYHUODSSHG SDUW LW LV LPSRVVLEOH WR XVH WKH OHYHO PDWULFHV IRU WKH PXOWLSOLFDWLRQ ZLWK WKH V\VWHP PDWUL[   7KH UHVXOWLQJ GLVDGYDQWDJH RI GRXEOLQJ WKH PHPRU\ UHTXLUHPHQWV PD\ EH RYHUFRPH E\ WZR UHPHGLHV ¿UVW ORZ DFFXUDF\ LV VXI¿FLHQW IRU WKH OHYHO PDWULFHV VLQFH WKH\ DUH XVHG LQ D SUHFRQGLWLRQHU 6HFRQG DVVHPEOLQJ RI WKH V\VWHP PDWUL[ PD\ EH UHSODFHG E\ LQWHJUDWLQJ WKH UHVLGXDO LPPHGLDWHO\ WKLV DSSURDFK PD\ EH HYHQ PRUH WLPH HI¿FLHQW LQVLGH D QRQOLQHDU LWHUDWLRQ ZKHUH RQO\ IHZ OLQHDU UHVLGXDOV ZLWK WKH VDPH PDWUL[ DUH QHHGHG

6.4. LDG

151


2 3 4 5 6 7 8

 8 9 9 9 9 9 9

 , ?   N   0.053 1.275 11 0.058 1.235 12 0.062 1.210 12 0.062 1.208 12 0.062 1.209 12 0.062 1.211 12 0.061 1.213 12

 , ?  N 0.117 0.139 0.143 0.144 0.143 0.143 0.143

 , ?     N  0.932 9 0.054 1.268 0.858 9 0.070 1.158 0.846 9 0.072 1.142 0.843 9 0.071 1.150 0.843 9 0.070 1.155 0.844 9 0.069 1.162 0.845 9 0.068 1.169

7DEOH  &RQWUDFWLRQ DQG FRQYHUJHQFH UDWHV RQ ORFDOO\ UH¿QHG JULGV

6.3.23 ,Q 7DEOH  ZH VKRZ WKDW WKH SUHFRQGLWLRQHU LV ZHOO VXLWHG IRU ORFDOO\ UH¿QHG PHVKHV DQG ORFDO VPRRWKLQJ 7KH FRQWUDFWLRQ QXPEHUV GLVSOD\HG LQ DUH H[DFWO\ WKH VDPH DV LQ WDEOHV  DQG  5HVXOWV ZHUH REWDLQHG ZLWK WKH PHVKHV LQ )LJXUH $

6.4 LDG 6.4.1 In this section, we will continue the investigations of Section 6.2 and develop precondiWLRQLQJ PHWKRGV IRU WKH /'* PHWKRG 7KLV ZLOO EH DFKLHYHG LQ WZR VWHSV )LUVW D SUHFRQGLWLRQHU IRU WKH 6FKXU FRPSOHPHQW RI WKH PL[HG OLQHDU V\VWHP LV GHULYHG ,Q D VHFRQG VWHS D EORFN SUH FRQGLWLRQHU LV FRQVWUXFWHG IRU WKH FRPSOHWH V\VWHP 6.4.2 8VLQJ EDVHV DFFRUGLQJ WR 'H¿QLWLRQ  RQ SDJH  IRU WKH VSDFHV 0 and . , the GLVFUHWH V\VWHP  EHFRPHV D OLQHDU V\VWHP RI HTXDWLRQV RI WKH W\SH 


  4 with


%

7   

)

4



% ) L 4 



% ) ; 5 



+HUH OLNH LQ SUHYLRXV FKDSWHU D PDWUL[ HQWU\ DW ORFDWLRQ %4 * !4 $ LV FRPSXWHG E\ DSSO\LQJ WKH FRUUHVSRQGLQJ ELOLQHDU IRUP WR EDVLV IXQFWLRQV ZLWK WKHVH LQGLFHV )RU LQVWDQFH OHW <, be a basis function of . and C, D YHFWRUYDOXHG EDVLV IXQFWLRQ RI 0 , then ,,  2 <, 4 C, 5  'H¿QLWLRQ $VVRFLDWHG ZLWK WKH PDWUL[
is its Schur complement ! ZKLFK LV WKH PDWUL[     7   4

ZKLFK LV JHQHUDWHG IURP
E\ EORFNHOLPLQDWLRQ RI WKH YDULDEOH L.



CHAPTER 6. LINEAR SOLVERS

152

6.4.4 Remark: %\ GH¿QLWLRQ  LV V\PPHWULF LI 7 and  DUH V\PPHWULF ZKLFK LV WKH FDVH IRU ERWK YHUVLRQ RI WKH /'* PHWKRG )XUWKHUPRUH L4  LV D VROXWLRQ RI  LI DQG RQO\ LI  is a solution of   

  ;   7  4



 and L   7   )LQDOO\ VLQFH 7 LV SRVLWLYH GH¿QLWH  SRVLWLYH VHPLGH¿QLWH DQG WKH V\VWHP LV XQLTXHO\ VROYDEOH /HPPD   WKH PDWUL[  LV SRVLWLYH GH¿QLWH

6.4.5 7KH ELOLQHDU IRUP  54 5 DVVRFLDWHG ZLWK WKH PDWUL[  FRUUHVSRQGV WR D SULPDO IRUPX ODWLRQ RI WKH /'* PHWKRG 7KLV SULPDO IRUPXODWLRQ LV GHVFULEHG " & DQG DQDO\]HG LQ >$%&0 $%&0@ :H ZLOO XVH WKH HTXLYDOHQFH RI  54 5 and -IP  54 5 VWDWHG LQ &RUROODU\  WR FRQVWUXFW HI¿FLHQW SUHFRQGLWLRQHUV IRU . 6.4.6 Remark: 7KH PDWUL[ 7 consists of  GLDJRQDO EORFNV ZKLFK DUH ERWK VFDOHG YHUVLRQV RI WKH PDVV PDWUL[ 6LQFH WKH GLVFUHWL]DWLRQ LV GLVFRQWLQXRXV WKHVH EORFNV GHFRXSOH IXUWKHU LQWR VPDOOHU EORFNV FRQVLVWLQJ RI VFDOHG FHOO PDVV PDWULFHV IRU HDFK PHVK FHOO ,I ZH FKRVH RUWKRJRQDO SRO\QRPLDOV RQ % ZLWK UHVSHFW WR WKH ZHLJKW 6D  7 ZRXOG EH D GLDJRQDO PDWUL[ VHH >&DV@  7KHUHIRUH HYDOXDWLRQ RI WKH 6FKXU FRPSOHPHQW LV SDUWLFXODUO\ VLPSOH 7 can be inverted in a FKHDS ZD\ EHIRUH WKH LWHUDWLYH VROXWLRQ DQG PXOWLSOLFDWLRQ ZLWK  7 EHFRPHV DERXW DV H[SHQVLYH DV D PXOWLSOLFDWLRQ ZLWK .  'H¿QLWLRQ 7ZR V\PPHWULF SRVLWLYH GH¿QLWH ELOLQHDU IRUPV -54 5 and 254 5 RQ WKH VSDFH . are called spectrally equivalent LI WKH HVWLPDWH  -'4 '  2'4 '   -'4 '4



KROGV ZLWK FRQVWDQWV  and  IRU DQ\ '  . . 6.4.8 Lemma: Let - 54 5 and  54 5 EH WZR V\PPHWULF SRVLWLYH GH¿QLWH DQG VSHFWUDOO\ HTXLY alent bilinear forms with associated matrices  and  (both with respect to the same basis). Let furthermore be a uniform preconditioner for . Then, is also a uniform preconditioner for . Proof. If LV D XQLIRUP SUHFRQGLWLRQHU IRU  WKHQ WKHUH DUH FRQVWDQWV  and  such that for DQ\    (    . )           4 RU HTXLYDOHQWO\            5 6LQFH IURP  IROORZV             4 WKH OHPPD IROORZV E\ FRPELQLQJ WKH ODVW WZR HVWLPDWHV

6.4. LDG

153 
1 2 3 4 5 6 7

 10 14 14 15 15 15 15



  1.363 12 0.763 14 0.715 14 0.708 14 0.701 14 0.701 14 0.701 14



  .864 13 .758 15 .752 16 .743 16 .742 16 .741 16 .741 16

  .794 16 .702 17 .653 17 .645 17 .643 17 .642 17 .642

  .648 .606 .592 .592 .591 .592

! Table 6.10: Conjugate gradient convergence rates for standard LDG with preconditioner .

   

M 2 4 4 4

? 4 9 13 13

Table 6.11: Values of stabilization parameters yielding optimal convergence

!
DV GH¿QHG LQ $OJRULWKP  RQ SDJH  LV 6.4.9 Lemma: The multilevel preconditioner a uniform preconditioner for the LDG Schur complement matrix 
on level
. Proof. This result" is &an immediate consequence of Lemma 6.4.8 the equivalence (3.31) of the bilinear forms -IP  54 5 and  54 5. !
6.4.10 The following numerical results give evidence to the fact that the preconditioner for the matrix 
is not only uniform , but that the resulting condition numbers are in fact VXI¿FLHQWO\ VPDOO WR UHFRPPHQG WKLV PHWKRG IRU SUDFWLFDO FRPSXWDWLRQ 6.4.11 Like in the previous chapter, we begin by investigating the convergence properties on Cartesian grids (see A.1.1 on page 167), D   and the right hand side chosen according to example A.2.2 on page 170. Table 6.10 shows convergence rates of the cg method for the !
for different polynomial degrees on LDG Schur complement 
using the preconditioner Cartesian grids. As for the interior penalty method, the convergence rate is independent of the mesh level as soon as  LV VXI¿FLHQWO\ VPDOO )XUWKHUPRUH LW GHWHULRUDWHV RQO\ VORZO\ ZLWK WKH polynomial degree. The choice of parameters can be found in Table 6.11. These parameters ZHUH IRXQG RSWLPDO E\ H[SHULPHQW $ WKHRUHWLFDO MXVWL¿FDWLRQ IRU WKHLU XQH[SHFWHG GHSHQGHQFH on the polynomial degree is still lacking. ! has two free parameters, namely M 6.4.12 The preconditioner  and ?. In paragraph 3.2.10 on page 75, we observed that M  GRHV QRW LQÀXHQFH WKH GLVFUHWL]DWLRQ DFFXUDF\ RYHU D ZLGH UDQJH RI YDOXHV )LJXUH  VXJJHVWV WKDW LWV LQÀXHQFH RQ WKH SUHFRQGLWLRQHU LV PXFK PRUH important. The graphs suggest that M  should be chosen close to the optimal value to achieve JRRG FRQYHUJHQFH 6WLOO WKH SHDN RI WKH FRQYHUJHQFH UDWHV LV TXLWH ÀDW DQG D IDFWRU RI  FKDQJHV WKH FRQYHUJHQFH UDWHV E\ OHVV WKDQ  )XUWKHUPRUH WKH RSWLPDO YDOXH LV ODUJHU WKDQ WKH SRLQW

CHAPTER 6. LINEAR SOLVERS

154

0.15 0.3 0.45 0.6 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

2-6

2-4

2-2 20 Guu

22

24

6

2

0

2

4

6

20 18 16 14 12 10 8 K

0.15 0.3 0.45 0.6 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 60 50 -6

2

40 2-4

2-2 20 Guu

30 2

2

20 4

2

K

10 26 0

Figure 6.7: Multigrid convergence rates for LDG with IP preconditioning,  (left) and  (right) elements

6.4. LDG

155
1 2 3 4 5 6 7

!





12 6 27 14 39 20 59 29 82 50 114 85 155 143

Table 6.12: Iteration counts  for superconvergent LDG 
1 2 3 4 5 6 7

 17 31 37 38 39 40 40

  .874 28 .497 38 .418 41 .402 43 .394 44 .392 45 .393 45

   .548 38 .403 46 .376 50 .363 52 .358 53 .357 54 .357 54

  .408 .333 .309 .301 .299 .298 .298

7DEOH  3HUIRUPDQFH RI FJ PHWKRG IRU PRGL¿HG VXSHUFRQYHUJHQW /'* where the accuracy of the discretization levels off in Figure 3.1 on page 76. Similarly, the convergence rates do not deteriorate fast if ? is chosen larger than the optimal value.

Preconditioning the superconvergent scheme 6.4.13 We apply the same preconditioning techniques to the superconvergent method, even if (3.31) is not proven for this case. The results    are in Table 6.12. They exhibit # with M ! a growth of the iteration counts with about 
with the interior penalty preconditioner (?  ). We compare this preconditioner with the genuine LDG preconditioner , where the smoothing is performed on the matrix  itself and conclude that both solvers are not satisfactory.

6.4.14 The performance of the superconvergent LDG method can be improved by using the ÀX[  RQ WKH 'LULFKOHW ERXQGDU\ VHH DOVR 5HPDUN   ,QGHHG 7DEOH  VKRZV WKDW the multilevel interior penalty scheme is a uniform preconditioner for the resulting method.

Solving the saddle-point system 6.4.15 Preconditioners for (stabilized) saddle-point systems can be derived using the following result due to [MGW00]: % ) % ) ) %        5       

CHAPTER 6. LINEAR SOLVERS

156

If we could compute the inverse on the left, the spectrum of the matrix would consist of the single value 1, and Krylov-space methods would converge in very few steps. Unfortunately, this includes the exact inversion of , which we have to replace by a reasonable approximate, !
or
 7KXV ZH GH¿QH WKH SUHFRQGLWLRQHU IRU WKH the uniform preconditioner , either saddle-point system % )    ) 4 (6.31)  M  where we introduced the additional tuning parameter M 9 . 6.4.16 Lemma: The matrix
 has only real eigenvalues and its spectrum is ! "  5 L
   LM !

(6.32)

Proof. (from [Kan03b]) The proof is based on the proof to [Kla98, Lemma 3.1]. Instead of computing the spectrum of
 , we consider the transpose system and solve the generalized eigenvalue problem
 0  A 04 or, splitting into components, L     AL  A 4 !  5 L    AM


(6.33)

If A   WKH ¿UVW HTXDWLRQ LV YRLG ,Q WKLV FDVH D QRQWULYLDO VROXWLRQ WR !     4 L    M  
  

 is singular, choose a non-zero pair L4      is an eigenvector to the eigenvalue 1. If   arbitrarily. If   is regular, choose L -  arbitrarily and  )    L. Therefore, A   is   eigenvalue. For A - , we compute the Schur complement of (6.33), namely, !  5        AM
!
from the left, we see that A is an eigenvalue of M !
! and  its associated Multiplying with M eigenvector. Finally, we remark that the eigenvalues of
 DUH HTXDO WR WKRVH RI 
 . 6.4.17 Remark: Since the matrix
 is not normal, convergence results for Krylov space methods cannot be applied solely based on the previous lemma. In [Kla98], normality of the PDWUL[ LV HVWDEOLVKHG ZLWK UHVSHFW WR DQRWKHU V\PPHWULF DQG SRVLWLYH GH¿QLWH PDWUL[ 8QIRUWX nately, the theoretical results yield a number of iteration steps growing like 
, a growth not observed in the numerical computations below.

6.4. LDG

157


1 2 3 4 5 6 7

Schur system     10 1.438 11 1.226 14 0.768 15 0.717 14 0.721 15 0.670 15 0.710 16 0.665 15 0.703 16 0.660 15 0.703 16 0.660 15 0.702 16 0.660

Table 6.14: GMRES performance for the preconditioned LDG Schur complement and the preconditioned system ( -elements)

GMRES steps

64

Q1 Q2 Q2

32

16 0.01

0.1

1 Gu

10

100

Figure 6.8: GMRES steps for the standard LDG system depending on the scaling factor M .

6.4.18 Since the preconditioned matrix
 is non-symmetric, we apply GMRES iteration and compare to GMRES applied to the preconditioned Schur complement in Table 6.14. The setup of the test calculations is the same as in the previous sections. The table shows that the GMRES convergence rates on the Schur complement are exactly as the corresponding values for the cg iteration in Table 6.10 on page 153. Applied to the saddle point system, the GMRES method needs exactly one more iteration than for the Schur complement, possibly due to the additional eigenvalue according to Lemma 6.4.16. :H FRQFOXGH WKDW WKH *05(6 PHWKRG RQ WKH VDGGOH SRLQW V\VWHP LV DQ HI¿FLHQW DOWHUQDWLYH if computation of the Schur complement is not feasible. Still, we remark that this approach suffers from vector lengths being 3 times (4 times in three dimensions) as large as for the Schur complement. !
! with respect to the addi6.4.19 The tuning parameter M serves to shift the spectrum of tional eigenvalue 1 of the system. Figure 6.8 exhibits that the convergence speed of the preconditioned GMRES method is rather insensitive to M within a wide range. This is expected IURP /HPPD  VLQFH WKH VLQJOH HLJHQYDOXH  GRHV QRW LQÀXHQFH WKH FRQYHUJHQFH RI .U\ORY space methods critically, even if isolated. Therefore, we will choose M  4

CHAPTER 6. LINEAR SOLVERS

158 D
2 3 4 5 6 7 8



1.6 1.4 1.3 1.2 1.2 1.2 1.2





















1.5 1.4 1.3 1.3 1.2 1.2 1.2

1.6 1.4 1.3 1.3 1.3 1.3 1.2

2.3 1.7 1.5 1.3 1.3 1.3 1.3

3.2 2.3 1.7 1.5 1.5 1.3 1.3

4.2 3.2 2.4 1.6 1.3 1.5 1.5

5.4 4.4 3.4 2.4 1.6 1.2 1.3

6.6 5.6 4.5 3.5 2.5 1.7 1.1

7.8 6.8 5.9 4.7 3.7 2.6 1.7

9.0 8.0 7.1 6.1 5.2 3.8 2.7

10.1 9.2 8.3 7.3 6.3 5.4 4.0

Table 6.15: Convergence rates for GMRES with multilevel preconditioner using downwind block-Gauß-Seidel smoothing

in all further computations.

6.5 Advection Diffusion 6.5.1 $VVXPH WKDW WKH DGYHFWLRQ ¿HOG 2 in equation (5.10) does not contain loops (see paragraph 6.5.5 below for the other case). In paragraph 1.4.19, we remarked" that & the mesh cells can be ordered in such a way that the matrix associated with the form E 54 5 is upper blocktriangular and the diagonal blocks correspond to the cell matrices. Therefore, the block-Gauß6HLGHO VPRRWKHU GH¿QHG LQ  LV D GLUHFW VROYHU 6.5.2 Based on these observations, we construct a multilevel preconditioner for the discretized DGYHFWLRQGLIIXVLRQ HTXDWLRQ  XVLQJ DOJRULWKP  ZLWK WKH PRGL¿FDWLRQ WKDW ZH GR not use transposed smoothing operators. We also will allow different numbers of pre and post smoothing steps. 6.5.3 We claim that the preconditioner constructed this way is robust with respect to the diffusion parameter, that is, a GMRES iteration applied to the preconditioned matrix converges LQ RQH VWHS LI WKH GLIIXVLRQ FRHI¿FLHQW D vanishes, like in the self-adjoint case, if D +  and the transition between the two extremal states is monotonous. Indeed, Table 6.15 shows this behavior. The problem solved was example A.3.2 with  elements. On the left side of the table, we see the same convergence rates as in Table 6.5 on page 142. They correspond to gaining more than one decimal in each iteration. To the right, these rates increase monotonously until up to 10 decimals are gained in a single step. This is the case where the smoother is nearly an exact solver. The convergence rates remain constant on the diagonals with D6  .

6.5. ADVECTION DIFFUSION

159 D


2 3 4 5 6 7 8



1.6 1.4 1.3 1.3 1.3 1.3 1.3



















1.5 1.3 1.3 1.3 1.3 1.3 1.3

1.3 1.1 1.2 1.2 1.2 1.3 1.3

1.2 0.9 0.9 1.0 1.1 1.2 1.2

1.1 0.8 0.6 0.6 0.8 1.0 1.1

1.1 0.7 0.5 0.3 0.4 0.6 0.8

1.2 0.7 0.4 0.2 0.2 0.2 0.4

1.2 0.7 0.4 0.2 0.2 0.1 0.2

1.1 0.7 0.4 0.2 0.1 0.1 0.1

1.1 0.7 0.4 0.2 0.1 0.1 0.1



1.1 0.7 0.4 0.2 0.1 0.1 0.1

Table 6.16: Performance of GMRES with multilevel preconditioner using upwind block-GaußSeidel smoothing

Figure 6.9: Numbering in case of a vortex

6.5.4 6LQFH ¿QGLQJ D GRZQZLQG RUGHULQJ LV QRW D VLPSOH WDVN LQ JHQHUDO ZH KDYH WR FKHFN LI LW LV UHDOO\ LPSRUWDQW ,Q 7DEOH  ZH VROYH WKH VDPH SUREOHP DV LQ 7DEOH  EXW WKLV WLPH VRUWLQJ WKH GHJUHHV RI IUHHGRP LQ RSSRVLWH GLUHFWLRQ +HUH WKH FRQYHUJHQFH UDWHV GURS as D WHQGV WR ]HUR UHÀHFWLQJ D JDLQ RI DFFXUDF\ RI OHVV WKDQ D GHFLPDO LQ  VWHSV LQ WKH ODVW FROXPQ 7KHUHIRUH ZH FRQFOXGH WKDW DW OHDVW D UHDVRQDEOH QXPEHULQJ DOJRULWKP LV FUXFLDO IRU the performance of the preconditioner. 6.5.5 ,I WKH ÀRZ ¿HOG KDV FORVHG LQWHJUDO FXUYHV VRPHZKHUH LQ WKH GRPDLQ D GRZQZLQG QXP EHULQJ FDQQRW EH IRXQG DQ\PRUH ,Q WKLV FDVH D QXPEHULQJ VFKHPH GHVFULEHG LQ >%: +3@ FDQ EH HPSOR\HG $VVXPH WKDW WKH YHFWRU ¿HOG FRQWDLQV D YRUWH[ OLNH LQ )LJXUH  7KHQ WKLV YRUWH[ LV FXW IURP LWV FHQWHU WR LWV ERXQGDU\ PRUH RU OHVV SHUSHQGLFXODU WR WKH YHFWRU ¿HOG $IWHU ZDUGV PHVK FHOOV DUH QXPEHUHG VWDUWLQJ DW WKLV FXWOLQH LQ GRZQZLQG GLUHFWLRQ XQWLO WKH FXWOLQH is reached from the opposite side. 6.5.6 ,Q 7DEOH  ZH VKRZ WKH HIIHFW RI VXFK D QXPEHULQJ LQ WKH FDVH RI WKH VLPSOH URWDWLQJ ÀRZ ¿HOG RI H[DPSOH $ 6LQFH WKH PDWUL[ LV QRW EORFNWULDQJXODU ZH GR QRW H[SHFW VROXWLRQ

CHAPTER 6. LINEAR SOLVERS

160 D
2 3 4 5 6 7 8



1.4 1.3 1.3 1.3 1.3 1.3 1.3



















1.5 1.4 1.3 1.3 1.3 1.3 1.3

1.4 1.3 1.3 1.3 1.3 1.3 1.3

1.7 1.3 1.3 1.3 1.3 1.3 1.3

1.7 1.4 1.1 1.2 1.2 1.3 1.3

1.9 1.4 1.0 1.0 1.0 1.1 1.3

2.2 1.4 1.0 0.7 0.8 0.9 1.0

2.4 1.4 1.0 0.6 0.5 0.6 0.7

2.8 1.4 1.0 0.6 0.4 0.3 0.4

3.1 1.4 1.1 0.6 0.4 0.2 0.2



2.2 1.3 1.0 0.6 0.3 — 0.1

Table 6.17: Convergence rates for GMRES with multilevel preconditioner using downwind block-Gauß-Seidel smoothing in a vortex

in a single step if D  . Even more, we know from the solution obtained by the method of characteristics that the equation looses stability if D   (see Remark 1.4.18). Therefore, ZH REVHUYH WKDW WKH FRQYHUJHQFH UDWHV RQ ¿QHU JULGV EUHDN GRZQ GUDPDWLFDOO\ LI D 7   5   $V ORQJ DV WKH GLIIXVLRQ FRHI¿FLHQW VWD\V DERYH WKLV WKUHVKROG WKH SHUIRUPDQFH RI WKH SUHFRQGLWLRQHU GHSHQGV RQO\ YHU\ ZHDNO\ RQ WKH GLIIXVLRQ FRHI¿FLHQW 7KHUHIRUH ZH FRQFOXGH that the proposed method is robust with respect to dominant advection in a wide range.

6.6 Stokes 6.6.1 Following paragraph 6.4.2, we derive the matrix formulation of the Stokes system 4.16 as
  4

(6.34)

with     7      4
  

  L     4

  ;      5 

(6.35)

The matrices  7 ,  and  are block diagonal multiples of the LDG matrices 7 ,  and , respectively in the sense that e.g.     4 5 5 5 4  with  diagonal blocks.  'H¿QLWLRQ The Schur complement  of the matrix
is   4     

(6.36)

where  is the block diagonal matrix   4 5 5 5 4  with  diagonal blocks and  the LDG 6FKXU FRPSOHPHQW GH¿QHG LQ   /HPPD ,I WKH FRHI¿FLHQWV RI WKH PHWKRG DUH FKRVHQ DV LQ  WKH FRQGLWLRQ QXPEHU of  is bounded independent of the viscosity D and the mesh size .

6.6. STOKES

161

Proof. Following [Bec95], we estimate the minimal eigenvalue by the Rayleigh quotient for the Schur complement, namely            
             
           
       #4    4  
  5     

A 

 

due to the symmetry of  . Here,         and  is the image of  in . . Since         4 # LDG , where # is the image of  in . , we have by the inf-sup condition (4.37)

 5 D For the maximal eigenvalue, we follow the same path A 


      #4    4  5        

A   

By Friedrichs inequality,

    4   D    5

Furthermore by the equivalence (3.31),

yielding

       #4   D  #   D   4 A  

 4 D

which concludes the proof, since A  6A 
does not depend on  and D. 6.6.4 Corollary: An accurate solution of the Schur complement equation of (6.34) can be achieved by the conjugate gradient method preconditioned with the mass matrix in a number of iteration steps bounded independent of  and D. 6.6.5 Following the same path as in paragraph 6.4.15, we construct a block preconditioner for ! for the LDG Schur complement and the Stokes saddle-point system using the preconditioner the mass matrix for the Stokes Schur complement   7     (6.37)  

CHAPTER 6. LINEAR SOLVERS

162 
1 2 3 4 5 6 7

 27 31 34 35 35 35 36

  .379 29 .330 33 .296 37 .288 37 .288 37 .287 38 .283 38

  .350 .305 .271 .274 .271 .270 .269

Table 6.18: GMRES performance for the Stokes system

6.6.6 Since the accurate inversion of the Laplacian in each step of the outer iterative scheme LV YHU\ WLPHFRQVXPLQJ ZH LPSURYH WKH HI¿FLHQF\ RI WKH 6WRNHV SUHFRQGLWLRQHU UHSODFLQJ LW E\ ! LQ HDFK YHORFLW\ FRPSRQHQW 7KH SHUIRUPDQFH RI WKLV SUHFRQGL WKH PXOWLOHYHO SUHFRQGLWLRQHU tioner can be seen in Table 6.18. We see that the number of iteration steps grows by about 50% FRPSDUHG WR WKH ³H[DFW´ SUHFRQGLWLRQHU EXW UHSODFLQJ DERXW  LQQHU LWHUDWLRQ VWHSV IRU VROYLQJ the Poisson problem in each component by a single one. &RPSDULQJ WKHVH UHVXOWV ZLWK WKH SUHYLRXV FKDSWHU ZH FRQFOXGH WKDW WKH VROXWLRQ RI WKH 6WRNHV problem requires less than twice as many iteration steps than the Poisson equation with the same DPRXQW RI ZRUN SHU VWHS DQG FRPSRQHQW 7KHUHIRUH LW FDQ EH FRQVLGHUHG YHU\ HI¿FLHQW 6.6.7 7KH FRQGLWLRQ QXPEHU RI WKH 6FKXU FRPSOHPHQW DQG WKHUHIRUH WKH FRQYHUJHQFH UDWH RI WKH VDGGOHSRLQW VROYHU QDWXUDOO\ GHSHQGV RQ M  and M   ,Q )LJXUH  ZH VKRZ WKLV GHSHQGHQFH IRU VROYLQJ WKH 3RLVVHXLOOH ÀRZ H[DPSOH $ ZLWK  and  HOHPHQWV 'LVSOD\HG DUH LVR lines close to the optimal convergence rates   PD[ as well as 6, 6 and 6'. We see a GHSHQGHQFH RQ M  VLPLODU WR WKH /'* PHWKRG IRU 3RLVVRQ¶V HTXDWLRQ WKH DUHD ZLWK FRQYHUJHQFH rates better than 6 of the optimum stretches by a factor of about  . Remarkably, M  has a VWURQJ LQÀXHQFH RQ WKH VROYHU HYHQ LI LW GRHV QRW DIIHFW VROXWLRQ DFFXUDF\ VHH SDUDJUDSK  on page 98). Here, convergence rates above 6 RI WKH RSWLPXP DUH REWDLQHG RQO\ LQ D VSDQ RI  .

6.7 Oseen equations 6.7.1 $V VRRQ DV DGYHFWLRQ LV DGGHG WR WKH 6WRNHV RSHUDWRU WKH SHUIRUPDQFH RI WKH OLQHDU VROYHU EHFRPHV GHSHQGLQJ RQ WKH YLVFRVLW\ D LQ D FUXFLDO ZD\ ,W WXUQV RXW WKDW WKH SUHFRQGLWLRQHU GHYHORSHG LQ WKH SUHYLRXV VHFWLRQ GRHV QRW \LHOG VDWLVIDFWRU\ UHVXOWV 6.7.2 $QDORJRXV WR HTXDWLRQ   WKH PDWUL[ RI WKH /'* GLVFUHWL]DWLRQ RI 2VHHQ HTXDWLRQV has the form     7     4
  

6.7. OSEEN EQUATIONS

163 24 22 20 Gpp 2-2 2-4

2-4

2-3

2-2

2-1

20

21

22

23

Guu

24

2-6 0.035 0.07 0.14 0.28

Figure 6.10: GMRES convergence rates depending on M   and M  LVROLQHV  3RLVVHXLOOH ÀRZ ZLWK  elements

ZKHUH QRZ WKH  FRQVLVWV RI WKH MXPS WHUPV RI WKH 6WRNHV RSHUDWRU  DQG WKH DGYHF " PDWUL[ & tion form E 54 5 FI HTXDWLRQ   $ SUHFRQGLWLRQHU RI WKH IRUP   7   4    

(6.38)

LV FRQVWUXFWHG DQDORJRXV WR   +HUH DQ HI¿FLHQW SUHFRQGLWLRQHU IRU WKH 2VHHQ 6FKXU FRP SOHPHQW  PXVW EH IRXQG 6.7.3 2QH VFKHPH VXJJHVWHG E\ .D\ DQG /RJKLQ FI >./:@  UHSODFHV WKH PDVV PDWUL[ DV D SUHFRQGLWLRQHU IRU WKH 6FKXU FRPSOHPHQW E\ DQ RSHUDWRU PRUH DGDSWHG WR WKH DGYHFWLRQ GRPLQDWHG FDVH $VVXPH WKDW WKH ³FRPPXWDWLRQ´ UHVXOW        KROGV 6LQFH  LV WKH PDWUL[ DVVRFLDWHG ZLWK WKH '* GLYHUJHQFH RSHUDWRU
  LV D PDWUL[ IRU D GLVFUHWH /DSODFLDQ RQ WKH SUHVVXUH VSDFH   7DNLQJ IRU LQVWDQFH WKH 6WRNHV FDVH 2  ), ZH REWDLQ D  WLPHV LGHQWLW\ 6LQFH WKH PDVV PDWUL[ ZDV VKRZQ WR EH D JRRG SUHFRQGLWLRQHU IRU WKH ORZHU ULJKW EORFN ZH DUULYH DW WKH IROORZLQJ RSHUDWRU IRU SUHFRQGLWLRQLQJ WKH 2VHHQ 6FKXU FRPSOHPHQW 
4 ZKHUH  LV WKH PDWUL[ RI D GLVFUHWL]DWLRQ RI WKH RSHUDWRU D+  2 ( RQ WKH VSDFH  .

CHAPTER 6. LINEAR SOLVERS

164


3 4 5 6 7

 31 35 38 40 41

- 33 36 39 45 51

 44 39 39 42 48

D - 72 65 49 43 45

 119 137 110 71 50

- 228 361 290 217 106

 377 1183 945 774 397

Table 6.19: Convergence of exact Kay/Loghin preconditioner for two dimensional Poisseuille ÀRZ ZLWK  elements

6.7.4 Remark: In [KLW02] it was shown that the commutation result postulated above holds LQGHHG LQ WKH FDVH RI D FRQVWDQW DGYHFWLRQ ¿HOG DQG LI 1   . 6.7.5 We choose interior penalty discretizations to obtain the matrices  and
for simplicity (an LDG discretization would require additional variables for the pressure gradient just for preconditioning). Experiments showed that the method converged best with a stabilization parameter slightly larger than for the LDG preconditioner. 6.7.6 ,Q 7DEOH  ZH SUHVHQW UHVXOWV IRU WKH 3RLVVHXLOOH ÀRZ SUREOHP H[DPSOH $ RQ  Cartesian grids. Following the results of Section 6.6, we chose M    and M   6$ with  HOHPHQWV 7KH WDEOH VKRZV WKDW RQ ¿QH JULGV WKHUH LV RQO\ D PRGHUDWH LQFUHDVH RI LWHUDWLRQ VWHSV up to a Reynolds number of 1000. In Particular, these numbers are not much higher than for the pure Stokes case with mass matrix preconditioning (see Table 6.18 on page 162). Then, for higher Reynolds numbers, the number of steps grows fast. Since this growth starts with higher 5H\QROGV QXPEHUV RQ ¿QHU JULGV LW VHHPV WKDW D FHUWDLQ UHVROXWLRQ RI WKH SRVVLEOH ERXQGDU\ layers is required to make the method work. Here, the discontinuous Galerkin schemes may face a disadvantage compared to continuous methods, where a considerable, mesh dependent DPRXQW RI DUWL¿FLDO GLIIXVLRQ LV LQGXFHG E\ VWDELOL]DWLRQ RI WKH DGYHFWLRQ WHUP 6.7.7 In the previous paragraph, we considered the ideal but not feasible case of an exact solution of the Poisson and advection-diffusion problems arising in the block preconditioner (6.38). In fact, exact solution means to an accuracy of two orders of magnitude more accurate than the desired solution of the Oseen problem in each VWHS XQOHVV D YDULDQW RI *05(6 IRU ÀH[LEOH preconditioning is used. Therefore, we replace the exact inversion by one or two V-cycles. The ¿JXUHV LQ WKH WDEOH VKRZ WKDW WKH QXPEHU RI LWHUDWLRQ VWHSV LV QRW PXFK KLJKHU WKDQ IRU WKH H[DFW solver using a single V-cycle only. In particular, the dependence on the viscosity D is the similar WR WKH H[DFW SUHFRQGLWLRQHU VR WKDW QR UREXVWQHVV LV ORVW 7KHUHIRUH WKLV VLPSOL¿FDWLRQ LV WKH PRVW HI¿FLHQW YHUVLRQ RI WKLV SUHFRQGLWLRQHU 6.7.8 )LQDOO\ ZH WHVW WKLV SUHFRQGLWLRQHU LQ WKH SUHVHQFH RI YRUWLFHV XVLQJ WKH ÀRZ ¿HOG RI example A.3.3. Again, we use the clockwise ordering of grid cells introduced in Section 6.5. A PRUH JHQHUDO QXPEHULQJ DOJRULWKP IRU DUELWUDU\ WZRGLPHQVLRQDO ÀRZ FDQ EH IRXQG LQ >+3@ Solving Oseen equations yields the convergence shown in Table 6.21. Here, we used Bicgstab iteration instead of GMRES to avoid a nonfeasible number of auxiliary vectors. Therefore, the iteration counts with large viscosity are lower than above, since Bicgstab performs a double step on symmetric matrices. Again, we see a very moderate increase of iteration steps for lower

6.7. OSEEN EQUATIONS

165






-

3 4 5 6 7

43 46 50 52 54

47 49 53 61 70

3 4 5 6 7

36 39 41 43 44

39 43 46 52 56

D  - one V-cycle 59 101 56 81 55 74 59 66 66 64 two V-cycles 48 84 47 67 47 60 50 55 56 55



-



181 201 134 107 82

294 507 388 294 181

440 1129 1226 964 615

141 165 121 88 67

263 410 323 244 129

411 1151 1022 823 428

7DEOH  &RQYHUJHQFH RI .D\/RJKLQ SUHFRQGLWLRQHU IRU WZR GLPHQVLRQDO 3RLVVHXLOOH ÀRZ with one and two V-cycles ( elements)

D
3 4 5 6 7

 23 24 26 27 28

- 30 30 28 31 31

 49 47 41 41 46

- 119 114 92 81 83

 376 365 308 222 215

- 3689 — 4080 3055 1680

Table 6.21: Iteration steps  .D\/RJKLQ SUHFRQGLWLRQHU IRU OLQHDU GULYHQ FDYLW\ ÀRZ ZLWK RQH variable V-cycles (Bicgstab,  elements)

166

CHAPTER 6. LINEAR SOLVERS

Reynolds numbers up to D  6. Then, the system is still solvable in reasonable time up to D  6. A stronger deterioration of convergence with decrease of D can be expected here, since the limit equation for D   is not well-posed anymore.

Appendix A Example problems Many of the model problems used to demonstrate the feasibility of the methods and analysis in this book appear in several places. Therefore, we collect these test examples in this appendix.

A.1 Meshes and domains A.1.1 Example: Most examples are computed on the unit square and unit cube #4 & in two and three dimensions, respectively, with Dirichlet boundary /  =1. Unless superconverJHQFH HIIHFWV DUH WR EH FDQFHOOHG &DUWHVLDQ JULGV PD\ EH XVHG RULJLQDWLQJ IURP WKH UH¿QHPHQW local or global, of a single cell constituting the mesh  . A.1.2 Example: 2Q WKH VDPH JULGV DV LQ H[DPSOH $ ZH ¿[ WKH 'LULFKOHW ERXQGDU\  3 7 /  0  =10    0   5

On the Neumann boundary /  =1  / , we prescribe homogeneous or inhomogeneous boundary conditions, depending on the approximated solution. A.1.3 Example: $ YDULDQW RI WKHVH JOREDOO\ UH¿QHG PHVKHV FDQ EH JHQHUDWHG E\ VKLIWLQJ HDFK YHUWH[ DIWHU UH¿QHPHQW $Q H[DPSOH ZKHUH WKH YHUWLFHV DUH VKLIWHG E\  RI WKH PLQLPDO diameter of its adjacent cells is shown in Figure A.1. A.1.4 Example: A second coarse mesh for the unit square is shown in Figure A.2. This mesh exhibits two topologically irregular points. Some superconvergence effects are cancelled there and optimality of
 error estimates can be observed. A.1.5 Example: 5REXVWQHVV ZLWK UHVSHFW WR ORFDO UH¿QHPHQW LV LQYHVWLJDWHG RQ D KLHUDUFK\ RI PHVKHV REWDLQHG E\ UH¿QLQJ WKH SRVLWLYH TXDGUDQW RI WKH VTXDUH #4 & in each step. Grid cells DGMDFHQW DUH UH¿QHG LI UHTXLUHG WR PDNH WKH PHVK RQH LUUHJXODU $ VHTXHQFH RI VXFK PHVKHV LV VKRZQ LQ )LJXUH $ , QRUGHU WR REWDLQ D PRUH SURQRXQFHG ORFDOLW\ RI UH¿QHPHQW ZH DOVR SHUIRUP FRPSXWDWLRQV RQ PHVKHV UH¿QHG RQ D VPDOO FLUFOH DURXQG WKH RULJLQ LQ )LJXUH $ 7KH VDPH W\SHV RI ORFDO UH¿QHPHQW DUH DOVR WHVWHG LQ WKUHH GLPHQVLRQV VHH )LJXUH $

168

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)LJXUH $ ,UUHJXODU VXEGLYLVLRQ RI WKH XQLW VTXDUH DQG VHFRQG UH¿QHPHQW

A.1. MESHES AND DOMAINS

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169

APPENDIX A. EXAMPLE PROBLEMS

170

Figure A.6: Subdivisions of a circle. Coarse mesh with  -mapping, with  -mapping and UH¿QHG PHVK ZLWK  -mapping

A.1.6 Example: Even if we do not cover smoothly curved boundaries theoretically, we present some results for this case. As a prototype for such a case, we choose a hierarchy of triangulations of a circle as in Figure A.6. Remark that the shape of the computational domain (and the quality of approximation of the real domain) depends on the order of the mapping between reference cell and actual grid cell. The two coarse grids in Figure A.6 use bilinear and biquadratic mapping on the left and on the right, respectively.

A.2 Poisson equation A.2.1 Example: $SSUR[LPDWH WKH ¿UVW HLJHQIXQFWLRQ RI WKH /DSODFLDQ RSHUDWRU RQ WKH XQLW square with homogeneous boundary values, namely #0 



  9 0 4

 0 



I  

  9 0 5



A.2.2 Example: A model problem with inhomogeneous boundary conditions is constructed using the exponential function. #0 



0 4

 0  





0 5



A.2.3 Example: Solutions with lower regularity are obtained on domains with reentrant corners. Our examples include the leading singularities for the L-shaped and the slit domain (see Figure A.7) with right-hand side    and homogeneous boundary values on the reentrant HGJHV 7KH YDOXHV RQ WKH UHPDLQLQJ SDUWV RI WKH ERXQGDU\ DUH VHW WR ¿W WKH VROXWLRQ 7KH VROX tions can be found in [Kon67]. Introducing polar coordinates, we name  the distance from the origin and < the angle computed counter-clockwise from the indentation into the unit square #4 & , we have in the L-shaped domain #4 <  

 


  <5

(a)

A.3. ADVECTION-DIFFUSION-REACTION

171

<

<

Figure A.7: L-shaped and slit domain

The solution in the slit domain is 

#4 <   


<5

(b)

A.3 Advection-diffusion-reaction A.3.1 Example: In order to investigate the behavior of discretizations in the case of reaction boundary layers, we solve D+#  #   in 1  #4 & #   if 0   = #   if   5 The solution of this problem is #04    




  





  

 



5

This solution exhibits an exponential boundary layer of width

(A.1) #

D at the boundaries 0  .

A.3.2 Example: 7KH VLPSOHVW DGYHFWLRQ ¿HOG LQ RXU LQYHVWLJDWLRQ LV D FRQVWDQW YHFWRU ) %  I6" 5 2


I6" :KLOH WKH YHFWRU ¿HOG LV FRQVWDQW LW LV REOLTXH ZLWK UHVSHFW WR WKH &DUWHVLDQ JULG FHOOV LQ H[DP ple A.1.1, avoiding a purely on-dimensional solution.

APPENDIX A. EXAMPLE PROBLEMS

172

A.3.3 Example: %HKDYLRU RI VROYHUV DQG VROXWLRQ LQ ÀRZ ¿HOGV ZLWK YRUWLFHV LV WHVWHG XVLQJ WKH WZRGLPHQVLRQDO YHFWRU ¿HOG FI >(6:@ #04  

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A.4 Stokes and Oseen equations $ ([DPSOH 3RLVVHXLOOH ÀRZ  $ VWDQGDUG H[DPSOH IRU ÀRZ SUREOHPV LV 3RLVVHXLOOH ÀRZ WKURXJK D VWUDLJKW SLSH /HW WKH GRPDLQ EH WKH VTXDUH #4 & RU WKH F\OLQGHU #4 &   . 7KHQ WKH ÀRZ ¿HOG # 04  

% )    4 

4 ZKHUH    and       LQ WZR DQG WKUHH GLPHQVLRQV UHVSHFWLYHO\ LV D VROXWLRQ WR WKH 6WRNHV HTXDWLRQV  ZLWK SUHVVXUH 04     D0  5 )XUWKHUPRUH LW VROYHV 2VHHQ HTXDWLRQV  ZLWK ÀRZ ¿HOG 2  # DQG WKHUHIRUH LW LV D VWDWLRQDU\ VROXWLRQ WR WKH LQFRPSUHVVLEOH 1DYLHU6WRNHV HTXDWLRQV :H VROYH WKLV SUREOHP ZLWK ERXQGDU\ FRQGLWLRQV #  # # =
#  
 

on =1 ! 0   on =1 !    on =1 ! 0   5

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A.4. STOKES AND OSEEN EQUATIONS

173

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APPENDIX A. EXAMPLE PROBLEMS

174 100

u p S

10

1

0.1

0.01

0.001 1

10

100

1000

10000

Figure A.9:
 -norms of Kovasznay solutions L, # and  depending on the Reynolds number

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% &RFNEXUQ * .DQVFKDW DQG ' 6FKŽRW]DX /'* PHWKRGV IRU 6WRNHV ÀRZ SURE OHPV ,Q ) %UH]]L $ %XIID 6 &RUVDUR DQG $ 0XUOL HGLWRUV Numerical Mathematics and Advanced Applications: ENUMATH 2001 SDJHV ± 0LODQR  6SULQJHU ,WDOLD

[CKS03b] B. Cockburn, G. Kanschat, and D. Sch¨otzau. The local discontinuous Galerkin PHWKRG IRU WKH 2VHHQ HTXDWLRQV Math. Comput.  ±  >&.6@

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[CKS07]

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/ 3 )UDQFD DQG 5 6WHQEHUJ (UURU DQDO\VLV RI VRPH *DOHUNLQ OHDVW VTXDUHV PHWK RGV IRU WKH HODVWLFLW\ HTXDWLRQV SIAM J. Numer. Anal. ± 

>*.D@

- *RSDODNULVKQDQ DQG * .DQVFKDW $SSOLFDWLRQ RI XQL¿HG '* DQDO\VLV WR SUH FRQGLWLRQLQJ '* PHWKRGV ,Q . - %DWKH HGLWRU Computational Fluid and Solid Mechanics 2003 SDJHV ± (OVHYLHU 

>*.E@

- *RSDODNULVKQDQ DQG * .DQVFKDW 0XOWLOHYHO SUHFRQGLWLRQHUV IRU WKH LQWHULRU SHQDOW\ PHWKRG ,Q ) %UH]]L $ %XIID 6 &RUVDUR DQG $ 0XUOL HGLWRUV Numerical Mathematics and Advanced Applications: ENUMATH 2001 SDJHV ± 0LODQR  6SULQJHU ,WDOLD

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>.-@

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>.5@

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>/5@

3 /H6DLQW DQG 3$ 5DYLDUW 2Q D ¿QLWH HOHPHQW PHWKRG IRU VROYLQJ WKH QHXWURQ WUDQVSRUW HTXDWLRQ ,Q & GH %RRU HGLWRU 0DWKHPDWLFDO DVSHFWV RI ¿QLWH HOHPHQWV in partial differential equations SDJHV ± 1HZ
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& /DVVHU DQG $ 7RVHOOL 2YHUODSSLQJ SUHFRQGLWLRQHUV IRU GLVFRQWLQXRXV *DOHU NLQ DSSUR[LPDWLRQV RI VHFRQG RUGHU SUREOHPV ,Q 1 'HELW 0 *DUEH\ 5 + : +RSSH - 3HULDX[ ' .H\HV DQG < $ .X]QHWVRY HGLWRUV Thirteenth International Conference on Domain Decomposition Methods ''0RUJ 

>0*:@ 0 ) 0XUSK\ * + *ROXE DQG $ - :DWKHQ $ QRWH RQ SUHFRQGLWLRQLQJ IRU LQGH¿QLWH OLQHDU V\VWHPV SIAM J. Sci. Comput., 21(6):1969–1972, 2000. >0RQ@

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>1LW@

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>57@

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>59:@

7 5XVWHQ 3 6 9DVVLOHYVNL DQG 5 :LQWKHU ,QWHULRU SHQDOW\ SUHFRQGLWLRQHUV IRU PL[HG ¿QLWH HOHPHQW DSSUR[LPDWLRQV RI HOOLSWLF SUREOHPV Math. Comput., 65(214):447–466, 4 1996.

>6DD@

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' 6FKŽRW]DX & 6FKZDE DQG $ 7RVHOOL '*)(0 IRU LQFRPSUHVVLEOH ÀRZV SIAM J. Numer. Anal. ± 

>7HP@

5 7HPDP 6XU O¶DSSUR[LPDWLRQ GHV VROXWLRQV GHV H TXDWLRQV GH 1DYLHU6WRNHV C. R. Acad. Sci. Paris S´er. I Math., 216:219–221, 1966.

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Index ,, 16
, 16 # 5&&, 21

5

, 20 active " &cell, 145 -IP  54 5 , 36

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INDEX relaxation parameter, 136 relaxed inf-sup condition, 95 restriction, 135 N, 142 Ritz projection, 46, 137, 138 %" , 101 Schur complement, 75, 151, 160 shape function, 136 shape function space, 17 shape functions, 17 shape regular, 18 simplicial cell, 15 Sobolev embedding, 19 Sobolev space, 68 spectral condition number, 133 spectrally equivalent, 75, 152 stability estimate, 25 stabilization, 68 VWDEOH ¿QLWH HOHPHQW SDLU  standard LDG, 70, 71, 93 static condensation, 75, 98, 112 Stokes equations, 89 6WRNHV ÀRZ  superconvergence, 31 symmetric, 137 , 15 % , 15 %% , 45 tensor product cell, 15, 17 tensor product polynomials, 16  , 16 < , 16 
< , 146 < , 146 topologically irregular point, 167 trace estimate, 22, 35 triangulation, 15 # , 21 # , 21 %: , 44 uniform preconditioner, 135 XSZLQG  XSZLQG ÀX[    XSZLQGLQJ SURMHFWLRQ  V-cycle, 137 local, 148

183 variable V-cycle, 136, 137 variational problem, 67, 68, 90 W-cycle, 137 3 ", , 20 ," , 18  , 17 ,"  ," , 17 Young inequality, 23