Numerical Methods for Nonlinear Variational Problems

CHAPTER I Generalities on Elliptic Variational Inequalities and on Their Approximation 1. Introduction An important an...

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CHAPTER I

Generalities on Elliptic Variational Inequalities and on Their Approximation

1. Introduction An important and very useful class of nonlinear problems arising from mechanics, physics, etc. consists of the so-called variational inequalities. We consider mainly the following two types of variational inequalities, namely: 1. elliptic variational inequalities (EVI), 2. parabolic variational inequalities (PVI). In this chapter (following Lions and Stampacchia [1]), we shall restrict our attention to the study of the existence, uniqueness, and approximation of the solution of EVI (PVI will be considered in Chapter III).

2. Functional Context In this section we consider two classes of EVI, namely EVI of the first kind and EVI of the second kind. 2.1. Notation • V: real Hilbert space with scalar product (•, •) and associated norm || • ||, • V*: the dual space of V, • a(-,-): V x V -» U is a bilinear, continuous and V-elliptic form on V x V. A bilinear form a{-, •) is said to be V-elliptic if there exists a positive constant a such that a(v, v) > oc\\v\\2, V v e V. In general we do not assume a{-, •) to be symmetric, since in some applications nonsymmetric bilinear forms may occur naturally (see, for instance, Comincioli [1]). • L: V -> U continuous, linear functional, • K is a closed convex nonempty subset of V, • j(-): V -* U = U \J {ao} is a convex lower semicontinuous (l.s.c.) and proper functional (;'(•) is proper ifj(v) > — oo, V v e V and j ^ + oo).

2

I Generalities on Elliptic Variational Inequalities and on Their Approximation

2.2. EVI of the first kind Find u e V such that u is a solution of the problem a(u, v - u) > L(v - u),

V»eJ(,

ueK.

(Pt)

2.3. EVI of the second kind Find u e V such that u is a solution of the problem a(u, v-u)+

j(v) - j(u) >L{v-u),

VveV,

ue V.

(P 2 )

2.4. Remarks Remark 2.1. The cases considered above are the simplest and most important. In Bensoussan and Lions [1] some generalization of problem ( P J called quasivariational inequalities (QVI) are considered, which arises, for instance, from decision sciences. A typical problem of QVI is: Find ue V such that a(u, v - u) > L(v — u),

V P G K(U),

Ue

K(u),

where v -> K(v) is a family of closed convex nonempty subsets of V. Remark 2.2. If K = V and j = 0, then problems ( P J and (P 2 ) reduce to the classical variational equation a(u, v) = L(v),

VveV,

ueV.

Remark 2.3. The distinction between (P x ) and (P 2 ) is artificial, since (P x ) can be considered to be a particular case of (P 2 ) by replacing _/(•) in (P2) by the indicator functional IK of K denned by O tiveK, + 00 if v $ K. Even though ( P J is a particular case of (P 2 ), it is worthwhile to consider (Pj) directly because in most cases it arises naturally, and doing so we will obtain geometrical insight into the problem. EXERCISE

2.1. Prove that IK is a convex l.s.c. and proper functional.

2.2. Show that ( P ^ is equivalent to the problem of finding ue V such that a(u, v — u) + IK(v) — IK(u) > L{v - U ) , V P G F . EXERCISE

3. Existence and Uniqueness Results for EVI of the First Kind

3

3. Existence and Uniqueness Results for EVI of the First Kind 3.1.

A theorem of existence and uniqueness

Theorem 3.1 (Lions and Stampacchia [1]). The problem (P t ) has a unique solution. PROOF. We first prove the uniqueness and then the existence. (1) Uniqueness. Let ux and u2 be solutions of (P^. We then have a(uu v — ut) > L(v — «i),

VveK,

ux e K,

(3.1)

a(u2, v-u2)>

VneiC,

u2 e K.

(3.2)

L(v - u2),

Taking v = u2 in (3.1), v = ut in (3.2) and adding, we obtain, by using the F-ellipticity a | | « 2 - " i l l 2 ^ a ( " 2 - « i , «2 - « i ) ^ 0,

which proves that ut = u2 since a > 0. (2) Existence. We use a generalization of the proof used by Ciarlet [l]-[3], for example, for proving the Lax-Milgram lemma, i.e., we will reduce the problem (Pj) to a fixedpoint problem. By the Riesz representation theorem for Hilbert spaces, there exist A e JS?(F, V) (A = A' if a{-, •) is symmetric) and / e V such that (Au, v) = a(u, v),

Vu,oeF

a n d L(v) = (/, v),

V c e V.

(3.3)

T h e n t h e p r o b l e m ( P i ) i s e q u i v a l e n t t o finding u e V s u c h t h a t (u - p(Au -

I) - u, v - u) < 0,

VDEX,

ueK,

p > 0.

(3.4)

This is equivalent to finding u such that u = PK(u — p(Au — /)) for some p > 0,

(3.5)

where P K denotes the projection operator from F to K in the || • |j norm. Consider the mapping Wp: V -* V defined by Wp(v) = PK(v - p(Av - /)).

(3.6)

Let i>!, v2 e V. Then since PK is a contraction we have — 2pa(v2 — vlt v2 — Vi). Hence we have HWp(oi) " Wp(v2)\\2

< ( 1 - 2pa + p2\\A\\2)\\v2 2

- vt\\2.

(3.7)

Thus W^, is a strict contraction mapping if 0 < p < 2a/\\A\\ . By taking pin this range, we have a unique solution for the fixed-point problem which implies the existence of a solution for (P t ). •

4

I Generalities on Elliptic Variational Inequalities and on Their Approximation

3.2. Remarks Remark 3.1. If K = V, Theorem 3.1 reduces to Lax-Milgram lemma (see Ciarlet [l]-[3]). Remark 3.2. If a(-, •) is symmetric, then Theorem 3.1 can be proved using optimization methods (see Cea [1], [2]); such a proof is sketched below. Let J: V -> R be defined by J(v) = Hv, v) - L(v).

(3.8)

Then (i) l i m | M H + 00 J(y) = +co since J(v) = \a{v, v) - L(v) > (a/2)|M| 2 - ||L|| \\v\\.

(ii) J is strictly convex. Since L is linear, to prove the strict convexity of J it suffices to prove that the functional v -> a(v, v) is strictly convex. Let 0 < t < 1 and u,veV

with u=£v; then 0 < a(v — u, v — u)

= a(u, u) + a(v, v) — 2a(u, v). Hence we have 2a(u, v) < a(u, u) + a(v, v).

(3.9)

Using (3.9), we have a(tu + (1 - t)v, tu + (1 - t)v) = t2a(u, u) + 2t(l - t)a(u, v) + (1 - t)2a(v, v) < ta(u, u) + (1 - t)a(v, v). (3.10) Therefore v ->• a(v, v) is strictly convex. (iii) Since a{-, •) and L are continuous, J is continuous. From these properties of J and standard results of optimization theory (cf. Cea [1], [2], Lions [4], Ekeland and Temam [1]), it follows that the minimization problem of finding u such that J(u) < J(v),

\/veK,

ueK

(n)

has a unique solution. Therefore (n) is equivalent to the problem of finding u such that (J'(u), v - u) > 0,

VveK,

ueK,

(3.11)

4. Existence and Uniqueness Results for EVI of the Second Kind

5

where J'(u) is the Gateaux derivative of J at u. Since (J'(u), v) = a(u, v) — L(v), we see that ( P J and (n) are equivalent if a( •, •) is symmetric. EXERCISE 3.1. Prove that (J'(u), v) = a(u, v) — L(v), V«, v e V and hence deduce that J'(u) = Au — /, V ue V.

Remark 3.3. The proof of Theorem 3.1 gives a natural algorithm for solving (P t ) since v -»P K (v — p(Av — /)) is a contraction mapping for 0 0, assuming that u" is known, define u"+1 by un+1

= PK(un - p(Au" - I)).

(3.13)

Then u" ->u strongly in V, where u is the solution of ( P ^ . In practice it is not easy to calculate / and A unless V = V*. To project over K may be as difficult as solving ( P J . In general this method cannot be used for computing the solution of ( P ^ if K =£ V (at least not so directly). We observe that if a(:, •) is symmetric then J'(u) = Au - I and hence (3.13) becomes u"+1 = PK(u" - P(J'(W)).

(3.13)'

This method is known as the gradient-projection method (with constant step p).

4. Existence and Uniqueness Results for EVI of the Second Kind Theorem 4.1 (Lions and Stampacchia [1]). Problem (P 2 ) has a unique solution. PROOF. AS in Theorem 3.1, we shall first prove uniqueness and then existence. (1) Uniqueness. Let ux and u2 be two solutions of (P 2 ); we then have a(uu v - » , ) + j(v) - j(ui) > L(v - « , ) ,

VceF,

uteV,

(4.1)

a(u2, v - u2) + j(v) - j(uz) > L(v - u2),

VceF,

u2 e V.

(4.2)

Since /(•) is a proper functional, there exists voeV Hence, for i = 1, 2,

such that — oo < j(v0) < oo.

- oo < X«i) < j(v0) - L(v0 ~ ">) + a{uh v0 - ut).

(4.3)

This shows that j(ut) is finite for i — 1,2. Hence, by taking v = u2 in (4.1), v = ul in (4.2), and adding, we obtain a|[«i - u2\\2 < a{ux - u2, «i - u2) < 0. Hence ux = u2.

(4.4)

6

I Generalities on Elliptic Variational Inequalities and on Their Approximation

(2) Existence. For each ueV defined as follows. Find w e V such that

and p > 0 we associate a problem (n") of type (P 2 )

(w, v — w) + pj(v) — pj(w) >{u,v — w) + pL(v - w) - pa(u, v - w),

VceF,

weV.

(4.5)

(np)

The advantage of considering this problem instead of problem (P 2 ) is that the bilinear form associated with (np) is the inner product of V which is symmetric. Let us first assume that (TI") has a unique solution for all u e V and p > 0. For each p define the mapping fp: V -* V by fp(u) = w, where w is the unique solution of (n"). We shall show that fp is a uniformly strict contraction mapping for suitably chosen p. Let uu u2e V and wf = //«;), i = 1, 2. Since j(-) is proper we haveX«i) finite which can be proved as in (4.3). Therefore we have (W1; W2 - Wt) + pj(w2) - pj(Wi) > («!, W2 - W,)

+ pL(w2 - Wj) - pa(«!, w2 - w j ,

(4.6)

(w2, wt - w2) + p;(wi) - p;(w2) > (« 2 . wi - w2) + pL{w^ - w2) - pa(u2, Wj - w2).

(4.7)

Adding these inequalities, we obtain < ((/ - pA)(u2 - «i), w 2 - Wj)

< !|/-pA|!![«2-w1||||w2-w1||.

(4.8)

Hence

It is easy to show that ||/ — pA\\ < 1 if 0 < p < 2a/\\A\\2. This proves that fp is uniformly a strict contracting mapping and hence has a unique fixed point u. This u turns out to be the solution of (P 2 ) since fp(u) = M implies (M, V - u) + pj(v) - pj(u) >{u,v - u) + pL(v — u) — pa(u, v — u), V v e V. Therefore a(u, v-u)+

j(v) - ; ( « ) > Up - «),

VoeK

Hence (P 2 ) has a unique solution.

(4.9) •

The existence and uniqueness of the problem {nup) follows from the following lemma. Lemma 4.1. Let b:Vx V ->• U be a symmetric continuous bilinear V-elliptic form with V-ellipticity constant p. Let LeV* and j : V -» U be a convex, l.s.c. proper functional. Let J(v) = jb(v, v) + j(v) — L{v). Then the minimization problem (n): Find u such that J(u) < J(v),

V v e V,

ue V

(n)

4. Existence and Uniqueness Results for EVI of the Second Kind

7

has a unique solution which is characterized by b(u, v - u) + j(v) - j(u) >L(v - u),

V v e V, ueV.

(4.10)

PROOF. (1) Existence and uniqueness ofu: Since b(v, v) is strictly convex, j is convex, and L is linear, we have J strictly convex; J is l.s.c. because i>(-, •) and L are continuous and j is l.s.c. Since; is convex, l.s.c, and proper, there exists X e V* and n e U such that j(v) > l(v) + n (cf. Ekeland and Temam [1]), then J(v) > ^ \\v\\2 - U\\ \\v\\ - \\L\\ \\v\\ + n

_ / [p

(imi + IILII) I2V

W2

2

(u\\ + \\L\\y

V/8/

2^

'

l

' '

Hence J(u)->+oo

as ||«|| -+ +oo.

(4.12)

1

Hence (cf. Cea [1], [2]) there exists a unique solution for the optimization problem (n). Characterization of u: We show that the problem (n) is equivalent to (4.10) and thus get a characterization of u. (2) Necessity of (4.10): Let 0 < t < 1. Let u be the solution of {%). Then for all v e V we have J(u) < J(u + t(v - u)).

(4.13)

Set J0(v) = jb(v, v) - L(v), then (4.13) becomes 0 < J0(u + t(v - uj) - J0(u) + j(u + t(v - u)) - j(u) < J0(u + t(v - u)) - J0(u) + t[M

~ Ml

Vve V

(4.14)

obtained by using the convexity of j . Dividing by t in (4.14) and taking the limit as t -> 0, we get 0 < (J'0(u), i > - « ) + j(v) - j(u),

VveV.

(4.15)

Vt,weK

(4.16)

Since b(-, •) is symmetric, we have (J'0(v), w) = b(v, w) - L(w), From (4.15) and (4.16) we obtain b(u, v - u) + j(v) - j(u) > L(v - u),

V v e V.

This proves the necessity. (3) Sufficiency of (4.10): Let u be a solution of (4.10); for v e V, J{v) - J(u) = #b(v, v) - b(u, «)] + j(v) - j(u) - L(v - u).

1

See also Ekeland and Temam [1].

(4.17)

8

I Generalities on Elliptic Variational Inequalities and on Their Approximation

But

b(v, v) = b(u + v — u,u + v — u)

= b(u, u) + 2b(u, v - u) + b(u - v,u - v).

Therefore J(v) - J(u) = b(u, v - u ) + j(v) - j(u) - L(v - u) + jb{v - u , v - u).

(4.18)

Since u is a solution of (4.10) and b(v — u, v — u) > 0, we obtain J(v) - J(u) > 0.

(4.19)

Hence u is a solution of (n). By taking b(-, •) to be the inner product in V and replacingX^) and L(v) in Lemma 4.1 by pj(v) and (u, v) + pL(v) — pa(u,v), respectively, we get the solution for (n"p). •

Remark 4.1. FromtheproofofTheorem4.1 we obtain an algorithm for solving (P 2 ). This algorithm is given by u° e V, arbitrarily given,

(4-20)

then for n > 0, u" known, we define M" +1 from u" as the solution of (w"+1, v - u"+1) + pj(v) - pj(u"+1) - pa(u", v -un+1),

Vv GK

> (un, v - M" + 1 ) + pL(v - M" + 1 ) un+1 eV.

(4.21)

If p is chosen such that 0 u strongly in V, where u is the solution of (P 2 ). Actually, practical difficulties may arise since the problem that we have to solve at each iteration is then a problem of the same order of difficulty as that of the original problem (actually, conditionning can be better provided that p has been conveniently chosen). If a(-, •) is not symmetrical the fact that (•, •) is symmetric can also provide some simplification.

5. Internal Approximation of EVI of the First Kind 5.1. Introduction

In this section we shall study the approximation of EVI of the first kind from an abstract axiomatic point of view.

5. Internal Approximation of EVI of the First Kind

9

5.2. The continuous problem The assumptions on V, K, L, and a{-, •) are as in Sec. 2. We are interested in the approximation of a(u, v-u)>L(v-u),

VveK,

ueK,

(PJ

which has a unique solution by Theorem 3.1.

5.3. The approximate problem 5.3.1. Approximation of V and K We suppose that we are given a parameter h converging to 0 and a family {Vh}h of closed subspaces of V. (In practice, the Vh are finite dimensional and the parameter h varies over a sequence). We are also given a family {Kh}h of closed convex nonempty subsets of V with Kh a Vh, V h (in general, we do not assume Kh a K) such that {Kh}h satisfies the following two conditions: (i) If {vh}h is such that vhe Kh,V h and {vH}h is bounded in V, then the weak cluster points of {vh}h belong to K. (ii) There exists x c V, x = K and rh: % -*• Kh such that limA^0 rhv = v strongly in V, V v e xRemark 5.1. If Kh a K, V h, then (i) is trivially satisfied because K is weakly closed. Remark 5.2. f]h Kh a K. Remark 5.3. A useful variant of condition (ii) for rh is (ii)' There exist a subset x<^V such that x = K and rh:x~* Vh with the property that for each vex, there exists h0 = ho(v) with rhv e Kh for all h < ho(v) and lim h ^ 0 rhv = v strongly in V. 5.3.2. Approximation o/(P 1 ) The problem (P : ) is approximated by a(uh, vh - uh) > L(vh - uh),

V c j e Kh, uh e Kh.

(Plft)

Theorem5.1. (Plh) has a unique solution. PROOF. In Theorem 3.1, taking V to be Vh and K to be Kh, we have the result. Remark 5.4. In most cases it will be necessary to replace a(-, •) and L by ah(-, •) and Lh (usually defined, in practical cases, from a(-, •) and L by a numerical integration procedure). Since there is nothing very new on that

10

I Generalities on Elliptic Variational Inequalities and on Their Approximation

matter compared to the classical linear case, we shall say nothing about this problem for which we refer to Ciarlet [1, Chapter 8], [2], [3]. 5.4. Convergence results Theorem 5.2. With the above assumptions on K and {Kh}h, we have lim,,^0 \\uh — u\\v = 0 with uh the solution o / ( P u ) and u the solution of PROOF. For proving this kind of convergence result, we usually divide the proof into three parts. First we obtain a priori estimates for {uh}h, then the weak convergence of {uh}h, and finally with the help of the weak convergence, we will prove strong convergence. (1) Estimates for uh. We will now show that there exist two constants Ct and C2 independent of h such that U w J ^ C J u J +C2,

Vh.

(5.1)

V vh e Kh

(5.2)

Since uh is the solution of (P l t ), we have a(uh, vh - uh) > L(vh - uh), i.e., a(uh, uh) < a{uh, vh) - L(vh - uh). By V-ellipticity, we get « I M 2 < \\A\\ IkH K | | + IILIKKU + HIIJX

V vh e Kh.

(5.3)

Let voe x and vh = rhvoe Kh. By condition (ii) on Kh we have rhv0 -» v0 strongly in V and hence ||i;J is uniformly bounded by a constant m. Hence (5.3) can be written as ! M 2 < - {{m\\A\\ + \\L\\)\\uh\\ + \\L\\m} = C M a

+ C2,

where C, = (\/a)(m\\A\\ + ||L||) and C2 = (m/a)||L||; then (5.1) implies \\uh\\ < C, V h. (2) Weak convergence of {uh}h. Relation (5.1) implies that uh is uniformly bounded. Hence there exists a subsequence, say {uh.}, such that uh. converges to u* weakly in V. By condition (i) on {Kh}h, we have u* e K. We will prove that u* is a solution of (PJ. We have a(uhl,uh)<

a(uht, vh) - L(vh. - «„.),

Vc,.eX,,

(5.4)

Let v 6 x and vh. = rh. v. Then (5.4) becomes a(uhi, %,.) < a(uhl, rhlv) - L(rh.v - uh).

(5.5)

Since rh.v converges strongly to v and uhi converges to u* weakly as hi -> 0, taking the limit in (5.5), we obtain lim inf a(uhi, uh) < a(u*, v) — L(v — u*),

Vce/.

(5.6)

5. Internal Approximation of EVI of the First Kind

11

Also we have 0 < a(uh. - u*, uh. - u*) < a(uhl, uh) - a(uhi, u*) - a(u*, uh) + a(u*, u*) i.e., a(uht, u*) + a(u*, uh) - a(u*, u*) < a(uh., uh). By taking the limit, we obtain a(u*, «*) < lim inf a{uh., uh).

(5.7)

/ii->0

From (5.6) and (5.7), we obtain a(u*, u*) < lim inf a{uhi, uh) < a(u*, v) — L(v — u*),

V v e /.

Therefore we have a(u*, v-u*)>

L(v - u*\

Vvex,

u* e K.

(5.8)

Since x is dense in K and a(-, •), L are continuous, from (5.8) we obtain a(u*, v - u*) > L(v - u*),

VuelC,

u* e K.

(5.9)

Hence u* is a solution of (P x ). By Theorem 3.1, the solution of (P x ) is unique and hence u* = u is the unique solution. Hence u is the only cluster point of {uh}h in the weak topology of V. Hence the whole {uh}h converges to u weakly. (3) Strong convergence. By F-ellipticity of a( •, •), we have 0 < cc\\uh - u\\2 < a(uh - u,uh-

u) = a(uh, uh) - a(uh, u) - a(u, uj + a(u, «),

(5.10)

where uh is the solution of (Plh) and u is the solution of (P t ). Since uh is the solution of (Plft) and rhv e Kh for any vex, from (P1(1) we obtain a(uh, uh) < a(uh,rhv) - L(rhv - uh),

VUG/.

(5.11)

Since limh^0 uh = u weakly in V and limft^0 rhv = v strongly in V [by condition (ii)], we obtain (5.11) from (5.10), and after taking the limit, V v e x, we have 0 < a lim inf \\uh - u\\2 < a lim sup||u ft - u\\2 < a(u, v - u) - L(v - u).

By density and continuity, (5.12) also holds for VveK; we obtain

(5.12)

then taking v = u in (5.12),

lim | K - u\2 = 0, i.e., the strong convergence.

•

Remark 5.5. Error estimates for the EVI of the first kind can be found in Falk [1], [2], [3], Mosco and Strang, [1], Strang [1], Glowinski, Lions, and Tremolieres (G.L.T.) [1], [2], [3], Ciarlet [1], [2], [3], Falk and Mercier [1], Glowinski [1], and Brezzi, Hager, and Raviart [1], [2]. But as in many nonlinear problems, the methods used to obtain these estimates are specific to the particular problem under consideration (as we shall see in the following

12

I Generalities on Elliptic Variational Inequalities and on Their Approximation

sections). This remark also holds for the approximation of the EVI of the second kind which is the subject of Sec. 6. Remark 5.6. If for a given problem, several approximations are available, and if numerical results are needed, the choice of the approximation to be used is not obvious. We have to take into account not only the convergence properties of the method, but also the computations involved in that method. Some iterative methods are well suited only to specific problems. For example, some methods are easier to code than others.

6. Internal Approximation of EVI of the Second Kind 6.1. The continuous problem With the assumptions on V, a(-, •), L, andj(-) as in Sec. 2.1, we shall consider the approximation of a(u, v-u)+

j(v) - j(u) >L(v-u\

VceF,

ueV

(P2)

which has a unique solution by Theorem 4.1.

6.2. Definition of the approximate problem Preliminary remark: We assume in the sequel that j : V -»• U is continuous. However, we can prove the same sort of results as in this section under less restrictive hypotheses (see Chapter 4, Sec. 2). 6.2.1. Approximation of V Given a real parameter h converging to 0 and a family {Vh}h of closed subspaces of V (in practice, we will take Vh to be finite dimensional and h to vary over a sequence), we suppose that {Vh}h satisfies: (i) There exists U c V such that U = V, and for each h, a mapping rA: U-+Vh such that lim^o rhv = v strongly in V, V v e U. 6.2.2. Approximation of j(-) We approximate the functional^-) by {jh}h where for each h,jh satisfies jh- Vh -* $&•. jh is convex, l.s.c, and uniformly proper in h.

(6.1)

The family {jh}h is said to be uniformly proper in h if there exist 1 e V* and ji e IR such that jh(vh) > Kvh) + V,

VvheVh,\/h.

(6.2)

6. Internal Approximation of EVI of the Second Kind

13

Furthermore we assume that {jH}h satisfies: (ii) If vh -* v weakly in V, then lim Mjh(vh) > j(v), Jl->0

(iii) limh^0 jh(rhv) = j(v),

VveU.

Remark 6.1. In all the applications that we know, if ;(•) is a continuous functional, then it is always possible to construct continuous jh satisfying (ii) and (iii). Remark 6.2. In some cases we are fortunate enough to have jh(vh) — j(vh), V I),,, V ft, and then (ii) and (iii) are trivially satisfied. 6.2.3. Approximation o/(P 2 ) We approximate (P 2 ) by a(uh, vh - uh) + jh(vh) - jh(uh) > L(vh - uh),

V vheVh,

uheVh.

(P 2 h )

Theorem 6.1. Problem (P2J,) has a unique solution. PROOF. In Theorem 4.1, taking V to be Vh,j{-) to bejh(-), we get the result. Remark 6.3. Remark 5.4 of Sec. 5 still holds for (P 2 ) and (P2h).

6.3. Convergence results Theorem 6.2. Under the above assumptions on {Vh}h and {jh}h> we have lim \\uh — u\\ = 0, (6.3)

\imjh(uh)=j(u). PROOF. AS in the proof of Theorem 5.2, we divide the proof into three parts.

(1) Estimates for uh. We will show that there exist positive constants Ct and C2 independent of h such that \\uh\\2
V*.

(6.4)

Since uh is the solution of (P2j,), we have a(uh, uh) + jh(uh) < a(uh, vh) + jh(vh) - L(vh - uh),

VvheVh.

By using relation (6.2), we obtain «KII 2 < I|A|| ||«J| + \n\ + \\A\\ K | | H^ll + \jh(vh)\ + \\L\$\\vh\\ + ||uj|).

(6.5) (6.6)

14

I Generalities on Elliptic Variational Inequalities and on Their Approximation

Let voe U and vh = rhv0. By using conditions (i) and (iii), there exists a constant m, independent of h, such that \\vh\\ < m and \jh(vh)\ < m. Therefore (6.6) can be written as

KH 2 < -(Ml + \\A\\m + IILIDRH + - ( 1 + ||L||) + ~ a a a = C J u J + C2 where C, = - (||A|| + \\A\\m + \\L\$ a

and

C, = - (1 + ||L||) + — a a

and (6.4) implies

Kll < c,

v fc,

where C is a constant. (2) Weak convergence of {uh}h. Relation (6.4) implies that uh is uniformly bounded. Therefore there exists a subsequence {uh.}h. such that uh. -» «,, weakly in V. Since «A is the solution of (Plfc) and rkv e Vh, V h and V v e U, we have a("/,,, «*,) + AX"*,) ^ a(uht> rh,v) + Jh,(rh,v) - L(rhlv - uh).

(6.7)

By condition (iii) and from the weak convergence of {uh.}, we have

limm{[_a(uhi,uh)+jhl(uh)-]
VveU.

(6.8)

»,—o

As in (5.7), and using condition (ii), we obtain a(u*, u*) + j(u*) < lim inf [a(«fcf, uh) + ;»,(«*,)]•

(6'9)

/ij->0

From (6.8) and (6.9), and using the density of U, we have a(u*, v-u*)+

j(v) - j(u*) > L(v - u*),

V v e V, u* e V.

This implies that u* is a solution of (P 2 ). Hence u* = u is the unique solution of (P 2 ), and this shows that {uh}h converges to u weakly. (3) Strong convergence of {uh}h. From the F-ellipticity of a(-, •) and from (P2/l) we have all"* - "II2 + ./*(«*) ^ a("» - « , « » - « ) + A(M») = a(« h , « 0 - a(u, uh) - a(uh, u) + a(u, u) + jh(uh) < a(uh, rhv) + jh(rhv) - L(rhv - uh) - a(u, uh) - a(uh, u) + a(u, u), V u e U.

(6.10)

The right-hand side of inequality (6.10) converges to a{u, v — u) + j(v) — L{v — u) as h -> 0, V v e U. Therefore we have lim inf jh(uh) *->0

< lim inf \a\\uh - u\\2 + jh{uh)~] h-0

< lim sup [a||uh - u|| < a(u, v-u)+

j(v) - L(y - u),

V v 6 U.

(6.11)

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

15

By the density of U, (6.11) holds, V c e K Replacing v by u in (6.11) and using condition (ii), we obtain j(u) < lim infjh(uh) < lim sup [a||t< - uh\\2 + jh(uky] < j(u).

This implies that limjh(uh)=j(u) and lim \\uh — u\\ = 0.

This proves the theorem.

•

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind 7.1. Synopsis In this section we would like to discuss the approximation of elliptic variational inequalities of the first kind by penalty methods. In fact these penalty techniques can be applied to more complicated problems as shown in Lions [1], [4] (see also Chapter VII, Sec. 4, where a penalty method is applied to the solution of transonic flow problems).

7.2. Formulation of the penalized problem Consider the EVI problem. Find ue K ( c F ) such that a(u, v -u)>L(v

-u),

\/ve K,

(7.1)

where the properties of V, a(-, •), L(-), and K are those given in Sec. 2.1. Now suppose that a functional;': V -> U has the following properties: j is convex, proper, l.s.c,

(7.2)

j(v) = 0oveK,

(7.3)

j(v) > 0,

VceK.

(7.4)

Let e > 0; we define jE: V -> U by

J. = -eJ.

(7.5)

16

I Generalities on Elliptic Variational Inequalities and on Their Approximation

The penalized problem associated toy'(") is defined b y : Find ue e V such that a(ue, v - u

s

)+ ; »

- ./.(M.) > L(v -

ue),

V C E K

(7.6)

(7.6) is definitely an EVI of the second kind, and from the properties of V, a{-, •), L(-), and;(-X it has (see Sec. 4) a unique solution according to Theorem 4.1. Remark 7.1. Suppose that j £ is differentiable; the solution uc of (7.6) is then characterized by the fact that it is the unique solution of the following nonlinear variational equation: a(u£,v) + a(Klv)

= L(v),

MveV, uEeV,

(7.7)

where j'e(v) (e V*, the topological dual space of V) denotes the differential of je at v, and where < •, • > is the duality pairing between V* and V. That differentiability property (if it exists) can be helpful for solving (7.6), (7.7) by efficient iterative methods like Newton's method or the conjugate gradient method (see Chapter IV, Sec. 2.6 and Chapter VII for references and also some applications of these methods).

7.3. Convergence of {H E } E

Concerning the behavior of {ue}e as e -> 0, we have the following: Theorem 7.1. If the hypotheses on V, K,a{-,-), and 7.2, we have lim \\uE-u\\=

L(•),./(•) are those of Sees. 2.1 0,

(7.8)

8->0

lim j e (t0 = 0,

(7.9)

where u (resp., ue) is the solution o/(7.1) (resp., (7.6)). PROOF. This proof looks very much like the proof of Theorems 5.2 and 6.2. (1) A priori estimates. From (7.6) we have a(K, tO + MuJ < a(ue, v) - L(v - ue) + jE(v), V v e V.

(7.10)

l

Since jc(v) = s~ j(v) = 0, V v e K [property (7.3)], we have, from (7.10), a(u., iO + Muc) < a(uE, v) - L(v - u8),

V»eK.

(7.11)

Consider voe K (since K # 0, such a v0 always exists). Taking v = v0 in (7.11), from the properties of a{-, •) and from (7.4), (7.5) we obtain «II«J 2 < U\\ ||u,|| ||»oll + IH-IKIM + boll),

(7-12)

0 <j(ut) < E ( M | | | U J K | | + ||L||(|«.|| + ||»0|D)

(7.13)

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

17

(a is the ellipticity constant of a(-, •))• Then it clearly follows from (7.12) that we have \\K\\
Va>0,

(7.14)

which combined with (7.13) implies 0<j(u£)
(7.15)

where in (7.14), (7.15), Cx and C 2 denote two constants independent of s. (2) Weak convergence. It follows from (7.14) that we can extract from {us}t a subsequence—still denoted {uje—such that lim uE = u* weakly in V,

(7.16)

where u* e V. It then follows from (7.15), (7.16) and from the weak lower semicontinuity of j( •) that j(u*) = 0 => u* e K [from (7.3)].

(7.17)

To prove that u* = u, we observe that from (7.10) we have a(uE, ut) < a(ue, v) - L(v - ue),

V v e K,

(7.18)

which implies, at the limit as e —> 0, a(u*, u*) < lim inf a(uE, uE) < a(u*, v) - L(v - u*),

V v e K.

(7.19)

Combining (7.17) and (7.19), we finally obtain a(u*, v-u*)>

L(v - «*),

V v e K, u* e K;

(7.20)

we have thus proved that u* = u and that the whole {uE}t converges weakly to u. (3) Strong convergence. From (7.3), (7.4), and (7.6) we have 0 < a\\us - u\\2 + js(us) < a{uc -u,uE-u)+ j£uE) < a(ue, ue) + jt(uc) — a(u, us) — a(ue, u) + a(u, u) < a{uc, v) - L(v - ue) - a(u, ue) - a(uE, u) + a(uy u),

V t i e K.

(7.21)

The weak convergence of {ue}E to u implies that at the limit in (7.21) we have 0 < lim inf [a||« £ - u|| 2 + j£uj] E->0

< a(u, v-u)-

< lim sup [a||uE - u\\2 + jE(uJ\ E->0

L{v-u),

V i ) 6 K.

(7.22)

Taking v = u in (7.22), we obtain lim[a||u E -u|| 2 +j £ (M E )] = 0 £->0

which clearly implies the convergence properties (7.8) and (7.9).

•

Remark 7.2. If a(-, •) is symmetric, then the penalized problem (7.6) is equivalent to the minimization problem: Find uEe V such that Je(ue)<Je(v), VteF, (7.23)

18

I Generalities on Elliptic Variational Inequalities and on Their Approximation

where

EXERCISE 7.1. Prove, if a(-, •) is symmetric, the equivalence between problems (7.6) and (7.23).

7.4. Some examples in finite dimension 7.4.1. Generalities In this section we discuss, in some detail, applications of the penalty method to the solution of some simple model problems in UN. In Sec. 7.4.2 we consider (resp., Sec. 7.4.3) a situation in which K is denned from linear equality constraints (resp., convex inequality constraints). In the following A is a N x JV real matrix, positive definite, possibly nonsymmetric, and b e UN. To A and b we associate a:UN x UN -*U

and

L:MN->U

denned by a(v, w) = (Av, w), L(v) = (b, v),

V v, w e UN, V v e UN,

where (•, •) denotes the usual scalar product of UN, i.e., (v, w) = £ vtwh

V v = {»,}?= t ,

w = {Wi}f=1.

(7.25)

We denote by || • || the norm associated to (7.25), i.e., ||v|| = (v, v) 1/2 ,

VveR*.

The form L(-) is clearly linear and continuous on UN; similarly a(-, •) is bilinear and continuous on UN x UN. Since A is positive definite, «(•>•) is R^-elliptic, and we have a(v,v)>A 0 ||v|| 2 ,

VveR",

(7.26)

where Ao is the smallest eigenvalue of the symmetric positive-definite matrix A + A'/2 (with A' the transpose matrix of A). 7.4.2. A first example Let B e i?(R N , R M ); B can be identified to a M x JV matrix. We define R(B) (the range of B) by R(B) = {q|q e RM, 3 v e R* such that q = Bv}

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

19

and then X c i " b y K = {v 6 UN, Bv = c}

(7.27)

c e K(BX=> K * 0).

(7.28)

where, in (7.27), we have

From the above properties of a(-, •)> L(-), and K, the EVI problem: Find ue K such that (Au, v - u) > (b, v - u),

Vv e K

(7.29)

N

has a unique solution since we can apply (with V = U ) Theorem 3.1 of Sec. 3.1. Remark 7.3. If A = A', then problem (7.29) is equivalent to the minimization problem: Find a e K such that J(u) < J(v),

VveX,

(7.30)

where J(v) = KAv, v) - (b, v). Before going on to the penalty solution of (7.29), we shall prove some properties of the solution u of (7.29); more precisely we have the following proposition. Proposition 7.1. The solution u of (7.29) is characterized by the existence of p e UM such that Au + Brp = b, (7.31) Bu = c. PROOF. (1) (7.29) implies (7.31). Let u be the solution of (7.29); we have U + Y/EK,

V w e Ker(B),

(7.32)

N

where Ker(B) = {v e U , Bv = 0}. Taking v = u + w in (7.22), we obtain (Au - b, w) > 0,

V w e Ker(B),

(Au - b, w) = 0,

V W E Ker(B)

(7.33)

and (7.33) clearly implies

i.e., Au - b 6 Ker(B) x . Since (it is a standard result) Ker(B)-1 = R(B<), we have the existence of p e RM such that (7.31) holds.

(7.34)

20

I Generalities on Elliptic Variational Inequalities and on Their Approximation

(2) (7.31) implies (7.20). The second relation (7.31) implies that u e K. Letting v e X , w e then have v - u e Ker(B),

(7.35)

and from (7.35) and from the first relation (7.31) it follows that (b, v - u) = (Au, v - u) + (B'p, v - u) = (Au, v - u) + (p, B(v - u)) = (Au, v - u). We have thus proved that (7.31) implies (7.29).

(7.36) •

Remark 7.4. Suppose that A = A'; then the vector p of Proposition 7.1 is a Lagrange multiplier vector for the problem (7.30), associated with the linear equality constraint Bv — c = 0 defining K. The following proposition and its corolaries state results quite easy to prove, but of great interest in studying the behavior of the solution of the penalized problem to be defined later on. Proposition 7.2. Problem (7.31) has a unique solution in UN x jR(B) if {b, c j e l f x R(B). Let us denote by {u, p} this solution; then all solutions of (7.31) can be written {u, p + q}, where q is an arbitrary element o/Ker(Br). Corollary 7.1. The above vector p has the minimal norm among all the p e UM such that {u, p} solves (7.31). Corollary 7.2. The Hnear operator A B' B 0 is an isomorphism from UN x R(B) onto UN x R(B). EXERCISE EXERCISE

7.2. Prove Proposition 7.2 and Corollaries 7.1 and 7.2. 7.3. Prove that p = 0 if and only if c = BA" J b.

In order to apply the penalty method of Sec. 7.2 to the solution of (7.29), we define;: R*-> U by ;(v) = i | B v - c | 2 ,

(7.37)

where |-| denotes the usual Euclidean norm of UM. We can easily see thatj(-) obeys (7.2)-(7.4); moreover, j(-) is a C°° functional whose differential / is given by /(v) = B'(Bv - c).

(7.38)

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

21

The penalized problem associated with (7.29) and (7.37) is defined by: Find u£ e UN such that

(An,, v - u£) + jjy) - j£(uE) > (b, v - u£),

VveB",

(7.39)

where je = (1/e)/ (with e > 0). From Remark 7.1 and (7.38) the penalized problem (7.39) is equivalent to the linear system A + - BB|u £ = - B e + b (7.40) e / e whose matrix is positive definite (and symmetric if A is symmetric). It follows from Theorem 7.1 (see Sec. 7.3) that lim ||ue - u|| = 0,

(7.41)

£->0

where u is the solution of (7.29). We have, in fact, ||u£ — u|| = O(s); several methods can be used to prove this result; we have chosen one of them based on the implicit function theorem. Define p£ e UM by p£ = i (Bu£ - c).

(7.42)

Problem (7.40) is then equivalent to the following system: Au£ + B'p£ = b, - B u £ + ep£ = - c ,

(7.43)

whose matrix

isaJV + MbyJV + M positive-definite matrix. EXERCISE

7.4. Prove that s/e is positive definite.

Since c e R(B), we have (from (7.42)) p8 e R(B). About the behavior of {u£, p£} as s -> 0, we have the following theorem. Theorem 7.2. Let u£ be the solution of (139) and let p£ be defined by (7.42); we then have I] u£ - i. = O(s),

(7.45)

| p £ - p | = O( £ ),

(7.46)

where u is the solution of (7.29) and p has been defined in Proposition 7.2.

22

I Generalities on Elliptic Variational Inequalities and on Their Approximation

PROOF. Define F: RN x RM x R -> UN x UM by < 1 -b\ - B v + eq +

(7.47)

still with c e R(B). We observe that F also maps RN x K(B) x R into RN x R(B). We have F(u,p,0) = 0;

(7.48)

moreover, we have

(n,M,, + g (

Since si is an isomorphism from RN x R(B) onto RN x R(B), we can apply the implicit function theorem in the space RN x R(B) x R to define, from F(v, q, s) = 0 (i.e., from (7.43)), uE ( = u(e)) and p8 ( = p(e)) as C°° functions of e (in fact, they are analytic functions of s) in the neighborhood of e = 0. We have {u(0), p(0)} = {u, p} and also ue = u(s) = u + gu(0) + • • •, p£ = p(e) = p + £p(0) + • • •

(7.50)

(with {u, p} = {du/ds, dp/de}); (7.50) implies (7.45), (7.46). The pair {u(0). p(0)} is clearly the unique solution in RN x R(B) of Au(0) + B'p(0) = 0, Bu(0) = - p .

l

' '

We have {u(0), p(0)} J= {0, 0}, unless p = 0, which corresponds (see Exercise 7.3) to the trivial situation u = A~: b. This proves that the estimates (7.45) and (7.46) are of optimal order in general. • EXERCISE 7.5. Prove that |p £ | < \p\, V e > 0.

Remark 7.5. It follows from Theorem 7.2 that ue and pE will be good approximations of u and p, respectively, provided that we use a sufficiently small s. But in this case the condition number v(A£) of the matrix A£ = A + - B'B

(7.52)

occurring in (7.40) will be large; we indeed have (we suppose A = A' for simplicity2)

v(AJ = ||A;1 |l IIAJ = I8—— (1 + m\ :

We also suppose that Ker(B) i= 0 (which is the usual case).

(7-53)

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

23

where, in (7.53), lim 8 ^ 0 j8(e) = 0, p(B'B) is the spectral radius of B'B (i.e., the largest eigenvalue of B'B) and where a is defined by a =

. , inf

(Av,v) |2

vsKer(B)-{0(

For small s it clearly follows from (7.53) that Ae is ill conditioned. Actually that ill-conditioning property that we pointed out for the model problem (7.29) is the main drawback of penalty methods. An elegant way to overcome this difficulty has been introduced by Hestenes [1] and Powell [1]: the so-called augmented Lagrangian methods in which the combined use of penalty and Lagrange multiplier methods allow larger e and moreover produces the exact solution u instead of an approximated one. 3 In Chapter VI, we will discuss the solution of a particular class of variational problems by these augmented Lagrangian methods; for more details and a substantial bibliography, see Fortin and Glowinski [1] and Gabay [1]. EXERCISE

7.6. Prove (7.53).

7.4.3. A second example Let G: UN ->(1) M ; we then have G = {gt}^=1, where gt are functionals from UN to U. We suppose that the following properties hold: V i = 1 , . . . , M, gt is a convex, l.s.c, and proper functional; the convex set K = {\ e UN, gt(\) < 0, V i = 1 , . . . , M} is nonempty.

(7.54) (7.55)

Suppose that the properties of a(-, •) and L(-) are those of Sec. 7.4.1. From these properties and from (7.55), the EVI problem: Find u e K such that (Au, v - u) > (b, v - u),

VveX

(7.56)

has a unique solution [it suffices to apply (with V = UN) Theorem 3.1 of Sec. 3.1]. Remark 7.6. If A = A1, problem (7.56) is equivalent to the minimization problem: Find u e K such that J(u) < J(v),

Vv e K

(7.57)

with J(v) = i(Av, v) - (b, v). 3

See Chapter VII, Sec. 5.8.7.3.3 for an application of augmented Lagrangian methods to the solution of (7.31) (and (7.29)).

24

I Generalities on Elliptic Variational Inequalities and on Their Approximation

Remark 7.7. If G obeys some convenient conditions (usually called qualification conditions), we can generalize Proposition 7.1 and associate to (7.56) the so-called F. John-Kuhn-Tucker multipliers; we shall not discuss this matter here 4 (the interested reader may consult Rockafellar [1], Cea [1], [2], Ekeland and Temam [1], and Aubin [1]). In order to apply the penalty method of Sec. 7.2 to the solution of (7.56), we define;': UN —• IR by

fi>) = i

(7.58)

X>J|0I + (V)I 2 >

where, in (7.58), a; are strictly positive and gf = sup(0, gt). Since ;'(•) satisfies (7.2)-(7.4), the associate penalized problem (with;'£ = (1/e)/, s > 0) is defined by: Find u£ e UN such that (An,, v - u£) + ;£(v) -; £ (u £ ) > (b, v - u£),

VveR".

(7.59)

Remark 7.8. Suppose that gt e C 1 , V i = 1 , . . . , M. We then have |gf(+ | 2 e C \ V i = 1 , . . . , M, implying that j e C 1 . We have

/(v) = £ aigt(y)g'i(y),

(7.60)

and from Remark 7.1, (7.59) is equivalent to the nonlinear system in R":

Au£ + i / ( i g = b.

(7.61)

It follows from Theorem 7.1 (see Sec. 7.3) that lim ||u£ - u|| = 0

(7.62)

£">0

where u is the solution of (7.56). To illustrate the above penalty method, we consider its application to the solution of & discrete obstacle problem (see Chapter II, Sec. 2 for a mathematical and mechanical motivation). For example (with M = N), we have 0.(v) = a - vt,

Vi=l,...,JV,

(7.63)

where c = {ct}f=1, v = {vt}?=1; we take at = 1, V i = 1 , . . . , JV, in (7.58). For this simple problem the equivalence property of Remark 7.8 holds and (7.61) reduces to the nonlinear system A u £ - - ( c - u £ ) + =b, e 4

See, however, Exercise 7.7

(7.64)

7. Penalty Solution of Elliptic Variational Inequalities of the First Kind

25

where

problem (7.64) can be solved by the methods described in Chapter IV, Sec. 2.6 and also in Chapter VI, Sec. 6.4. 7.7. Prove that the solution u of (7.56), with G defined by (7.63), is characterized by the existence of p = {p;}f= x e IR^ such that

EXERCISE

Au — p = b, Pi>0,

Vi = 1

JV,

(7.65)

Also prove that lim c ^ 0 (l/e)(c — u £ ) + = p. Hint: Observe that: (i) if v e UN+ = {v e UN, vt > 0, V i = 1 , . . . , N}, then u + v e K; (ii) c e K. The vector p in (7.65) is precisely a F. John-Kuhn-Tucker multiplier.

7.5. Further comments For more details on penalty methods applied to the solution of variational problems, we refer the reader to Lions [1], [4], Cea [1], [2], and also to G.L.T. [1], [3], Oden and Kikuchi [1], and Ohtake, Oden, and Kikuchi [1], [2]. Actually similar ideas can be applied to the solution of EVI of the second kind; for example, we can replace the solution of : Find u e V such that a(u, v-u)+

j(v) - j(u) > L ( v - u ) ,

VveV

(7.66)

(where V, a(-, •), L(-),j(-) obey the hypotheses of Sec. 2.1) by the solution of: Find ME e V such that a(us, v-us)+

je(v) - j£(ue) > L(v - uE),

V»eK,

(7.67)

where je is an "approximation" of; which is more regular. For example, if we suppose that j is nondifferentiable, it may be interesting from a computational point of view to replace it by jE differentiate. Such a process is called—for obvious reasons—a regularization method. If 78 is differentiate, (7.67) is clearly equivalent to the variational equation: Find ue e V such that a(uE, v) + (j'e(ue), v) = L(v),

V v e V,

(7.68)

26

I Generalities on Elliptic Variational Inequalities and on Their Approximation

where j'e(v) denotes the differential of;£ at v. An application of these regularization methods is given in Chapter II, Sec. 6.6; we refer to G.L.T. [1], [2], [3] for further details and other applications of these regularization methods.

8. References For generalities on variational inequalities from a theoretical point of view, see Lions and Stampacchia [1], Lions [1], Ekeland and Temam [1], Baiocchi and Capelo [1], [2], and Kinderlherer and Stampacchia [1]. For generalities on the approximation of variational inequalities from the numerical point of view, see Falk [1], G.L.T. [1], [2], [3], Strang [1], Brezzi, Hager, and Raviart [1], [2], Oden and Kikuchi [1], and Lions [5]. For generalities and applications of the penalty and regularization methods discussed in Sec. 7, see Lions [1], [4], Cea [1], [2], G.L.T. [1], [2], [3], and Oden and Kikuchi [1] (see also Chapter II, Sec. 6.6 and Chapter VII, Sec. 4 of this book). Some additional references will be given in the following chapters.

CHAPTER II

Application of the Finite Element Method to the Approximation of Some Second-Order EVI

1. Introduction In this chapter we consider some examples of EVI of the first and second kinds. These EVI are related to second-order partial differential operators (for fourth-order problems, see Glowinski [2] and G.L.T. [2], [3]). The physical interpretation and some properties of the solution are given. Finite element approximations of these EVI are considered and convergence results are proved. In some particular cases we also give error estimates. Some of the results in this chapter may be found in G.L.T. [1], [2], [3]. For the approximation of the EVI of the first kind by finite element methods, we also refer the reader to Falk [1], Strang [1], Mosco and Strang [1], Ciarlet [1], [2], [3], and Brezzi, Hager and Raviart [1], [2]. We also describe iterative methods for solving the corresponding approximate problems (cf. Cea [1], [2] and G.L.T. [1], [2], [3]).

2. An Example of EVI of the First Kind: The Obstacle Problem Notations All the properties of Sobolev spaces used in this chapter are proved in Lions [2], Necas [1], and Adams [1]. Usually we shall have Q: a bounded domain in IR2, F = <9Q, x = {xu x2}, a generic point of Q, V =_{d/8xu d/dx2}, Cm(Q): space of m-times continuously differentiable real valued functions for which all the derivatives up to order m are continuous in Q, • CQ(Q) = {v e Cm(Q) | Supp(u) is a compact subset of Q}, • IML, P> n = L«l<m \\D«v\\LP(n){or v e Cm(Q), where a = {cc1,a2};x1,oc2aie non-negative integers, |a| = otj + oc2 and D" = d^/dxf dx22, • • • . •

28

II Application of the Finite Element Method

• Wm'p(Q): completion of Cm(U) in the above norm, • WS'"(Q,): completion of CQ(Q) in the above norm, • Hm(Q.)= Wm-2(Q),

2.1. The continuous problem Let V = tf£(Q) = {v e H\Q)\v\r

= trace of v on T = 0}

(cf. Lions [2] and Necas [1] for a precise definition of the trace), a(u, v) =

VM

• Vv dx,

Jn

where _

du dv

• Vv = —- —

+

du

dv

dx2 dx2

L(v) = i> a.e. on Q}. Then the obstacle problem is a particular ( P J problem defined by: Find u such that a(u, v-u)>

L(v - w),

V»eK,

ueK.

(2.1)

The physical interpretation of this problem is as follows: Let an elastic membrane occupy a region Q in the xu x2 plane; this membrane is fixed along the boundary T on Q. If there is no obstacle, from the theory of elasticity, the vertical displacement u, obtained by applying a vertical force F, is given by the solution of the following Dirichlet problem: —AM

= /' in Q, (2.2)

where / = F/t, t being the tension of the membrane. If there is an obstacle, we have a free boundary problem, and the displacement M satisfies the variational inequality (2.1) with \\i being the height of the obstacle. Similar EVI also occur, sometimes with nonsymmetric bilinear forms, in mathematical models for the following problems: • Lubrication phenomena (cf. Cryer [1]). • Filtration of liquids in porous media (cf. Baiocchi [1] and Comincioli [1]). • Two-dimensional irrotational flows of perfect fluids (cf. Brezis and Stampacchia [1], Brezis [1], and Ciavaldini and Tournemine [1]). • Wake problems (cf. Bourgat and Duvaut [1]).

2. An Example of EVI of the First Kind: The Obstacle Problem

29

2.2. Existence and uniqueness results For proving the existence and uniqueness of the problem (2.1), we need the following lemmas stated below without proof (for the proofs of the lemmas, see, for instance, Lions [2], Necas [1], and Stampacchia [1]). Lemma 2.1. Let Qbe a bounded domain in UN. Then the seminorm on H1(Q) \l/2

\Vv\2dx\ i

/

is a norm on HQ(Q) and it is equivalent to the norm on Hj(Q) induced from Hl(Q). The above Lemma 2.1 is known as the Poincare-Friedrichs lemma. Lemma 2.2. (Stampacchia [1]). Let f:U-*M be uniformly Lipschitz continuous (i.e., 3 k > 0 such that \f(t) - f(t')\ f(v) is a continuous map into H1(Q). Similar results hold for # £ ( Q ) whenever / ( 0 ) = 0. Corollary 2.1. / / v+ and v~ denote the positive and the negative parts of v for veH1(Q.) (respectively, HQ(Q)), then the map v -* {v+ , v~} is continuous from H\Q) -> H\Q) x H\Q) (respectively, H j ( Q ) -+ ff£(fi) x Hj(Q)). Also v -* \v\ is continuous. Theorem 2.1. Problem (2.1) has a unique

solution.

PROOF. In order to apply Theorem 3.1 of Chapter I, we have to prove that a(-, •) is F-elliptic and that K is a closed convex nonempty set. The F-ellipticity of a( •, •) follows from Lemma 2.1 and the convexity of K is trivial. (1) K is nonempty. We have ¥ e Hl(Q) n C°(Q) with *P < 0 on F. Hence, by Corollary 2.1, >?+ e H^Q). Since VP | r < 0, we have W+ | r = 0. This implies that I*"1" e HQ(£1); since

we have f+ e K. Hence K is nonempty. (2) K is closed. Let vn -* v strongly in Hj(n), where vne K and v e HJ(Q). Hence vn—>v strongly in L2(Q). Therefore we can extract a subsequence {vn.} such that vn. -* v a.e. on Q. Then vn. > T a.e. on Q implies that v > ¥ a.e. on Q; therefore v e K. Hence, by Theorem 3.1 of Chapter I, we have a unique solution for (2.1).

•

30

II Application of the Finite Element Method

2.3. Interpretation of (2.1) as a free boundary problem From the solution u of (2.1), we define

Q + = {x\xeQ,u(x)> Q° = {x\xeQ,u(x) +

»F(x)}, = V(x)},

y = dQ n dQ°;u

+

= w| n+ ;M 0 = u\n0.

Classically, problem (2.1) has been formulated as the problem of finding y (the free boundary) and u such that -Au=/inQ+,

u+\y

(2.3)

u = V on Q°,

(2.4)

u = 0 on T,

(2.5)

= u°\y.

(2.6)

The physical interpretation of these relations is the following: (2.3) means that on Q + the membrane is strictly over the obstacle; (2.4) means that on Q° the membrane is in contact with the obstacle; (2.6) is a transmission relation at the free boundary. Actually (2.3)-(2.6) are not sufficient to characterise u since there are an infinite number of solutions for (2.3)-(2.6). Therefore it is necessary to add other transmission properties: for instance, if *P is smooth enough (say *P E H 2 (Q)), we require the "continuity" of Vw at y (we may require Vu e H1(Q)

Remark 2.1. This kind of free boundary interpretation holds for several problems modelled by EVI of the first and second kinds. 2.4. Regularity of the solutions We state without proof the following regularity theorem for the solution of problem (2.1). Theorem 2.2 (Brezis and Stampacchia [2]). Let Qbe a bounded domain in U2 with a smooth boundary. If L(v) = f fv dx and

with f E Z/(Q),

1 < p < + oo

*Pe^2'"(Q),

then the solution of the problem (2.1) is in W2'p(il).

(2.7)

(2.8)

2. An Example of EVI of the First Kind: The Obstacle Problem

31

Remark 2.2. Let Q c R * have a smooth boundary. We know that WS'\Q) <= C(Q) if s > — + k (2.9) P (cf. Necas [1]). It follows that the solution u of (2.1) will be in C^U) if/ e Z/(Q), ¥ G W2'"(Q) with p > 2 (take s = 2, AT = 2, fc = 1 in (2.9)). The proof of this regularity result will be given in the following simple case: L(v) = [ fv dx,

f e L2(Q),

T-OonQ.

(2.10) (2.11)

2

Before proving that (2.10), (2.11) imply u e H (Q), we shall recall a classical lemma (also very useful in the analysis of fourth-order problems). Lemma 2.3. Let Q be a bounded domain of UN with a boundary F sufficiently smooth. Then ||Au||L2(n) defines a norm on H 2 (Q) n Hl(il) which is equivalent to the norm induced by the H2(Q)-norm. 2.1. Prove Lemma 2.3 using the following regularity result due to Agmon, Doughs, and Nirenberg [1]: If w e L2(Q) and if r is sufficiently smooth, then the Dirichlet problem

EXERCISE

— Av = w in Q,

has a unique solution in HQ(Q) n H2(Q) (this regularity result also holds if Q is a convex domain with F Lipschitz continuous). We shall now apply Lemma 2.3 to prove the following theorem using a method due to Brezis and Stampacchia [2]. Theorem 2.2*. / / F is smooth enough, if *F = 0, and if L(v) = jQfv dx with feL2(Q) then the solution u of the problem (2.1) satisfies ueKn

H2(Q),

\\Au\\LHa)

< \\f\\ma)-

(2-12)

PROOF. With L and \j/ as above, it follows from Theorem 2.1 that problem (2.1) has a unique solution u. Letting e > 0, consider the following Dirichlet problem: -eAu £ + ut = u in Q,

ut\Y = 0.

(2.13)

Problem (2.13) has a unique solution in H\{Q), and the smoothness of T implies that us belongs to H2(Ci). Since u > 0 a.e. on Q, by the maximum principle for second-order elliptic differential operators (cf. Necas [1]), we have ue > 0. Hence

uceK.

(2.14)

32

II Application of the Finite Element Method

From (2.14) and (2.1), we obtain a(u, uc-u)>

L{uE - u) = j f(uc - u) dx.

(2.15)

The F-ellipticity of o(-, •) implies

a(uc, u£ — u) = a(ut — u, uz — u) + a(u, us — u) > a(u, ue — u), so that a(ue, us-u)>

\ f(us - u) dx.

(2.16)

By (2.13) and (2.16), we obtain E

Vu£ • V(Au£) dx > s \ fAus dx,

so that, \u. • v(A«.) dx >

f Au, dx.

(2.17)

By Green's formula, (2.17) implies -

f(AtO 2 *c> f fAu£dx. •>n -la

Thus IIAwJum < I I / I I L ^ , ,

(2.18)

2

using Schwarz inequality in L (ii). By Lemma 2.3 and relations (2.13), (2.18) we obtain lim ut = u weakly in H2(Q),

(2.19)

(which implies that lim ue = u strongly in HS(Q), for every s < 2 (cf. Necas [1])), so that u e H 2 (fi) with ||Au||L2(n) < ||/||L2(Q).

(2.20)

• 2.5. Finite element approximations of (2.1) Henceforth we shall assume that Q is a polygonal domain of U2. Consider a "classical" triangulation 2Th of Q, i.e. 2Th is a finite set of triangles T such that TcQ ^ 0 ^

= 0

VTef», V Tu T2 e ^

U T = Q, and

Tx ^ T2.

(2.21) (2.22)

33

2. An Example of EVI of the First Kind: The Obstacle Problem Figure 2.1

MiT

m2T

MlT

m 3T

Moreover V r 1 , r 2 e J , and Ti # T2, exactly one of the following conditions must hold (1)

T,nT2 = 0,

(2)

Ti and T2 have only one common vertex,

(3)

Tt and T2 have only a whole common edge.

(2.23)

As usual h will be the length of the largest edge of the triangles in the triangulation. From now on we restrict ourselves to piecewise linear and piecewise quadratic finite element approximations. 2.5.1. Approximation of V and K • Pk: space of polynomials in xt and x2 of degree less than or equal to k, • l.h = {P e U, P is a vertex of T e $~h}, . IJ, = {P e Q, P is the midpoint of an edge of T e ^"J,

Figure 2.1 illustrates some further notation associated with an arbitrary triangle T. We have m,T e Sj,, M , T e Z t . The centroid of the triangle T is denoted by GT. The space K = Hj(Q) is approximated by the family of subspaces (V%)h with k = 1 or 2, where F£ = to e C°(O), wfc | r = 0 and vh\TePk,\/Te

3Th},

k = 1,2.

It is clear that the V\ are finite dimensional (cf. Ciarlet [1]). It is then quite natural to approximate K by K\ = K e V\, vh(P) > Y(P), V P £ S£}, fe = 1, 2. Proposition 2.1. T/ien Xj /or fc = 1, 2 are closed convex nonempty subsets of

34

II Application of the Finite Element Method

EXERCISE

2.2. Prove Proposition 2.1.

2.5.2. The approximate problems For k = 1, 2, the approximate problems are defined by a(ukh, vh - u\) > L(vh - uk),

VvheKkh,

ukhsKkh.

(P\h)

From Theorem 3.1 of Chapter I and Proposition 2.1, it follows that: Proposition 2.2. (P\h) has a unique solution for k = 1 and 2. Remark 2.3. Since the bilinear form a(-,-) is symmetric, (P*A) is actually equivalent to (cf. Chapter I, Remark 3.2) the quadratic programming problem

Min rjafo, vh) - L(vh)l

(2.24)

vh e

2.6. Convergence results In order to simplify the convergence proof, we shall assume in this section that ¥ e C°(Q) n H\Q)

and

*P < 0 in a neighborhood of F.

(2.25)

Before proving the convergence results, we shall describe two important numerical quadrature schemes which will be used to prove the convergence theorem. 2.3. With notations as in Fig. 2.1, prove the following identities for any triangle T: EXERCISE

w dx = m e a 3 S ( r ) f w(MiT), vae r wdx

=

^l

£ w(miT),

V we P l s V we P2.

(2.26) (2.27)

Formula (2.26) is called the Trapezoidal Rule and (2.27) is known as Simpson's Integral formula. These formulae not only have theoretical importance, but practical utility as well. We have the following results about the convergence of u\ (solution of the problem (Pklh)) as h -» 0. Theorem 2.3. Suppose that the angles of the triangles of 2Th are uniformly bounded below by 90 > 0 as h -» 0; then for k = 1,2, lim ||u\ -

M || H4(n)

= 0,

where u\ and u are the solutions of(P\h) and (2.1), respectively-

(2.28)

2. An Example of EVI of the First Kind: The Obstacle Problem

35

PROOF. In this proof we shall use the following density result to be proved later: nK

= K.

(2.29)

To prove (2.28) we shall use Theorem 5.2 of Chapter I. To do this we have to verify that the following two properties hold (for k = 1,2): (i) If (vh)h is such that vhe Kjj, V h and converges weakly to v as h -* 0, then v e K. (ii) There exist x, I = K and r\: x -> K\ such that lim^o r^v = v strongly in V, V v e xVerification of (i). Using the notation of Fig. 2.1 and considering e 3>{Q) with <j> > 0, we define <j>h by cf>h = ^ T S ^ h <^(GT)^T, where ^r is the characteristic function1 of T and GT is the centroid of T. It is easy to see from the uniform continuity of <j> that lim (/>„ = (j> strongly in L°°(n).

(2.30)

Then we approximate *P by *Pfc such that (2.31) ^ ( P ) = ¥(P),

VPeZj.

This function *Ph satisfies lim «Ph = ¥ strongly in L°°(Q).

(2.32)

h->0

Let us consider a sequence (vh)h, vhe Kl,V h such that lim vh = v weakly in V. Then l i m ^ 0 vh = v strongly in L2(Q) (cf. Necas [1]) which, using (2.30) and (2.32), implies that

lim If {v (vhh -- ¥„)(£„ VM>k dx dx== If (v (v --

VW dx,

(2.33)

(actually, since , -> ^> strongly in Lco(fi), the weak convergence of vh in L2(Q) is enough to prove (2.33)). We have

[ (vh - Vh)h dx= X
(2.34)

From (2.26), (2.27), and from the definition of T ft , we obtain for all Te#~h, f (t>* - «P») dx = m e a '

f (^ - Tft) dx = ^

l

(r)

(T)

Xi{x) = 1, V x e T , XT(X) = 0 if X <£ T.

I K(M, T ) - W M f f ) ]

if /c = 1,

(2.35)

E [ O ^ T ) - W«ir)]

if * = 2.

(2.36)

36

II Application of the Finite Element Method

Using the fact that 4>h > 0, the definition of K\ and relations (2.35) and (2.36), it follows from (2.34) that

1

(ffi — ^hi^h dx > 0,

V (j)

a

so that as h -»• 0 I (t> - 4*)^ dx > 0, •In

V cf> € &(Q),

4>>0

which in turn implies v > T a.e. in £2. Hence (i) is verified. Verification of (ii). From (2.29) it is natural to take x = ®(P) <"> K. We define

as the "linear" interpolation operator if k = 1 and "quadratic" interpolation operator iffe = 2,i.e, r*hveVkh,

V v e Hj(fi) n C°(fi), (2-37) V P e SJ forfc= 1, 2.

On the one hand it is known (cf., for instance, Ciarlet [1], [2] and Strang and Fix [1]) that under the assumption made on 9~h in the statement of Theorem 2.3, we have \\rkhv-v\\v
Voe©(QX

k = 1, 2,

with C independent of h and u. This implies that l i m \\riv - v\\v = 0 ,

V t; £ x,

k=l,2.

On the other hand, it is obvious that r&eKl,

VveKnC°(n),

so that rkhveKkh,

Vvex

f o rfe= 1, 2 .

In conclusion, with the above x and r£, (ii) is satisfied. Hence we have proved the Theorem 2.3 modulo, the proof of the density result (2.29). D Lemma 2.4. Under the assumptions (2.25), we have 3}(£l) n K = K. PROOF. Let us prove the Lemma in two steps. Step 1. Let us show that Jf = {ti e K n C0(H), v has a compact support in Q}

(2.38)

is dense in K. Let v e K; K a H},(Q) implies that there exists a sequence {„}„ in 3(0.) such that lim 4>n = v strongly in V.

2. An Example of EVI of the First Kind: The Obstacle Problem

37

Define vn by vn = maxCP, 4>n)

(2.39)

so that vn = %£¥ + <$>„) + I ^

~n\l

Since ve K, from Corollary 2.1 and relation (2.39), it follows that lim vn = \\_QV + u) + | v P - i ; | ] = max(*P, v) = v strongly in V.

(2.40)

From (2.25) and (2.39), it follows that each vn has a compact support in Q,

(2.41)

vn 6 K n C°(Q).

(2.42)

From (2.40)-(2.42) we obtain (2.38). Step 2. Let us show that: For every v e Jf, there exists a sequence {vm}m such that vm 6 ®(fi) n K,

V OT and

lim |[um — u[|HJ(n) = 0.

From Step 1 this proves that S>(Q) n K is dense in K. Let pn be a sequence of mollifiers, i.e., pn e 2(U2), f Supp(pn) = {0},

Pn{y)

pn > 0,

(2.43)

dy = 1,

(2.44)

{Supp(pn)} is a decreasing sequence.

n=l

Let u e JT. Let v be an extension of v defined by (v(x) ifxefi, then t5 e H^U2). Let {!„ = v * pn, i.e.,

o.W = f P^x - y)C(y) dy,

(2.45)

then 5B e @(R2), Supp(Cn) <= Supp(y) + Supp(pJ, lim vn = v strongly in

(2.46)

2

H^R ).

n~* oo

Hence from (2.41) and (2.46), we have Supp( | »„ |) <= 12 for n large enough.

(2.47)

38

II Application of the Finite Element Method We also have (since supp(0) is bounded) lim vn = v strongly in LCO(R2).

(2.48)

Define vn = vn\n; then (2.46)-(2.48) imply vn e

(2.49)

lim vn = v strongly in #o(Q) n C°(Q); v e Jf and *P < 0 in a neighborhood of F imply that there exists a 5 > 0 such that v = 0,

¥ < 0 on ad,

(2.50)

where Qj = {x e fi|d(x, F) < 5} (d(x, F) = distance from x to F). From (2.48) and (2.50) it follows that V e > 0, there exists an n0 = no(e) such that V n > no(e) u(x) - £ < t)B(x) < v{x) + e,

V x e f i - Q, /2 , (2.51)

yn(x) = t)(x) = 0 for x e fls/2 Since Q — Q.s/2 is a compact subset of Q, there exists a function 0 (cf., for instance, H. Cartan [1]) such that 0 e S>(fi),

0 > 0 in Q (2.52)

Finally, define w^ = un + £0. Then from (2.49), (2.51), and (2.52), we have w* e 3){Q),

lim wcn = v strongly in Ho(fl),

with W^(JC) > v(x) > T(x), V x e Q, so that Step 2 is proved.

•

Remark 2.4. Analyzing verification (i) in the proof of Theorem 2.3, we observe that if for k = 2 we use, instead of K\, the convex set {vheV2h,vh{P)

±

then the convergence of u\ to u still holds provided 2Th obeys the same assumptions as in the statement of Theorem 2.3. EXERCISE 2.4. Extend the previous analysis if Q. is not a polygonal domain. 2

EXERCISE 2.5. Let Q be a bounded domain of U and let F o be a "nice" subset

of F (see Fig. 2.2). Define V by V = {v e H\Q.), v \TQ = 0}. Taking the bilinear form a(-, •) as in (2.1), and L e F*, study the following EVI:

a(u, v — u) > L(v — u),

VueK,

u e K,

2. An Example of EVI of the First Kind: The Obstacle Problem

39

Figure 2.2

where K = {veV, v>¥ a.e. in O} and ¥ £ C°(fi) n H\O), ¥ < 0 in a neighborhood of F o . Also study the finite element approximation of the above EVI. _ Hint: Use the fact that if F and F o are smooth enough, then -V = V, where (see Fig. 2.2), iT = {v e C°°(Q), t; = 0 in a neighborhood of F o }.

2.7. Comments on the error estimates We do not emphasize this subject too much since this is done in detail in Ciarlet [1, Chapter 9], [2, Chapter 5] and G.L.T. [3, Appendix 1], at least for piecewise linear approximations. 2.7.1. Piecewise linear approximation Using piecewise linear finite elements and assuming that feL2(Q) and \ji, ueH2(Q), 0(h) estimates for \\u — uh\\HHn) have been obtained by Falk [1], [2], [3], Strang [1], Mosco and Strang [1], and Brezzi, Hager, and Raviart [1]. We also refer to Ciarlet [1, Chapter 9], [2, Chapter 5] (resp., G.L.T. [3, Appendix 1]) in which the Falk (resp., Brezzi, Hager, and Raviart) analysis is given. 2.7.2. Piecewise quadratic approximation Assuming more regularity for / , *F, and u than in the previous case (also assuming some smoothness hypotheses for the free boundary, an O(/i3/2~E) estimate for \\uh — u|| H i (n) has been obtained by Brezzi, Hager, and Raviart [1] and Brezzi and Sacchi [1] for an approximation by piecewise quadratic finite elements, similar to the one described in Sec. 2.6.

40

II Application of the Finite Element Method

2.8. Iterative solution of the approximate problem Once the continuous problem has been approximated and the convergence proved, it remains to effectively compute the approximate solution. In the case of the discrete obstacle problem, this can be done easily by using an over-relaxation method with projection as described in Cea and Glowinski [1], Cea [2], and also in Chapter V, Sec. 5 of this book. Let us justify the use of this method. It follows from Remark 2.3 that the discrete problem is of the following type: Min &(Av, v) - (b, i>)]

(2.53)

veC

where (-, •) denotes the usual inner product in UN and v = {vu..., A = (a 0 ),

1 4 ' i ) l < i < N}.

(2.55)

Since C is the product of closed intervals of U, the over-relaxation method with projection on C can be used. Let us describe it in detail: u° e C, ii° arbitrarily chosen in C (u° = {^*l5..., ^JV} may be a good guess), (2.56) then u" being known, we compute «" +1 , component by component using for

\

where Pt(x) = Max(x, ¥,),

V x e K.

(2.59),

From Chapter V, Sec. 5 (see also Cea and Glowinski [1], Cea [2] ,and G.L.T. [1], [3]) it follows that: Proposition 2.3. Let {«"}„ be defined by (2.56)-(2.59). Then for every u° e C and V 0 < to < 2, we have l i m , , ^ u" = u, where u is the unique solution of (2.53). Remark 2.5. In the case of the discrete obstacle problem, the components of u will be the values taken by the approximate solution at the nodes of t.h if k = 1 and th n l'h if k = 2. Similarly, ^ will be the values taken by ¥ at the nodes stated above, assuming these nodes have been ordered from 1 to N.

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

41

Remark 2.6. The optimal choice for co is a critical and nontrivial point. However, from numerical experiments it has been observed that the so-called Young method for obtaining the optimal value of co during the iterative process itself leads to a value of co with good convergence properties. The convergence of this method has been proved for linear equations and requires special properties for the matrix of the system (see Young [1] and Varga [1]). However, an empirical justification of its success for the obstacle problem can be made, but will not be given here. Remark 2.7. From numerical experiments it has been found that the optimal value of co is always strictly greater than unity.

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem 3.1. Formulation. Preliminary results Let SI be a bounded domain of IR2 with a smooth boundary F. With the same definition for V, a(-, •), and L(-) as in Sec. 2.1 of this chapter, we consider the following EVI of the first kind: a(u, v - u) > L(v - u),

VceK,

u e K,

(3.1)

where K = {ve Hj(fi), |Vv\ < 1 a.e. in Q}.

(3.2)

Theorem 3.1. Problem (3.1) has a unique solution. PROOF. In order to apply Theorem 3.1 of Chapter I, we only have to verify that K is a nonempty closed convex subset of V. K is nonempty because 0 e K, and the convexity of K is obvious. To prove that K is closed, consider a sequence {vn} in K such that vn -> v strongly in V. Then there exists a subsequence {vn.} such that lim Vvn. = Vv a.e. Since | Vvn \ < 1 a.e., we get | Vv \ < 1 a.e. Therefore v e K. Hence K is closed.

•

The following proposition gives a very useful property of K. Proposition 3.1. K is compact in C°(Q) and I v(x) | < d(x, F),

V x e Q andM veK,

where d(x, F) is the distance from x to F.

(3.3)

42 EXERCISE

II Application of the Finite Element Method

3.1. Prove Proposition 3.1.

Remark 3.1. Let us define ux and M_X by M^OC) M.JX)

= d(x, T), = -d(x, T).

Then w^ and w_ „ belong to K. We observe that ux is the maximal element of X and M_x is the minimal element of K. Remark 3.2. Since a(-, •) is symmetric, the solution u of (3.1) is characterized (see Sec. 3.2 of Chapter I) as the unique solution of the minimization problem J(u) < J(v),

VveK,

ueK,

(3.4)

with J(v) = ^a(v, v) - L(v).

3.2. Physical motivation Let us consider an infinitely long cylindrical bar of cross section Q, where Q is simply connected. Assume that this bar is made up of an isotropic elastic perfectly plastic material whose plasticity yields is given by the Von Mises Criterion. (For a general discussion of plasticity problems, see Koiter [1] and Duvaut and Lions [1, Chapter 5]). Starting from a zero-stress initial state, an increasing torsion moment is applied to the bar. The torsion is characterised by C, which is defined as the torsion angle per unit length. Then for all C, it follows from the Haar-Karman Principle that the determination of the stress field is equivalent (in a convenient system of physical units) to the solution of the following variational problem:

Minji f |VP| 2 dx - C f v dx\.

(3.5)

This is a particular case of (3.1) or (3.4) with L(i>) = C [vdx. (3.6) Jn The stress vector a in a cross section is related to u by a = Vu, so that u is a stress potential, and we can obtain a once the solution of (3.5) is known. Proposition 3.2. Let us denote by uc the solution of (3.5) and let, as before, ux = d(x, F); then lim c ^ + x uc = ux strongly in HQ(£1) n C°(fi). PROOF. Since uc is the solution of (3.5), it is characterized by f V«c • V(y -uc)dx>C

j (u - uc) dx,

V v e K, uc e K.

(3.7)

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

43

Since ux e K, from (3.7) we have f Vuc • V(Uoo - uc) dx > C \ {ux - uc) dx,

(3.8)

i.e. C I («„ - uc) dx + ) | Vuc\2 dx< \ Vum • Vuc dx < f | Vuw | • | VMC I dx < meas(Q). (3.9) From (3.3) we have u^ — uc> 0, so that (3.9) implies llMoo ~ UC\\LHCI) ^ C~l meas(Q) which in turn implies lim \\um - uc\\LHn) = 0.

(3.10)

C—+oo

From the definition of K and from the Proposition 3.1, we find that K is bounded and weakly closed in V and hence weakly compact in V. Furthermore, K is compact in C°(ii). Relation (3.10) implies lim uc = u , strongly in C°(Q), lim uc = uM weakly in K It follows from (3.8) that |V(uM - uc)\2 dx + C \ (um - u c ) dx

Vum • Viu^ - « c ) > *il

*Q

*£2

= \\uc — u^Wy + C\\ux — uc\\Li(a).

(3.12)

It follows easily from (3.11) and (3.12) that lim C\\u^ ~ uc\\LHa) = °> lim \\um — uc\\v = 0.

D

Remark 3.3. In the case of a multiply connected cross section, the variational formulation of the torsion problem has to be redefined (see Lanchon [1], Glowinski and Lanchon [1], and Glowinski [1, Chapter 4]).

3.3. Regularity properties and exact solutions 3.3.1. Regularity results Theorem 3.2 (Brezis and Stampacchia [2]). Let ube a solution of (3.1) or (3.4) and L(v) = §n fv dx.

44

II Application of the Finite Element Method

1. If SI is a bounded convex domain ofU2 with F Lipschitz continuous and if / e LP(Q) with 1 < p < + oo, then we have ueW2-p(Q).

(3.13)

2. IfQ is a bounded domain ofU2 with a smooth boundary F and with 1 < p < +oo, then u e W2-"(Q).

iffeLp(€i)

Remark 3.4. It will be seen in the next section that, in general, there is a limit for the regularity of the solution of (3.1) even if F and / are very smooth. Remark 3.5. It has been proved by H. Brezis that under quite restrictive smoothness assumptions on F and / , we may have

3.3.2. Exact solutions In this section we are going to give some examples of problems (3.1) for which exact solutions are known. EXAMPLE 1. We take Q = {x| 0 < x < 1} and L(v) = c JJ v dx with c > 0. Then the explicit form of (3.1) is f u'(v' -u')dx>c\(v-u)dx, Jo Jo

V v e K,

we K,

(3.14)

where K = {v e Hj(O), | v' \ < 1 a.e. on Q} and v' = dv/dx. The exact solution of (3.14) is given by u(x)

= C-x(l - x),

Vx,

if c < 2;

(3.15)

if c > 2, we have x

if 0 < x < 2

1 - x

, c

if- + - < x < 1. 2 c

EXAMPLE 2. In this example we consider a two-dimensional problem. We take fi=

{x|xf + x\

L(v) = c

v dx with c > 0.

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

45

Then setting r = (x\ + x|) 1 / 2 , the solution u of (3.1) is given by u{x) = C-(R2-r1) 4

ifc<|; K

(3.17)

if c > 2/R, then 2 ii-
u(x) = R-r

u(x) = - \(R2 - r2) - ( R - -

ifO
(3.18)

These examples illustrate Remark 3.4. We see that for c sufficiently large, we have u e W2' °°(J2) n Hj(Q),

u $ H 3 (Q).

(3.19)

In fact we have u e HS(Q.),

Vs < | .

3.2. Verify that the u given in Examples 1 and 2 are exact solutions of the corresponding problems.

EXERCISE

3.4. An equivalent variational formulation In H. Brezis and M. Sibony [1] it is proved that if Q is a bounded domain of IR2 with a smooth boundary T and if L(v) = c

v(x) dx

(c > 0 for instance),

Jsi

then the solution of (3.1) is also a solution of

a(u, v — u) < c \ (v — u) dx,

VceX,

Jil

uek=

(3.20)

{ve Ho(Q), | v(x) | < d(x, T) a.e.}.

Problem (3.20) is very similar to the obstacle problem considered in Sec. 2 of this chapter. Since a(-, •) is symmetrical, (3.20) is also equivalent to J(u) < J(v),

VveK,

uek,

with J(v) = ja(v, v) — c

v(x) dx.

(3.21)

46

II Application of the Finite Element Method

The numerical solution of (3.20) and (3.21) is considered in G.L.T. [1, Chapter 3] (see also Cea [2, Chapter 4] and Chapter V, Sec. 5.5 of this book). 3.3. Study the numerical analysis of (3.21). 3.4. Assume c > 0 in (3.20). Then prove that the solution u of (3.20) is also the solution of the EVI obtained by replacing K by {v e HQ(Q.), V(X) < d(x, T) a.e.} in (3.20). EXERCISE EXERCISE

3.5. Finite element approximation of (3.1) In this section we consider an approximation of (3.1) by first-order finite elements. From the viewpoint of applications in mechanics (in which / = c), it seems that, given the equivalence of (3.1) and (3.20), it is sufficient to approximate (3.20) (using essentially the same method as in Sec. 2). However, in view of other possible applications, it seems to us that it would be interesting to consider the numerical solution of (3.1), working directly with K instead of K. For the numerical analysis of (3.20) by finite differences, see G.L.T. [3, Chapter 3] and Cea, Glowinski, and Nedelec [1]. 3.5.1. Approximation of V and K We use the notation of Sec. 2.5 of this chapter. We assume that Q is a polygonal domain of U2 (see Remark 3.8 for the nonpolygonal case), and we consider a triangulation 2Th of Q satisfying (2.21)-(2.23). Then V and K, respectively, are approximated by Vh = H e C°(Q), vh = 0 on r , vh \T e Pu V T e STh}, Kh = Kn

Vh.

Then one can easily prove: Proposition 3.3. Kh is a closed convex nonempty subset of Vh. Remark 3.6. If vhe Vh, then Vvh is a constant vector on every T e STh. 3.5.2. The approximate problem The approximate problem is defined by: Find uh e Kh such that a(uh, vh - uh) > L(vh - uh),

V vheK.

(3.22)

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

47

One can easily prove: Proposition 3.4. The approximate problem (3.22) has a unique solution. In Sec. 7 of this chapter one may find practical formulae related to finite element approximation. Using these formulae, (3.22) and the equivalent problem (3.23) can be expressed in a form more suitable for computation. Remark 3.7. Since a(-, •) is symmetrical, (3.22) is equivalent to the non linear programming problem (3.23) v h 6 A'j,

The natural variables in (3.23) are the values taken by vh over the set t,h of the interior nodes of ^ , . Then the number of variables in (3.23) is Card(Lh). The number of constraints is the number of triangles, i.e., Card ( ^ ) , and each constraint is quadratic with respect to these variables since VuJ < 1 if and only if | Vvh | 2 < 1 over T.

(3.24)

Remark 3.8. If O is not polygonal, it is always possible to approximate Q by a polygonal domain Qh in such a way that all vertices of Fh = dQh belong to T. Then, instead of defining (3.22) over Q, we define it over Qh. 3.5.3. Remarks on the use of higher-order finite elements In this book only an approximation of (3.1) by first-order finite elements has been considered. This fact is justified by the existence of a regularity limitation for the solution of (3.2), which implies that even with very smooth data one may have u $ V n H3(Q) (see the examples of Sec. 3.3.2). We refer to G.L.T. [3, Chapter 3] and Glowinski [1, Chapter 4, Sec. 3.5.3] for further discussions on the use of finite elements of order < 2.

3.6. Convergence results. General case

In this section we take L(v) =
= K.

(3.25)

48

II Application of the Finite Element Method

PROOF. We use the notation of Lemma 2.4. Let ve K and e > 0; define vc by e)+ - (v + £ ) " .

vc = (v-

(3.26)

1

Then we have vEeH (Q) with |V«J < 1 a.e. in Q. From the inclusion X c K = {v e V, \v(x)\ < d(x, F) a.e. in Q}, it follows that vE(x) = 0 if d(x, T) < e, (3.27) I vF(x) | < d(x, F) — £ elsewhere so that from (3.27) it follows that vse K and has a compact support in Q.

(3.28)

From Corollary 2.1 we have lim ve = v strongly in V.

(3.29)

E->0

From (3.28) and (3.29) it follows that if Jf = {v e K, v has a compact support in Q}, then 3f = K. Thus, to prove the lemma, it suffices to prove that any v e Jf can be approximated by a sequence (vn\ of functions in ®(fl) n K. Let (pn)n be a mollifying sequence as defined in Lemma 2.4 of this chapter. Let v e JT. Denote by v the extension of v to U2 by taking zero outside Q. Then 5 e H'(IR2). Let vn = v * pn so that Cn(x) = f pB(x - y)i5Cy) dy,

(3.30)

VDn(x)= I" A(x - yJVeO-) dy.

(3.31)

Then »„ 6 S>(U2) and

lim Cn = v strongly in H^tt2).

(3.32)

Since Supp S c Q, from (3.30) we have Supp »„ c Q for n sufficiently large.

(3.33)

Define vn = vn\n for n sufficiently large. Then (3.32) and (3.33) imply vn e @(£2),

lim vn = v strongly in V.

(3.34)

From (3.31), pn > 0, JR2 pndy = 1 and |VC(}>)| < 1 a.e. on U2, we obtain | Vvn{x) | = | Vvn{x) I < f I Vt>0;) | A (x - y) dy < 1, which completes the proof of the lemma.

Vx e

fl,

(3.35) •

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

49

3.6.2. A convergence theorem Theorem 3.3. Suppose that the angles of the triangles of STh are uniformly bounded by 90 > 0 as h > 0. Then lim uh = u strongly in Vn C°(U),

(3.36)

where u and un are, respectively, the solutions of (3.1) and (3.22). PROOF. TO prove the strong convergence in V, we use Theorem 5.2 of Chapter I, Sec. 5. To do this, one has to verify the following properties. (i) If (vh)h, vh e Kh, V h, converges weakly to v, then v e K, (ii)' There exists x and rh with the following properties: 1- X = K, 2. rh: X^Kh,y h; 3. for each v e x, we can find h0 = ho(v) such that for all h < ho(v), rhveKh and lim^o rhv = v strongly in V. Verification of (i). Since Kh <= K and K is weakly closed, (i) is obvious. Verification of (ii)'. Let us define x by

Then by Lemma 3.1 and from limA^ 1 Iv — v strongly in V, V v e V, it follows that x = K. Define rh: V n C°(U)-+ Vh by rhveVh,

yVeVnC°(U),

(rhv)(P) = v(P),

\/Peth.

(3.37)

Then rhv is the "linear" interpolate of » on 5 i . From the assumptions on S~h we have (cf. Strang and Fix [1] and Ciarlet [1], [2]) I V(r_ v - v) | < Ch || v ||m, „<„, a.e.,

V v e &(Q),

(3.38)

with C independent of h and v. This implies lim ||rfct? — tj||K = 0,

Vi;eZ,

\Vrhv(x)\ < \Vv(x)\ + Ch\\v\\W2.«,m a.e..

(3.39) (3.40)

Since v e x, it follows from (3.40) that we have | Vrh v{x) | < 1 a.e. for h < ho(v). This implies rhv e Kh. This completes the verification of (ii)' and hence, by Theorem 5.2 of Chapter I, we have the strong convergence of uh to u in V. The strong convergence of uh to u in the L°°-norm follows from the convergence in V and from the compactness of K in C°(Q) (see Proposition 3.1). •

50

II Application of the Finite Element Method

3.7. Error estimates From now on we assume that / e Lp for some p > 2. In Sec. 3.7.1 we consider a one-dimensional problem (3.1). In this case if / e L 2 ( Q ) we derive an O(h) error estimate in the F-norm. In Sec. 3.7.2 we consider a two-dimensional case with feL",p>2, and Q convex; then we derive an 0(/i 1/2 ~ 1/p ) error estimate in the F-norm. 3.7.1. One-dimensional case We assume here O = {x e K|0 < x < 1} and that feL2(Q). (3.1) can be written as

f o

Jo

dx \dx ueK

Then problem

dx =
< la.e. in£H.

dx

(3.41)

Let AT be a positive integer and h = l/N. Let x{ = ih for i = 0 , 1 , . . . , N and e, = [xi-1,x&

i=l,2,...,N.

Let Vh = {vh E C°(U), vh(0) = vh(l) = 0, vh\ei e Pui

=

1,2,...,N},

Kh = K n Vh = {vh e Vh, \ vh(Xi) — vh(xt-. J | < h for i = 1, 2 , . . . , N}. The approximate

problem is defined by Uh)dX

'

^V"€K"'

u eK

» >-

(142)

Obviously this problem has a unique solution. Now we are going to prove: Theorem 3.4. Let u and uh be the solutions of (3.41) and (3.42), respectively. If fe L2(Q), then we have Ik -

M|| K

= 0{h).

PROOF. Since uhe Khcz K, from (3.41) we have a(u, uh-u)>

f(uh - u) dx.

(3.43)

Jo Adding (3.42) and (3.43), we obtain a{uh - u,uh-

u) < a(vh - u,uk-

u) + a(u, vh - u) -

f(vh - u) dx,

V vh e Kh

- u\\2v < i\\vh - ufv + J d£ i^ - ~ J dx - ^ f(vh - u) dx,

Vvhe KH.

which in turn implies

(3.44)

51

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem Since u e K n H2(0, 1), we obtain

0

du d — — (vh - u) dx = dxdx Ik* - " I I L * -

But we have

< fl/IL-

d~x~2

(3-45)

Therefore (3.44) becomes ill«» - M||K < JIIP* - M|IF + 2||/||i21|«* - u\\Li,

Vi),eK A .

(3.46)

Let v e K. Then the usual linear interpolate rhv is defined by rhveVh,

{rhv)(xd = ifai).

i = 0, 1 , . . . , JV.

(3.47)

We have d dx

=

vixd-vjx^,) h

11 p = h J.

dv —dx.

Hence we obtain < 1 since

< 1 a.e. in Q.

(3.48)

Thus rhve Kh. Let us replace vh by rhu in (3.46). Then I l k - u\\2v < i\\rhu - u\\l + 2||/|| L2(n) ||r ft u - u\\ma).

(3.49)

From (3.45) and standard approximation results, we have \\rhu - u\\v < Ch\\u\\HHn) < Ch\\f\\LHa), 2

2

\\rku ~ u\\LHn) < Cfc ||«||H1(n) ^ Ch \\f\\LHai,

(3.50) (3.51)

where C denotes constants independent of u and h. Combining (3.49)-(3.51), we get | k - U\\y = O(h). This proves the result.

•

EXERCISE 3.5. Prove (3.45).

3.7.2. Two-dimensional case In this subsection we shall assume that Q is a convex bounded polygonal domain in IR2 and that feLp(Q) with p > 2. The latter assumption is quite reasonable since in practical applications in Mechanics we have / = constant.

52

II Application of the Finite Element Method

Theorem 3.5. Suppose that the angles of 3Th are uniformly bounded by 60 > 0 as h —• 0; then with the above assumptions on Q and f, we have

where u and uh are the solutions of (3.1) and (3.22), respectively. PROOF. Since / e Z/(Q) with p > 2 and il is bounded, from Theorem 3.2 of this chapter we have u e W2-"(Q). Then, as in proof of Theorem 3.4 and using Kh a K, we obtain i K - W||K < i l k - u\\r + a(u, vh-u)-

f(vh - u) dx

^ i l k * - a l l * - U-Au-fXvh-u)dx,

V»,e4

(3.52)

Then, using Holder's inequality, it follows from (3.52) that i l k - u\\y 2. If W2-P(T) c WU"(T), it follows from Ciarlet [2] and the Sobolev imbedding theorem (W2'P(T) <= W1'X(T) a C°(T)) that V T e $~h and V v e W2'"(T), we have ||V(o - nTv)\\LnT)XL,m

< Ch'r+^-^WvW^.w

(3.54)

In (3.54) nTv is the linear interpolate of i> at the three vertices of T, hT is the diameter of T, and C is a constant independent of T and v. Let v e W2-"(Q.) and let nh: V n C°(Q) -> Vh be denned by nh v e Vh,

V v e HJ(Q) n C°(fi),

(nhv)(P) = v(P),

VPe£ fe .

2

Since p > 2 implies W -"(Q) c= C°(U), one may define nhv, but unlike the onedimensional case, usually nhv$Kh

for v e W2-"(Q) n K.

Since W2'p(il) c Wu "(H) for p > 2, it follows from (3.54) that a.e. \V(nhv

- »X*)I ^ r h l - 2 " \ \ v \ \ w 2 , P ( i i ) ,

Vve

W2--(£l)

which in turn implies that a.e. \V(nhv)(x)\ < 1 + rh'-^-WvW^,^,

V t e X n W2-"(Q).

The constant r occurring in (3.55) is independent of v and h. Let us define

(3.55)

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

53

by

rhv =

rT

•

(3-56)

VveW2-p(Q)nK.

(3.57)

1 2/p

1 +

rh - \\v\\W2,r(a)

It follows from (3.55) and (3.56) that rhveKh, 2 P

Since u e W - (Q) n K, it follows from (3.57) that we can take vh = rhu in (3.53) so that -u\\2v

- 2 ll r * U

~ U\\2y +• {||Au|\LP •+

>}lk*« « | | T O - Il/L

(3.58)

We have 71* M -

u — rl, 1 - 2 / " N i l .

which implies

Ik*"

-U\\y

<

||7thM

- U\\y +

i — u\\ LP'lil) 5; \\nhu - u \\Lp, P

" 2/ "ll" ll»"2,p||u|| K ,

(3.59)

I-2/,|| U

(3.60)

2

1<

II»' .J'(!2)II IILJ''-

p

Since p > 2 we have L (Q) c L '(ii), and from standard approximation results (see Strang and Fix [1] and Ciarlet [1], [2]) it follows that under the above assumption on 2Th we have \\nhu - u\\v < Ch\\u\\W2,P(il), 2

\\nhu - u\\LP>m < Ch \\u\\W2,nn), ll2 llp

with C independent of h and u. Then the O(h ~ ) Theorem 3.5 follows directly from (3.58)-(3.62).

(3.61)

(3-62)

error estimate of the statement of •

Remark 3.9. It follows from Theorem 3.5 that if / = constant (which corresponds to application in mechanics) and if Q is a convex polygonal domain, we have "practically" an O(^h) error estimate. Remark 3.10. In Falk [1] one may find an analysis of the error estimate for piecewise linear approximations of (3.1) when Q is not polygonal. Remark 3.11. In Falk and Mercier [1] (see also G.L.T. [3, Appendix 3]) we may find a different piecewise linear approximation of (3.1). Under appropriate assumptions this approximation leads to an O(h) error estimate for \\uh — u\\v. In G.L.T., loc. cit., a conjugate gradient algorithm for solving these new types of approximate problems is also described. 3.8. A dual iterative method for solving (3.1) and (3.22)

There are several iterative methods for solving (3.1), and (3.22), and the reader who is interested in this direction of the problem may consult G.L.T. [3, Chapter 3] (see also Cea, Glowinski, and Nedelee [1]). In this section we shall

54

II Application of the Finite Element Method

use the material of Cea [2, Chapter 5, Sec. 5] to describe an algorithm of Uzawa type which has been successfully used to solve the elasto-plastic torsion problem. Another method will be described in Chapter VI, Sec. 6.2. 3.8.1. The continuous case Following Cea [2] and G.L.T. [1, Chapter 3], [3, Chapter 3], we observe that K can also be written as K = {veV, |Vu|2 - 1 < 0 a . e . } . Hence it is quite natural to associate with (3.1) the following Lagrangian functional i£ defined on Hj(Q) x L°°(Q) by

,}i) = \ [\Vv\2dx-(f,vy+\

[ n(\Vv\2 -

dx.

It follows from Cea [2] and G.L.T. [1], [3] that if j§? has a saddle point {u, X}eH10(Q) x L?(Q)(L?(Q) = {q e Lx(Q),q > Oa.e.}), then u is a solution of (3.1). Thus X appears as an infinite-dimensional multiplier (of F. John Kuhn-Tucker type) for (3.1). The existence of such a multiplier in L + has been proved by H. Brezis [2] in the physical case (i.e., / = constant), but in more general cases the existence of such a multiplier in L+(Q) is still an open problem. Following Cea and G.L.T., loc. cit., it is then natural to use a saddle point solver like the following algorithm of Uzawa type for solving (3.1): X° e L ? ( Q ) arbitrarily given (for example, A0 = 0); +l

then, by induction, assuming X" known, we obtain u" and X" $£(un, X") < se{v, X"), A"

+1

n

Vve H&Q), 2

= [X + p(|V«"| - 1)]

+

by

u" e Hj(Q), with P>0.

(3.63)

(3.64) (3.65)

Let us analyze (3.64) in detail; actually (3.64) is a linear Dirichlet problem, whose explicit form is given (in the divergence form) by - V - ( ( l + A")Vu")=/inQ, (3.66) M »| r

= 0.

Problem (3.66) has a unique solution in HJ(fi) whenever X" e L + (Q). Since we are not generally certain of the existence of a multiplier in L+(Q), the above algorithm is purely formal in general. 3.8.2. The discrete case In this section we shall follow G.L.T. [3, Chapter 3, Sec. 9.2]. Define Vh and Kh as in Section 3.5.1 of this chapter. Define Lh (approximation of LCO(Q)) and

3. A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem

55

A,, (approximation of L+) by

Teirh

where XT ls the characteristic function of T, and \ h = {pL e Lh, n > 0 a.e. in Q}. It clearly folio ws that for vh e Vh, Vvh e Lh x Lh, and forvhe Kh,l — \Vvh\2 e Ah. Define the Lagrangian £C on Vh x Ln as in Sec. 3.8.1; then we have: Proposition 3.5. The Lagrangian Jz? has a saddle point {uh, AJ in F , x A , where uh is the solution of (3.22), Xh(\Vuh\2

(3.67)

- I) = 0.

(3.68)

PROOF. Since Vh and Lh are finite dimensional, (3.67) and (3.68) will follow from Cea [2, Chapter 5] (cf. also Rockafellar [1, Chapter 28]) if we can prove that there exists an element of Vh in the neighborhood of which the constraints are strictly satisfied. Let us show that there exists a neighborhood jVh of zero in Vh such that Vc,€ Jfh, | Vvh | 2 — 1 < 0. In order to show this, observe that the functional given by vh -> |Vvh\2 — 1 is C°°, and at zero it is equal to — 1. Hence the assertion follows. •

To conclude Section 3, let us describe an algorithm of Uzawa type which is the discrete version of (3.63)-(3.65): 2° e Ah arbitrarily chosen (for instance, 2° — 0),

(3.69)

then, by induction, once AJ| is known, we obtain u\ and X1+1 by < <e(vh, AD,

VvheVh,

|2-l)]+

unheVh,

withp>0.

(3.70) (3.71)

We observe that if X\ is known, then u\ is the unique solution of the following approximate Dirichlet problem (given in variational form)

L

dx =
VvheVh,

unheVh.

(3.72)

From Cea [2, Chapter 5] and G.L.T. [3, Chapter 2], it follows that for p > 0 and sufficiently small, we have rim,,^^ u\ = uh, where uh is the solution of (3.22). Remark 3.12. The computations we have performed seem to prove that the optimal choice for p is almost independent of h for a given problem. Similarly, the number of iterations of Uzawa's algorithm for a given problem is almost independent of h.

56

II Application of the Finite Element Method

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem Most of the material in this section can be found in G.L.T. [1, Chapter 4], [3, Chapter 4].

4.1. The continuous problem: Existence and uniqueness results

As usual, let Q. be a bounded domain of 1R2 with a smooth boundary T. We define V = H\Q), (4.1) f a(u, v)= \ Vu-Vvdx L(i>) = ,

C +

uv dx,

feV*

(4.2) (4.3)

K = {ve H^Q), yv > 0 a.e. on T},

(4.4)

where yv denotes the trace of v on F. We then have the following: Theorem 4.1. The variational

inequality

a(u, v - u)> L(v -u),

V v e K,

ueK

(4.5)

has a unique solution. PROOF. Since the bilinear form a(-, •) is the usual scalar product in H1(il) and L is continuous, it follows from Theorem 3.1 of Chapter 1 that (4.5) has a unique solution provided we show that K is a closed convex nonempty subset of V. Since Oe K (actually, Hj(fi) c: K), K is nonempty. The convexity of K is obvious. If (!)„)„ a K and vn-*v in Hi(Q), then yvn -> yv, since y: H^fl) -> L 2 (F) is continuous. Since vn e K, yvn > 0 a.e. on T. Therefore yv > 0 a.e. on T. Hence t> e K which shows that K is closed. •

Remark 4.1. Since a(-, •) is symmetric, the solution M of (4.5) is characterized (see Chapter 1, Section 3.2) as the unique solution of the minimization problem J(u) < J(v), where J(v) = %a(v, v) - L{v).

VveK,

ueK,

(4.6) (4.7)

Remark 4.2. Actually (4.5) or (4.6) is a simplified version of a problem occurring in elasticity, called the Signorini problem for which we refer to Duvaut and Lions [1, Chapter 3] and to the references therein. We also refer to Duvaut and Lions, he. cit., Chapters 1 and 2 for other physical and mechanical interpretations of (4.5) and (4.6).

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

57

Remark 4.3. Assuming that Q is bounded (at least in one direction of IR2), we consider V= {veH

l

(Q), i; = 0

a.e. on T o },

a(u, v)= f VM •VJ; dx,

(4.8) (4.9)

L(v) = if, v} with / e V*,

(4.10)

K = {v e V,yv > g a.e. on

(4.11)

where F o and Fj are "good "subsets of F such that F x n F o = 0,T = T1 u F o (see Fig. 4.1). Assuming that the measure of F o is positive and that g is sufficiently smooth, it can be proved that the following variant of (4.5), a(u, v-u)>L(v-u),

Vuel,

ueK,

(4.12)

has a unique solution. In the proof of this result, one uses the fact that a(v, v) defines a norm on V which is equivalent to the norm induced by H 1(Q). EXERCISE

4.1. Prove that a(v, v) defines a norm equivalent to the norm induced

by H^Q).

4.2. Regularity of the solution Theorem 4.2. (H. Brezis [3]). Let Qbe a bounded domain of U2 with a smooth boundary F (or Q is a convex polygonal domain). If L(v) = ^nfv dx with fe L2(Q), then the solution u of (4.5) is in H2(Q).

Figure 4.1

58

II Application of the Finite Element Method

4.3. Interpretation of (4.5) as a free boundary problem Let us recall some definitions and results related to cones. Definition 4.1. Let X be a vector space, C <= X and x e C; then C is called a cone with vertex at x if for all y e C, t > 0 implies x + t(y — x) e C. Lemma 4.1. Let H be a real Hilbert space, let b{-, •) be a bilinear form on H x H, let Xbe a linear form on H, and let C be a convex cone contained in H with vertex at 0. Then every solution of b(u, v - u) > Mv - u),

VDEC,

ueC

(4.13)

is a solution of b(u, v) > X(v),

V v e C, ( 4 - 14 )

b(u, u) = MM), ueC, and conversely. EXERCISE

4.2. Prove Lemma 4.1.

Proposition 4.1. Assume that

L(c)= f fvdx + [gyvdT,

(4.15)

with f and g sufficiently smooth. Then the solution u of (4.5) is characterised by —Aw + u = f a.e. in Q, yu > 0, •— > g a.e. on T, on (8u \ yu\g] = 0 a.e. on i. \dn )

(4.16)

PROOF. First we will prove that (4.5) implies (4.16). Since K is a convex cone with vertex at 0, it follows from Lemma 4.1 that a(u,v)>L(v),

VveK,

(4.17)

a(u, u) = L(u).

(4.18)

Since 3>(Ci) <= K, from (4.17) we have f Vu • V(/> dx + f u(j> dx = f f4> dx,

V (j) e S>(Q).

(4.19)

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

59

From (4.19) it follows that - A u + u = f a.e. in

fi.

(4.20)

Let ve K. Multiplying (4.20) by v and using Green's formula, it follows that

a(u,v)= [ fvdx + fyu — dT,

V v e K.

(4.21)

J r on

JQ

From (4.17) and (4.21) we obtain

17--

g)yvdT>0,

V u e K.

(4.22)

J r \dn

Since the cone yK is dense in L+(F) = {«e L 2 (F), u > 0 a.e. on F}, from (4.22) it follows that — - g > 0 a.e. on F. 3n Taking v = u in (4.21) and using (4.18), we obtain

(d± -g)dT = 0.

(4.23)

(4.24)

Since yu > 0, and using (4.23), we obtain yu(du/dn — g) = 0 a.e. on F. This shows that (4.5) implies (4.16). (2) Let us show that (4.16) implies (4.5). Starting from (4.20) and using Green's formula, one can easily prove (4.17) and (4.18). These two relations, in turn, imply, from Lemma 4.1, that u is the solution of (4.5). •

Remark 4.4. Similar results may be proved for the variant (4.12) of (4.5) (see Remark 4.3). Remark 4.5. From the equivalent formulation (4.16) of (4.5), it appears that the solution u of (4.5) is the solution of a free boundary problem, namely: Find a sufficiently smooth function u and two subsets Fo and F + such that

r 0 u r + = r,

r 0 n r + = 0,

(4.25)

— AM + u = / in Q, yu = 0 on F o , — > g onT0, on

(4.26)

yu>0onT+,-~ = gonr+. on 4.4. Finite-element approximation of (4.5) In this section we consider the approximation of (4.5) by pieeewise linear and piecewise quadratic finite elements. We assume that Q is a bounded polygonal domain of U2, and we consider a triangulation 2Th of Q obeying (2.21)-(2.23)

60

II Application of the Finite Element Method

(see Sec. 2.5, Chapter II); we use the notation of Sees. 2.5.1 and 3.6 of this chapter. 4.4.1. Approximation of V and K The space V = H1(Q) may be approximated by the spaces V\, where V\ = {vh e C°(H), vh\TePk,VTefh},

k = 1, 2.

Define yh = {P e X, n T} = £„ - th,

[ yh if* = l j y f c u y i iffe = 2.

k 7/1

Then we approximate K by K\ = {«, e F j , vh(P) > 0, V P 6 y*}. We then have the obvious: Proposition 4.2. For k = 1, 2, t/ze X^ are closed convex nonempty subsets of V\ and K\ c X, V /J. 4.4.2. T/ie approximate problems For fe = 1, 2, the approximate problems are defined by a(ukh,vh-4)>L(vh-ukh),

VvheKkh,

U\BK\.

(P^)

Then one can easily prove: Proposition4.3. The problem (P*;,) (k = 1, 2) has a unique solution. Remark 4.6. Since a(-, •) is symmetric, (Pi A ) is equivalent (see Chapter I, Sec. 3.2) to the quadratic programming problem mm[ia(vh,vh)

-

L(vh)l

vheK«h

Remark 4.7. Using the formulae of Sec. 7, one may express (4.5) and the equivalent quadratic problem in a form more suitable for computation. 4.5. Convergence results 4.5.1. A density lemma To prove the convergence results of Sec. 4.5.2, we shall use the following: Lemma 4.2. Under the above assumptions on Q, we have K n C°°(Q) = K.

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

61

PROOF. Since r is Lipschitz continuous, we have (see Necas [1])

using the standard decomposition i> = v+ — v~, from Corollary 2.1 it follows that veK

if and only if v' e

tfj(fi).

(4.27)

Since ®(Q) = ffj(fi) in the //'(^-topology, from (4.27) it follows that we have only to prove K n C°°(fl) = K,

(4.28)

where X = { c e H^Q), v > 0 a.e. in Q}. Since F is Liptchitz continuous, Q has (see Lions [2] and Necas [1]) the so-called 1-extension property which implies V v e H^Q), 3 v e Hl(U2)

such that v \a = v.

(4.29)

Let ve K and let v e H1(U2) be an extension of v obeying (4.29). From v > 0 a.e. in Q and Corollary 2.1, it follows that \v\ is also an extension of v obeying (4.29). Therefore if v 6 K, it always has an extension v > 0 a.e. obeying (4.29). Consider such a non-negative extension i> and a mollifying sequence pn (as in Lemma 2.4 of this Chapter) Define vn by vn = v*Pn.

(4.30)

We have vn € ®(R2),

lim vn = v strongly in H\U2).

(4.31)

From pn > 0 and 5 > 0 a.e., we obtain, from (4.30), vn(x) > 0,

V x e IR2.

Define vn by vn = v.\a;

(4.32)

from (4.31) a n d (4.32), it follows that vn e C°°(n), lim vn = v strongly in Hl(il), rt-*

vn>0

a.e. in fi.

00

This proves the lemma.

•

4.5.2. Convergence theorem Theorem 4.3. Suppose that the angles of ^~h are uniformly bounded below by 60 > 0ash -> 0; then lim u\ = u strongly in H\Q,), where u, u\ are the solutions of (4.5) and (Pklh), respectively, for k = 1,2.

(4.33)

62

II Application of the Finite Element Method Figure 4.2

PROOF. TO prove (4.33), we use Theorem 5.2 of Chapter I. To do this we only have to verify that the following two properties hold : (i) If (vh)h, vh e K\, converges weakly to v, then v e K. (ii) There exist / cz K and rkh: x -> K\ such that x = K and lim^o T\V = v strongly

in V,V v exVerification of (i). If k = 1, then (i) is trivially satisfied, since K\ c K. If k = 2, using the notation of Fig. 4.2, we consider <\> e C°(T),

0, and we define ^ = I«Mitl,2toH/2,

(4-34)

where Xi+i/i denotes the characteristic function of the open segment ]M f , Mi+ j[. Then 4>h > 0 a.e. on F,

lim || ^ - 01|L . ( r ) = 0.

(4.35)

fc->0

Let us consider a sequence (vh)h,vheK^,'ih,

such that

lim vh = v weakly in V. (4.36) 2 From (4.36) (see Necas [1]) it follows that limft_»0 yvh = yv strongly in L (F). This in turn implies that lim f y i v k d r =

\yvdT.

(4.37)

From Simpson's rule it follows that \yvh<j>hdT = - £ lAf.Mj+'.WAf

J f o M ) + 4vh(Mi+ll2) Vi)»eKj ! ,

+ vh(Mi+J]

V^eC°(F),

From (4.37) and (4.38) we obtain (* yv dT > 0,

V (j) e C°(F),

•Jr

which implies yy > 0 a.e. on F. This proves (i).

4> > 0,

> 0,

(/> > 0.

(4.38)

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

63

Verification of (ii). From Lemma 4.2, it is natural to take x = K n C°°(Q). Define r\: H\a) n C°(fi) -^ Fj by r\veV\,

y v e H\a)

t*v{P) = i>(P),

n

CiH),

V P e S j , k = 1, 2.

(4.39)

On one hand, under the assumptions made on 9~h, we have (see Strang and Fix [1]) llrju - »||F < Chk\\v\\Hk,Hii),

V i; £ C M (a),

k = 1, 2,

(4.40)

with C independent of /i and y. This implies Z)

fc = 1,2.

(4.41)

On the other hand, it is obvious that r\v e Kj, V i; e K n C0(Q), so that rju e Kj, V « e z, k = 1, 2. In conclusion, with the above % and rkh, (ii) is satisfied. •

Remark 4.8. For error estimates in the approximation of (4.5) by piecewise linear finite elements, it has been shown by Brezzi, Hager, and Raviart [1] that we have

assuming reasonable smoothness for u on Q.

4.6. Iterative methods for solving the discrete problem We shall briefly describe two types of methods which seem to be appropriate for solving the approximate problems of Sec. 4.4. 4.6.1. Solution by an over-relaxation method The approximate problems (P%) are, for k = 1,2, equivalent to the quadratic programming problems described in Remark 4.6. By virtue of the properties of K\ (see Sec. 4.4.1), for the solution of (P%), we can use the over-relaxation method with projection, which has already been used in Sec. 2.8 to solve the approximate obstacle problem and which is described in Chapter V, Sec. 5. From the properties of our problem, the method will converge, provided 0 < co < 2. 4.6.2. Solution by a duality method We first consider the continuous case. Let us define a Lagrangian J? by

-L(v)-

[qyvdT, •IT

(4.42)

64

II Application of the Finite Element Method

and let A be the positive cone of L 2 (F), i.e., A = {q € L2(T), q>0 a.e. on F}. Then we have:

Theorem 4.4. Let L(v) = | n fv dx + j r gyv dT with f and g sufficiently smooth. Suppose that the solution u of (4.5) and (4.6) belongs to H 2 (O); then {u, du/dn — g} is the unique saddle point of£? over Hl(Q) x A. PROOF. We divide the proof into two parts. In the first part we will show that {u, du/dn — g] is a saddle point of <£ over Hi(Q) x A, and in the second part we will prove uniqueness. (1) Let p = du/dn — g. From the definition of a saddle point, we have to prove that &(u, q) < y{u, p) < y(v, p), 2

V{v,q}eV

l2

x A, {«, p} e V x A .

(4.43)

2

Since u e H (Q.), we have du/dn e H ' (T) c= L (T) (see Lions and Magenes [1]). Then if g is smooth enough, we have p = du/dn — g e L2(T). From Proposition 4.1 we have du p=

#>0onr, 8n

(4.44) pyu = 0 a.e. on F. This implies that we have {u,p} 6 H1(Q) x A and j r pyu dT = 0. Since yu > 0 on F, we have {qyudT>0,

V q e A.

(4.45)

From (4.44) and (4.45) it follows that &{u, q) = ia(u, u) - L{u) - \ qyu dT < \a(u, u) - L(u) = |a(u, u) - L{u) - [ pyu dT = JS?(«, p),

Vq 6 A

•Jr

which proves the first inequality of (4.43). To prove the second inequality of (4.43), we observe that the solution u* of the minimization problem Se{u", p) < £C(v, p),

V c e H'(fi),

u* e H^O)

(4.46)

is unique and is actually the solution of the linear variational equation a(u*, v) = L(v) + I pyv dT,

V v e H^Q),

u* e H^Q).

(4.47)

Jr Since L(v) = \afv dx + J r gyv dT, u* is actually the solution of the Neumann problem - A M * + u* = / i n f i , du* du — =p + , = - o n F .

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

65

Since from Proposition 4.1 we obviously have —AM + u — fin Q, du

du*

y- = -x- on T, on on it follows from the uniqueness property of the Neumann problem (4.48) that u — u*. Using (4.46) and u = u*, we obtain the second inequality in (4.43). This proves that {u, p} is a saddle point of i f over /^(Q) x A. (2) Uniqueness. Let {«*, p*} be a saddle point of if over H^Q) x A. We will show that u* = u,p* = p. From (4.42) and (4.43) it follows that Up-

q)yu dT < 0,

V q e A.

(4.49)

V <j 6 A.

(4.50)

Similarly, we have f (p* - <j)yu* dF < 0,

Taking q = p* (respectively, q = p) in (4.49) (respectively, (4.50)), we obtain (p* - p)y(u* - u) dT < 0. r From the second inequality of (4.43) it follows that u is the solution of a(u, v) = L(v) + (* pyv dT,

(4.51)

V c e H\Q),

u e H'iSi),

(4.52)

V v e H^Q),

u* e H^Q).

(4.53)

and similarly, a(u*,v) = L(v)+

\p*yvdT, •JT

Taking v = u* — u (respectively, v = u — «*) in (4.52) (respectively, (4.53)), we obtain a(u* -u,u*

- M) = [ (p* - p)7(u* - M) dr. •'r

(4.54)

Using the F-ellipticity of a(-, •), from (4.51)-(4.54) it then follows that u* = u and

I (p* - p)y^rfr = o,

VceHHS),

•'r

which implies that p* = p. Hence {u, p] is the unique saddle point of i f over H1(Q) x A.

•

66

II Application of the Finite Element Method

From Theorem 4.4 it follows that we can apply Uzawa's algorithm to solve (4.5) (see Cea [2, Chapter 5] and G.L.T. [1, Chapter 2], [2, Chapter 4, Sec. 3.6], [3]). In the present case this algorithm is written as follows: p° e A is arbitrarily chosen (for instance, p° = 0);

(4.55)

then p" being known, we compute, for n > 0, {un, pn+1} by &{un, pn) < <£{y>, p"),

V c e H^Q),

u" e H\Q),

(4.56) (4.57)

2

2

where PA is the projection operator from L (F) to A in the L (F) norm, and p > 0. From (4.56) it follows that u" is in fact the solution of the Neumann problem -AM" + u" = / i n Q , dif

l)n~

(4.58) r

The projection PA is given by PM

= q+ ,

V<jeL 2 (F).

(4.59)

Since y: H1(Q) -» L 2 (F) is a continuous linear mapping, we have \\yv\\LHr) < \M\ • \\v\\HHn),

V v e H\O).

(4.60)

From Cea and G.L.T., loc. cit., it then follows that lim M" = u strongly in H\Q),

(4.61)

where u is the solution of the problem (4.5) provided that 0 < p < 2/||y||2. Let us give a direct proof for this convergence result. This proof will use the characterization (4.5) given in Proposition 4.1 (even if a(-, •) is not symmetrical, the same result follows). It will be convenient to take (4.56), (4.58) in the following equivalent form: a(u", v) = L(v) + f p"yv dY, Jr

V U G H^Q),

U" e H\C1).

(4.62)

Let u be the solution of (4.5) and p = du/dn — g. From Proposition 4.1 it follows that a(u, v) = L(v) + f pyv dY, Jr

I

(q - p)yu dY > 0,

r

V c £ Hl{Q),

V«eA,

u e H\Si),

p e A.

(4.63)

(4.64)

4. A Third Example of EVI of the First Kind: A Simplified Signorini Problem

67

Relation (4.64) can also be written as (i — P)(P — Pyu — p) dT < 0,

V q e A, p > 0 and p e A,

which is classically equivalent to p=PA(p-pyu).

(4.65)

Let us consider u" = u" - u,

p" = p" - p.

Since J°A is a contraction, from (4.57) and (4.65) we have \\pn+1\\mr)<\\pn-pyan\\^r)-

(4.66)

From (4.66) it follows that n n

p dT - p2||yu"||£2(r).

(4.67)

Taking v = u" in (4.62) and (4.63), we obtain a{u",u")=

\pnyundY. Jr

(4.68)

From (4.67) and (4.68) it then follows that IIPi£i,n - IIP"+1ll£>(r) > P(2 - PIMI2)II«1^(Q)-

(4.69)

2

If 0 CO

Since u" = u" — u, we have proved the convergence. Similarly we can solve the approximate problems (P\h), k = 1,2, using the discrete version of algorithm (4.55)-(4.57). We shall limit ourselves to k = 1, since the extension to k = 2 is trivial. Here we use the notations of Sec. 4.1. Assume that yh = EA — t,h has been ordered. Let yh =

68

II Application of the Finite Element Method

We approximate A and 5£ by Aj|= {qh\qh= { 0} and ^l(vh, qh) = Hvh, vh) - L(vh) + qi+1vh(Mi+1)l

(4.70)

We can prove that JSfjJ has a unique saddle point {uh, ph}, where ph is a F. John-Kuhn-Tucker vector for (T[h) over V\ x A/J and uh is precisely the solution of (P\h). The discrete analogue of (4.55)-(4.57) is then pJeAj, n

^l(u h,

n

P h) < ^l(vh,

p"h),

(4.71) VvheVl,

Pl+Y = ip"-pul(MdT,

u"heVl

Vi, p > 0 .

(4.72) (4.73)

One can prove that if 0 < p < /?, with /? small enough, then limn^ + 00 «^ = wh, where uh is the solution of (P\h). In G.L.T. [3, Chapter 4] one may find numerical applications of the above iterative methods for piecewise linear and piecewise quadratic approximations of (4.5). EXERCISE

4.3. Extend the above considerations to (Pfh).

5. An Example of EVI of the Second Kind: A Simplified Friction Problem 5.1. The continuous problem. Existence and uniqueness results Let Q be a bounded domain of R2 with a smooth boundary F = <9Q. Using the same notations as in Sec. 4, we define V = H\Q), a(u, v) =

VM

Jn

• Vv dx + uv dx, Jn

L(c) = , j(v) = g We have then the following:

\yv\dT,

feV*, where g > 0.

(5.1) (5.2)

(5.3) (5.4)

5. An Example of EVI of the Second Kind: A Simplified Friction Problem

69

Theorem 5.1. The variational inequality a(u, v-u)+

j(v) - j(u) >L(v-u),

VveV,

ue V,

(5.5)

has a unique solution. PROOF. In order to apply Theorem 4.1 of Chapter I, it is enough to verify thatj(-) is convex, proper, and l.s.c. Actually j(-) is a seminorm on V. Therefore, using the Schwarz inequality in L2(T) and the fact the y e J5?(ff'(fi), L2(F)), we have \j(u) - j(v)\ < \j(u -v)\<

3(meas(r))1/2Hy(t> - «)|| L2(r) < C\\u - v\\v

(5.6)

for some constant C. Hence j( •) is Lipschitz continuous on V, so that j( •) is l.s.c. ;j{ •) is obviously convex and proper. Hence the theorem is proved. •

Remark 5.1. If g = 0, it is easy to prove that (5.5) reduces to the variational equation a(u, v) = L(v),

VceF,

u e V.

This is related to the variational formulation of the Neumann problem. Remark 5.2. Since a( •, •) is symmetrical, the solution u of (5.5) is characterized, using Lemma 4.1 of Chapter 1, as the unique solution of the minimization problem J(u) < J(v),

V v e V, ueV,

(5.7)

where J(v) = \a(v, v) + j(v) - L(v). Remark 5.3. Problem (5.5) (and (5.7)) is the simplified version of a friction problem occurring in elasticity. For this type of problem we refer to Duvaut and Lions [1] and the bibliography therein. EXERCISE

5.1. Let us denote the solution of (5.5) by ug. Then prove that lim ug = u strongly in H J (Q), ff~* + 00

where u is the unique solution of a(u, v) = Hy),

V D G ffJ(Q), u e Hj(Q).

5.2. Regularity of the solution Theorem 5.2 (H. Brezis [3]). If (I is a bounded domain with a smooth boundary and if L(v) = \n fv dx with f e L2(Q), then the solution u of (5.5) is in H2(Q).

70

II Application of the Finite Element Method

5.3. Existence of a multiplier Let us define A by

A = {^ e L\T), \ix{x)\ < 1 a.e. on T). Then we have: Theorem 5.3. The solution u of (5.5) is characterized by the existence of X such that a(u, v) + g f Xyv dT = L(v), Jr X e A,

V v e V,

ueV,

Ayw = | yu | a.e. on F.

(5.8) (5.9)

PROOF. We will prove first that (5.5) implies (5.8) and (5.9). Taking v = 0 and v = 2M in (5.5), we have a(u, u) + j(u) = L(u).

(5.10)

It follows then from (5.5), (5.10) that L(u) - a(u, v) < j(v),

VCEF,

which implies \L(v)

- a(u, v)\ < j(v) = g \ \yv\ dT,

V c e V.

(5.11)

We have H 1 ^ ) = #J(Q) © [^o(fi)]1, where [Hj(Q)]1 is the orthogonal complement of Hl(Q) in H^Q). Since y: [//J(Q)] 1 -» H 1 / 2 (r) is an isomorphism, it follows from (5.11) that L(v) - a(u, v) = l(yv),

MveV,

(5.12)

1I2

where /(•) is a continuous linear functional on H (T). It then follows from (5.11), (5.12) that IWI<0MlL.(n,

V^6H"!(I>

(5.13)

Since H 1 / 2 (r) ez Ll(T), it follows from (5.13) that we can apply to /(•) the HahnBanach Theorem (see, for instance, Yosida [1]) to obtain the existence of -16 L co (r), I A(x) | < 1 a.e. on T such that (5.14) Therefore it follows from (5.12) and (5.14) that a(u, v) + g f Xyv dT = L(v), which proves (5.8).

V v e V,

5. An Example of EVI of the Second Kind: A Simplified Friction Problem

71

Taking v = u in (5.8), we obtain a(u, u) + g \ Xyu dT = L{u). •Jr

Using (5.10) and the above equations, we obtain U\yu\ - Xyu) dT = 0. •>r

(5.15)

Since | X \ < 1 a.e., we have \yu\-

Xyu> 0a.e..

(5.16)

From (5.15) and (5.16) it follows that | yu | = Xyu a.e.. This completes the proof of (5.8) and (5.9). Assuming (5.8) and (5.9), we will show that (5.5) holds. Let {u, X} be a solution of (5.8), (5.9). From (5.8) it follows that a(u, v — u) + g

Xy(v — u) dT = L(v — u),

V v e V,

which can also be written as a(u, v - u ) + g \ XyvdY Jr From (5.9) and (5.17) we obtain a(u, v-u)

- g [Xyu dT = L(v - u ) , •'r

VceK

+ g [xyvdT - g f \yu\ dT = L(v - u), Jr Jr

VoeF.

(5.17)

(5.18)

But since Xyv < | yv \ a.e. in T, it follows from (5.18) that a(u, v - u) + j(v) - j(u) > L(v - u),

VteF.

This proves the characterization.

D

Remark 5.4. Assuming that

L(v)= f fovdx+ Ja

[j\yvdT, Jr

with/o, / i sufficiently smooth, we can express (5.8) by — AM + H = / O in Q

^ + gX = j \ a.e. on r . on From (5.19) it follows that X is unique. EXERCISE 5.2. Prove that X is unique.

(5.19)

72

II Application of the Finite Element Method

5.4. Finite element approximation of (5.5) Let Q be a bounded polygonal domain of R2. The notation used here is largely the same as in Sec. 4.4 of this chapter. 5.4.1. Approximation of V We use the piecewise linear and piecewise quadratic approximations of V = H^Q) described in Section 4.4.1 of this chapter. 5.4.2. Approximation ofj( •) We use the notation of Figure 4.2. Then we approximate _/(•) by

H(vh) = § £ IMji i + ! |(|yvh{Md| + |yvh(Mi+ 01), = lY. \MiMi+l\(\yvh(Mi)\ + |y»»(Afi+1)|),

V vh e Vlh,

(5.20)

+ 4\yvh(Mi+1/2)\

V» t eK ( 2 .

In (5.20) and (5.21) we have Mt e yh and Mi+1/2

(5.21)

e yj,.

Remark 5.5. Clearly (5.20) and (5.21) are obtained from j(-) by using trapezoidal and Simpson's numerical integration formulae, respectively. 5.4.3. The approximate problem For k = 1, 2, problem (5.5) is approximated by fl(«J, t;* - «J)+;£(»,,)-jJ(«!D>M» f c -"ID.

Vi) t eV{,

«j€Vt

(P*2fc)

Then: Proposition 5.1. Problem (P^h) has a unique solution. Remark 5.6. Since a(-, •) is symmetrical, (Pk2h) is equivalent to the nonlinear programming problem Min[_Hv h , vh) + ; > * ) - L(vh)l

(5.22)

Remark 5.7. Using (5.20), (5.21), and (7.1)-(7.4) of Sec. 7 of this chapter, we may express (P^) and (5.22) in a form more suitable for computations.

5. An Example of EVI of the Second Kind: A Simplified Friction Problem

73

5.5. Convergence results Theorem 5.4. Suppose that the angles of 2Th are uniformly bounded below by 60 > 0 as h -> 0; then lim u\ — u strongly in H1(Q),

(5.23)

where u and u\ are, respectively, the solutions of (5.5) and (P2/,) far k = 1, 2. PROOF. TO prove (5.23) it suffices to verify the following properties (see Theorem 6.3 of Chapter I). (i) There exists U c V, U = V and r\: U ->• V\, such that lim r\v = v strongly in V, (ii) Vvk-*v

Vcel/.

weakly in V, then lim Mfh(vh) > j(v).

(iii) \imjkh(rkhv)=j{v),

V veU.

Verification of(i). Since F is Lipschitz continuous, we have (see Necas [1]) C 0 0 ® = H\n).

(5.24)

m

Therefore it is natural to take U = C (U). Define r\ by (4.39) of Theorem 4.3, Chapter II; under the above assumptions on ,fh, it follows from Strang and Fix [1] that \\A.v-v\\v
VoeK,

(5.25)

where C is a constant independent of h and v. This implies (i). Verification of (ii) (1) Case k = 1. We again use the notation of Fig. 4.2. Since the trace of vh restricted to [Mj, Mi+l~\ is affine, it follows that yvh(M) =

,

(\MMi+1\yvh(Mi)

+

\MMi\yvh{Mi+l)),

VvheVl,

VMe[Mi,M i + 1 ].

(5.26)

Since

the convexity of £ -> \£\ implies

I yvh(M) I < - = 4 |MM |M f M| \yvh(Mi+l)\),

VvheVl

¥Me[M,Mitl],

(5.27)

II Application of the Finite Element Method

74 Integrating (5.27) on MtMi+,, (

we obtain i+1

\yvh\dT<—'

which implies that, V vh € F^, we have

f

v- f

Thus we have proved j(vh)
VvheVl

(5.28) 2

Let uft -> f weakly in K Then limA^0 yuh = 71; strongly in L {Y), which implies \imj{Vh)=j{v).

(5.29)

fi->0

It then follows from (5.28) and (5.29) that lim infft^0 jl(vh) > j(v), which proves (ii) iffc = 1. (2)Casek = 2. Let us define M i + 1 / 6 , M 1 + 5 / 6 by (see Fig. 5.1)

Then we define qh: C°(r) ^ L°°(r) by (5.30)

where Z , (respectively, Xi+ll2) is the characteristic function of M;_ 1 / 6 MjM i + 1 / 6 (respectively, M i + 1 / 6 A? j + 5 / 6 ). We then have the following obvious properties: lim qh(ji) = n strongly in L co (r),

V n e C0(T),

= g\\qkyvk\\LHr),

.II w , <

w < C 2 ||7^|| L2(r) ,

V «„ e Kft2, V !>* e F,2,

1/2

Figure 5.1

(5.31) (5.32) (5.33)

5. An Example of EVI of the Second Kind: A Simplified Friction Problem

75

where in (5.33), Cj and C 2 are positive constants independent of vh, h, and T (values for C t and C2 may be found in G.L.T. [3, Chapter 4]). Now, taking n e C°(F), we define sh(n) by sh(n) e L°°(F),

s,(/i)|]Mi,M( + lI = MM ; + 1 / 2 ).

(5.34)

Then lim sh(n) = n strongly in L°°(r),

V / J E C°(r),

(5.35)

/i->0

and from Simpson's integration formula, we have \ sh{ii)qhyvhdY = Lh(n)yvhdr,

V /i e C°(r),

Vt.eF,2.

(5.36)

Let t)ft -> u weakly in V, vh e V\, V h; then lim yvh = yu strongly in L2(T).

(5.37)

On the one hand, it follows from (5.33) that > < C,

(5.38)

where C is independent of h. On the other hand, (5.31), (5.35)-(5.38) imply that lim f sh(fi)qhyvh dT = \ nyv dT, h^o •'r

V fi e C°(T).

(5.39)

Jr

In turn, (5.38) and (5.39) imply that lim qhyvh = v weakly in L2(T).

(5.40)

Since the functional n -^ !|/u||Li(n) is convex and continuous on 1/(1"), it follows from (5.40) that lim inf \\qhyvh\\LHr) > ||yt!||L.(n.

(5.41)

/i^O

Combining (5.41) with (5.32), we obtain lim inffc^0 j2(vh) > j(v) which proves (ii) when k = 2. Verification of (iii). Let veU = C^fi). From (5.25) and from the uniform continuity of yv on F, it follows almost immediately that \imfh(rkhvh)=j(v),

/c = l,2.

Since conditions (i), (ii), and (iii) are satisfied, the strong convergence of uh to u follows from the Theorem 6.2 of Chapter I. D Remark 5.8. It is proved in G.L.T. [3, Chapter 4] that vh -* v weakly in V, vh e V\, implies l i m ^ 0 j£Oj,) = j(v), fe = 1, 2.

76

II Application of the Finite Element Method

Since the proof of this result is rather technical we have used a simpler approach in this book, from which it follows that lim Mfh(vh) > j(v),

k = 1, 2.

As we have seen before, this result was sufficient for proving Theorem 5.4. 5.6. Iterative methods for solving (P*fc) In this section we briefly describe some iterative methods which may be useful for solving the approximate problems (P|fc). 5.6.1. Solution of(Pk2h) by relaxation methods It follows from (5.20)-(5.22) (see Remark 5.6) that (Pk2h), k= 1,2, are particular cases of Min f(v), (5.42) where, with v = {vu ..., vN), N

f(v) = \{Av, v) - ( b , v ) + X on I v, I •

(5.43)

In (5.43), (•, •) denotes the usual inner product of UN, A is a N x JV symmetrical positive-definite matrix, and at > 0, V i = 1 , . . . , N. It then follows from Chapter V, Sec. 5, Cea [2, Chapter 4], Cea and Glowinski [1], and G.L.T. [3, Chapter 2] that we can use a relaxation method for solving (5.42). Actually, from the computations we have done, it appears that the introduction of an over-relaxation parameter at, w > 1, will increase the speed of convergence. Finally the algorithm we used is the following: u° arbitrarily given in UN;

(5-44)

then for i = 1, 2 , . . . , N, f(u\+ \...,

utl,

M?+ \ «?+1, • • •) < / «

\ • • •, utl,

vh u"i+ u ...),

VVi e U, (5.45),

If co = 1, (5.44)-(5.46) reduces to a relaxation method. Numerical solutions of (5.5) using (5.44)-(5.46) are given in G.L.T. [2, Chapter 4], [3, Chapter 4]. Remark 5.9. If a; > 0, u"+1 is the solution of a single variable nondifferentiable minimization problem which can be exactly solved by simple calculations. EXERCISE

5.3. Express u"+1 as a function of A, b, u", un+1.

5. An Example of EVI of the Second Kind: A Simplified Friction Problem

77

5.6.2. Solution o/XP^) by a duality method We first examine the continuous case. Define a Lagrangian £C:H1(Q) x L2(T) - D3 by

Se(v, p) = fr(v, v) - L(v) + g [nyv dY.

(5.47)

Then, using the notation of Sec. 5.3, the following theorem follows from Theorem 5.3. Theorem 5.5. Let {u, X) be a solution of (5.8), (5.9); then {u, X} is the unique saddle point of ^ over H1(Q) x A. EXERCISE

5.4. Prove Theorem 5.5.

From Theorem 5.5 it follows that to solve (5.5) we can use the following Uzawa algorithm: X° e A arbitrarily chosen (for instance, X° = 0); +1

then by induction, knowing X", we compute u" and X" J?(w", X") < £?{v, X"), n+1

X

V » 6 H\O),

= PA(X" + pgyu"),

(5.48)

by U" e V,

p > 0.

(5.49) (5.50)

The minimization problem (5.49) is actually equivalent to the Neumann variational problem a(u", v) = L(v) - g \ X"yv dY,

V u e H\n\

u" e H\O).

(5.51)

In (5.50), P A is the projection operator from L2(Y) to A in the L 2 -norm; then PA(A<) = Sup(-l,Inf(l,AO),

V^eL2(r).

(5.52)

Using Cea [2] and G.L.T. [1, Chapter 2], [3, Chapter 2], it follows that for 0 < p < 2/g2\\y\\2 we have lim u" = u strongly in H1(Q);

u is solution of (5.5), (5.7).

As in Sec. 4.6.2, a direct proof of the convergence of (5.48)-(5.50) can be given; however, it will use the results of Theorem 5.3. 5.5. Using Theorem 5.3, give a direct proof of the convergence of (5.48)-(5.5O).

EXERCISE

78

II Application of the Finite Element Method

The adaptation of (5.48)-(5.5O) to the discrete problems (P|A), k = 1, 2, is straightforward (see G.L.T. [3, Chapter 4]), since it is a simple variant of the discrete algorithm described in Sec. 4.6.2. EXERCISE

5.6. Study the discrete analogues of (5.48)-(5.5O) related to (P^),

k = 1, 2.

6. A Second Example of EVI of the Second Kind: The Flow of a Viscous Plastic Fluid in a Pipe Most of the material in this section can be found in G.L.T. [2, Chapter 5], [3, Chapter 5] and in Glowinski [1], [3].

6.1. The continuous problem. Existence and uniqueness results

Let Q be a bounded domain of U2 with a smooth boundary T. We define V = Hj(fi), a(u, v) =

VM • Vv dx,

L(v) = a\v}, j(v)=

feV*,

\\Vv\dx. Jn

Let n, g be two positive parameters; then: Theorem 6 . 1 . The variational

inequality

Ha(u, v - u) + gj(v) - gj(u) > L(v has a unique

u),

VueF,

ueV

(6.1)

solution.

PROOF. I n o r d e r t o a p p l y T h e o r e m 4.1 o f C h a p t e r I, w e o n l y h a v e t o verify t h a t }{•) is c o n v e x , p r o p e r , a n d l.s.c.. It is o b v i o u s t h a t j(•) is c o n v e x a n d p r o p e r . L e t u,veV; then \j(v) - j(u)\ < y m e a s ^ H u - v\\v; hence XO is l.s.c. This proves the theorem. EXERCISE 6.1. Prove that j(-) is a n o r m on V.

(6.2)

•

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

79

Remark 6.1. If we take g = 0 in (6.1), we recover the variational formulation of the Dirichlet problem — fiAu = f in Q, u = 0 on F. Remark 6.2. Since a{-, •) is symmetrical, the solution u of (6.1) is characterized (using Lemma 4.1 of Chapter I) as the unique solution of the minimization problem J(u) < J(v),

VceF,

ueV,

(6.3)

where J(v) = (ji/2)a(v, v) + gj(v) — L(v).

6.2. Physical motivation If L(v) = C\nv dx (for instance, C > 0), as proved in Duvaut and Lions [1, Chapter 6], (6.1) models the laminar stationary flow of a Bingham fluid in a cylindrical pipe of cross section Q, u(x) being the velocity at x e Q (we refer to Prager [1], Germain [1], and Duvaut and Lions [1, Chapter 6] for the definition of Bingham fluids). The constant C is the linear decay of pressure and H, g are, respectively, the viscosity and plasticity yield of the fluid. The above medium behaves like a viscous fluid (of viscosity /J.) in Q + = {xefi, | and like a rigid medium in fi° = {x e Q, VM(X) = 0}. We refer to Mossolov and Miasnikov [1], [2], [3] for a detailed study of the properties of Q + and Q°. We observe that (6.1) also appears as a free boundary problem.

6.3. Regularity properties Theorem 6.2 (H. Brezis [4]). / / L(v) = \Qfv dx,feL2(Q), of (6.1) satisfies

then the solution u

u e V n H 2 (Q), and ifQ. is convex, we have

IMI

<

(6-4)

80

II Application of the Finite Element Method

6.4. Further properties

Let us denote by a the following quantity: a -

inf

J

^

(6.5)

v

Then a > 0. From this we derive the following Proposition 6.1. Let u be the solution of (6.1) and fe L°°(Q); then u = 0 if

tl/IL-w) ^ 0«-

(6-6)

PROOF. By taking v = 0, v = 2« in (6.1), we obtain /«J(M, U) + gj(u) =

fu dx.

(6.7)

•>n It then follows from (6.5) and from \nfu dx < ||/||L=o(n)l|M|!L!(fi)tnat Ha(u, u) + (got - ||/||f.=o(n))||M||t,(n) < 0.

(6.8)

If / obeys (6.6), it then follows from (6.8) that u = 0.

•

We also have:

Proposition 6.2. Let u be the solution of (6.1) and f e L"(Q), p > 1. Then if f > 0, we have u > 0. EXERCISE

6.2. Prove Proposition 6.2.

Proposition 6.3. Let u be the solution of (6.2) and f = c, a constant. Then u = 0 if and only ifc < go. and u / 0 if c > ga. EXERCISE

6.3. Prove Proposition 6.3.

Proposition 6.4. Let u be the solution of (6.1) and feL2(Q);

then u = 0 if

where -

inf

PROOF. From a result of L. Nirenberg and M. Strauss (cf. Strauss [1]), it follows that j8>0. By taking v = 0 and v = u in (6.1), we obtain Ha(u,u) + gj(u)=

f fu dx. Jo

(6.10)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe an

Using \afu dx < \\f\\L2(n)\\u\\L2m

81

d /? > 0, we obtain

im(u, u) + (gp - ||/|| L 2 (n) )||«|| L2(Q) < 0.

(6.11)

Hence, if/satisfies (6.9), from (6.11) we have a(u, u) = 0. This implies u = 0, hence, the proposition is proved. •

6.5. Exact solutions 6.5.1. Example 1 We take Q = ]0, 1[ a n d / = c, c a positive constant. In this case the solution of H Jo

u(v' - u')dx + g \ \v'\dx - g \ \u'\ dx > c \ (v - u)dx, Jo Jo Jo VveH0(Q),

ueH0(Q), (6.12)

(where v' = dv/dx) is given by u = 0ifg>C-.

(6.13)

If g < c/2,

= —

-I

2\i \2

if

cj

< x <- + -, 2

c

2

(6.14)

c

u(x) = ^ - x ( l - x ) - - ( 1 - x) if- + - < JC < 1. We observe that if g < c/2, u e #J(Q) n PF2> 0O(Q), but u ^ H 3 (Q). 6.5.2. Example 2 Let Q = {x|xf + x\ < R2}, j ' — C, C a positive constant. Then the solution of (6.1) is given by w= 0 ifg(>—,

(6.15)

If g < CR/2, then ifR'
«W=|^r^ llT(* + *')-2ff|

ifO
82

II Application of the Finite Element Method

where

We also observe that tig < CR/2, then u e H^Q) n W2' °°(Q), but u $ H3(Q) (we actually have u e HS(Q), s < §). EXERCISE 6.4. Verify that (6.13), (6.14) and (6.15), (6.16) are solutions of (6.12)

and (6.1), respectively.

6.6. Existence of multipliers Let us define A by A = {q\q e L2(Q) x L2(Q), \q{x)\ < 1 a.e.} with \q{x)\ = yfq\{x) + q22(x); then we have: Theorem 6.3. The solution u of (6.1) is characterized by the existence of p such that Lia(u,v) + g f p • VP dx = , p-V« = |Vw|a.e.,

V»eK,

u 6 V,

p e A.

(6.17) (6.18)

PROOF. There are classical proofs of the above theorem using Min-Max or HahnBanach Theorems (see, for instance, Cea [2, Chapter 5], G.L.T. [1, Chapter 1], and Ekeland and Temam [1]). In the sequel following G.L.T. [3, Chapter 5] we shall give an "almost constructive" proof, making use of a regularization technique. (1) We first prove that (6.17), (6.18) imply (6.1). It follows from (6.17) that Ha(u, v — u) + g

p • V(v — u) dx = na(u, v — u) + g = ,

p - V v dx — g

Vt>eK

p • V « dx (6.19)

It follows from (6.18) that [p-Vudx=

f \Vu\dx,

(6.20)

and from the definition of A that \ p-Vvdx< •In

( \p\\Wv\dx< Jo

\ \Vv\dx, Jfi

VueF.

(6.21)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe 83 Then, from (6.17), (6.19)-(6.21), we obtain

/M(U, v — u) + gj(v) — gj(u) >
VneF, u e V.

Thus (6.17) and (6.18) implies (6.1). (2) Necessity of (6.17) and (6.18). Taking u = 0 and v = 2u in (6.1), we obtain Ha(u, u) + gj(u) = .

(6.22) 2

2

Let e > 0. Regularize j(-) by jE(-) defined by ;,.(«) = Jn^/e + |V«| rfx.Since j £ (-) is convex and continuous on V, the regularized problem lia(uc,v-ut)

+ gje{v)-gjr(ur)>L{v-ur),

V v e V, u£eV,

(6.23)

has a unique solution. Let us show that limE^0 uc = u strongly in V. From (6.1) and (6.23) it follows that lia(uc, u - ur) + gjc{u) - gje(ue) > L(u - uE), Ha(u, uE-

u) + gj(ut) - gj(u) > L(uc - u).

Adding these inequalities we obtain Ha{uc -u,u,-

u) + g(jr(us) - j(uj) < g(jF{u) - j(u)).

(6.24)

From 0 <- Jt2

+ F2 - \ t \ =

< £, ^

VfeIR,

'

it follows that 0 < j£v) - j(v) < e meas(fi), V' v e V, which using (6.24), implies that lia(iiz — u, ut — u) < ge meas(fi), so that / (q V / 2 II«r - u\\v < v m e a s ^ ) - e .

(6.25)

V i From (6.25) we obtain lim us = u strongly in V.

(6.26)

Since j e ( ) is differentiable on V, problem (6.23) is equivalent (see, e.g., Cea [1]) to the following nonlinear variational equation: Ha(uc, v) + gij'lu), v} = L(v),

VceF,

u,eV,

(6.27)

with
C Jfl^

Vw-Vv VwVv 2

- dx,

TvV| |V|2

V v, w e V.

(6.28)

If we define pt by p, = _

^

£

_

(6.29)

then PceA.

(6.30)

84

II Application of the Finite Element Method From (6.27)-(6.30) we obtain

lia(ue,v) + g | ps • Vv dx = L(v),

VveF, uceV.

(6.31)

Since A is a bounded closed convex subset of L2(Q) x L2(Q), it is weakly compact, so that from (p£)e we can extract a subsequence, still denoted by (p£)£, such that lim pc = p weakly in L2(O) x L2(£l),

p e A.

(6.32)

Actually we have pe -> p in L°°(n) x L°°(O) weakly *. Taking the limit as e -> 0 in (6.31), from (6.26) and (6.32) we have /M(U, v) + g \p-Vvdx

= L(v),

VtieF,

u e V,

(6.33)

so that (6.17) is proved. To complete the proof of the theorem, we need only prove that p-Vu = |V«|a.e..

(6.34)

Taking v = u in (6.33) and comparing with (6.22), we obtain

[\Vu\dxJJJ

[p-S/udx= JQ

[ (|Vu| -p-Vu)dx

= 0.

(6.35)

JQ

Since p e A, it follows from Schwarz inequality in U2 that p • Vu < | V« | a.e.. Combining (6.35) and this inequality we obtain p • Vu = |VM| a.e.. This proves (6.18) and, hence, the theorem.

•

Remark 6.3. The function p occurring in (6.17), (6.18) is not unique if Q e R 2 ; this is shown in G.L.T. [2, Chapter 5], [3, Chapter 5]. Remark 6.4. Relation (6.17) implies -nAu-gV-p=fmQ (6.36) u| r = 0. 6.7. Finite element approximation of (6.1) In this section we follow G.L.T. [3, Chapter 5]. For the sake of simplicity, we shall assume that Q is a polygonal domain of U2. 6.7.1. Definition of the approximate problem Let 9~h be as in Sec. 2 of this chapter. We approximate V by Vh = {vh e C°(Q), vk = 0onr,vh\TeP1,

V T G 3Th)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

85

and (6.1) by Ha(uh, vh - uh) + gj(vh) - gj(uh) >
V vheVh,

uheVh. (6.37)

Then: Theorem 6.4. The approximate problem (6.37) has a unique solution. Remark 6.5. So far, only an approximation by piecewise linear finite elements has been considered. This fact is justified by the existence of a regularity limitation for the solutions of (6.1), which implies that even with very smooth data we may have u £ H3(Q) (see Sec. 6.5). Nevertheless, in Fortin [1], Bristeau [1], G.L.T. [3, Chapter 5], and Bristeau and Glowinski [1], one may find applications of piecewise quadratic finite elements, straight or isoparametric, for solving (6.1). The numerical results which have been obtained seem to prove that for the same number of degrees of freedom the accuracies at the nodes are of the same order for the finite elements of order 1 and 2. From the above works it also appears that the second-order finite elements are much more costly to use (storage, computational time, etc.). We must also note that when using first-order finite elements, | n | Vvh | dx can be expressed exactly with respect to the values of vh at the nodes of 3Th, while with secondorder finite elements we need a numerical integration procedure. Remark 6.6. From the symmetry of a( •, •), (6.37) is equivalent to the minimization problem J(uh)<J(vh),

VvheVh,

uheVh,

(6.38)

where J(vh) = % a(vh, vh) + g f | \ v h \ dx -
(6.39)

6.7.2. Convergence of the approximate solutions (general case) We use the notations of the previous sections. Theorem 6.5. / / , as h -> 0, the angles of STh are bounded from below uniformly in h, by 90 > 0, then lim \\uh- u\\v = 0, where u and uh are, respectively, the solutions of (6.1) and (6.37).

(6.40)

86

II Application of the Finite Element Method

PROOF. In order to prove (6.40), we use Theorem 6.3 of Chapter I. Here we have to verify that the following three properties hold: (i) There exist U <= V, U = V, and rh:U^Vh

such that limft^0 rhv = v strongly

in V, V v 6 U. (ii) If Vi, -> v weakly in V as h -> 0, then lim inf ji,(vi,) > j(v).

(iii) lim^o jh(rhv) = j(v), VveU. Verification of (i). We take U = @(Q). Then 0 = tfj(fi) = V. Define rhv by rhveVh,

V v e tf'(O) n C°(Q), (6-41)

Then since rftu is the "linear" interpolate of v on 9~h, it follows from Ciarlet [1], [2], and Strang and Fix [1] that under the above assumptions on STh, we have .»(fi)

(6.42)

Then from (6.42) we find that limfc_»0 rhv = v strongly in #J(fl), V » e U . Verification of (ii). Since7h(yft) = X^X V », £ F t , (ii) is trivially satisfied. Verification of (iii). Since jft(yft) = ;(;;,,), V yA £ 1^ and from the continuity of }{•) on V, (iii) is trivially satisfied. Hence from (i), (ii), and (iii), it follows that uh-*u

strongly in V.

•

6.7.3. Convergence of the approximate solutions ( / e L 2 (Q)) From the regularity theorem, Theorem 6.2 of this chapter, we have H lliP(n) H«ll

<

llll

if Q is convex and T is sufficiently smooth. This property still holds if Q is a convex polygonal set. In this section we will always assume that Q is a convex polygonal domain. We have the following: Theorem 6.6. With assumptions on 3~h as in Theorem 6.5, we have \\uh - u\\v = O(h112).

(6.44)

PROOF. From (6.1) and (6.37) it follows that fia{u, uh — u) + gj(uh) - gj(u) > (/, uh - u), Ha(uh, vh - uh) + gj(vh) - gj(uh) > ( / , vh - uh), V vh e Vh, which imply, by addition, - u,uh-

u) < na(uh - u, vh - u) + gj(vh) - gj(u) + fia(u, vh-u)~

( / , vh - u),

V»,6F».

(6.45)

Since j( •) is a norm on V, using the Schwarz inequality in V and (6.45), we obtain - I k - u\\l < - \\vh - u\\l + gj(vh -u)

+ /M(U, vh-u)-

( / , vh - u),

V vh e Vh. (6.46)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

87

Since u e HQ(Q.) n H2(Q), we have a(u, v) = —I Auv dx,

VceF.

Hence, from (6.46) we have - I k - u\\v < - \\vh - u\\v + gj{vh - u) + 2.

Z

(-/iAw - f)(vh - u) dx,

V vh e Vh,

JJJ

so that - \\uh - u\\2v < - \\vh - u\\v + gj(vh - u) \\LHai)\\vh -

(6.47)

u\\LHa),

Since ||A«||L2(n) is a norm equivalent to the // 2 (fl)-norm over H2(Q) n //J(Q), it follows from (6.2), (6.43) and (6.47) that ^\\Uk -

U \ \ y < ^\\Vh

- U\\y

mM

,)

V « , E Fh.

(6.48)

Since F is Lipschitz continuous, we have (cf. Necas [1]) H2(Q) a C°(Q). The solution u of (6.1) 6 H2(ii) cz C°(Q); using the above inclusion, we can define rhu by rhueVh,

(rhu)(P) = v(P),

VPeS,.

From the above assumptions on ,Th, we have (cf. Ciarlet [1], [2] and Strang and Fix [1]) \\rhu - u\\v < y^Httll^n,, 2

\\rku - u\\LHn) < y2h \\u\\Hzin),

(6.49) (6.50)

where yt and y2 are constants independent of h and u. Taking vh = rhu in (6.48), it follows from (6.43), (6.49), and (6.50) that (6.51) with 70 = yo(Q) and M(Q) = meas(Q). Hence, from (6.51) we have I k - ti|| = O(h"2). This proves the theorem.

•

Remark 6.7. If Q is an open interval of R, i f / e L 2 (Q), and if piecewise linear elements are used, then an O(h) estimate can be obtained for the approximation error; the proof of the above estimate is very close to the proof of Theorem 3.4 of Sec. 3.7.1 and is given in detail in G.L.T. [3, Appendix 5].

88

II Application of the Finite Element Method

6.8. The case of a circular domain with / = constant In this section we consider a particular case of the general problem (6.1) by taking O = {x e IR2, jx\

+ x\ < R},

L(v) = C f vdx,

(6.52)

C > 0.

(6.53)

Jn

6.8.1. Exact solutions and regularity properties The solution of (6.1) corresponding to (6.52), (6.53) is given in Sec. 6.5.2 of this chapter. We recall that if g < CR/2, then ueVnW2'

a

(Q),

u$Vn

H3(Q).

(6.54)

In the sequel we assume that g < CR/2. 6.8.2. Approximation by finite element of order 1 Let &h be a finite triangulation of Q satisfying (2.22), (2.23) of Sec. 2.5, Chapter II and TCZQ.

(6.55)

Define Qh and Fh by

Then Qh <= Q, and in the sequel we assume that Th satisfies: all the vertices of Th belong to T.

(6.56)

Then we approximate V by Vh = {vh e C°(nh), vh = 0 on rh, vh\T ePltVTe

3Th}.

Now Vh can be considered to be a subspace of V, obtained by extending vh e Vh to Q by taking zero in Q — Q,,. It is then possible to approximate (6.1) by r

Ha{uh, vh - uh) + gj(vh) - gj(uh) > C

Jn

(vh - uh) dx,

V»,eFt,

uh e Vh. (6.57)

This is a finite-dimensional problem which has a unique solution. 6.8.3. Error estimate In this section we will obtain an error estimate of order hyj — log h. The following three lemmas play an important role in obtaining the above error estimate.

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

89

Lemma 6.1. Let p, q e IR2 - {0}. Then 2

IPl I q l

^

.

(6.58)

PROOF. We have

But

IPl

|q|2-2p-q = | p -

Iql

Consequently, j

^ <2|p-q|

IPl I q l

•

which obviously implies (6.58).

Remark 6.8. In (6.58), 2 is the best possible constant (take p = - q ) . Moreover, (6.58) is also true in UN, V N. Lemma 6.2. Let u and uh be the solutions o/(6.1) and (6.57), respectively. Let p satisfy (6.17), (6.18); then we have fia(uh - u,uhV vh e Vh,

u) < na{uh - u,vhand

V ph e A

u) + g

(ph - p) • V(vh - u) dx, •in

such that ph • Vvh = | Vvh | a.e..

(6.59)

PROOF. We shall prove (6.59) with feV*. From (6.45) we have Ha(uh -u,uh-u)<

iM(uh -u,vh-u)

+ gj(vh) - gj{u) + JM(U, vh - u)

Taking into account (6.17), we obtain lia(uh

- u , u

h

- u)< na(uh

- u, vh - u) - 9J((u) - g \ p-V{vh-u)dx, •In

(6.60)

Let vh e Vh and ph e A such that ph •Vvh = \ Vvh \ a.e.; such a ph always exists. Substituting this in (6.60) and using the relations •/(«») =

\ Ph • Vvh dx, •In

j{u)= I \Vu\dx> I ph-Vudx, we obtain (6.59). This proves the lemma.

(6.61) (6.62)

•

90

II Application of the Finite Element Method

Let u be the solution of (6.1) and 5 > 0. Define Q? c Q by Qs = {xeQ,

|Vw(x)| > 3}.

In the case of the problem (6.1) associated with (6.52), (6.53) (assuming g < CR/2), we have: Lemma 6.3. We have the following identity

(6.63) PROOF. From (6.16) we obtain du ~dr so that r dr

which implies (6.63).

D

From the above lemmas we deduce: Theorem 6.7. Let u be the solution of the problem (6.1) associated with (6.52), (6.53). Let uh be the solution of the problem (6.57) with S~h satisfying (6.55), (6.56). Assume that as h -> 0, the angles of3~h are bounded from below uniformly in hby 60 > 0. Then we have II"* — "IIK = OQiJ-logh).

(6.64)

PROOF. Starting from Lemma 6.2, from (6.59) we obtain ^.\\uh - u\\l < ~\\rhu - u\\l + g \ \ph2 2 Jn

p\\V{rhu - u)\dx,

V^eA

such that ph • Vrh u = | Vrh u \,

(6.65)

where rhu is defined by rhueVh,

(rhu)(P) = u(P),

V P e vertex of erh.

We have rhu = 0 on fi — Qh so that f — u\\y =

C \V(rhu — u)\2 dx =

C \Vu\2 dx +

\V(rhu — u)\2 dx. (6.66)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

91

Let us define

\Vu\2dx, \V{rhu - u)\2 dx.

= ~{ 1

•'fih

It is easily shown that meas(Q - Qft) < - h2.

(6.67)

Furthermore, (6.16) implies |Vu(x)| < —\R - —I, 2/i \ CJ From (6.67) and (6.68) it follows that

Vx e a

^

2

2

.

(6.68)

(6.69)

Since u e W2< °°(Q), on each triangle T 6 f , , w e have (cf. Ciarlet and Wagshal [1]) IVfou - u)(x)\ < - 7 ^ - \\p(D2u)\\LX(n).

(6.70)

sin t/Q

where D2 u(x) is the Hessian matrix of u at x, defined by d2u

/ ___M

I D2u(x) =

8x\

3xj ox2

I 52u \ 3 5 ~ (x)

fe)x

52M

Tl. (x

yjXi dx2

8x2

and p{D2u{x)) is the spectral radius of D2u(x). We have 2o Z)2w(x) = 0 i f O < r < ~ ,

(6.71)

so that p(D2 u(x)) = 0, and it is easily verified that 7

C

9/7

2n

C

< R.

(6.72)

Then (6.70)-(6.72) imply (6.73) So it remains to estimate the termg \n\ph — p\\V(rhu — u)\ dx in (6.65). We have fi = U Q i;

(6.74)

92

II Application of the Finite Element Method

where

n 3 = a - nft, Q4 = {x e ilh, r>2g/C + h}, Q5 = {x e Qft, 2#/C - h
f

—J'

(6.75)

We have rhu = 0 over Q — Qfe, so that in (6.65) we can take ph = 0 o v e r n - Q , , .

(6.76)

From (6.67), (6.68), (6.70), and p e A, it follows that

I V« | dx < ^ ffcC« - ^ ) ^ 2 -

*3 < 0 f

(6-77)

From (6.70)-(6.72), and since p, ph e A, we have

so that X

5

< ^ ^ - ^ . /i sin 0O

(6.78)

From the definition of h(h = maximal length of the edges of T e f J , we have rhu = u = constant over Q6, so that X6 = 0.

(6.79)

It remains to estimate XA. Since the equipotential lines of u in fi4 are circular, for h sufficiently small, from (6.16) we have rhu\T # constant,

VTe^,,

T a Q4,

so that V^Mj-# 0,

VTe^,

T c H4.

(6.80)

Taking into account (6.80), it follows from (6.65) that Vrhu

VTe/,,

TcQ4.

(6.81)

Furthermore, we observe that (6.16) implies — >0, 2H

VxeQ4.

(6.82)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe 93 This, in turn, implies VxeO 4 . I Vu(x) I Applying Lemma 6.1 to the pair {Vrhu, Vu} and Lemma 6.3 with d = Ch/2[i, it follows from (6.80)-(6.82) that p

nu

Lwu ,|Vt +

where From (6.70) and (6.82) it follows that C 2 ( ChVc

dx

which, using (6.63), implies

or, more simply, XA<—
1 \R

ysin OQJ \

log — ]. C

(6.83)

RJ

Taking (6.65) into account, the estimate (6.64) is obtained by addition of the Xh i — 1,..., 6. More precisely, for sufficiently small h, Ik - u\\v < 4^ ^/n—- V-log *. sin 99 0 H sin

(6.84)

a 6.8.4. Generalization From the numerical experiment we have done, it seems that in a great number of cases (important from the point of view of application) we have the following properties for u: (1) we V n W2'X{£1), (2) Q o = {x | VM(X) = 0} is a compact subset of Q with a smooth boundary, (3) Q o has a finite number of connected components. Moreover, it seems that in the above cases we can conjecture that for S > 0, we still have

J n «|V«(x)| With these properties we can easily prove the following error estimate: \\uh-u\\v

= OQiJ - log h).

94

II Application of the Finite Element Method

Remark 6.9. Using an equivalent formulation of (6.1) (less suitable for computations) Falk and Mercier [1] have obtained an 0(h) estimate for ||«2 — u||jji(n) f° r a piecewise linear approximation (see also G.L.T. [3, Appendix 5]). 6.9. Iterative solution of the continuous and approximate problems by Uzawa's algorithm We begin with the continuous problem (6.1). Let us define if: V x H -> U by £C(v, q) = ~ a(v, v) - L(v) + g f q-Vvdx, *•

V»eF,

V qeH,

-In

where H = L2(Q) x L2(Q). Let {u, p} be the solution of (6.17), (6.18). Then we have: Theorem 6.8. The pair {«, p) is a saddle point of' $£ over V x A if and only if {u, p} satisfies (6.17) and (6.18). EXERCISE

6.5. Prove Theorem 6.8.

From Cea [2, Chapter 5] (see also G.L.T. [3, Chapter 5]) it follows that to solve (6.1) we can use the following Uzawa algorithm: p° e A arbitrarily chosen (for example, p° = 0):

(6.86)

then, by induction, knowing p", we compute u" and pn+1 by Ha(u", v) = < / » -

g

f / • Vv dx,

V v e V, u" e V,

(6.87)

Pn+1 = PA(p" + pgVW),

(6.88)

where PA: H —* A is the projection operator in the H-norm, defined by P (a) =

q

AW

Sup(l,|
V " ~ +d ' (6.89) «-| r = 0. We shall give a direct proof for the convergence of (6.86)-(6.88) based on Theorem 6.3 of Sec. 6.6.

Theorem 6.9. Let un be the solution of '(6.87). Then if

0 < p< ^ ,

(6.90)

6. A Second Example of EVI of the Second Kind: Flow of Viscous Plastic Fluid in a Pipe

95

we have lim | K - u\\v = 0,

(6.91)

n~* oo

where u is the solution of (6.1). PROOF. Let {u, p} satisfies (6.17) and (6.18). Then (6.18) implies V = PA(P + pgVu)-

(6-92)

We define u" = u" — u, p" = p" — p. Using the fact that P A is a contraction mapping, and from (6.88), (6.92), we obtain |p" + 1 1 2 < |p"| 2 + 2f>g f p" • Vu" dx + p2g2 I* |Vu"|2 dx, •'si

(6.93)

Jn

where

From (6.17) and (6.87) it follows that \xa(un, v) + g f p" • Vt; dx = 0, •in

V v e V.

(6.94)

Replacing v by «" in (6.94), we obtain \m(un, u") + g f p" • Vu" dx = 0.

(6.95)

From (6.93) and (6.95) we have |p"| 2 - | p " + T > p(2/i - pg2)\\u"\\2v.

(6.96)

2

If 0 < p < 2p/gf , then using standard reasoning, we obtain lim ||u"||F = 0, n-* ao

which proves the theorem.

•

Let us describe the adaptation of (6.86)-(6.88) to the approximate problem (6.37). W e define Lh c L 2 (Q) x L2(Q) by

where XT is the characteristic function of T. It is then clear that V « k e Fh, Vvh e Lh. We also define Ak by Aft = A n L,. We can easily prove that

Then (6.86)-(6.88) is approximated by: p° 6 Aft arbitrarily chosen,

(6-97)

96

II Application of the Finite Element Method

by induction, knowing pi, we obtain u\ and pl+1 by Ha(unh, vh) = L(vh) - g f p"h • Vvhdx,

VvheVh,

unheVh,

pl+1 = PA(pl + PgV<).

(6.98) (6.99)

2

Then for 0 < p < 2^/g we obtain the convergence of u\ to uh. EXERCISE

6.6. Study the convergence of (6.97)-(6.99).

Remark 6.10. The above methods have been numerically applied for solving (6.1) in Cea and Glowinski [2], Fortin [1], Bristeau [1], Bristeau and Glowinski [1], and G.L.T. [3, Chapter 5]. They appear to be very efficient and particularly well suited for taking nondifferentiable functionals like J n | Vv \ dx into account.

7. On Some Useful Formulae Let T be the triangle of Fig. 7.1. We denote by M(T) the measure of T. Let i; be a smooth function defined on T. We define v{ and vJk by vt = v(Mi),

vjk = v(Mjk).

Then we have the following formulae:

1

uv dx =

M(T)

+ v3)

12 V u, v e Pu

M«3 1 2 {\M2M3\2v\ vj 4M(T)2 + 2M2M\

+ 2M3M[

• M3M[V1V2

\\2v\ + \WjA*2\2v23 M2 •

+

M2M3v3v1 (7.2)

-MXM2V2V3},

M

M,

(7.1)

M2

Figure 7.1

7. On Some Useful Formulae

97

J \v\2 dx = —— {Joivi + x>\ + vj) + rs(v2l2 + v223 + v231) - v23v31 + v31v12) v3vl2)}, "' 2 dx

VveP2,

(7.3)

=

+ 2(t)12 + v23 - v31)M3M[\2 + \vxM2M\ + v2M3M[ - v3M1M2 + 2(v23 + v31 - v12)M^M*2\2 + | -vlM2M3 + v2M3M\ + v3MtM2 + 2(o31 + v12 -

v23)M2M'3\2}, \/veP2.

(7.4)

The above formula may be useful for expressing the approximations of the problems of this chapter, in a form suitable for computations.

CHAPTER III

On the Approximation of Parabolic Variational Inequalities

1. Introduction: References In this chapter we would like to give some indications of the approximation of parabolic variational inequalities (PVI) (mostly without proof). For a detailed reference, see Fortin [1], Bristeau [1], Bristeau and Glowinski [1], C. Johnson [1], and A. Berger [ I ] . 1 See also Richtmyer and Morton [1], Kreiss [1], and Lascaux [1] for the numerical analysis of time-dependent equations.

2. Formulation and Statement of the Main Results Let H and V be two real Hilbert spaces such that V c: H, V = H. Assuming that H = H*, we have V <= H c V*. The scalar product in H (resp., in V) and the corresponding norms are denoted by (•, •), H (resp., ((•, •)), II-ID- Moreover, we also use (•, •) for the duality between V* and V. We now introduce: • A time interval [0, T] with 0 < T < oo, a bilinear form a: V x V -> R, continuous and elliptical in the following sense: 3 a > 0 and X > 0 such that a(v, v) + /l|u| 2 > <x\\v\\2, V c e F , • fe L2(0, T; V*), u° e H (for the definition of L 2 (0, T; Z), see Lions [1],

[3]); • K: closed convex nonempty subset of V, • j:V -> U convex, proper, l.s.c. We then consider the following two families of PVI: Find u(t) such that tdu , v - u } + a(u, v -u)>(f,v -u), V v e K, a.e. t e ]0, T[, dt ) u(t) e K a.e. t e ]0, T[, w(0) = u0. 1

See also G.L.T. [3, Appendix 6] and the references therein.

(2.1)

3. Numerical Schemes for Parabolic Linear Equations

99

Find u(t) such that —, v - u I + a(u, v - u) + j(v) - j(u) >(f,v-

«), VveV, a.e. t e ]0, T[,

u(t) £ V a.e. t e ]0, T[, u(0) =

M0 .

(2.2)

Remark 2.1. If K = F a n d ; = 0, then (2.1) and (2.2) reduce to the standard parabolic variational equation: Y, v j + a(u, v) = (/, u),

M veV a.e. in t e ]0, T[,

«(t) e F a.e. t e ]0, T[,

M(0)

= u0.

(2.3)

Under appropriate assumptions on u 0 , K, and ;'(•), it is proved that (2.1), (2.2) have unique solutions in L 2 (0, T; V) n C°([0, T], if). For the proof we refer to Brezis [4], [5], Lions [1], and Duvaut and Lions [1]. In the following sections of this chapter, we would like to give some discretization schemes for (2.1) and (2.2), and then, in Sec. 6, study the asymptotic properties over time of a specific example, for the continuous and discrete cases.

3. Numerical Schemes for Parabolic Linear Equations Let us assume that V and H have been approximated (as h -* 0) by the same family (Vh)h of closed subspaces of V (in practice, Vh are finite dimensional). We also approximate (-,-)> a( •. •) by (•, • \ , ah{ •, •) in such a way that ellipticity, symmetry, etc. are preserved. We also assume that u0 is approximated by (uOh)h such that uOh e Vh and l i m ^ 0 uOh = u0 strongly in H. We now introduce a time step At; then, denoting u"h the approximation of u at time t = nAt (n = 0,1, 2,...), we approximate (2.3) using the classical step-by-step numerical schemes (i.e., we describe how to compute u"h+i if ui and u\~^ are known). 3.1. Explicit scheme (Euler's scheme)

n = 0, 1,... • u° = u

(3-1)

Stability: Conditional (See Lascaux [1] for the terminology). i>: O(At) (we consider only the influence of the time discretization).

100

III On the Approximation of Parabolic Variational Inequalities

3.2. Ordinary implicit scheme (backward Euler's scheme)

(<+1 ~< A +ah(uV\vh) = (fl+\vh)h, \ At 7»

\/vheVh,

n = 0,1,2,...;

u°h=uQh.

(3.2)

u°h = uOh.

(3.3)

Stability: Unconditional. Time accuracy: O(At).

3.3. Crank-Nicholson scheme 1

n+l n Uh — Uh

= 0,1,2,...;

Stability: Unconditional. Time accuracy: O(\At\2).

3.4. Two-step implicit scheme

= (/Z +1 ,«>*)*,

V^eF,,

n=l,2,...;

«? = u Oh , uj given.

(3.4)

Stability: Unconditional. Time accuracy: O(\At\2). Unlike the three previous schemes, this latter scheme requires the use of a starting procedure to obtain u\ from u° = uOh; to compute u\ we can use, for example, one of the schemes (3.1), (3.2), or (3.3); we recommend (3.3), since it is also an O(\ At|2)-scheme. Similarly, the generalizations of the scheme (3.4) discussed in Sees. 4 and 5 will require the use of a starting procedure which can be the corresponding generalization of schemes (3.1), (3.2), or (3.3).

3.5. Remarks Remark 3.1. The function /jj (or f"h+1/2) occurring in the right-hand sides of (3.1)-(3.4) is a convenient approximation of / at t = nAt (or t = (n + j)At). In some cases it may be defined as follows (we only consider / J since the technique described below is also applicable to f n h + 1 / 2 ) .

4. Approximation of PVI of the First Kind

101

First we define / " e V* by / " = / ( « At) if/eC°[0, T;V*l In the general case, it is defined by

!"+m)Atf(t)dt

f"=~ A r

ifn>l.

J(n-l/2)6a

Then, since (•, -)h is a scalar product on Vh, one may define f\ by (f"

u.V

= f f

v , )

V i). p V f

p

V

In some cases we have to use more sophisticated methods to define f"h. Remark 3.2. At each step (n + 1), we have to solve a linear system to compute u"h+1; however, if we can use a scalar product (•, • \ leading to a diagonal matrix, with regard to the variables defining vh, then the use of the explicit scheme will only require us to solve linear equations in one variable at each step. Remark 3.3. We can also use nonconstant time steps Atn. Remark 3.4. If we are interested in the numerical integration of stiff phenomenon or in long-range integration, we can briefly say: • Schemes (3.1), (3.2) are too dissipative; moreover the stability condition in (3.1) may be a serious drawback. • Scheme (3.3) is, in some sense, not sufficiently dissipative. • Scheme (3.4) avoids the above inconveniences and is highly recommended for "stiff" problems and long-range integration. In most cases the extra storage it requires is not a serious drawback. Remark 3.5. There are many works related to the numerical analysis of parabolic equations via finite differences in time and finite elements in space approximations. We refer to Raviart [1], [2], Crouzeix [1], Strang and Fix [1], Oden and Reddy [1, Chapter 9], and the bibliographies therein.

4. Approximation of PVI of the First Kind We assume that K in (2.1) has been approximated by (Kh)h, Kh <= Vh, V h, as in the elliptic case (see Chapter I). We also suppose that the bilinear form a(•, •) is possibly dependent on the time t and has been approximated by ah(t; uh, vh).

102

III On the Approximation of Parabolic Variational Inequalities

4.1. Explicit scheme + ah{n

\ >(flvh-u"h+\,

At;

u l Vh

-

u h+1)

"

u"h+leKh,

VvheKh,

n = 0,1,2,...;

u°h = uoh.

(4.1) Stability: Conditional (see G.L.T. [3, Chapter 6]). This scheme is almost never used in practice, since it is conditionally stable and the computation of u"h+i will generally, require the use of an iterative method, even if the matrix corresponding to (•, • )h is diagonal. 4.2. Ordinary implicit scheme ,n + 1 *

n *

' ,vh - unh+1 1 + ah((n + 1) A t ; unh+ \ vh -

u"h + leKh,

VvheKh,

u"h+1)

n = 0, 1, 2 , . . . ; «° = «ofc.

(4.2)

Stability: Unconditional. At each step we have to solve an EVI of the first kind related to Kh to compute u1+1. This scheme is often used in practice. 4.3. Crank-Nicholson scheme

+ a^n + *> At; "* Un 4-

u",+ll2sK, u

t

h

Jv

un+1

ul+112 /i'

u

h

—

«-012 j

'

"

—

u

>

1

'

z

• u° - u -' • • • '

u

h

—

"Oft-

(4.3)

Stability: Unconditional. Since (u",+ 1 - wJD/At = K + 1 / 2 - «J)/(Ar/2), we observe that at each step we have to solve an EVI of the first kind to compute unh+lj2. We also observe that possibly unh4 Kh. We do hot recommend this scheme if the regularity over time of the continuous solution is poor. 4.4. Two-step implicit scheme \ul+1 — 2ll"h + 2MJT1

At

n=

1,2,...;

u°h = uOh, u\ given.

(4.4)

5. Approximation of PVI of the Second Kind

103

Stability: Unconditional. At each step we have to solve an EVI of the first kind in Kh to compute u"k+1. Remark 3.4 also applies to this scheme.

5. Approximation of PVI of the Second Kind 5.1. Explicit scheme u"h+1) + jh(vh) - jh(u"h+1)

- ah(n At;u"h,vh>(flvh-u"h+\,

VvheVh,

ul+1eVh, n = 0,1,2,...;

u°h=uOh.

(5.1)

Stability: Conditional. This scheme also, is, almost never used in practice since it is conditionally stable and the computation of M£ + 1 will require the solution of an EVI of the second kind in Vh (in general, by an iterative method) even if the matrix corresponding to (•, •),, is diagonal. 5.2. Implicit scheme n+l _ *

>(fl

> Vh - K+1 ) + ah((n + 1) At; u\+ \ vh - unh+ J ) + jh(vh) +

- jh(u"h+ x )

+1

\vh-ul+\,

VvheVh, u"h"+1eVh, n = 0,1,2,...;

u°h=uOh.

(5.2)

Stability: Unconditional. At each step we have to solve in Vh an EVI of the second kind to compute

5.3. Crank-Nicholson scheme + ah((n + ^)At;utll2,vh ,vh-u"h+1'2\, U h+ h

" f

-;

VvheVh,

, n = 0,1,2,...;

Wh+l'2eVh, uho = uOh.

(5.3)

104

III On the Approximation of Parabolic Variational Inequalities

Stability: Conditional. Since (u"h+1 - u%)/At = (unh+112 - u"h)/(At/2), we observe that at each step we have to solve an EVI of the second kind to compute unh +1/2. If the regularity over time of the solution is poor, we do not recommend this scheme.

5.4. Two-step implicit scheme

2Uh 2 + 2Uh

-2

v

+ ah((n + 1) At; u\+ \ vh - u"h+1) > ( / J + 1 , vh - unh+1), VvheVh,

n = 0,1,2,...;

u°h = uOh, u\ given. (5.4)

We use one of the above schemes (5.1)—(5.3) to compute u\, starting from < = uOh. Stability: Unconditional. At each step we have to solve, in Vh, an EVI of the second kind to compute w£+1. Remark 3.4 applies to this scheme as well. 5.5. Comments The properties of stability and convergence of the various schemes of Sees. 4 and 5 are studied in the references given in Sec. 1. In some cases, error estimates also have been obtained. In Fortin [1] and G.L.T. [2, Chapter 6], [3, Chapter 6 and Appendix 6], applications to more complicated PVI than (2.1), (2.2) are also given. For the numerical analysis of hyperbolic variational inequalities, see G.L.T. [2, Chapter 6], [3, Chapter 6] and Tremolieres [1].

6. Application to a Specific Example: Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe Following Glowinski [4], we consider the time-dependent problem associated with the EVI of Chapter II, Sec. 6 and study its asymptotic properties.

6.1. Formulation of the problem. Existence and uniqueness theorem Let Q be a bounded domain of U2 with a smooth boundary F. We consider: • V = ffj(fi), H = L2(Q), V* = H-^Q), • a(u, v) = j n VM • Vv dx, • A time interval [0, T], 0 < T < oo,

6. Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe

105

. / G L 2 ( 0 , T ; V*),u0eH, •700 = $n\Vv\dx, • n > 0, g > 0. We then have the following: Theorem 6.1. The PVI (du

,v - u\ + \ia(u, v - u) + gj(v) - gj(u) >(f,v-

u),

V v e V a.e. t e ]0, T[, u(x, 0) = uo(x), (6.1) has a unique solution u such that u E L2(0, T;V)n

C°([0, T ] ; if),

- ^ e L 2 (0, T, F*)

a«d tfcjs V u0 6 # , V / e L2(0, T; F*). For a proof, see Lions and Duvaut [1, Chapter 6].

6.2. The asymptotic behavior of the continuous solution Assume that / is independent of t and that / € L2(Q). We consider the following stationary problem: Ha(u, v - u) + gj(v) - gj(u) > (/, v - u),

V v e V, u e V.

(6.2)

In Lions and Duvaut [1, Chapter 6] (see also Chapter II, Sec. 6 of this book), it is proved that u = 0 if#j3> ||/|| L 2 ( O ) ,

(6.3)

B = inf . v,v Mmm

, , .. (6-4)

where

t>#0

Then we can prove the following: Theorem 6.2. Assume that f e L2(Q) with ||/|| L 2 ( n ) < fig; then if u is the solution of (6.1), we have u(t) = 0 for t > A

oA*

\

py -

117 IILV (6-5)

where Ao is the smallest eigenvalue of — A in H{,(£1) (2.0 > 0).

106

III On the Approximation of Parabolic Variational Inequalities

PROOF. We use | -1 for the L2(fi)-norm and || • || for the Ho(fl)-norm. Since

it follows from Theorem 6.1 that the solution of (6.1) is defined on the whole IR + . We now observe that if gfl > | / 1 , then zero is the unique solution of (6.2); it then follows from Theorem 6.1 that if u(t0) = 0 for some 10 > 0, then u(f) = 0,

Vt>f0.

(6.6)

Taking v = 0 and v = 2w in (6.1), we obtain '8u \ —, u I + fia(u, u) + gj(u) = ( / , u) a.e. in f.

(6.7)

But since u e L 2 (0, T; V), »' e L 2 (0, T ; V*) implies (this is a general result) that t -> |u(t)l 2 is absolutely continuous with (d/dt)\v\2 = 2(dv/dt, v), from (6.7) we obtain - - | u | 2 + ixa(u, u) + gj(u) = (/, u) < \ f \ \ u \ a.e. in t. Since a(v, v) > ko\v\2, obtain

(6.8)

V c e V, and j(v) > f}\v\, V n e F (from (6.4)), from (6.8) we

|u| 2 + H ) M 2 + (gP - \f\)\u\

< 0 a.e. in teU+.

(6.9)

2 dt Assume that u(t) # 0, V t > 0; since t -> | u(i) \2 is absolutely continuous with | u(t) | > 0, it follows that t -> | u(t) | is also absolutely continuous. Therefore from (6.9) we obtain - I / I ) < 0a.e. te R + .

(6.10)

< -nk0

(6.11)

From (6.10) it follows that

d/dt\u(t)\

a.e. t e R+.

Define y by y = (g/S — \f\ )//U 0 ; then y > 0. By integrating (6.11) it then follows that |«(f)l + y < ( | « 0 | + y)e~"Xo\

Vt6R+;

(6.12)

(6.12) is absurd for t large enough. Actually we have u(t) = 0 if

i.e., (6.13)

6.1. Let fe L2(Q), possibly with | / 1 > gfi. Let us denote by u^ the solution of (6.2); then prove that EXERCISE

where u(t) is the solution of (6.1).

6. Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe

6.3.

107

On the asymptotic behavior of the discrete solution

We still assume that / e L2(fi). To approximate (6.1) we proceed as follows: Assuming that Q is a polygonal domain, we use the same approximation with regard to the space variables as in Chapter II, Sec. 6 (i.e., by means of piecewise linear finite elements, see Chapter II, Sec. 6). Hence we have ah(uh, vh) = a(uh, vh),

\/uh,vheVh,

and from the formula of Chapter II, Sec. 7, we can also take («*,

vh)h = (uh, vh),

M

uh,vheVh.

Then we approximate (6.1) by the implicit scheme (5.2) and obtain n+l

.n

gj(vh)

At -gj(wh+1)>(fh,vh-unh+1), u"h+leVh;

VvheVh,

n = 0,1,2,...;

u°h = uOh.

(6.14)

We assume that uOheVh,^ h and lim uOh = M0 strongly in L2(Q).

(6.15)

Similarly, we assume that / is approximated by (fh)h in such a way that (/*» vh) can be computed easily and lim fh = / strongly in L2(Q). Theorem 6.3. Let \f\ < fig. If (6.15) and (6.16) /JOW, small, we have wj| = 0 for n large enough.

(6.16) then if h is sufficiently

PROOF. AS in the proof of Theorem 6.2, taking vh = 0 and u,, = 2i(j + : in (6.14), we obtain \-^—.

-, u"h+1)

+ / M \ V u " h + l \ 2 dx + g

| V K £ + 1 | dx =

fhu"h+1

dx,

Vn> 0; (6.17)

2

using the Schwarz inequality in L (Q), it follows from (6.17) that At——

|Mft+1l + M ) K + T + {gP - l/J)l"i!+1l ^ o,

Vn > o. (6.18)

2

Since fh-* f strongly in L (il), we have gP ~ I/*I > 0 for h sufficiently small

(6.19)

108

III On the Approximation of Parabolic Variationai Inequalities

From (6.18), (6.19) it then follows that u"h° = 0 => u"h = 0 for n > n0 if h is small enough.

(6.20)

Assume that uj ^ 0, V n; then (6.18) implies V»>0.

(6.21)

We define •/,, by yh = g(3 - \ fh |; then yh > 0

for h small enough and

lim yh — y = g[l — \ f |.

(6.22)

/i->0

From (6.21) it follows that

which implies that Vi

Since yh > 0 for h small enough, (6.23) is impossible for n large enough. More precisely, we shall have u\ = 0 if I'll

^

/1

which implies: vl

/ / h is small enough, then u"h = 0 if n >

.

(6.24)

Log(l + lon At) Relation (6.24) makes the statement of Theorem 6.3 more precise. Moreover, in terms of time, (6.24) implies that u"h is equal to zero if (6.25)

Log(l + Xoii At) We observe that Log[l + Xoii{\u°\lyJ\

1

/

|«°|

hm Af — - — — — = - — Log 1 + X0fi Log(l +XolxAt) Aoyu \ y h^0

Hence, taking the limit in (6.25), we obtain another proof (assuming that unh converges to u in some topology) of the estimate (6.5) given in the statement of Theorem 6.2. EXERCISE 6.2. Let u£° be the solution of the time-independent problem associated with/,,, possibly with \fh\ > fig; then prove that

Arr" / 2 K - K\,

V« > 0.

6. Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe

109

6.4. Remarks Remark 6.1. We can generalize Theorem 6.2 to the case of a Bingham flow in a two-dimensional bounded cavity. Remark 6.2. In Glowinski [4], Bristeau [1], and Begis [1], numerical verifications of the above asymptotic properties have been performed and found to be consistent with the theoretical predictions. Remark 6.3. In H. Brezis [5], one may find many results on the asymptotic behavior of various PVI as t -> + oo.

CHAPTER IV

Applications of Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations

1. Introduction For solving some nonlinear elliptic equations, it may be convenient, from the theoretical and numerical points of view, to view them as EVFs. In this chapter we shall consider two examples of such situations: (1) a family of mildly nonlinear elliptic equations; (2) a nonlinear elliptic equation modeling the subsonic flow of an inviscid compressible fluid.

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

2.1. Formulation of the continuous problem Let Q be a bounded domain of UN (N > 2) with a smooth boundary T. We consider . V = tf i(Q); • a: V x V -> U bilinear, continuous, and F-elliptic with a > 0as ellipticity constant; a(-, •) is possibly nonsymmetric; • <j>: U. -> U, 4> e C°(R), nondecreasing with 0(0) = 0. We then consider the following nonlinear elliptic equation (P) defined by: Find u e V such that a(u, v) + <(«), v} = L(v),

VceK,

From the Riesz Representation Theorem it follows that there exists A e Se(V, V*)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

111

such that a{u, v) = {Au, v}, V « , c e F . Therefore (P) is equivalent to Au + (j)(u) =f,

ueV,

(2.1)

X

EXAMPLE 2.1. Let us consider a function a0 e L (Q) such that ao(x) > a. > 0 a.e. in Q.

(2.2)

a(u, v) = | ao(x)Vw • Vu dx + ) j8 • VM t?dx,

(2.3)

Define a(-, •) by

where y? is a constant vector in UN. From the definition of ao(-\ and using the fact that j n )S • Vv vdx = 0, V v e HQ(Q), we clearly have a(»,tO 2: a||t>||£.

(2.4)

Au = -V-(a 0 Vw) + p- Vu.

(2.5)

From (2.3) we obtain

Hence, in this particular case, (2.1) becomes - V • (a 0 Vu) + P • Vu + (j>(u) = / ,

u G F, (/.(«) e L J (Q).

(2.6)

Remark 2.1. IfN = 1, we have Ho(fi) <= C°(O). Because of this inclusion there is no great difficulty in the study of one-dimensional problems of type (P). If N > 2, the main difficulty is precisely related to the fact that ffo(fi) is not contained in C°(Q). Remark 2.2. The analysis given below may be extended to problems in which either V = H1(Q) or V is a convenient closed subspace of H^Q).

2.2. A variational inequality related to (P) 2.2.1. Definition of the variational inequality Let

=f

(2.7)

Jo

(2.8) The functional;: L 2 (Q) -> U is defined by j(v) = f O(D) rfx if
j(f) = + oo if ®(v) $ L\Q).

(2.9)

112

IV Applications of Elliptic Variational Inequality Methods

Instead of studying the problem (P) directly, it is natural to associate with (P) the following EVI of the second kind: a(u, v — u) + j(v) — j(u) > L(v — u),

V v e V, ueV.

(n)

If a(-, •) is symmetrical, a standard method for studying (P) is to consider it to be the formal Euler equation of the following minimization problem encountered in the calculus of variations: J(u) < J(v),

V v e V, ueV,

(2.10)

where J(v) = %a(v, v) + I 3>(u) dx - L(v). EXERCISE

(2.11)

2.1. Prove that D(
2.2.2. Properties of ){•) Since : U -» U is nondecreasing and continuous with (/>(0) = 0, we have $eC'(R),

O(t) > 0, V t e R .

(2.12)

The properties ofXO are given by the following: Lemma 2.1. The functional j(-) is convex, proper, and l.s.c. over L2(Q). PROOF. Since j(v) > 0, V v e L2(£l), it follows that ;(•) is proper. The convexity of ;(•) is obvious from the fact that O is convex. Let us prove that;'(•) is l.s.c. Let {vn}n, vn e L2(Ji), be such that lim i!n = v strongly in L2(Q). Then we have to prove that lim inf ;(*;„) >7(i>).

(2.13)

If lim inf,,^ j(vn) = + oo, the property is proved. Therefore assume that lim inf j(vn) = I < +co. Hence we can extract a subsequence {vnu}nk such that ) = /,

(2.14)

vnk -* v a.e. in Cl.

(2.15)

lim O(iO =
(2-16)

l

Since O e C (R), (2.15) implies

Moreover, 0 a.e. and (2.14) imply that {®(vnt)}k is bounded in L\Q).

(2.17)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

113

Hence, by Fatou's Lemma, from (2.16) and (2.17), we have

lim inf | ®(vnj dx > ) O(u) dx.

(2.18)

From (2.14) and (2.18) we obtain (2.13).

This proves the lemma.

•

Corollary 2.1. The functional j(-) restricted to V is convex, proper, and l.s.c.. 2.2.3. Existence and uniqueness results for (n) Theorem 2.1. Under the above hypotheses on V, a(-, •), L(-), and (•), problem (n) has a unique solution in V n PROOF. Since V, a(-, •), L(-), and j(-) have the properties (cf. Corollary 2.1) required to apply Theorem 4.1 of Chapter I, Sec. 4, the EVI of the second kind, (n), has a unique solution u in V. Let us show that u 6 D(9). Taking v = 0 in (n), we obtain a(u,u) +j(u) < L(u) <\\f\\Ju\\v.

(2.19)

Since j(u) > 0, using the ellipticity of a{-, •), we obtain Nlv£—. a

(2-20)

j(u)<^-.

(2.21)

which implies

a This implies u e D().

•

Remark 2.3. If a(-, •) is symmetric, (TT) is equivalent to (2.10).

2.3. Equivalence between (P) and in) In this section we shall prove that (P) and (n) are equivalent. First we prove that the unique solution of (n) is also a solution of (P). In order to prove this result, we need to prove that (M) and u(j>(u) belong to L^Q). Proposition 2.1. Let u be the solution of (n). Then ucj)(u) and (j)(u) belong to L\Q). PROOF. Here we use a truncation technique. Let n be a positive integer. Define

Kn = {veV,\v(x)\
114

IV Applications of Elliptic Variational Inequality Methods

Since Kn is a closed convex nonempty subset of V, the variational inequality a(un, v - un) + j(v) - j(un) > L(v - Mn),

V v e Kn,

uneKn,

(nn)

has a unique solution (in order to apply Theorem 4.1 of Chapter I, we need to replace j byj + lKn, where IKn is the indicator functional of Kn). Now we prove that lim,,.,^ un = u weakly in V, where u is the solution of (n). Since Qe Kn, taking v — 0 in (nn), we obtain (as in Theorem 2.1 of this chapter) \\un\\v <

=,

(2.22)

ex (2.23)

From (2.22) it follows that there exists a subsequence {unk}nk of {«„}„ and u* e V such that lim unH = u* weakly in V.

(2.24)

k->oo

Moreover, from the compactness of the canonical injection from HQ(Q) to L2(Q) and from (2.24), it follows that lim unk = u* strongly in L2(Q).

(2.25)

Relation (2.25) implies that we can extract a subsequence, still denoted by (unjnk, such that lim unk = u* a.e. in il.

(2.26)

Now let v e V n L ^ f l ) ; then for large k, we have v e Knk and «(«»„> "»„) + JKJ < a{unk, v) + j(v) - L{v - uj.

(2.27)

Since lim infk^^ a(unk,unk) > a(u*,u*) and lim inf^..^ _/(wBk) > j(w*), it follows from (2.24) and (2.27) that a(u*, «*) + j(«*) < a(u*, v) + j(v) - L(v - «*),

V t; e L ^ f i ) n K,

M* 6 V,

which can also be written as

a(u*,v-u*)+j(v)-j(u*)>L(v-u*),

V v e V n L°°(ii), «* e K (2.28)

For n > 0, define rn: F -» Kn by rnv = Inf(n, Sup(-n, y))

(2.29)

(see Fig. 2.1). Then, from Corollary 2.1 of Chapter II, Sec. 2.2, we have lim tn v = v strongly in V, (2.30) lim xnv = v a.e. in Q. n-* a:

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

115

Figure 2.1 Moreover, we obviously have Tnu(x)| < |u(x)|a.e.,

(2.31)

v(x)inv(x)>0a.e..

(2.32)

From (2.30)-(2.32) and from the various properties of (Tnp) < d>(u) a.e.,

(2.33)

limO>(Tny) = ®(v)a.e..

(2.34)

n-» oo

Since xnv e L°°(n) n V, it follows from (2.28) that a(u*, xnv-u*)+

j(xnv) - j(u*) > L(xnv - u*\

V ve V, u* e V. (2.35)

If v $ D(Q>), then by Fatou's lemma, \imj'(Tnv) = +oo. n-* co

If v e D(
From these convergence properties and from (2.30), and by taking the limit in (2.35), it follows that a(u*, v -u*)+ ){v) - j(u*) > L(v - u*\

VveV,

u* e V.

(2.36)

Then u* is a solution of (n), and from the uniqueness property, we have u* = u. This proves that lim,,^ un = u weakly in V. Let us show that <j>(u), u
dx>L(v-un),

VveKn. (2.37)

116

IV Applications of Elliptic Variational Inequality Methods

Since 0> e C\U) and , we have hm

* ( « „ + t(p - «„)) - O(u B )

= <£(«„)(» - un) a.e..

(2.38)

Moreover, since <J> is convex, we also have, V t e ]0, 1], «0O0

n)

*

From (2.38), (2.39), and using Lebesgue's dominated convergence theorem in (2.37), we obtain a(un,v-un)+

\(un){v-un)dx>L{v-un),

VceX,.

(2.40)

Then, taking v = 0 in (2.40), we have a(un, «„) +

4>{un)un dx < L(un),

which, using (2.22), implies II f\\2

C

UkK
Jn

(2.41)

a

But 4>{v)v > 0, V v e V. Hence 4>(un)un is bounded in L1 (Q). Moreover, for some subsequence {«»Jnt of {«„}„, we have («„>«„ - *
a

-

e

-

i n

a

X

Then by Fatou's lemma we obtain u<j>{u) e L {Q), and this completes the proof of the proposition, since u(j>(u) e L J (Q) obviously implies that 4>(u) e L^Q). D Incidentally, when proving the convergence of (un)n to u, we have proved the following useful lemma: Lemma 2.2. The solution u of (n) is characterized by a(u, v — u) + j(v) — j(u) > L(v — w), VueFnL^Q),

ueV, <&{u)eL\Q).

(2.42)

In view of proving that (71) implies (P), we also need the following two lemmas: Lemma 2.3. The solution u of (n) is characterized by a(u, v — u) +

(j>(u)(v — u) dx > L(v — u), V i ) € L°°(fi) n V, ueV, u<j>(u) e L\Q).

(2.43)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

117

PROOF. (1) (7c) implies (2.43). Let v e L°°(f2) n V. Then v e £>(), and since D(), V t e ]0, 1]. Replacing v by u + t(v — u) in (7c) and dividing by t, we obtain V t e ]0, 1], a(u,v-u)+

\ (2-44)

Since O e C 1 and is convex, we have <Wu + t(v — M)) — <1>(M)

lim —

>1

\1

=

^(U)(,, _ M) a .e.,

(2.45)

f

t->o 00

^HXo _ H) < ^ + ^ - ».)) -
(2 46)

By Proposition 2.1, we have <^(M), u(j>(u)e L\Q). Hence ^(u)(f - tOeL^fi) and (u) — O(M) e L'(£2), Vu e Lcc(f2) n K Then, using the Lebesgue dominated convergence theorem, it follows from (2.45) and (2.46) that r O(u + t(y - «)) - 4>(M)

f

Using the above relation and (2.44), we obtain (2.43). This proves that (7c) => (2.43). (2) We will now prove that (2.43) => (n). Let u be a solution of (2.43). Since $ is convex, it follows that - O ( M ) = O(0) - 0>(M) > <j){u)(O - u) = -0(«)u. This implies 0 < 0>(w) < «(«) and $(u) e L^fl). Let w 6 L^fl) n K Then, from the inequality 4>(u)(v — u) <
j(v) - j(u),

which combined with (2.43) and (u)v dx = L(v),

V v e L°°(Q) n F,

« e V,

by <j)(u) e

L^O). (2.47)

PROOF. (1) (TC) implies (2.47). Let v e V n L°°(fi). If u is the solution of (71), then u is also the unique solution of (2.43).

118

IV Applications of Elliptic Variational Inequality Methods

Let xn be denned by (2.29). Then xnu e V n Lm(Q). Replacing v by xnu + v in (2.43), we obtain a(u, v) +

4>(u)v dx + a(u, xnu — u) + J JJ

(f>(u)(xnu — u) dx > L(v) + L(xnu — u),

JQ

V c e F n L°°(n).

(2.48)

From (2.29)-(2.32) it follows that lim a(u, xnu — u) = 0, (2.49) lim L(xnu — u) — 0, lim 4>(u)(xnu -u)

= Q a.e.,

(2.50)

0 < 4>(u)(u - xnu) < u(j)(u) a.e..

(2.51)

Then, by the Lebesgue dominated convergence theorem and (2.50), (2.51), we obtain lim 4>(u)(xnu - u) = 0 strongly in L^fi).

(2.52)

n-* ao

Then (2.48), (2.49), and (2.52) imply a(u, v) + I 4>(u)v dx > L(v),

V t e F n LCO(O).

Since the above relation also holds for — v, we have a(u, v) + I* 4>(u)v dx = L(v),

V v e V n L°°(Q).

(2.53)

By Proposition 2.1 we have 4>{u) e L^fi); combining this with (2.53) we obtain (2.47). This proves that (TC) => (2.47). (2) (2.47) implies (n). We have a(u, v) + J <j)(u)v dx = L(v),

Vv e V n

Lx(il).

Then a(u, xnu)+

I 4>{u)xnu dx = L(xnu),

V n.

(2.54)

Since Tn« -> u strongly in V, {|n cf>(u)xnu dx]n is bounded. But (u)xnu > 0 a.e.. Hence we find that 4>(u)xnu is bounded in L'(fi). We also have lim,,^ xnu(j)(u) = u(f>(u) a.e.; hence, by Fatou's lemma, we have u4>(u) g Ll(il). But now we observe that 0 < 4>(u)xnu < ucf>(u).

(2.55)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

119

Hence, by the Lebesgue dominated convergence theorem, lim

4>(u)Tnu dx =

4>(u)u dx,

which along with (2.54) yields a(u, u) + f (p(u)u dx = L(u).

(2.56)

•In

Then by substracting (2.56) from (2.47) we obtain a(u, v — u) +

<j>iu){v — u) dx = L(v — u), V c e F n L°°(fi),

u e V, u(j)(u) e L^fi), (2.57)

and obviously (2.57) implies (2.43). This completes the proof of the lemma.

•

Corollary 2.2. / / u is the solution of (n), then u is also a solution of (P). PROOF. We recall that V* = H ^ ^ Q ) c S>'(°-) and that a(u, v) = (Au, v), V u, v e V and L(v) =
f (u) u)v dx = {f,

-'a -'a

t>>,

v e ®(fi).

(2.58)

From (2.58) it follows that Au + 0(«) = / i n @'(O.);

(2.59)

since Au a n d / e F*, we have 0(u) e F*. Hence
= V, there exists a sequence {vn}n,vn e ®(Q),

lim vn = v strongly in V.

(2.60)

n-* oo

Let us define wn by (see Fig. 2.2) wn = min(u+, yn+) - min(«", v~).

(2.61)

120

IV Applications of Elliptic Variational Inequality Methods

Figure 2.2 Then wn has a compact support in SI,

(2.62) (2.63)

IKIIt-tffl ^ Mli/»(n>> and it follows from Chapter II, Corollary 2.1 that lim wn = v strongly in V.

(2.64)

From (2.63) and (2.64) we obtain lim,,^ vvn = v for the weak* topology of L°°(fl). Thus we have proved that * = ( c e L°°(fi) n V, v has compact support in Q} m

is dense in L (fi) n V for the topology given in the statement of the lemma. Let ve"U and (pn)n be a mollifying sequence (see Chapter II, Lemma 2.4). Define vn by v(x)

if x e fi,

(2.65) (2.66)

Vn = Pn * V.

Then vn e 3>(UN),

lim SB = v strongly in i/1([RN),

Dn has a compact support in Q, for n large enough.

(2.67) (2.68)

Let vn = vn\n; then for n large enough, ;>„ e <3(Q), and lim,,-,^ vn = v strongly in V. ||L=o(n), it follows from (2.66) that Since IICIIi^duK) •)n(x)\ < \ pn(x-

y)\v(y)\dy<

\\v\\

(2.69)

From this it follows that

< INI

(2.70)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

121

Summarizing the above information, we have proved that V t e L™(£2) n V, there exists a sequence (vn\, vn e @{Q), V n, such that lim vn = v strongly in V, K I I L - W , ^ Nli-.

Vn.

(2.71) (2.72)

Hence from (2.71) and (2.72) we find that vn -> v in L°°(Q) weak *. This completes the proof of the lemma. • Theorem 2.2. Under the above hypothesis on V, a(-, •), L(-), arcd $(•)» problems (n) and (P) are equivalent. PROOF. We have already proved that (n) implies (P). We need only prove that (P) implies (n). From the definition of (P) we have a(u, v) + <0(tO, »> = L(v),

ViieF,

u e V, <j>{u) 6fl"'(Q) n L^fl).

(2.73)

From (2.73) it follows that . a(u, v ) +

f (t>(u)v dx = L ( D ) ,

V c e ®(0).

(2.74)

If u e F n Lco(f2), from Lemma 2.5 we know that there exists a sequence (vn\, vn e such that lim vn = v strongly in V,

(2.75)

n-*oo

lim vn = v in L*(n) weak *.

(2.76)

n-* cc

Since vn e 3i(Q), from (2.74) we have a{u,vn)+ U(u)vndx = L(vn).

(2.77)

From (2.77) it follows that lim,,^ a(u, vn) = a(u, v), limn^^ L(vn) = L(v), and since 4>{u) e L 1 ^ ) , (2.76) implies lim

4>(u)vn dx =

(j>(u)v dx.

Thus, taking the limit in (2.77), we obtain a(u, v) + f 4>(u)v dx = L(y),

V t e F n L°°(Q).

Therefore (P) implies (2.47) which in turn implies (;t). This completes the proof of the theorem. • EXERCISE 2.2. In IR2, find a function v such that v e H~ 1(Q) n L ^ Q ) , v $ L"(Q), V p > 1, where Q is some bounded open set in IR2.

122

IV Applications of Elliptic Variational Inequality Methods

EXERCISE 2.3. Prove that if u > 0 a.e., then
2.4. Some comments on the continuous problem We have studied (P) and (n) with rather weak hypotheses, namely, <j) e C°(U) and nondecreasing, and / e V*. The proof we have given for the equivalence between (P) and (7r) can be shortened by using more sophisticated tools of convex analysis and the theory of monotone operators (see Lions [1] and the bibliography therein). However, our proof is very elementary, and some of the lemmas we have obtained will be useful in the numerical analysis of the problem (P). Regularity results for problems a little more complicated than (P) and (TT) are given in Brezis, Crandall, and Pazy [1]; in particular, for 2 / G L 2 ( Q ) and with convenient smoothness assumptions on A, the H (Q)regularity of u is proved.

2.5. Finite element approximation of (n) and (P) 2.5.1. Definition of the approximate problem Let O be a bounded polygonal domain of U2 and let &], be a triangulation of Q. satisfying (2.21)-(2.23) of Chapter II. We approximate V by Vh = {vh e C°(Q), vh\r = 0, vh\T e i \ ,

VTefJ.

Then it is natural to approximate (P) and (n), respectively, by

a(uh,vh)+

I 4>{vh)vh dx = L(vh),

V»,eF», uheVh

(PJ)

and a(uh, vh - uh) + j(vh) - j(uh) > L(vh - uh),

V vheVh,

uheVh

{nf)

with j(vh) =

®(vh) dx.

Obviously (P%) and (n%) are equivalent. From a computational point of view, we cannot generally use (Pf) and (n%) directly, since they involve the computation of integrals which cannot be done exactly. For this reason we shall have to modify (n*) and (P*) by using some numerical integration procedures. Actually we shall have to approximate a(-, •), L(-), andj(0- Since the approximation of (•, •) and L(-) is studied in Ciarlet [1, Chapter 8], we shall assume that we still work with a(-, •) and L(-), but we shall approximate j(-).

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

123

Figure 2.3

M17

To approximate X 0 w e shall use the two-dimensional trapezoidal method. Hence, using the notation of Fig. 2.3, we approximate j ( 0 by

JM=

I

" " ^ ^ Z*(»»(Mir)),

VvheVh.

(2.78)

Actually 7fc(uA) may be viewed as the exact integral of some piecewise constant function. Using the notation of Chapter II, Sec. 2.5, assume that the set I.h of the nodes of 3~h has been ordered by i = 1, 2 , . . . , Nh, where Nh = Card(Lh). Let Mt e 2,h. We define a domain Q ; by joining, as in Fig. 2.4, the centroids of the triangles, admitting Mt as a common vertex, to the midpoint of the edges admitting Mt as a common extremity (if M ; is a boundary point, the modification of Fig. 2.4 is trivial). Let us define the space of piecewise constant functions

Lk = - L l ^ = t n,xt, /<« e R, i = 1, 2,..., Nh\

(2.79)

where %t is the characteristic function of Q ; . We then define qh: C°(Q) n #J(Q) -> Lft by

9»i; = £ v(Mt)Xi.

(2.80)

i=i

Then it follows from (2.79) and (2.80) that =

\

h)

dx,

Figure 2.4

V^K,

(2.81)

124

IV Applications of Elliptic Variational Inequality Methods

implying that MVH) = j(qhvh),

VvheVh.

(2.82)

Then we approximate (P) and (n) by a{uh, vh) +

(t>(qhuh)qhvh dx = L(vh),

V vh e Vh, uheVh

(Pft)

Jn and a(uh, vh - uh) + jh(vh) - jh(uh) > L(vh - uh),

V vheVh,

uheVh.

(nh)

Then: Theorem 2.3. Problems (Pft) and (nh) are equivalent and have a unique solution. EXERCISE

2.4. Prove Theorem 2.3.

2.5.2. Convergence of the approximate solutions Theorem 2.4. If, as h-> 0, the angles of ^h are uniformly bounded below by 90 > 0, then lim \\uh — u\\r = 0, where u and uh are, respectively, the solution of(P) and (Ph). PROOF. Since j(-) is not continuous on V, the result of Chapter I, Sec. 6 on the approximation of the EVI of the second kind cannot be applied directly. However, the proof of convergence follows the same lines as in Theorem 6.3 of Chapter I. (1) A priori estimates for uh. Taking vh = 0 in (;tft), we obtain (2.83) 0<

J

®(qhuh)dx<^l*.

Jn

(2.84)

a

(2) Weak convergence ofuh. From (2.83) and from the compactness of the injection of V in L2(Q), it follows that we can extract from (uh)h a subsequence, still denoted by (uh)h, such that uh -> u* weakly in V, (2.85) uh -> u* strongly in L2(Q), uh -* u* a.e. in

(2.86) ft.

(2.87)

Admitting, for the moment, the following inequality: Uh% - VHWLHSI) ^ -j IIVf ll^o) x u.(n).

VcjePi,

V p with 1 < p < oo, (2.88)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

125

it follows from (2.83) and (2.86) that strongly in L2(Q).

qhuh^u*

(2.89)

Then, modulo another extraction of a subsequence, we have qhuh-> u* a.e. mil,

(2.90)

from which it follows that d>((jh uh) -»
(2.91)

Then taking v e S>(£2), it follows from Ciarlet [1], [2] and Strang and Fix [1] that under the assumptions on 2Th of the statement of the theorem, we have ^

)

(

(2.92)

)

2

IIrhv - v||L.(n) < Ch 1|v||„,,„<„,,

V v e @(Q),

(2.93)

where C is a constant independent oft) and h and rh is the usual linear interpolation operator over &~h. Moreover, (2.88) with p = + oo, (2.92), and (2.93) imply hm \\qhrhv - v\\L.m = 0,

V » e S>(fi).

(2.94)

Taking yh = rhv in (7th), we obtain a(uh, uh) +

-'n

®(qhuh) dx < a(uh, rhv) +

•'a

V c e 2(Q). (2.95)

F r o m (2.85), (2.89), a n d L e m m a 2.1 we have a(u*, u*) + (" (qhrhv) dx = f o •'n •'n

V»£

Then in the limit in (2.95) we obtain a(u*,

u*) + j{u*) < a(u*, v) + j(v) - L(v - u*\

V n e 3>(Q).

(2.96)

From Fatou's lemma applied to (2.84) and (2.91) we obtain
(2.97)

Then from (2.96) and (2.97) it follows that u* satisfies a(u*, v-u*)+

j(v) - j(u*) > L(v - u*),

V » e @(Q), u* e V, d>(u*) 6 L'(fl). (2.98)

We now take v e V n L co (n); from Lemma 2.5 it follows that there exists a sequence (»„)„, vn e @(ti), such that lim vn = v strongly in V, (2.99) lim vn = v in Lm(Q) weak *.

(2.100)

126

IV Applications of Elliptic Variational Inequality Methods

From (2.98) we have a(u*, vn - «*) + j{vn) - j(u*) > L{vn - u*),

V n, u* e V, O(u*) 6 L\Q).

(2.101)

From (2.99) we obviously have lim a(u*, vn — u*) = a(u*, v — u*), n-*oo

lim L(vn — u*) = L(v — u). Since un -* v in the weak* topology of L°°(£X), we have a constant C such that K l U n , < C,

Vn.

(2.102)

Moreover, for some subsequence, (2.99) implies lim vn = v a.e. in O.

(2.103)

From (2.103) we obtain <$(vn) -> cc

n) dx =

n—w •'n

Therefore, taking the limit in (2.101), we obtain a(u*, v-u*)+

j(v) - j(u*) > L(v - «)*,

'iveVn

L°°(n), u* e V, <£(«*) e L\O).

(2.104)

Since from Lemma 2.2 we know that (2.104) is equivalent to (n), we have proved that u* = u, where u is the solution of (71). From the uniqueness of the solution of (n), it follows that the whole sequence (uh)h converges to u. (3) Strong convergence of(uh)h. From the K-ellipticity of a(-, •) and from the variational inequality satisfied by uh, we obtain all"* - "IIK + ;»(«*) ^ «(«* - «. «* - «) + jh(uh) < -a(uh, u) + a(u, u) + a(uh, uh) -a(u, uh) +jh(uh) < - a(uh, u) + a(u, u) + a(uh, rhv) + jh(rhv) - L(rhv - uh) -a(u,uh), V»e®(Q). (2.105) Using the various convergence results of part (2), from (2.105) we obtain X") ^ lim inf jh(uh) < lim inf(a||uft - u\\l + jh(uh)) < l i m s u p ( a | | « f c - u\\y + jh(uh)) < a(u, v - u) + }(v) - L(v - u),

V c e ®(Q). (2.106)

As in part (2), using the density of &{il) in L°°(Q) n V (for the strong topology of V and the weak * topology of L^fi)), we find that (2.106) also holds for all v e VnLx(£T). Then j(u) < lim inf jh(uh) < lim inf(a||wft - u\\v + jh(uh)) < lim sup(a||Mft - u\\r + jh(uh)) < a(u, xnv - u) + j(znv) - L{xnv - v),

VPEF. (2.107)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

127

Using the properties of xnv, it then follows from (2.107), by taking the limit as n -» co, that (2.106) also holds for all v e V. Hence, by taking v = u, we obtain j(u) < lim inf jh(uh) < lim i n f ( a K - u\\v + jh(uh)) < lim sup(a||u h - u\\v + jh(uh))

<j(u\

which implies lira jh(uh)

=j(u),

lim \\uh - u\\v = 0.

This proves the theorem, modulo the proof of (2.88).

D

We now prove (2.88). Lemma2.6. We have Vp, 1 < p < oo, \\qhvh - vh\\Lm) < f/i||V^ ILp^xt^n) where qh, vh are as before. PROOF. We use the notation of Sec. 2.5.1. We have (see Fig. 2.5) | vh(M) - qh vh(M) | = | vh(Md ~ vh(M) |,

V M e ilt n T.

(2.108)

But since vh\T e Pv we have vh(M) = v^Mt) + M~M -\7vh,

y

Meilt^T,

from which, combined with (2.108), it follows that \qhvh(M) - vh(M)\ < | But from the definition offt,we have \MtM\ < fft, V T, so that we finally have \qhvh(x) - vh(x)\ <±h\Vvh(x)\ a.e. in Q,V vheVh. This implies hhVh - vh\\ma)

This proves the lemma.

<

ih\\Vvh\\mil)xLP(il).

•

Remark 2.4. We have not considered the problem of error estimates. This problem is discussed in Glowinski [5].

Figure 2.5

M,

128

IV Applications of Elliptic Variational Inequality Methods

Remark 2.5. The numerical analysis of problems like (P), but with much stronger hypotheses on a( •, •), (f>, and / , is considered in Ciarlet, Schultz, and Varga [1].

2.6. Iterative methods for solving the discrete problem 2.6.1. Introduction In this section we briefly describe some iterative methods which may be useful for computing the solution of (Ph) (and (nh)). Actually most of these methods can be extended to other nonlinear problems. Many of the methods to be described here can be found in Ortega and Rheinboldt [1]. A method based on duality techniques will be described in Chapter VI. 2.6.2. Formulation of the discrete problem Here we are using the notation of the continuous problem. Taking as unknowns the values of uh at the interior nodes of 3~h, the problem (P,,) reduces to the finite-dimensional nonlinear problem Au + D0(u) = f,

(2.109)

where A is a N x N positive definite matrix, D is a diagonal matrix with positive diagonal elements dt and where u = {uu ..., uN} e UN, f e UN, 0(u) e UN with ((u)); = , f, we can see that problem (2.109) has a unique solution. 2.6.3. Gradient Methods The basic algorithm with constant step (see Cea u°

u"

+1

= u" --

N

eU

pS~\Au" +

Clj) is given by

given,

D(Xu") --f),

(2.110) P > 0.

(2.111)

In (2.111), S is a symmetric positive-definite matrix; a canonical choice is S = identity. But in most problems it will yield a slow speed of convergence. If A is symmetrical, the natural choice is S = A, and if A # A*, we can take S = (A + A*)/2. For the convergence of u" to u (where u is the solution of (2.109)), it is sufficient to have (v) — f is the gradient at v of the functional J(v) = |(Av, v) + £f = 1 d^v^ - (f, v), where (-, •) denotes the usual inner product of UN and O(f) = f0 <j)(x) dx.

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

129

Remark 2.7. In each specific case, p has to be determined; this can be done theoretically, experimentally, or by using an automatic adjustment procedure which will not be described here. Remark 2.8. Let us define g" by g" = Au" + D
(2.112)

Suppose A = A*; then if we use (2.110), (2.112) with pn defined by J(u--pI1S-1r)^J(u"-pS-1g")>

VpeR,

pneU,

(2.113)

the resulting algorithm is, for obvious reasons, called the steepest descent method. This algorithm is convergent for £ C°(K). We observe that at each iteration the determination of pn requires the solution of a one-dimensional problem; for the solution of such one-dimensional problems, see Householder [1], Polak [1], and Brent [1]. Remark 2.9. At each iteration of (2.110), (2.111) or (2.110), (2.112), we have to solve a linear system related to S. Since S is symmetric and positive definite, this system can be solved using the Cholesky method, provided the S = LL* factorization has been done. From a practical point of view, it is obvious that the factorization of S will be made in the beginning once and for all. Then at each iteration we only have to solve two triangular systems, which is a trivial operation. 2.6.4. Newton's method The Newton's algorithm is given by (for sufficient conditions of convergence, see Ortega and Rheinboldt [1]) u° £ UN given, u"

+1

= (A + D f t O r ^ D ^ u " ) ! ! " - D0(u") + f),

(2.114) (2.115)

where '(v) denotes the diagonal matrix

Since ' > 0. This implies that A + D'(u") depends on n, this method may not be convenient for large N; for this reason a large number of variants of the Newton's method

130

IV Applications of Elliptic Variational Inequality Methods

which avoid this drawback have been designed. In this regard, let us mention, among others, Broyden [1], [2], Dennis and More [1], Crisfield [1], Matthies and Strang [1], O'Leary [1], and also the references therein; most of the above references deal with the so-called quasi-Newtori's methods. Remark 2.11. The choice of u° is very important when using Newton's method. 2.6.5. Relaxation and over-relaxation methods. In this section we shall discuss the application of relaxation methods for solving the specific problem (2.109); the algorithms to be described below are, in fact, particular cases of a large family of algorithms to be discussed in Chapter V. In this section we use the following notation:

f = {/i,/2, • • • , / * } •

Since A is positive definite, we have au > 0, V i = 1, 2 , . . . , N. Here we will describe three algorithms. ALGORITHM 1

u°eUN given;

(2.116)

n+1

then with u" known, we compute u +i

auu?

1

, component by component, using

= ft - E«u"" + 1 - l>u«".

+ d^mr )

u 1 + 1 = u1 + co(u1+1

-«?),

i=l,2,...,N.

(2-117) (2.118)

Since an > 0, e C°(U) and is a nondecreasing function, we can always solve (2.117), and the solution is unique. If co = 1, we recover an ordinary relaxation method (see Cea [1]). If A = A* and since is C° and nondecreasing, the solution u" of (2.116)—(2.118) converges to the solution u of (2.109). If, in (2.109), A is nonsymmetrical and co # 1, there are certain sufficient conditions for the convergence of u" to u, where u" is given by (2.116)—(2.118) and where u is the solution of (2.109) (see Ortega and Rheinboldt [1] and S. Schechter [1], [2], [3]). 2. This algorithm is the variant of (2.116)—(2.118) obtained by replacing (2.117) and (2.118) by

ALGORITHM

anu1+1 + Jf, \

- I fly«J+' - £ ayMj) for i = 1, 2,..., JV. ]
}>i

I

(2.119)

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

131

Remark 2.12. If co = 1 and/or cj> is linear, the two algorithms coincide. In the general case the convergence of (2.116), (2.119) seems to be an open question. However, from numerical experiments it seems that the second algorithm is more "robust" that the first, perhaps because it is more implicit. Furthermore, it can be used even if cf> is defined only on a bounded or semibounded interval ]a, /?[ of U such that (<*) = - oo, (/?) = + oo; in such a case, if c/> e C°(]a, /?[) and (j) is increasing, (2.109) has still a unique solution, but the use of (2.116)— (2.118) with co > 1 may be dangerous. Remark 2.13. If c/> e C\U), an efficient method for computing u"+ i in (2.117) and M"+ 1 in (2.119) is the one-dimensional Newton's method. Let g £ C^IR). In this case, Newton's algorithm for solving the equation g(x) = 0 is x° £ U given,

(2.120)

x "+i = x "_^gl.

(2.121)

If in the computation of u"+1 and M"+ 1, we use only one iteration of Newton's method, starting from u", then the resulting algorithms are identical and we obtain: u°eUN given,

(2.122)

au + fl; CO (M; )

(2.123) In S. Schechter [1], [2], [3], sufficient conditions for the convergence of (2.122), (2.123) are given. Remark 2.14. If co > 1 (resp., co = 1, co < 1), the previous algorithms are overrelaxation (resp., relaxation, under relaxation) algorithms. Remark 2.15. In Glowinski and Marrocco [1], [2], we can find applications of relaxation methods for solving the nonlinear elliptic equations modeling the magnetic state of electrical machines. 2.6.6. Alternating-direction methods. In this section we take p > 0. Here we will give two numerical methods for solving (2.109). First method u°eUN given;

(2.124)

132

IV Applications of Elliptic Variational Inequality Methods

knowing u", we compute u" + 1 / 2 by ) + f; then we calculate u" pu"

+1

+1

(2.125)

by + D
(2.126)

For the convergence of (2.124)-(2.126), see R. B. Kellog [1]. Second method u°eUN given;

(2.127)

knowing u", we compute u" + 1 / 2 fey pu »+i/2 +

Au n + 1 / 2 = pu" - D0(u") + f;

(2.128)

= pu" - Au"+ 1/2 + f.

(2.129)

then we calculate u" +1 by

Using the results of Lieutaud [1], we can prove that, for all p > 0, u" + 1 / 2 and u" converge to u if we suppose that A and satisfy the hypotheses given in Sec. 2.6.2; the same convergence result holds if we exchange the role of A and in (2.128), (2.129). Remark 2.16. At each iteration we have to solve a linear system whose matrix is constant, since we use a constant step p. This is an advantage from the computational point of view (cf. Remark 2.9). We also have to solve a nonlinear system of N equations, but in fact these equations are independent of each other and reduce to N nonlinear equations in one variable, which can be solved easily. 2.6.7. Conjugate gradient methods. In this section we assume A = A* (if A ^ A*, we can also use methods of conjugate gradient type). For a detailed study of these methods, we refer to Polak [1], Daniel [1], and Concus and Golub [1]. If the functional J defined in Remark 2.6 is not quadratic (i.e., cf> is nonlinear), several conjugate gradient methods can be used. Let us describe two of them, whose convergence is studied in Polak [1]. Let J be given by J(v) = i(Av, v) + X dtdXvd - (f, v), i=i

where CE>(t) = f/0 C/>(T) di, (j) being a nondecreasing continuous function on U with (0) = 0. Let S be a N x N symmetric positive-definite matrix.

2. Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations

133

First method (Fletcher-Reeves) u°eUN given; g° = S" W

(2.130)

+ D0(n°) - f), w° = g°.

(2.131) (2.132)

Then assuming that a" and w" are known, we compute un+1 by u" + 1 = u " - p n w " ,

(2.133)

where pn is the solution of the one-dimensional minimization problem J(u" - pnwn) < J(u" - pw"),

V p e U, pneU.

(2.134)

Then g"+1 = S'HAu""1"1 + D0(u"+1) - f),

(2.135)

and compute w" +1 by w" +1 = g" +1 +2 n w",

(2.136)

where (Sen+1 K

s" +1 ) (2137)

(sg",g«)

Second method (Polak-Ribiere). This method is like the previous method, except that (2.137) is replaced by g"+1g"+1-g") )

•

(2138)

Remark 2.17. For the computation of pn in (2.134), see Remark 2.8 (and also Shanno [1]). Remark 2.18. From Polak [1] it follows that if (f> is sufficiently smooth, then the convergence of the above algorithms is superlinear, i.e., faster than the convergence of any geometric sequence. Remark 2.19. The above algorithms are very sensitive to roundoff errors; hence double precision may be required for some problems. Moreover, it may be convenient to periodically take w" = g" (see Powell [2] for this restarting procedure problem). Remark 2.20. At each iteration we have to solve a linear system related to S; Remark 2.9 still applies to this problem.

134

IV Applications of Elliptic Variational Inequality Methods

2.6.8. Comments. The methods of this section may be applied to more general nonlinear systems than (2.109). They can be applied, of course, to finite-dimensional systems obtained by discretization of elliptic problems like — V • (ao(x)Vj<) + /? • VM + (j)(x, u) = f in Q,

plus suitable boundary conditions

where, for fixed x, the function t -> 4>(x, t) is continuous and nondecreasing on U.

3. A Subsonic Flow Problem 3.1. Formulation of the continuous problem Let Q be a domain of UN (in applications, we have N = 1,2,3) with a sufficiently smooth boundary T. Then the flow of a perfect compressible irrotational fluid (i.e., V x v = 0, where v is the velocity vector of the flow) is described by OinQ,

[(y + D/(y -

(3.1)

J

with suitable boundary conditions. Here: • • • • •

(j> is a potential and V0 is the velocity of the flow; p{4>) is the density of the flow; p0 is the density at V = 0; in the sequel we take p0 = 1; y is the ratio of specific heats (y = 1.4 in air); C^ is the critical velocity.

The flow under consideration is subsonic if < C # everywhere in Q.

(3.3)

If | V(j> | > C^ in some part of Q, then the flow is transonic or supersonic, and this leads to much more complicated problems (see Chapter VII for an introduction to the study of such flows). Remark 3.1. In the case of a subsonic flow past a convex symmetrical airfoil and assuming (see Fig. 3.1) that v^ is parallel to the x-axis (Q is the complement of the airfoil in IR2 and d(j)/dn\r = 0), H. Brezis and Stampacchia [1] have proved that the subsonic problem is equivalent to an EVI of the first kind in the hodograph plane (see Bers [1] and Landau and Lifschitz [1] for the hodograph transform). This EVI is related to a linear operator, and the corresponding convex set is the cone of non-negative functions.

3. A Subsonic Flow Problem

135

Figure 3.1

In the remainder of Sec. 3 (and also in Chapter VII), we shall work only in the physical plane, since it seems more convenient for the computation of nonsymmetric and/or transonic flows. For the reader who is interested by the mathematical aspects of the flow mentioned above, see Bers [1] and Brezis and Stampacchia [1]. For the physical and mechanical aspects, see Landau and Lifschitz [1]. Additional references are given in Chapter VII.

3.2. Variational formulation of the subsonic problem

Preliminary Remark. In the case of a nonsymmetric flow past an airfoil (see Fig. 3.2) the velocity potential has to be discontinuous, and a circulation condition is required to ensure the uniqueness (modulo a constant) of the solution of (3.1). If the airfoil has corners (as in Fig. 3.1), then the circulation condition is related to the so-called Kutta-Joukowsky condition from which it follows that for a physical flow, the velocity field is continuous at the corners (like 0 in Fig. 3.2). For more information about the Kutta-Joukowsky condition, see Landau and Lifschitz [1] (see also Chapter VII). For the sake of simplicity, we shall assume in the sequel that either Q is simply connected, as is the case for the nozzle of Fig. 3.3, or, if Q is multiply connected, we shall assume (as in Fig. 3.1) that the flow is physically and geometrically symmetric, since in this case the Kutta-Joukowsky condition is automatically satisfied. In the sequel we assume that the boundary conditions associated with (3.1), (3.2) are the following: g0overTr0,p-^

l

= 9,

Figure 3.2

(3-4)

136

IV Applications of Elliptic Variational Inequality Methods Figure 3.3

n

where F o , F t c F with F o n F x = 0 , F o u F t = F. Then the variational formulation for the flow problem (3.1), (3.2), (3.3), (3.4) is • Vv dx

gtvdr,

VveV0,

(3.5)

where (3.6)

Vto={veH\n),v\ro

= g0}.

(3.7)

If g0, g1 are small enough, we can prove that (3.5) has a solution such that <M

C # a.e..

When solving a practical flow problem, we may not know a priori whether the flow will be purely subsonic or not. Therefore, instead of solving (3.5), it may be convenient to consider (and solve) the following problem:

~4>)dx > f dl{v - 4>)dT,

VveKd,

4>eKd, (3.8)

where (3.9) The variational problem (3.8), (3.9) is an EVI of the first kind, but we have to observe that unlike the EVI's of Chapters I and II, it involves a nonlinear partial differential operator, namely, A defined by A{4>) = - V Remark 3.2. In practical applications we shall take 6 as close as possible to C*. Remark 3.3. Problem (3.8), (3.9) appears as a variant of the elasto-plastic torsion problem of Chapter II, Sec. 3.

3. A Subsonic Flow Problem

137

3.3. Existence and uniqueness properties for problem (3.8) In this section we shall assume that meas(T 0 ) > 0. To prove that (3.8) is well posed, we will use the following: Lemma 3.1. The function

is convex if ^ E [0, C^\, concave if

and strictly convex if £ £ [0, C^[_. EXERCISE 3.1. Prove Lemma 3.1.

We can now prove: Theorem 3.1. Assume that Q is bounded and that g0, gx are sufficiently smooth and small. Then (3.8) has a unique solution. PROOF. Since Q is bounded, and if g0 is sufficiently smooth and small, we observe that K6 is a closed, convex, and nonempty bounded subset of H1(Q.) (consisting of uniformly Lipschitz continuous functions). Define J ( ) by

From Lemma 3.1 it follows that J(-) is strictly convex over Ks. It is easy to check that J(-) is continuous and Gateaux-differentiable over Kd with (J'(v), w) = f p(v)Vv -Vwdx-

\ QiWdT.

(3.11)

Since Kd is a closed convex nonempty subset of Hl(Q) and J ( ) is continuous and strictly convex over Kt, from standard optimization theory in Hilbert spaces (see, e.g., Cea [1], [2]), it follows that the minimization problem J(u)<J(v),

VceXj,

ueKd,

(3.12)

has a unique solution. Moreover, since J(-) is differentiable, the unique solution of (3.12) is characterized (see Cea [1], [2]) by (J'(«), v - u) > 0,

VceKj,

from (3.11), this completes the proof of the theorem.

UEKS;

138

IV Applications of Elliptic Variational Inequality Methods

Remark 3.4. Suppose that F o = 0. Then defining K5 by Ks={ve

H'iQ), \Vv\<3
a.e., v(x0) = v0}

with x 0 e Q and v0 arbitrarily given, we can prove that if Q is bounded and gfx is sufficiently smooth, then

f p()V • V(«> - { g i ( v - ) dT,

VveKd,

e K6, (3.13)

Jn Jr has a unique solution (if (f> is a solution of (3.13); then cf> + C is the unique solution of the similar problem obtained by replacing v0 by v0 + C). EXERCISE

3.2. Prove the statement of Remark 3.4.

Remark 3.5. In all the above arguments we assumed that Q. is bounded. We refer to Ciavaldini, Pogu, and Tournemine [1] which contains a careful study of the approximation of subsonic flow problems on an unbounded domain Q^ by problems on a family (£!„)„ of bounded domains converging to Qx (actually they have obtained estimates for (p^ — „). The above EVFs will have a practical interest if we can prove that in the cases where a purely subsonic solution exists, then for 3 large enough it is the solution of (3.8); actually this property is true and follows from: Theorem 3.2. Assuming the same hypotheses on Q, g0, qx as in Theorem 3.1, and that (3.1), (3.2), (3.4) has a unique solution in H\Q.) with |V0|<<5 O
(3.14)

then (f> is a solution o/(3.8), (3.9), V 3 e [<50, C^Q Conversely, if the solution of (3.8), (3.9) is such that \ V $ | < 60 < 3 < C* a.e., then (f> is a solution of (3.1), (3.2), (3.4). PROOF. (1) Let <j> e H^Q) satisfy (3.1), (3.2), (3.4), and (3.14). If v e Vo, then using Green's formula, it follows from (3.1), (3.2), (3.4) that f p(0)V0 • Vv dx = f QivdT,

MveVo.

(3.15)

From (3.4), (3.15) and from the definition of Vgo it follows that f p{4>)V4> • V(v - 4 > ) d x = \ 9 l ( v - e Kd a Vgo, V 5 e [c50, C # [, it follows from (3.16) that

V v e VSo.

J

V(B - ) dx > f

9i(v

- <j>) dT,

VveKs,

<j> e Kd

*'ri

if 5 e [50, C^C; therefore 0 is the solution of the EVI (3.8), (3.9), V 5 e [<50, CJ_.

(3.16)

3. A Subsonic Flow Problem

139

(2) Define U <= Vo by U = {ve C°°(fi), v = 0 in a neighborhood of F o }. Then, if we suppose that F is sufficiently smooth, we find that the closure t/ fll(n) of U in H\il) obeys t7 HI(fi) = Vo.

(3.17)

We assume that for 5 < C^, (3.8) has a solution such that \V
(3.18)

Then V t> e [/, and for f > 0 sufficiently small, <$> + tv e Kd. Then replacing t> by <> / + ft) in (3.8) and dividing by t, we obtain f p(<^>)V0 • Vv dx > \ c,

r,

•lr1

JQ

V»E[/,

which implies f

f p(4>)V4> • Vv dx =

gtv dT,

V v e U.

(3.19)

Since &(Q.) a U, it follows from (3.19) that f p(<^))V0 • Vt; Jx = 0,

V v e @(Q),

(3.20)

i.e., which proves (3.1). Assuming (3.1) and using Green's formula, we obtain f pW>)V • VP dx = f p — v dT,

\/veU.

(3.21)

Using (3.17) and comparing with (3.19), we obtain

P

tn

i.e., (3.4), which completes the proof of the theorem.

Remark 3.6. A similar theorem can be proved for the problem mentioned in Remark 3.4. 3.4. Comments

The solution of subsonic flow problems via EVFs like (3.8) or (3.13) is considered in Ciavaldini, Pogu, and Tournemine [2] (using a stream function approach) and in Fortin, Glowinski, and Marrocco [1], Iterative methods for solving these EVFs may be found in the above references and also in Chapter VI of this book.

CHAPTER V

Relaxation Methods and Applications1

1. Generalities The key idea of relaxation methods is to reduce, using some iterative process, the solution of some problems posed in a product space V = YW= 1 ^ (minimization of functional, solution of systems of equations and/or inequalities, etc.) to the solution of a sequence of subproblems of the same kind, but simpler, since they are posed in the Vt. A typical example of such methods is given by the classical point or block Gauss-Seidel methods and their variants (S.O.R., S.S.O.R., etc.). For the solution of finite-dimensional linear systems by methods of this type, we refer to Varga [1], Forsythe and Wasow [1], D. Young [1] and the bibliographies therein. For the solution of systems of nonlinear equations, we refer to Ortega and Rheinboldt [1], Miellou [1], [2], and the bibliographies therein. For the minimization of convex functionals by methods of this kind, let us mention S. Schecter [1], [2], [3], Cea [1], [2], A. Auslender [1], Cryer [1], [2], Cea and Glowinski [1], Glowinski [6], and the bibliographies therein. The above list is far from complete. The basic knowledge of convex analysis required for a good understanding of this chapter may be found in Cea [1], Rockafellar [1], and Ekeland and Temam [1].

2. Some Basic Results of Convex Analysis In this section we shall give, without proof, some classical results on the existence, uniqueness, and characterization of the solution of convex minimization problems. Let (i) V be a real reflexive Banach space, V* its dual space, (ii) K be a nonempty closed convex subset of V, 1

In this chapter we follow Cea and Glowinski [1] and Glowinski [6].

2. Some Basic Results of Convex Analysis

141

(iii) J o : V -> U, be a convex functional Frechet or Gateaux differentiable2 on V, (iv) Jt: V -+M, be 3 a proper l.s.c. convex functional. We assume that K n D o m ^ ) # 0 , where Don^Jj) = {v\v e V, J^v) e K}. We define J:V^U

by J = J0 + Ji and assume that lim

J(v)=

+oo.

(2.1)

|M|-> + 00 VEK

Under the above assumptions on K and J, we have the following fundamental theorem. Theorem 2.1. The minimization problem J(u) < J(v),

VceK,

ueK,

(2.2)

has a solution characterized by <J'o(u), D - u> + JiOO - Jtiu) > 0,

VceJC,

M e X.

(2.3)

4

77HS solution is unique if J is strictly convex.

Remark 2.1. If K is bounded, then (2.1) may be omitted. Remark 2.2. Problem (2.3) is a variational inequality (see Chapter I of this book). Let us now recall some definitions about monotone operators. Definition 2.1. Let A: V -> V*. The operator A is said to be monotone if

0,

Vu,veV,

and strictly monotone if it is monotone and

(A(v) - A(u), v - u) > 0,

Vu,veV,

u # v.

Let F: V -> R; the Gateaux-differentiability property means that

where <•, •> denotes the duality between V* and V and F ^ e f 7 * ; F'(v) is said to be the Gateaux-derivative (or G-derivative) of F at v. Actually, we shall very often use the term gradient when referring to F'. !

R = Ru { + 00} u {-oo}.

* i.e., J(tv + (1 - t)w) < U(v) + (1 - t)J(w), V t e ]0,1[, V v, w e Dom(J), v * w.

142

V Relaxation Methods and Applications

We shall introduce the following proposition which will be very useful in the sequel of this chapter. Proposition 2.1. Let F: V —> R be G-differentiable. Then there is equivalence between the convexity of F (resp., the strict convexity of F) and the monotonicity (resp., the strict monotonicity) of F'. To prove Theorem 2.1 and Proposition 2.1, we should use the following: Proposition 2.2. If F is G-differentiable, then F is convex if and only if F(w) - F(v) >
Vv,weV.

(2.4)

3. Relaxation Methods for Convex Functionals: Finite-Dimensional Case 3.1. Statement of the minimization problem. Notations With respect to Sec. 2, we assume that V = UN, with v = {»!,..., vN},

Vi

eU,

1 < i < N.

The following notation will be used in the sequel: JV

(M, V) = £ UiVh

\\V\\ = J(V, V).

i= l

We also assume that K = {v\v e UN, Vi e Ki = lat, bj, at < bh 1 < i < N},

(3.1)

where the a{ (resp., theft,-)may take the value — oo (resp., + oo); K is obviously a nonempty closed convex subset of RN. Furthermore, we assume that J(v) = J0(v)+ YhiVil

(3-2)

where Jo e C\UN) and is strictly convex and where the j , e C°(U) and are convex, 1 < i < N. We note that we are not assuming the differentiability of the ji. Finally, we assume that lim

J(v) = +oo.

(3.3)

|| V || -* + 00

Then, under the above assumptions and from Theorem 2.1, it follows that the problem J(u) < J(v) VveK, ueK, (3.4)

3. Relaxation Methods for Convex Functionals: Finite-Dimensional Case

143

has a unique solution characterized by (J'o(u), v-u)+

£ (/i(»i) -;<(«>)) 2: 0,

VceK,

u e X,

(3.5)

t=i

where J'0(v) denotes the gradient of Jo at v.

3.2. Description of the relaxation algorithm To solve (3.4) we can use the following relaxation algorithm: «° € K arbitrarily given;

(3.6)

n+1

then, u" being known, we compute u

, component by component, by

<J{u\+\...,utl,vhu1^,..),

VvteKh

ur'eK,,

(3.7)

where 1 < i < N. From the above assumptions on K and J, and from Theorem 2.1, we obtain: Proposition 3.1. Each subproblem (3.7) has a unique solution which is characterized by ^ « \ . . . , u r \ « ?

+

j , • • .X»( - «?+ x) + ;,<»i) -7i(«7 + ') > o,

u1+1eKi.

VveK(,

(3.8)

Remark 3.1. To compute u"+1, we can proceed as follows. First we compute u" + 1 by solving J^H!

,...,«,-_!,«,-

, Ui+1, . . . ) S Jyu-L

, . . . , M ; - i , Vt, Ui+1, . . . ) ,

\fvteU,

M?+16lR.

(3.9)

Then we project u"+1 on Kt to obtain w"+ i; so «7+

L

= PKi(u1+ J ) = Max(a ; , Minfo, S?+ x)).

(3.10)

If 7; is differentiable, then w"+x is the solution of the single-variable equation ^ ( M - 1 + 1 , . . . , « ? _ + 1 1 , S ? + 1 ) H 7 + ! , . . . ) + ^ ( M ? + 1 ) = 0.

(3.11)

Equation (3.11) can be solved by various methods (see, for example, Householder [1]).

144

V Relaxation Methods and Applications

3.3. A lemma on the monotonicity of the gradient of strictly convex functionals To prove the convergence of algorithm (3.6), (3.7), we shall use the following: Lemma 3.1. Assume that F e C^IR^) and is strictly convex. Then F is uniformly convex on the bounded sets of UN, i.e., V M > 0, 3 5M: [0, 2M] -> U+, continuous, strictly increasing, and such that dM(0) = 0, (F'(v) - F'(u), v-u)> N

Vu, veR ,

(3.12) 3M(\\v -

\\u\\ < M,

M||)

|M|<M,

(3.13)

and F(v) > F(u) + (F'(u), v-u) \fu,veUN,

\\u\\ < M,

+ &M(\\v

- M||),

\\v\\ < M.

(3.14)

PROOF. Let BM = {v\ve UN, \\v\\ < M}. For x e [0, 2M], we define d^, by ^(t)=

inf

(F(v) ~ F'(u), v - u).

||v—u|| = x u,veB\f

(?•*••>)

From the definition of 5% it follows that S°M(0) = 0

(3.16)

and (F'(v)

- F'(u),

v - u ) > 5°M(\\v

- u\\),

Vu,ve BM.

(3.17)

£

Let T 2 ] 0 , 2M]. From the continuity of {u, v} -* (F'(v) — F'(u), v — u) and from the compactness of BM x BM, it follows that there exists at least one pair {u2, v2} realizing the minimum in (3.15). Then <52f(T2) = (F'(v2) - F'(u2), v2 - u2\ and <5°,(T2) > 0

from the strict monotonicity o f f (cf. Sec. 2, Proposition 2.1). Let x^ 6 ]0, T 2 [. We define w e ] u 2 , f 2 [ by w = u2 -\

(v2 — u2).

Since 0 < x1/x2 < 1, from the strict monotonicity o f f it follows that (F'(v2) - F'(u2), v2 - u2) > [F'(u2 + -(V2\

\

T

2

U2)) - F'(u2), v2 - u2). /

/

3. Relaxation Methods for Convex Functional: Finite-Dimensional Case

145

This implies (F'(v2) - F'(u2), v2 - u2) > - (F'(w) - F\u2), w - u2) > (F'(w) - F'(u2), w - u2). (3.18) Since j|w — u2\\ = z1, (3.18) in turn implies

Applying (3.17) to {u + t(v - u), u}, it follows that (F'(u + t(v - u)), v-u)>

(F'(u), « - « ) + | ^Ktlli; - u\\),

V t e ]0, 1]. (3.19)

From the continuity of F , it easily follows that lim - 5°M(?) = 0.

(3.20)

Then from (3.20) it follows that (3.19) could be extended at t = 0. Integrating (3.19) on [0, 1], it follows that F(v) - F(u) > ( F ( » ) , v-u)+

f 5°M(t\\v - ii||) - . Jo

(3.21)

'

We also have F(u) - F(v) > (F'(v), u-v)+

f d°M(t\\v - u||) - .

(3.22)

Then, by summation of (3.21), (3.22), we obtain (F'(v) - F'(u), v - u) > 2 f d°Jt\\v - M||) Jo t (•\\v-u\l

= 2 Jo

j

SUs)~s

0.23)

Therefore the function 5M defined by 5M{x) = 2 CdUs) Jo s

(3-24)

has the required properties. Furthermore, (3.14) follows from (3.21) and from the definition of 5M. • Remark 3.2. The term forcing function is frequently used for functions such as dM (see Ortega and Rheinboldt [1]).

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V Relaxation Methods and Applications

3.4. Convergence of algorithm (3.6), (3.7) We have: Theorem 3.1. Under the above assumptions on K and J, the sequence (u")n defined (3.6), (3.7) converges, V u° e K, to the solution u of (3.4). PROOF. For the sake of simplicity, we have split the proof into several steps. Step 1. We shall prove that the sequence J(u") is decreasing. We have J(u") - J(u"+1) = £ ( J « 1 , . . . , uill

ul...)-

J « \ ..., uT \ u?+!,.. •)). (3.25)

Since u? e Kt, it follows from (2.4), (3.8) that, V i = 1 , . . . , N, we have J(ul+ \ . . . , utl

J(u\+ \...,

ul,...)-

uT \ u% „ ...)

+

J0(u\ \ ..., « T \ u1+ u ...) + jiui) - JM+1)

= J 0 ( « r ' , . . . , utl, ul...)-

(3.26)

Then (3.26) combined with (3.25) implies J(u")> J(un+l),

Vn>0.

(3.27)

Moreover, since J satisfies (3.3), it follows from (3.27) that there exists a constant M such that IMI <M, \\u"\\ <M, ||{M-1+1.---,«I

+ 1

V«,

,«?+I,...}II^M,

V(=l,...,Af,

VH.

(3.28)

Step 2. F r o m (3.8), (3.14), (3.25), (3.26), and (3.28), it follows that

J(u") - J(u"+!) > \ X 5M{ | a? + 1 - H? |).

(3.29)

The sequence J(u") is decreasing and bounded below by J(u), where w is the solution of (3.4). Therefore the sequence J(u") is convergent, and this implies lim (J(un) - J(un+1)) = 0.

(3.30)

From (3.29), (3.30), and the properties of SM, it follows that lim ( « " - u" + 1 ) = 0.

(3.31)

Step 3. Let u be the solution of (3.4). Then it follows from (3.15), (3.28) that C W + 1 ) - J' 0 (u), u"+i -u)>

5MQ\u"+1

- u\\),

which implies (J'0(u"+1) - J'0(u\ u"+l -u) + J^u"*1) - J^u) > JW+1) - JM

+ <5M(||u"+1 - M||). (3.32)

3. Relaxation Methods for Convex Functionals: Finite-Dimensional Case

147

Since u is the solution of (3.4) and un+' e K, we have (cf. (3.5)) (J'0(u), u" + l - a) + Jt{u"+i) - JM

> 0,

which, combined with (3.32), implies (io(«" + 1 ),«" + 1 - u) + JW+1)

- JM

> dM(\\u"+1 - till).

(3.33)

Relation (3.33) implies (J' 0 (u" +1 ), un+1 - u) + - W

N

) - ^(u)

tdl +1

+

1

-«II),

1

(3-34)

n

where fl? ' = { u f , . . . , uT \ u V i = 1 , . . . , N,

i+ u

...}. Since ut e Kh

+

^

it follows from (3.8) that,

' ) - yiC***) < o.

(3.35)

Therefore (3.34) and (3.35) show that

f (""+1}" af ("r 1 } r r * ~~Ui) - dMu"+1 ~u|l)- (3-36) «" +1 - u1+1\\ < \\un+1 - uunn\\\\, it follows from (3.31) that V i = 1 , . . . , N, we have Since ||«" lim ( w " + 1 - « r 1 ) = 0.

(3.37)

Since J'o 6 C°(RN), J'o is uniformly continuous on the bounded subsets of U". This property, combined with (3.37), implies, V i = 1 , . . . , N, = 0.

lim

(3.38)

n~* + oo

Therefore, from (3.28), (3.36), (3.38), and the properties of8M, it follows that lim ||u" - u\\ = 0, n-* oo

which completes the proof of the theorem.

•

3.5. Various remarks Remark 3.3. We assume that K = UN and that J = Jo (i.e., Jx == 0), where jo(v)

= ±(Av, v) - (b, v),

where beUN and

A is an N x N symmetrical positive-definite matrix.

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V Relaxation Methods and Applications

The problem (3.4) associated with this choice of J and K obviously has an unique solution characterized (cf. (3.5)) by Au = b.

(3.39)

If we apply the algorithm (3.6), (3.7) to this particular case, we obtain u° e UN, arbitrarily given;

- L.«y«r - I «u«"j.

(3.40)

1 ^ i < N.

(3-41)

The algorithm (3.40), (3.41) is known as the Ganss-Seidel method for solving (3.39) (see, e.g., Varga [1] and D. Young [1]). Therefore, when A is symmetric and positive definite, optimization theory yields another proof of the convergence of the Gauss-Seidel method through Theorem 3.1. Remark 3.4. From the above remark it follows that the introduction of overor under-relaxation parameters could be effective for increasing the speed of convergence. This possibility will be discussed in the sequel of this chapter. L e t F : F - + U. We define D(F)= {v\veV,\F(v)\ < +oo}.

(3.42)

If F is convex and proper, then D(F) is a nonempty convex subset of V. Remark 3.5. If in Sec. 3.1 we replace the conditions j , € C°(U) and jt convex, V i = 1 , . . . , N, by ji: U -> U is convex, proper, and l.s.c. and we assume Kt n DO',) ^ 0, V i = 1 , . . . , N, then the other assumptions being the same, (3.4) is still a well-posed problem and (3.5) still holds. Moreover, the algorithm (3.6), (3.7) could be used to solve (3.4), and the convergence result given by Theorem 3.1 would still hold. Remark 3.6. We can complete Remark 3.5 in the following way. We take jt as in Remark 3.5 and assume Jo strictly convex, proper, and l.s.c., D(J0) is an open set of UN and J o e C1(/>(J0))- T h e n , if D(J) n K ± 0 and if lini||i;||_ + 00 J(v) = +oo, problem (3.4) is well posed and (3.5) still holds. Moreover, algorithm (3.6), (3.7) could be used to solve (3.4). Remark 3.7. A typical situation in which algorithm (3.6), (3.7) could be used isK = UN, Jo as in Remark 3.3 and J^v) = Vf=1 a ; |i;,|,a; > 0 , V i = l,...,N.

3. Relaxation Methods for Convex Functionals: Finite-Dimensional Case

149

3.6. Some dangerous generalizations In this section we would like to discuss some of the limitations of the relaxation methods. 3.6.1. Relaxation and nondifferentiable functionals We consider K = U2 and J e C°(U2), strictly convex, defined by

J(v) = Wi + vl) + K - v2\ - (»! + v2); J is nondifferentiable on the line v1 = v2. The unique solution of J(u) = MinJ(v)

(3.43)

is obviously u = {1, 1}. Now, starting from u° = {0, 0}, let us apply (3.6), (3.7) to (3.43). Since u\ is the solution of Min (jv\ + \v11 — uO. we have u\ = 0. In a similar way we have u\ = 0, so that u" = {0, 0},

V n.

From the above result it follows that the algorithm (3.6), (3.7) does not converge to u. 3.6.2. Relaxation and nonfactorable convex sets In this section we assume that K is a closed convex subset of UN, such that N

If all the other assumptions of Sec. 3.1 hold, we can generalize algorithm (3.6), (3.7) as follows: u° e K, +

(3.44)! +

J(u\ \..., i d » r \ u?+ ! , . . . ) < J{u\ \..., utl

vt, M?+ 1 ; ...),

V{ti"1+1,...,«?+11,t>i,H7+1)...}eK,

{<'

«f+1,ai"+1)...}6K;

i = l,...,N.

(3.44)2

Since M° e K, it is very easy to prove that (3.44) is a well-posed problem, V n > 0. A very simple example will show that (3.44)!, (3.44)2 generally does not converge if K is nonfactorable. We take J = Jo denned by J(v) = v\ + v\ and K defined (see Fig. 3.1) by K = {v\v e U2, Vi > 0 , v2 > 0 , v,_ + v2> 1 } .

150

V Relaxation Methods and Applications Figure 3.1

The solution of

J(u) = MinJ(v)

(3.45)

veK

is obviously u = {%, %}. Then, starting from u° = {0, 1}, let us apply the algorithm (3.44)1;(3.44)2 to (3.45). It is easy to see that u" = {0,1} # u,V« > 0. Therefore we do not have convergence to the solution. Remark 3.8. Let's consider the "one-dimensional" torsion problem (3.46) of Chapter II, Sec. 3.7.1. In the sequel we assume that / e C ° [ 0 , 1]. Using the notation vh(xt) = vt, the above problem is approximated by the following variant of (3.41) (Chapter II, Sec. 3.7.1):

fduh/dvh_du, j0 dx \dx

uheKh,

dx

dx

where ft = f(xt),

(3.46)

i=l,...,N.

Problem (3.46) is equivalent to the minimization problem uheKh,

(3-47)

where ,v,

= {v J1 (fl v

h\ h)

\

OiV+1

h 2

N-l

Vi+1

= vN

- v,

N-l

-h^fiVt,

h

i=0

-o,

(3.48)

2

vi+i

-

h

Vi

£1,V/-*..

(3.49)

N-ll (3.50)

K,, is a nonfactorable closed convex subset of UN+1. However, it is proved in G.L.T. [1, Chapter 3], [3, Chapter 3] that (3.44),, (3.44)2 converges to the solution uh of (3.47) if/ > 0 on [0, 1] and if we start with u°h
4. Block Relaxation Methods

151

Since / > 0 implies M, > 0, V i = 1 , . . . , JV — 1, an obvious choice for ti° is u°h = 0.

4. Block Relaxation Methods In this section we take V = UN such that

V = f[Vt with V,= UNi,

(4.1)

where f; > 1 and

£ iV; = N-

(4-2)

If i> e F, then v = {vu ..., vq}, vt e Vt. We assume that K = f|9_

Kt, where

K; is a closed convex subset of Vt.

(4.3)

Then (4.3) implies that K is a closed convex subset of V. Finally we consider a convex functional J: V -> IR such that D(J) n X / 0 ,

lim

J(u) = + oo, J = J0 + Jl5

(4.4)

where J o G CX{V),

Jo strictly convex

(4.5)

and ,

(4.6)

where t h e ^ are convex, proper, and l.s.c. on V{. From Theorem 2.1 of Sec. 2 it follows that J(u) < J(v),

VoeK,

ueK,

(4.7)

has a unique solution (in D(J) n K) characterized by (J'0(U), » - u) + t 0"J(«I) - ./<(«<)) > 0,

V » 6 K,

ueK.

(4.8)

The generalization of the point relaxation algorithm (3.6), (3.7) is obviously u° e K, +

(4.9)

J « S . . . , Ultl u\ \ U"i+ ! , . . . ) < J ( M f \ . . . , «? + /, «,-, M?+l5 . . .), VvteKt, «-+1eXi; i = l , . . . , 9 . (4.10)

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V Relaxation Methods and Applications

Algorithm (4.9), (4.10) is a block-relaxation algorithm. Using a variant of the proof of Theorem 3.1, it may easily be proved that the sequence u" defined by (4.9), (4.10) converges to the unique solution of (4.2). Remark 4.1. Most of the remarks we have made on algorithm (3.6), (3.7) still apply to (4.9), (4.10). Unfortunately, Remark 3.1 does not apply in general. Remark 4.2. In Cea and Glowinski [1] and Glowinski [6], it is proved that (4.9), (4.10) could also be used in infinite-dimensional situations.5 In this case the assumptions Jo e C 1 and strictly convex are not sufficient to insure the convergence of the relaxation algorithm. The basic reason for this is that closed bounded sets are generally noncompact in Banach spaces of infinite dimension. Using the same notation, in the two references cited above, it is proved that sufficient convergence conditions are J o is C 1 with J'o Lipschitz continuous on the bounded sets of V;

(4.11)

Jo locally uniformly convex, i.e., <J'0(i>) - J'M,v

- "> > K(\\v - «ll),

V u, v,

\\u\\ < M,

\\v\\ < M, (4.12)

where SM is the same as in Sec. 3.3.

5. Constrained Minimization of Quadratic Functionals in Hilbert Spaces by Under and Over-Relaxation Methods: Application We follow Cea and Glowinski [1, Sec. 3]. 5.1. Statement of the minimization problem Let V = Y\f= i Vt, where Vt is a real Hilbert space, V i = l,...,N. Norm and scalar product on Vi are, respectively, denoted by || • ||; and ((•, •));• If v e V, then v=

{vu...,vN}

with VteVi,

i=l,...,N.

On V we define a scalar product and a norm by

((u,v))=

£««*.»«))«, N

' More precisely, in reflexive Banach spaces.

\l/2

(5.1)

5. Constrained Minimization of Quadratic Functionals in Hilbert Spaces

153

V is a Hilbert space for ((•, •)) and || • ||. Let K be a closed convex subset of V such that K=

\[Ki,Ki±0,

Vi = 1

JV,

where Kt is closed and convex in Vt. Let J: V -> U be defined by J(v) = $a(v, v) - ((/, v)\

(5.3)

where the bilinear form a(-, •) on V x V, is continuous, symmetrical, and F-elliptic, i.e., 3 a > 0 such that a(u, u) > a|M|2,

V»eF,

and where feV. Under the assumptions on V, K, and J, the optimization problem J{u)

< J(v),

VOEK,

ueK

(5.4)

has a unique solution. This solution is characterized by a(u, v - u ) - ( ( / , v - u ) ) > 0 ,

V » e K,

u e K.

(5.5)

5.2. Some preliminary results From the properties of V it follows that

J(v) = J(vu ...,vN)

=\

X aifa, vj) - t <Wi, »d)i,

(5-6)

where the atj are bilinear and continuous on Vt x J^- with a y = a*. The forms flj,- are 1^-elliptic (with the same constant a). Using the Riesz representation theorem, it is easily proved that there exists Ay e Sf(Vj, V^ such that di/v;, Vj) = i(Vi, AijVj))t,

(5.7)

Atj = A%.

(5.8)

Moreover, the AH are self-adjoint and are isomorphisms from Vt to Vt. In the sequel it will be convenient to use the norm defined by au on Vt, i.e.,

Ilk-Ill? = (tufa, vd = ((.AtlVi,

vt))i,

i=l,...,N.

(5.9)

The norms || • |j ; and 111 • 111; are equivalent. The projection from Vt to Kt in the 111 • 111; norm will be denoted by P{. Before giving the description of the iterative method, we shall prove some basic results on projections, useful in the sequel. Let: (i) if be a real Hilbert space, with scalar product and norm denoted by (•, •) and || • ||, respectively, (ii) &(•,•) be a bilinear form on H, continuous, symmetric, and H-elliptic (i.e., 3 j8 > 0 such that b(v, v) > 0\\v\\2, V v e H).

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V Relaxation Methods and Applications

Then from the Riesz representation theorem follows the existence of an isomorphism B: H -> H such that (Bu, v) = b(u, v),

V u,veH,

B = B*.

(5.10)

We denote by [-, •] and |-| the scalar product on H and the norm on H, respectively, denned by [w, D] = b(u, v),

Vu,veH,

(5.11)

\v\2 = b(v,v),

V»e/f.

(5.12)

The norms | • | and || • || are equivalent. Let (iii) C ^ 0 b e a closed convex subset of H and n be the projector from H ~* C in the | • | norm. (iv) j:H -» U be the functional defined by j(v) = $b(v,v)-(g,v),

Vcefl,

(5.13)

where g e H. Under the above assumptions, we have the following lemmas. Lemma 5.1. / / u is the unique solution of j(u)<j(v),

VveC,

usC,

(5.14)

then u = n(B-1g).

(5.15)

PROOF. The solution u of (5.14) is characterized by {Bu - g, v - u) > 0,

VceC, ueC.

(5.16)

From (5.16) it follows that (B(v-u), B-1g-u) = b(v-u,B-1g-u)<0,

VceC, u e C,

1

and (5.17) characterizes u as the projection on C of B~ g in the norm | • |.

(5.17) •

Lemma 5.2. Let uoe C and let u1 be defined by Uj = n(u0 + (a{B'lg - u0)),

co > 0.

(5.18)

Then 7(«o) - X « i ) >

2

-^

l«o — «i I2-

(5-19)

5. Constrained Minimization of Quadratic Functionals in Hilbert Spaces

155

PROOF. We have, V vu v2 e H,

Xvi) -J(P2) = Kloi - B" : 9l 2 - \v2 ~ B-W). i

(5.20)

1

Since uy — B~ g = u0 — B~ g + ux — u0, we have K - B~lg\2 = |«, - B - ^ l 2 + 2[H 0 - B " V « 0 - « J - |u 0 - u t | 2 .

(5.21)

From (5.18), and since u0 e C, it follows that

[w0 + a(B~lg - u0) - uu u0 - u{\ < 0. This implies ux\2 < co[u0 - B'ig,

\u0-

u0 - « J .

(5.22)

Then (5.19) clearly follows from (5.20)-(5.22).

5.3. Description of the algorithm For i = l,...,N, let the co, be positive numbers. Now let us consider the following algorithm: u° = {u°,...,

«$} arbitrarily given in K;

n+l

u" being known, we compute u

(5.23)

by

(5.24) Remark 5.1. It follows from (5.24) that the computation of u"+i can be achieved in three steps: Step 1. On Vt we minimize the functional

Hence we obtain a solution ,,"+1/3 _

d-l/f _ \

Step 2. We compute

M?+2/3

V J ,.»+! j
by

Step 3. At last we obtain u"+ i by We remark that if o); = 1, V i = l,...,N, then algorithm (5.23), (5.24) reduces to the block algorithm (4.9), (4.10) (see Remark 4.2).

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V Relaxation Methods and Applications

5.4. Convergence of algorithm (5.23), (5.24) Proposition 5.1. We have J(u") - J(un+1)

> £ — - — - \\\u"+1 - M"|||,2-

(5.25)

PROOF. We have N

J{u") - J(un+")

= £ {J(u\+ \ . . . , uT-\, u i . . . . ) - J{u\+',...,

u1+ \ u " i + j , . . . ) ) .

(5.26)

Then (5.25) clearly follows from (5.24), and from the application of Lemma 5.2 at each of the differences of the right-hand side of (5.26). Proposition 5.2. IfOi<2,\/i=

I,... ,N,we have

J(un)>J(un+1),

Vn

and lim (un -un+1)

= 0 strongly in V.

(5.27)

PROOF. Since 0 < a>; < 2 implies (2 - cod/a, > 0, V i = 1 , . . . , N, it follows from (5.25) that J(u") is a decreasing sequence. Since J(u") is bounded below by J(u), where u is the solution of (5.4), J(u") is convergent. This implies lim (J(u") - J(u"+1)) = 0.

(5.28)

n-* + oo

Then, from (5.25), (5.27) and from (2 - a^/co; > 0, V i = 1 , . . . , N, it clearly follows that lim \\u1+1 - ^ | |

j =

0,

V J = 1,...,JV.

This implies (5.27).

From these two propositions we deduce: Theorem 5.1. / / 0 < cot < 2, V i = 1 , . . . , N, then the sequence u" defined by (5.23), (5.24) satisfies lim u" = u, n-* + co

where it is the solution of (5.4). PROOF. The F-ellipticity of a(-, •) implies a(u"+i

- u,u"+l

-u)>

a | | « " + 1 - u\\2.

(5.29)

From (5.29) it follows that a(u"+\ M" +1 - u) - ((/, u" + 1 - «)) > a{u, un+1 - u) - ((/, un+1 - «)) + a||U"+1 + u\\2. (5.30)

5. Constrained Minimization of Quadratic Functionals in Hilbert Spaces

157

Since u is the solution of (5.4), and since u"+i e K, we have a(u, u"+1 - w) - ((/, w"+1 - «)) > 0, which, combined with (5.30), implies a(un+l, u"+1 - u) - ((/, u" +1 - u)) > a\\un+1 - u\\2.

(5.31)

The left-hand side of (5.31) could be written as follows:

a(u"+\ u"+> - ii) - ((/, «" +1 - «)) = I ff I ^««J + 1 - / „ «? +1 - «,)) • (5.32) Let u"+1 be the vector of Kt for which the functional

attains its minimum on Kt. From Lemma 5.1 it follows that u"+l —PI A Mj

—

'I

1

'•

l f — V A u " + ' — V A ii" I I \J'

L*

'jui

Z^ Aijuj

(S 331 vJiJJ^

I r

Moreover, from the usual characterization of the minimum we have (AH01+1

+ I AijU"j + l + X A,tf - ft,v, - uTl))

> 0,

V o, 6 X,.

(5.34)

It follows from (5.32) that a(u"+\ un+1 - u) - ((/, u"+i - u))

(

i=l\\j
j>i

Since u, e Kt, (5.34) implies that the last term on the right-hand side of (5.35) is <0. Therefore (5.31), (5.35) imply

i=l\\j
( « ! i=l\\j>i

/

))

(5.36)

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V Relaxation Methods and Applications

From (5.36) it follows that to prove the strong convergence of u" to u, it suffices to prove that the left-hand side of (5.36) converges to zero. Since the A{j are linear and continuous, the u1,u" are bounded uniformly in i and n, and limn^ + 00 \\u"+1 — u"\\ = 0, it follows from (5.36) that linv, + 0O ||fi" - u"\\ = 0 (where u" = {u"u ..., Wh ..., u"N}) will imply convergence. Let us prove the last property. Since a>; is >0, from (5.34) it follows that >uJ

))

(5.37)

From (5.24), (5.37), and since Pt is a contraction, it follows that

w s r ' - M r M i i ^ i i - ^ i i p r 1 -«?in,,

v »• = 1,2,..., AT.

(5.38)

Since 0 < Wj < 2 implies 0 < 11 - ct», | < 1, it follows from (5.38) that | | | t ? + 1 — wf"1"1 III,- < H i s r 1 - « ? l l l i .

Vi=l,...,iV.

(5.39)

F r o m the triangle inequality and (5.38), (5.39), it follows that III"? - n? + 1 |lli > III"? - S7 + I llli - lll"? + 1 - «? + 1 llli > ( 1 -ll-ffl.DIHfif+l-M-lll, > ( 1 _ | l _ 

(5.40)

where 0 < 1 — 11 —
Remark 5.2. The above theorem generalizes Cryer [2], and also generalizes a classical result in finite dimensions (without constraints) for which we refer to R. S. Varga [1] and D. M. Young [1]. Remark 5.3. If to,- > 1 (resp., to,- = 1, co,- < 1), V i = 1 , . . . , N, then algorithm (5.23), (5.24) is an over-relaxation algorithm (resp., relaxation, underrelaxation algorithm) with projection.

5.5. Application to an elastic-plastic torsion problem 5.5.1. Statement of the continuous problem We consider the elastic-plastic torsion problem described in Chapter II, Sec. 3 (we use the notation of Chapter II, Sec. 3): a(u, v -u)>C

Jn Jn

(v -u)dx, - u) dx,

VceX,

where a(u, v) = | n VM • Vt> dx, and where K = {v\v e H},(Q), \Vv(x)\ < 1 a.e.}.

we K,

(5.41)

5. Constrained Minimization of Quadratic Functional in Hilbert Spaces

159

From the equivalence result of Brezis and Sibony [1] mentioned in Chapter II, Sec. 3.4, it follows that the solution of (5.41) is also the solution of a{u, v - u) > C \ (v - u) dx, Jn ue£={v\ve

Vcel,

H £(Q), | v(x) | < d(x, F) a.e.}.

(5.42)

In this section we consider only the numerical analysis of (5.42). 5.5.2. A finite element approximation of (5.42) We consider piecewise linear approximations only. As mentioned in Chapter II, Sec. 3.4, (5.42) is a variant of the obstacle problem we considered in Chapter II, Sec. 2. We assume that Q is a polygonal domain and that 2Th is a standard triangulation of Q (see Chapter II). From Chapter II, Sec. 2 it follows that Hj(Q) and K may be, approximated respectively, by6 Vh = {vh\vh e C°(Q), vh = 0 on F, vh\T e Pu V

Te.rh},

Kh = {vh\vh e Vh, \vh(x)\ < d(x, F), V x e ±h}.

Then (5.42) is approximated by a(uh, vh-uh)>C

(vh - uh) dx,

V vh e Kh,

uheKh.

(5.43)

Jn

From the results of Chapters I and II it easily follows that: Proposition 5.3. The approximate problem (5.43) has a unique solution. This solution is also the unique solution of the minimization problem J(uh)<J(vh),

VvheRh,

uheKh,

(5.44)

where J(v) = ja(v, v) — C \nv dx. Moreover, using the methods of Chapter II, we may prove the following: Theorem 5.2. Suppose the angles of ,Th are uniformly bounded below by 60>0ash-+0. Then lim ||uh - M|| H i (n) = 0,

where u and uh are, respectively, the solutions of (5.42) and (5.43). 6

t,h: set of the interior nodes of ^1.

V Relaxation Methods and Applications

160

Figure 5.1

5.5.3. Solution of the approximate problem The natural unknowns of the approximate problems (5.43) and (5.44) are uh(P), P e i j . Since vh -* J(vh) is quadratic with respect to vh(P), it follows from the structure of Kh that algorithm (5.23), (5.24) could be used to solve (5.43), (5.44). 5.5.4. Application to the case Q = ]0, 1[ x ]0, 1[ When Q = ]0, 1[ x ]0, 1[, or, more generally, a rectangle, it is convenient to use a finite element approximation which is actually equivalent to a finite difference approximation. Let AT be a positive integer and h = l/N. On Q we use the triangulation 2Th of Fig. 5.1. We have £„ = {My | My = {ihjh}, i,j integers, 0 < i,j < N],

By analogy with the finite difference notation, it is convenient to denote vh(Mij) by vu. Then the approximate problems (5.43), (5.44) are easily expressed in function of vu, since 2

Vlj-l

%1

\ _

h

2

/„

C

where vkl = 0 if Mkl $ £*, and since Kh = {vh\vh e Vh, \viS\ < du, 1 < i, j < N - 1}, where dtJ = d{Mtj, Y).

_

„

\2

5. Constrained Minimization of Quadratic Functionals in Hilbert Spaces

161

Algorithm (5.23), (5.24) has been used with cou = co, V 1 < ;', j < N - 1. Therefore the explicit form of (5.23), (5.24) is u° e Kh arbitrary given,

(5.45)

and for 1 < i,j < N - 1,

ui;1/2 = (1 - co)ulj + I (u1+ tJ + unij+! + « r h + U"J+JI + ^ O

( 5 - 46 )

«"/ l = max( — dij, min(w"/ 1/2 , di})),

(5.47)

where u£| = 0 if Mkl $ l,h. From Theorem 5.1 it follows that uh converges to the solution of (5.43), (5.44)ifO
I

K+1-M?;|<10-4.

(5.48)

The number of iterations necessary to obtain the convergence is given below as a function of co: CO

1

1.4

1.5

1.6

1.7

1.8

1.9

290

138

118

98

93

108

188

2

Number of

Iterations The C.P.U. time on an IBM 360/91 was 7.33 sec for co = 1 and 2.51 sec for co = 1.7. Figure 5.2 shows the behavior of

as a function of n (when co = 1.7). In the Fig. 5.3 we can see the elastic part of Q (in white) and the plastic parts of Q (striped area). In the plastic parts we have | VH | = 1 and | u(x) \ = d(x, F). In the elastic part we have — Aw = C and | u(x) \ < d(x, F). Remark 5.4. The optimal value of co is very close to the optimal value of co corresponding to the solution, by S.O.R., of the Dirichlet problem — Aw = C in the elastic part of Q (with the same discretization step h). This follows directly from the fact that the plastic part of Q 7 is very quickly obtained. From 7

It is the part of Q in which the constraints \u\ < d{x, T) are active (i.e., \u\ = d(x, F)).

V Relaxation Methods and Applications Figure 5.2

this fact it follows that the computation time is mainly used to solve — Au = C in the elastic part of £1 Moreover, the plastic part increases with C; this fact explains that the optimal value of a> and the number of iterations needed to obtain the convergence are, for a given h, decreasing functions of C (as is the computational time). Remark 5.5. In Cryer [1], [2] we can find a discussion on the choice of a>, when using point S.O.R. with projection to minimize quadratic functionals on a product of intervals.

Figure 5.3

6. Solution of Systems of Nonlinear Equations by Relaxation Methods

163

6. Solution of Systems of Nonlinear Equations by Relaxation Methods 6.1. Statement of the problem In this section we very briefly describe some relaxation methods for solving systems of nonlinear equation such that fi{ui,u2,...,uN)

= 0,

fi(u1,u2,.':.,uN)

= Q,

where ft: UN -> R. We define F: UN -+ UN by F = {fu...

(6.1)

,fN}.

6.2. A first algorithm We consider the following algorithm: M° given; n+1

u" being known, we compute u

(6.2)

by

m

-uX),

(6.3)

1 U is a strictly convex C 1 function such that lim||i;||_ + 00 J(v) — +oo, then (6.3) has a unique solution (see Sec. 2). This solution is also the unique solution of J(u) < J(v),

Vc6R N ,

us UN.

(6.4)

Moreover, if co = 1, it follows from Theorem 3.1 that algorithm (6.2), (6.3) converges to the solution u of (6.1), (6.4). If F = VJ, co # 1, we refer to S. Schechter [1], [2], [3]. In these papers it is proved that under the hypothesis (i) J e C2(UN), (ii) (J'(w) - J'(v), w - v) > tx\\w - v\\2, V v, w; a > 0, (iii) 0 < co < coM, algorithm (6.2), (6.3) converges to the solution of (6.1), (6.4). Moreover, estimates of coM and of the optimal value of co are given. For the convergence of (6.2), (6.3) when F # VJ, we refer to Ortega and Rheinboldt [1], Miellou [1], [2], and the bibliography therein.

164

V Relaxation Methods and Applications

6.3. A second algorithm This algorithm is given by u° given; n+1

u" being known, we compute u

(6.5)

by

1 < i < N.

(6.6)

To our knowledge the convergence of (6.5), (6.6) for co ^ 1 and F nonlinear has not yet been considered. Remark 6.1. Algorithms (6.2), (6.3) and (6.5), (6.6) are identical if co = 1 and/or F is linear. Remark 6.2. In many applications, from the numerical experiments it appears that (6.5), (6.6) is faster than (6.2), (6.3), co having its (experimental) optimal value in both cases. Intuitively this seems to be related to the fact that (6.5), (6.6) is "more implicit" than (6.2), (6.3). For instance, (6.5), (6.6) could easily be used if F is only denned on a subset D on [R*; in such a situation, when using (6.2), (6.3) with co > 1, it could happen that {u"+ 1,...,u"+i, u"+ u .

6.4. A third algorithm In this section we assume that F e C^M"). A natural method for computing w" +1/2 in (6.3) or u"+l in (6.6) is Newton's method. We recall that Newton's method applied to the solution of the single-variable equation /(*) = 0 is basically: x° given;

(6.7) m

f(x }

4+1/2

(6-8)

+1

In the computation of u" in (6.3) or u" in (6.6) by (6.7), (6.8), the obvious starting value is u". Then obvious variants of (6.2), (6.3) and (6.5), (6.6) are obtained if we run only one Newton iteration. Actually, in such a case, (6.2), (6.3) and (6.5), (6.6) reduce to the same algorithm, which is u° given;

(6.9)

6. Solution of Systems of Nonlinear Equations by Relaxation Methods

165

In S. Schecter, loc. cit., the convergence of (6.9), (6.10) is proved, if F = VJ, under the same assumptions as in Sec. 6.2 for algorithm (6.2), (6.3) (with a different <x>M in general). Remark 6.3. In Glowinski and Marrocco [1], [2] and Concus [1], we can find comparisons between the above methods when applied to the numerical solution of the nonlinear elliptic equation modeling the magnetic state of ferromagnetic media (see also Winslow [1]). Applications of the first algorithm for solving minimal surface problems may be found in Jouron [1].

CHAPTER VI

Decomposition-Coordination Methods by Augmented Lagrangian: Applications1

1. Introduction 1.1. Motivation A large number of problems in mathematics, physics, mechanics, economics, etc. may be formulated as Min {F(Bv) + G(v)},

(P)

where • V, H are topological vector spaces, • F: ff -> R, G:V->U are convex proper l.s.c. functional. Let us give two examples taken from Chapter II. EXAMPLE 1 This is the Binghamflow problem of Chapter II, Sec. 6; we recall that with Q a bounded domain of U2, we consider the variational problem

Min 1^ f \Vv\2dx + g\ \Vv\dx - f fvdxl,

(1.1)

where v and g are positive constants. Then (1.1) is the particular problem (P) in which • • . . •

V = H J(Q), H = L2(Q) x L2(Q), £ = V, F(q) = {vl2)\n\q\2 dx + g \n\q\dx(\q\ G(«)= -\nfvdx.

Actually, we can also take • F(q) = gla\q\dx, • G(v) = 1

This chapter follows Fortin and Glowinski [2].

= Jq{ + q\),

1. Introduction

167

EXAMPLE 2 This is the elastic-plastic torsion problem of Chapter II, Sec. 3; with Q still bounded in U2, we consider

Minj^ f | V i ; | 2 ^ - f fvdxl,

(1.2)

where

K = {veHKCl Problem (1.2) is the particular problem (P) in which • V = tf £(fi), H = L2(Q) x L2(Q), . B = V, . G(t;) =

-\afvdx,

where /^ is the indicator functional of the convex set K={qeH, "0

M < 1 a.e.}, if qeK

We can also take = /*,

Synopsis. Problems of type (P) have a special structure, and in the sequel we shall introduce iterative methods of solution taking it into account. 1.2. Principle of the methods The decomposition-coordination methods to follow are based on the following obvious equivalence result. Theorem 1.1. (P) is equivalent to Min {F(q) + G(v)},

(n)

{vq}EW

where W = {{v, q} e V x H, Bv - q = 0}. In the sequel we shall assume that V and H are Hilbert spaces with inner products and norms denoted by ((•, •)), ||-|| and (•, •), \-\, respectively. We then define a Lagrangian functional .Sf associated with (n), by JSf (c, q, ft) = F(q) + G(v) + (p, Bv - q),

(1.3)

168

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

and for r > 0 an Augmented Lagrangian ^Cr is defined by Ser(v, q, n) = &(v, q,fi) +

r

-\Bv-q\2.

(1.4)

Remark 1.1. Augmented Lagrangian methods for solving general optimization problems have been introduced by Hestenes [1] and Powell [1]. Augmented Lagrangian methods for solving problems like (P) via (n) have been introduced by Glowinski and Marrocco [3] and also [4]-[6].

2. Properties of (P) and of the Saddle Points of <£ and <er 2.1. Existence and uniqueness properties for (P) Let us define J: V -> U by J(v) = F(Bv) + G(v). Then (P) can also be written J(u)<J(v),

\fveV, ueV.

(2.1)

Let j : X ->• U; we define the so-called domain ofj(-) by dom(j) ={xeX,

j(x) e R}.

Then, if dom(F o B) n dom(G) # 0 ,

(2.2)

/ is convex, proper, and l.s.c.. Therefore, sufficient conditions for (P) to have a unique solution are (cf. Cea [1], [2] and Ekeland and Temam [1]): |M| = +GO, • J strictly convex. Remark 2.1. If B is an injection from V to H, with R(B) ( = range of B) closed in H, then | J3i?| is a norm on V equivalent to \\v\\.

2.2. Properties of the saddle points of =£? and i£r We have: Theorem 2.1. Let {u, p, X} be a saddle point of $£ onV x H x H; then {u, p, X} is also a saddle point of J£r, V r > 0 and conversely. Moreover, u is a solution of (P) and p = Bu.

2. Properties of (P) and of the Saddle Points of if and <£,

169

PROOF, (i) {u, p, A} be a saddle point of if; then if(u, p, A) e IR and , p, /x) < i f (u, p, X) < ^{v, q, A), V {v, q, n) 6 V x H x tf, {«, p, A} e F x # x H.

(2.3)

From the first inequality (2.3) and from (1.3) it follows that (jt, Bu-p)<

(A, Bu-p),

V n 6 H,

which obviously implies that Bu = p.

(2.4)

From the second inequality (2.3) and from (1.3) and (2.4) it follows that J(u) = <£{u, p , X) < £C(v, q, X),

V{v,q}eV x H.

(2.5)

Taking q = Bv in (2.5), it follows from (1.3) that J(u)

< £C(v, Bv, X) = J(v),

V V E V ;

(2.6)

hence u is a solution of (P). Since p = Bu, we have <£r{u, p, n) = J?(u, p, n) = J(u),

MneH;

(2.7)

it then follows from (2.3), (2.7) that if r (u, p, n) = £?r(u, P, X) < &(v, q, X),

V {v, q, n) e V x H x H.

(2.8)

2

Since &,(», 9, fi) = ^f(", 1, n) + (r/2)\Bv - q\ , from (2.8) we obtain if r («, P, n) < yr(u, P, X) < £er(v, q, X),

V {v, q, n) e V x H x H,

(2.9)

which proves that {«, p, X} is also a saddle point of if,. onV x H x H.To conclude this part of the proof, we observe that from (2.3), {u, p} is a solution of i f (u, p , A) < &(v, q, X),

V {v, q}eVxH,

[u, p} e V x H,

(2.10)

from which it follows that {u, p} is characterized (see Cea [1], [2], Ekeland and Temam [1], and also Chapter V, Sec. 2) by F(q) ~ F(p) - (X, q - p) > 0, G(v) - G(u) + (X, B(v - «)) > 0,

V q e H, p e H,

(2.11)

V v e V, u e V.

(2.12)

(ii) Let {u, p, X] be a saddle point of ifr with r > 0. Then, as in part (i), this implies that p = Bu and that u is a solution of (P). Moreover, since {u, p, A} is solution of i f r ( u , p , X) <
V{v,q}eV

x H, {u, p} e V x H,

(2.13)

it is characterized by F(q) - F(p) + r(p - Bu,q - p) - (X,q - p)>0, G(v) - G(u) + r(Bu - p, B(v - «)) + (A, B(v - u)) > 0,

VqeH,

peH,

VueF,

ueV.

(2.14) (2.15)

But since Bu - p = 0, (2.14), (2.15) reduce to (2.11), (2.12), and this fact implies that {«, p, A} satisfies (2.10). It then follows from (2.7) that {u, p, A} satisfies (2.3), and this completes the proof of the theorem. •

170

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

3. Description of the Algorithms From Theorem 2.1 it follows that a method for solving (P) is to solve the saddle-point problem £er(u, P, H) < &r{u, p , X) < Ser{v, q, X), V {v, q, n} e V x H x H,

{u, p,X}eVxHxH.

(3.1)

To do this we shall use (see Cea [1], and G.L.T. [1, Chapter 2], [3, Chapter 2]) an algorithm of Uzawa type and a variant of it. 3.1. First algorithm We denote by ALG 1 the following algorithm: X°eH

given;

(3.2)

then, X" known, we define u", p", X"+' by jS?r(«"» V\ A") < <£r(v, q, X"),

V {v, q} e V x H, {«", p"}eVx

X»+1 = X" + pn(Bu" - p"),

Pn

> 0.

H, (3.3) (3.4)

Problem (3.3) is in fact equivalent to the following system of two coupled variational inequalities (of the second kind): G(v) - G(un) + (Xn, B(v - «")) + r(Bun - p", B(v - u")) > 0, \fveV, u"eV, (3.5) F(q) - F(p") - (Xn, q - p") + r(p" - Bit", q - p") > 0,

V q e H,

pneH. (3.6)

The convergence of ALG 1 will be studied in Sec. 4. 3.2. Second algorithm The main drawback of ALG 1 is that it requires the solution of the coupled EVI's (3.5), (3.6) at each iteration. To overcome this difficulty, it is natural to consider the following variant of ALG 1 (denoted ALG 2 in the following): {p°, Xl}eH

x Hgiven;

n

(3.7)

n+1

then, {p"~\ X"} known, we define {u , p", X } by G(v) - G(un) + (Xn, B(v - u")) + r(Bu" - pn~ \ B(v - w")) > 0, V v e V, u"e V, (3.8) n F(q) - F(p") - (X , q- p") + r(p" - Bit", q - p") > 0, V q £ H, p"eH, (3.9) +i

X"

n

= X" + pn(Bu -p"),

Pn>0.

The convergence of ALG 2 will be studied in Sec. 5.

(3.10)

4. Convergence of ALG 1

171

4. Convergence of ALG 1 4.1. General case In this subsection, V and H are possibly infinite dimensional; of course we assume that dom(F o B) n dom(G) # 0 ,

(4.1)

B is an injection and R(B) is closed in H.

(4.2)

and also

We also assume that /

lim I 4 I - + 00

^ ) = +oo,

(4.3)

\i\

F = F o + -f 1 with ^o? ^1 convex, proper, and l.s.c,

(4.4)

Fo is Gateaux-differentiable and uniformly convex on the bounded sets of H.

(4.5)

By definition we say that Fo is uniformly convex on the bounded sets of H if the following property holds: V M > 0, 3 SM: [0, 2M] -> U, continuous, strictly increasing with «5M(0) = 0, such that V q , p e H with \p\ < M, \q\ < M we have (F'0(q) - F'0(p), q - p) > 8M(\q - p \ ) , (4.6) where F'o = VF 0 is the G-derivative of Fo. From the above properties, (P) has a unique solution u, and we define p e H by p = Bu. EXERCISE

4.1. Prove that (P) is well posed if (4.1)-(4.5) hold.

On the convergence of ALG 1, we have: Theorem 4.1. We assume that =Sf has a saddle point {u, p, A} £ V x H x H. Then under the above assumptions on B, F, G and if 0 < a 0 < pn < «! < 2r,

(4.7)

the following convergence properties hold: u" -> it strongly in V,

(4.8)

p" -> p = Bu strongly in H,

(4.9)

n

X"+1 _ X ->• 0 stron^/y m H, n

X is bounded in H.

(4.10) (4.11)

172

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Moreover, if X* is a weak cluster point of {X"}n in H, then {it, p, X*} is a saddle point of !£r over V x H x H. PROOF. Since {u, p, X} is a saddle point of <£,, we have ^r{u,p,X)<^r(v,q,X),

V{v,q}eVxH,

{u,p}eVxH.

(4.12)

Therefore {u, p} is characterized by G(v) - G(u) + (X, B(v - «)) + r(Bu - p, B(v - u)) > 0, (F'o(p),q-p) + Fl(q)-F1(p)-(lq-p)

V v e V, ueV,

+ r(p-Bu,q-p)>0,

(4.13)

MqeH, peH. (4.14)

Moreover, from Theorem 2.1 we have Bu = p; therefore X = I + Pn(Bu - p).

(4.15)

Let us define u", p", 1" by u" = u" - u,

p" = p" - p ,

i " = X" - I.

From (3.4), (4.15) it then follows that 1" + 1 = I" + Pn(Bu" - p"), which implies |!»+i| 2 = |A"|2 + 2pn(l\ Bu" - p") + p2n\Bu" - p"| 2 or, more conveniently, \l"\2 - | I " + 1 | 2 = -2pn(X",Bu" - p") - pl\Bu" - p"\2.

(4.16)

Since {u", p"} is a solution of (3.3), it is characterized by G(v) - G(u") + {X", B(v - un)) + r(Bu" - p", B(v - «")) > 0,

VceF,

u" e V, (4.17)

(F'0(p"), q-p") + F.iq) - Fj(p") - (X", q - p") + r(p" - Bu", q - p") > 0, V q e H, p" e H. (4.18) Taking v = u (resp., v = u") in (4.17) (resp., (4.13)) and q = p (resp., q = p") in (4.18) (resp., (4.14)), we obtain, by addition, {I", Bu") + r(Bu" - p", Bu") < 0,

(4.19)

(F'0(p") - F'0(p), p") - (I", p") + r(p" - Bu" • p") < 0,

(4.20)

which imply, also by addition, (1", Bu" - p") + (F'0(pn) - F 0 (p), p") + r\Bu" - p"\2 < 0, i.e., - ( 1 " , Bu" - p") > (F'0(p") - F'0(p\ pf) + r\Bu"-p"\2.

(4.21)

Combining (4.16) and (4.21), we obtain |i"| 2 - U " + ' I2 > 2pn{F'0(p") - F'0(p), p") + Pn{2r - pn)\Bu" - p"| 2 > 0.

(4.22)

4. Convergence of ALG 1

173

Assuming that (4.7) holds, it follows from (4.22) that lim I B S " - p " | = 0,

(4.23)

n-> + oo

lim (F'0(p") - F'Q(p), p"-p)

= 0,

(4.24)

n-» + oo

A" is bounded in H.

(4.25)

Since p = Bu, it follows from (4.23) that lim \Bu" - p"\ = 0.

(4.26)

n-> + oo

Since F is proper, there exists poe H such that F(p0) e R; then, from the characterization (3.6), we have F(Po) - (*", Po) + r(p" - Bu", p 0 ) > F(p") - (A", p") + r(p" - Bu\ p").

(4.27)

Since A" and p" — £«" are bounded, (4.27) implies A) > F(p") - Pi\pn\,

(4.28)

where jS0, /?i are independent of n. From (4.3), (4.28) it then follows that p" is bounded in H, i.e., 3 M such that | p" | < M,

V «.

(4.29)

Then, using the uniform convexity property (4.5), (4.6) of F o , we obtain from (4.29) (assuming M > \p\) that ( W ) - F'0{p), f ~ p ) > 5M(\p"

- p\\

which combined with (4.24), implies, lim <5M(IP" - p|) = O o lim \p" - p\ = 0.

(4.30)

n~* + X

It then follows from (4.26), (4.30) that lim Bu" = p = Bu strongly in H.

(4.31)

n-* + oo

Since B is an injection with R(B) closed in H, then (4.31) implies that lim u" = u strongly in V.

(4.32)

n-* + oo

The convergence result (4.10) clearly follows from (4.7), (4.26). Let X* be a weak cluster point of {/!"}„ in H. Then, passing to the limit in (3.5), (3.6) and using the l.s.c. property of F and G, we have G(v) + (A*, B(v - u)) + r(Bu - p, B(v - u)) > lim inf G(u") > G(u), F(q) - (A*, q-p)

+ r(p-Bu,q-p)>

lim inf F(p") > F(p),

V v e V, VqeH,

ueV,

peH,

i.e., G{v) - G(u) + (A*, B(v - u)) + r(Bu - p, B{y - «)) > 0, F(q) - F(p) - (A*, q-p)

+ r(p-Bu,q-p)>Q,

VveV,

u e V, (4.33)

V q eH, peH.

(4.34)

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VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

As noted before (see (2.13)-(2.15)), (4.33), (4.34) is equivalent to &r(u,

p , k*) < &r(v, q, A*),

V {v, q} e V x H,

{ u , p}eV

x H.

(4.35)

Since, from p = Bu, we have ££r(u, p, jj) = £?{u, p, ix) = J(u),

V fieH,

we obtain Ser{u, p , fi) = £er(u, p , A*),

V fi e H.

(4.36) 2

It clearly follows from (4.35), (4.36) that {u, p, X*} is a saddle point of £Cr; this completes the proof of the theorem. •

4.2. Finite dimensional case If V and H are finite dimensional, we have convergence of ALG 1 with weaker assumptions on F, B, and G than in Sec. 4.1. The reasons for this are the following: (1) Since the constraint Bv — q = 0 is linear, if (P) has a solution, then if and j§^ have a saddle point (see Rockafellar [1] and Cea [1], [2]). (2) R(B) is always closed. (3) It follows from Cea and Glowinski [1] (and from Chapter V, Sec. 3.3) that Fo satisfies the uniform convexity property (4.5), (4.6) if Fo is C 1 and strictly convex. (4) If Fo is C 1 and strictly convex, then F'o is C° and strictly monotone, i.e., (F0(12) -

F

'o(iiX I2 - 0,

V qu q2 e H,

q1 /

q2.

Then if (P) has a solution, the property

lim ^ f = + ° o is not necessary. This is related to the following: Lemma4.1. Let H be finite dimensional and A:H^H be continuous and strictly monotone. Let {#"}„>0, Pn e H,V n, and p e H be such that lim (A(p") - A(p), p" - p) = 0;

(4.37)

H-» + 00

then lim f = p.

(4.38)

n~* + 00

2 Concerning the convergence of {X"}a, we have, in fact, a weak convergence of the whole sequence to a X* such that {u, p, A*} is a saddle point of i£r (see Sec. 7).

4. Convergence of ALG 1

175 Figure 4.1

PROOF. Assume that (4.38) does not hold. Then there exist 5 > 0 and a subsequence, denoted {pm}m, such that \pm-p\>6, Let S(p; 5/2) = {qeH,\q-p\

Vm.

(4.39)

= 6/2}. We define zm e S(p; 6/2) by 5 pm — p Z P

+

^

m

(440)

2^J\;

m

then z e ]p, p \_ a H (see Fig. 4.1). We introduce tm = 6/2\pm - p\; then zm = p + tm(pm - p),

(4.41)

0 < tm < \.

(4.42)

and from (4.39),

Since A is strictly monotone, we have {A{f) - A(p), pm - p) > (A(p + t(pm - p)) - A(p), pm - p),

Vt6]O, 1[; (4.43)

m

then, taking t = t in (4.43), we obtain (A(pm) - A(p), pm-p)>

(A{zm) - A(p), pm-p)>

0.

(4.44)

From (4.41), (4.42), and (4.44), it then follows that (A(pm) - A(p), Pm-P)>^,

(A(zm) - A(p), zm-p)>

2(A(zm) - A(p), z™ - p)

> (A(zm) - A(p), zm - p)> 0.

(4.45)

Since S(p, 8/2) is compact, we can extract from {zm}m a subsequence—still denoted {zm}m—such that lim zm = z,

z e s(p, - ) .

(4.46)

Since A is continuous, it follows from (4.37), (4.45), and (4.46) that (A(z) - A(p), z - p) = 0.

(4.47)

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VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

The strict monotonicity of A and (4.47) imply that z = p, which is impossible since \p — z\ = 6/2. Therefore (4.39) cannot hold => limn^ + 0O p" = p. •

From the above properties we can easily prove the following: Theorem 4.2. Assume that V and H are finite dimensional and that (P) has a solution u. We suppose that: • B is an injection, • G is convex, proper, and l.s.c, • F = Fo + Ft with Fj convex, proper, and l.s.c. over H and Fo strictly convex and C1 over H. Then (P) has a unique solution, and if 0 < a 0 < pn < ati < 2r holds, we have, for ALG 1, the following convergence properties: lim u" = u, -* + 00

lim p" = Bu, -> + oo

lim X"+ i

_ ;

X" is bounded in H. Moreover, if X is a cluster point of {/l"}n, then {u, p, X} is a saddle point of 5£r over V x H x H.

4.3. Comments on the use of ALG 1. Further remarks Assume that r is fixed and that we use a fixed value p for pn. Then, from our computational experience, it appears that the best convergence is obtained for p = r. Concerning the choice of r, it can be theoretically proved that the larger the r, the faster the convergence; practically, the situation is not so simple, for the following reasons. The larger the r, the worse the conditioning of the optimization problem (3.3) (or of the equivalent system (3.5), (3.6)). Then, since (3.3) is numerically (and not exactly) solved, at each iteration an error is made in the determination of {«", p"}. The analysis of this error and the effect of it on the global behavior of ALG 1 is a very complicated problem since we have to take into account the conditioning of (3.3), the stopping criterion of the algorithms (usually iterative) solving (3.3), roundoff errors, etc. Fortunately, it seems that the combined effect of all these factors is an algorithm which is not very sensitive to the choice of r (see Glowinski and Marrocco [5] and Fortin and Glowinski [1] for more details).

4. Convergence of ALG 1

177

From a numerical point of view, the only nontrivial part in the use of ALG 1 is the solution at each iteration of problem (3.3). Taking into account the particular structure of (3.3), it follows from Chapter V, Cea and Glowinski [1], and Cea [2] that a method very well suited to the solution of (3.3) is the block-relaxation method described below: All the problems (3.3) are of the following type: J2?,(K, p , H) < Ser{v,

q, n),

V {», q} e V x H , {u, p } e V x H , (4.48)

where \i is given. The minimization problem (4.48) is equivalent to the system

G(v) - G(u) + (n,B(v - u)) + r(Bu - p,B(v - u)) > 0,

VveV,ueV, (4.49)

F{q) - F(p) - {ii, q - p) + r(p - Bu, q - p) > 0,

V«-efl,

p e H. (4.50)

Then a block-relaxation method for solving (4.49), (4.50) is {u°,p0} given;

(4.51)

then, {um, pm} known, we obtain {um+ \ pm+1} from G(v) - G(« m+1 ) + (/i, B(v - wm+1)) + r(Bum+1 - pm, B(v - um+ x)) > 0, m+i

F(q) - F(j>

) -(n,q-

m+1

p

) + r(p

m+1

\fve V, um+1e V, m+1

- Bu

(4.52)

m+1

, q - p ) > 0, pm+1eH.

V<JEJFJ,

(4.53)

Sufficient conditions for the convergence of (4.51)-(4.53) may be found in Chapter V, Cea and Glowinski, and Cea, loc cit. In practice, when using (4.51)-(4.53), a stopping test of the following type will be used: Max(|| M m+1 - um\\, \pm+1 - pm\) < e. Another possibility is to stop after a fixed number of iterations. If, for instance, we stop after only one iteration of (4.51)-(4.53), and if at iteration n of ALG 1 we initialize with {M"" 1 , p"" 1 } the computation of {np, p"} by (4.51)-(4.53), then we recover ALG 2. Remark 4.1. Other relaxation methods can also be used; moreover, it can be worthwhile to introduce over-relaxation parameters to increase the speed of convergence of (4.51)-(4.53). Remark 4.2. The choice p = r may be motivated by the following proposition.

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VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Proposition 4.1. Suppose that F(q) = j\q\2 and that G is linear. Then V 1° e H, we have, for the sequence {«"}„ of A L G 1, convergence to the solution u of (P) in less than three iterations if we use pn = p = r,r given. Preliminary Remark. In the above situation we have (P) equivalent to B'Bu = f,

(4.54)

where G(v) = ((/, v)), V v e V. Therefore, using ALG 1 for solving (P) has no practical interest. But even in this trivial case we shall observe that the behavior of ALG 1 is "interesting" since the convergence of u" in a finite number of iterations does not imply a similar convergence property for p" and X". PROOF OF PROPOSITION 4.1 From (4.17), (4.18) it follows that in the particular case we are considering, ALG 1 reduces to

X° given in H;

(4.55)

rB'Bu" = rB'p" - B'X" + / ,

(4.56)

n

p" = X + r(Bu" - p"), A"

+1

(4.57)

= X" + r{Bu" - p").

(4.58)

We can easily prove that the unique saddle point of i f r over V x H x H is {u, Bu, Bu}, i.e., p = Bu, X = Bu; using the notation u" = u" — u, p" = p" — p, X" = A" — X, it follows from (4.56)^(4.58) that B'Xn + rB\Bu" - p") = 0, +1

n

X"

+

= p => X" * = p",

p" = X" + r(Bu" - p"),

V n > 0,

(4.59)

V n > 0,

(4.60)

V n > 0.

(4.61)

Multiplying (4.61) by B' and comparing with (4.59), we obtain B'p" = 0,

V n > 0.

(4.62)

Since (4.60), (4.61) imply pn+1 =p«

+ r(BU-+1

_ p»+i);

v«>0,

(4.63)

multiplying by B' and taking (4.62) into account, we obtain B'Bu"+ x = 0,

V n > 0 => Bu"+1 = 0,

V n > 0.

(4.64)

Since \Bv\ is a norm on V, (4.64) implies that u" = u, V n > 1. Hence the convergence of u" to u requires two iterations at most. Using (4.63), (4.64), we have "+'

1+r

p",

V n > 0.

(4.65)

From (4.65) it follows that the larger the r, the faster p" converges to p = Bu; for more details on the convergence of p", see Fortin and Glowinski [2]. •

5. Convergence of ALG 2

179

5. Convergence of ALG 2 5.1.

Synopsis

In this section we shall prove that under fairly general assumptions on F and G, we have convergence of ALG 2 if 0 < pn = p < [(1 + -v/5)/2]r. We do not know if this result is optimal, since in some cases (G linear, for example) the upper bound of the interval of convergence is 2r. Actually this question is rather academic, since in the various experiments we have done with ALG 2, the optimal choice seems to be p = r.

5.2. General case We study the convergence of ALG 2 with the same hypotheses on B, F, G as in Sec. 4.1. We then have: Theorem 5.1. We suppose that ^£r has a saddle point {u, p, X} over V x H x H. Then, if the assumptions on B, F, G are those of Sec. 4.1 and if

0
(5.1)

we have the following convergence properties: u" —> u strongly in V,

(5.2)

p" -*• p strongly in H,

(5.3)

X«+1 _ X« _• 0 strongly

in H,

(5.4)

X" is bounded in H.

(5.5)

Moreover, if X* is a weak cluster point of {X"}n, then {u, p, X*} is a saddle point of£eroverV x H x H. PROOF.

Let us again define u", p", 1" by u" = u" - u,

p" = p" - p,

I" = X" - X.

Since {u, p, X} is a saddle point of i£r over V x H x H, we have G(v) - G(u) + (X, B(v - u)) + r(Bu - p, B(v - «)) > 0,

V v e V,

(F'0(p), q - p) + Ft(q) - F\(p) - (X, q - p) + r(p - Bu, q - p) > 0 X = X + p(Bu - p).

(5.6)

V q e H, (5.7) (5.8)

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VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Moreover, (3.8)-(3.10) imply G(i>)- G(u") + (An, B(v - u")) + r(Bu" -p"-\B(v

- « " ) ) > 0,

V v e V,

( W X 9 - P") + F^q) - F.ip") - (A", q - f) + r(p" - Bun, q - p") > 0, V q e H, +1

r

(5.9) (5.10)

= r + P(Bu"-p").

(5.ii)

Taking v = u" (resp. t> = u) in (5.6) (resp., (5.9)) and q = p" (resp., q = p) in (5.7) (resp., (5.10)), we obtain, by addition, r(Bif - pn'\ Bun) + (1", BM") < 0,

(5.12)

(F'oipl ~ F'0(p), p") + r(p" - 5«", p") - (1", p") < 0.

(5.13)

By addition of (5.12) and (5.13), it follows that (F'0(p") - F'0(p), p"-p) + r\Bu" - p"\2 + (I", Bu" - p") + r(p" - p"~ \ Bit") < 0. (5.14) By subtracting (5.8) from (5.11), we obtain |!"| 2 - | I " + 1 | 2 = -2p(Bu" - p",X") - p2\Bu" - p"| 2 .

(5.15)

From (5.14), (5.15) it then follows that | A f - \X"+l\2 > 2p(F'0(p") - F'Q(p),p") + p(2r - p)\Bu" - p"\2 + 2pr(p" -

p"-\Bu"). (5.16)

Starting from Bu" = (Bu" - Bunl)

+ (Bu"~s - p"" 1 ) + p " - \

we obtain (Bu", pn - p"~') = (Bu" - Bu"~l,p" - p"~l) + (Bu"~l -p"-\p"

~ p"-1) + (p"-1,p"-p"-1).

(5.17)

Since (p"- 1 , p" - ff-1) = i ( | p f - Ip"-112 - \p" - p""112), it follows from (5.17) that 2pr(Bu", p" - p " " 1 ) = 2pr(Bu" - Bunl, p" - p " " 1 ) + 2pr(Bu"~1 + pr(\p"\2-\p"-1\2-\p"-r-1\2).

- p " ~ \ p" - p " " 1 ) (5.18)

Taking (5.10) at n — 1 instead of n, we have (F'0(p"- % q-p"~1) + FM - F^"^) + r(p"~1 - Bu"' \ q - f~l) > 0.

-(X"~\q-

p"~l) (5.19)

Taking q = p"~1 in (5.10) and q = p" in (5.19), we obtain, by addition, (F'0(p") - F 0 (p"-'), p" - p""') - (ln - l"~\ p" - p"-1) + r\p" - p-112 -r(Bu" - Bu"~ \ p" - p"- 1 ) < 0. (5.20)

5. Convergence of ALG 2

181

But since F'o is monotone, it follows from (5.20) that r\F ~ F~112 - (^ - I-"' \ p" - f-1)

- r(Bu" - Bu"~\ p" - F'1) < 0.

(5.21)

We have (from (3.10)) X" = A"" 1 + p(Bu"~l

-

p"~l),

1" - l"~1 = p(Bu"-'

- p"'1).

which implies that (5.22)

From (5.21), (5.22) it then follows that r\p" - p"" 1 ! 2 - p{Bu"-1 - p"~\ p" - p"'1) - r{Bu" - Bu"'1^"

- p"" 1 ) < 0,

i.e., r{Bu" - Bu"~ \ p" - p"'1)

p"'1 \2 - piBu"'1

>r\p"~

- p"~ \ p" - p " ~ ! ) .

(5.23)

From (5.18), (5.23) it then follows that 2pr(Bu",p" - p"-1) > pr(\p"\2 - |p"^| 2 ) + pr\P" -

F'1?

+ 2p(r - p)(Bu"-1 - p"-\ p" - p"~l).

(5.24)

Finally, combining (5.16) and (5.24), we obtain (\l"\2 + pr|p""M 2 ) - (U" + 1 | 2 + pr\T\2) > MF'O(P") - F'Q(p),jr) + p(2r - p)\Bu" - pn\2 + pr\p" - p"'1]2) + 2p(r - p)(Bu"-1 - pn~ \ p" - f ~ ' ) .

(5.25)

Then using, the Schwarz inequality, it follows from (5.25) that, V a > 0, we have (|Af + H p " " T ) - (U" +1 | 2 + pr\p"\2) > 2p(F'0(p") - F'0(p),p") + p(2r - p)\Bu" - p"\2 + pr\? - p"~'\2 - p\r - PlHBu"-1

-F~l\2

+ *\F-r-1\2\

(5-26)

If p = r, it is clear that, using the same method as in the proof of Theorem 4.1, we have (5.2)-(5.5). If 0 2p(F>0(p") - F'0(p), F) + pr\Bu - p"| 2 + p 2 |p" - p"" 1 1 2 > 0, which clearly implies (5.2)-(5.5). If p > r, we have \r — p\ — p — r, and then it follows from (5.26) that (5.2)-(5.5) holds if we have p < pM, where - pM) = - PM(PM

~r),

a ~ r). By elimination of a, it follows from (5.27) that PM

- ?PM - r2 = 0,

(5.27)

182

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

i.e. (since pu > 0),

_ 1 + V5 Pu —

2

r

'

Then, using basically the same method as in the proof of Theorem 4.1, we can easily prove, from (5.2)-(5.5), that {u, Bu, X*} is a saddle point of !£r over V x H x H if X* is a weak cluster point of {X"}n. •

5.3. Finite dimensional case Using a variant of the proof of Theorem 5.1, and Lemma 4.1, we can easily prove: Theorem 5.2. Assume that the assumptions on V, H, F, B, G are those of the statement of Theorem 4.2. Then if

the conclusions of the statement of Theorem 4.2 still hold.

5.4. Comments on the choice of p and r 5.4.1. Some remarks Remark 5.1. If G is linear, it has been proved by Gabay and Mercier [1] that ALG 2 converges if 0 < pn = P < 2r. The proof of this result is rather technical, and an open question is to decide if it can be extended to the more general cases considered here. Remark 5.2. If G is linear, we observe that the step (3.8) of ALG 2 is a linear problem related to the self-adjoint operator B'B. Therefore, in the finitedimensional case, assuming B injective, it will be convenient to factorize (by a Cholesky method, for example) the symmetric positive-definite matrix B'B once and for all, before starting the iterations of ALG 2. 5.4.2. On the choice of p and r If r is given, our computational experience seems to indicate that the best choice for p is p = r. The choice of r is not clear, and ALG 2 appears to be more sensitive to the choice of r than ALG 1. By the way, ALG 1 seems to be more robust on very stiff problems than ALG 2; this means that the choice of

6. Applications

183

the parameter is less critical and that the computational time for ALG 1 may become much shorter than for ALG 2. Remark 5.3. In Remark 4.2 we have seen that if JF(^) = j\q\2 and G is linear, the sequence {«"}„ related to ALG 1 converges in two iterations (at most) if we use p = r. If we use ALG 2 with the same hypotheses on F and G, then we have convergence of {«"}„ in two iterations at most only if p = r = 1 (for any choice of {p°, A1}). This fact also confirms the greater robustness of ALG 1.

6. Applications 6.1. Bingham flow in a cylindrical pipe

This is the problem considered in Chapter II, Sec. 6 and also in Sec. 1.1 of this chapter (we recall that Q is a bounded domain of U2):

Min \\ f | Vv\2 dx + g [ \Vv\dx-

[ fv dx\

(6.1)

Then (6.1) is a particular problem (P) corresponding to H = L 2 (Q) x L2(Q),

V = HJ(Q),

B = V,

(6.2)

F(q) = ^ j \q\2 dx • 2 Jn

(6.3)

G(V)=-

(6.4)

+ g j \ q \ dx,

r fv dx.

Moreover, we have F = Fo + Ft with f« . | / i ) — _

1 \n\

Jn

/TV

(6.5)

K(q) = vq,

i ^ ) = gr f |$|dx.

(6.6)

Jn

It then follows from (6.2)-(6.6) that the various assumptions required to apply Theorems 4.1 and 5.1 are satisfied. Therefore we can solve (6.1) by ALG 1 and ALG 2. Moreover, since G is linear, the result of Gabay and Mercier [1] holds (see Remark 5.1) and ALG 2 converges if 0 < pn = p < 2r. The augmented Lagrangian S£t to be used in this case is given by

JS?r(». 1, fi = I f k l 2 dx + g f \q\ dx - f fvdx+ ^ Jn

Jn

\\Vv-q\2dx. Jn

Jn

f

Jn

p.-(Vv-q)dx (6.7)

184

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Solution of (6.1) by ALG 1. When applying ALG 1 to the solution of (6.1), it follows from (3.2)-(3.4), (6.7) that we have X° e L 2 (Q) x L2(Q), arbitrarily given;

(6.8)

then for n > 0, -rAun=f

+ W-r -rW -pnonQ,

n

p (x) = 0 ifg>\X\x)

un\r = 0,

(6.9)

+ rWu\x)\,

A"+1 = A" + pn(Vw" -/?").

(6.11)

Solution of (6.1)ftyALG 2. We have to replace (6.8) by {p°, X1} arbitrarily given in (L 2 (Q)) 2 x (L 2 (fi)) 2 ,

(6.12)

-rAu" =f+V-Xn-rV-p"'1

(6.13)

and (6.9) by onD., u"\r = 0.

Remark 6.1. In practice, (6.8)-(6.11) and (6.12), (6.13), (6.10), and (6.11) will be applied to finite element or finite difference approximations of (6.1). It then follows from (6.9), (6.13) that it is easy to use either ALG 1 (combined with the block-relaxation method of Sec. 4.3) or ALG 2 once we have at our disposal an efficient program for solving approximate Dirichlet problems for -A. Bibliographical Comments. Numerical solutions of (6.1) by ALG 1 and ALG 2 may be found in Gabay and Mercier [1] and Fortin, Glowinski, and Marrocco [1]; in Fortin [2] we can also find an iterative method of solution of (6.1) close to ALG 2 but obtained by a different approach. 6.2. Elasto-plastic torsion of a cylindrical bar This is the problem of Chapter II, Sec. 3, also considered in Sec. 1.1 of this chapter (Q is a bounded domain of U2 in the sequel):

Min I \ f |Vv\2 dx - f fv dx , V<=K [2 J n

Jn

(6.14)

J

where K = {v\v e Hj(Q), |Vv\ < 1 a.e.}; (6.14) is a particular problem (P) corresponding to V = Hj(Q),

H = L2(Q) x L2(Q),

B = V,

(6.15)

G{v)=-[fvdx,

(6.16)

F = F0 + Fu

(6.17)

6. Applications

185

where

[

\ [q\2dx^F'0{q)

= q,

(6.18)

FM = h(q)

(6.19)

with K = {q e H, \ q \ < 1 a.e.} and /^ the indicator functional of K, i.e.,

°

^!f

(6.20)

+ oo ifq$K. Here, too, it follows from (6.15)-(6.20) that the various assumptions required to apply Theorems 4.1 and 5.1 are satisfied. Therefore we can solve (6.14) by ALG 1 and ALG 2. Moreover, from the linearity of G, we have the convergence of ALG 2 if 0 < pn = p < 2r. In the present case, i f r is given by = - I \q\2 dx + Ij>(q) -

f fvdx fvdx + + f /p-(Vv -

v-q\2dx.

q) dx (6.21)

Solution o/(6.14) by ALG 1. From (3.2)-(3.4), (6.21) it follows that when applying ALG 1 to (6.14), we obtain X° arbitrarily given in (L 2 (Q)) 2 ;

(6.22)

then for n > 0, -rAu" = f + V • X" - rV • p" on Q, (6.23) «"lr = 0,

X"+1 = X" + Pn(Vu" - p").

(6.25)

Solution of (6.14) by ALG 2. We have to replace (6.22) by (6.12) and (6.23) by (6.13). Remark 6.1 still applies to (6.14), and numerical solutions of (6.14) by ALG 1 and ALG 2 may be found in Fortin, Glowinski, and Marrocco [1] and Gabay and Mercier [1].

6.3. A nonlinear Dirichlet problem In this section we follow Glowinski and Marrocco [ 5 ] ; let us consider 1 < s < +oo and Wl-S(Q)

= mQ)wllSW

where Q is a bounded domain of UN.

= {ve WU%Q.),

v\T = 0 } ,

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VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Then we consider on Q the following nonlinear Dirichlet problem: -V-(|VM|S-2VW)=/,

(6.26) where feV' = W-Us'(O)(l/s + 1/s' = 1 => s' = s/(s - 1)). It can be proved (see, for instance, Glowinski and Marrocco [5]) that (6.26) has a unique solution which is also the solution of

Min

[- f | Vv\s dx - .

(6.27)

We observe that Wl's(Q.) is not a Hilbert space if s ^ 2; therefore we cannot apply Theorems 4.1 and 5.1 to the iterative solution of (6.27). Nevertheless, once (6.27) has been approximated by a convenient finite element or finite difference method, it is possible to apply the above theorems (or Theorems 4.2 and 5.2) to the iterative solution of the approximate problem. For the sake of simplicity, we shall confine our study to the continuous problem, since it has simpler notation. We have V = Wl's{£l\

H = (Ls(Q)f,

H' = (LS'(Q))JV,

B = V,

F(q) = F0(q) = - f \q\> dx, F'(q) = q\q\'~2, s Jn G(v)= - < . / » . We observe that lim

^ =

o o ,

+

where

\q\s = ( [ \q\'dx)1"

= \\q\\vm-

We also have V p, q e H: (F'(q) - F'(pl q - p) > \q - p\l if s > 2, ( F ( g ) - F'(p), q - p ) > * UP

IF'(q) ~ F ' ( p ) \ , < P(|p\,

\ q ~ Pls2_s is "r \q\s)

+ \q\sf~2\q 1

\F'(.q)-F'(p)\,
i f l < s < 2 ,

- p \ , if s > 2, if 1 < s < 2,

where a, j5 are independent of p, q and are strictly positive.

(6.28) (6.29)

(6.30) (6.31)

6. Applications EXERCISE

187

6.1. Prove (6.28)-(6.31).

We refer to Glowinski and Marrocco [5] for a detailed analysis, including error estimates, of a finite element approximation of (6.26), (6.27) (see also Ciarlet [2]). From our numerical experiments it appears that solving (6.26), (6.27) if s is close to 1 (say 1 < s < 1.3) or large (say s > 5) is a very difficult task if one uses standard iterative methods; to our knowledge the only very efficient methods are ALG 1 and ALG 2 (or closely related algorithms; see Glowinski and Marrocco, loc. cit., for more details). The augmented Lagrangian j§?r to be used for solving (6.26), (6.27) is defined by

&Jv,q,fi)

= - f \qfdx s Jn

- < / » + T- { \ V v - q \ 2 d x + [ p • (Vi> - q) dx. 2 Jn

Jn

(6.32) Solution of (6.26), (6.27) by ALG 1. From (3.2)-(3.4), (6.32) it follows that when applying ALG 1 to (6.26), (6.27), we obtain: A°e(Z/(Q)f;

(6.33)

then for n > 0, -rAu" = / + V • A" - rV • p" in Q, (6.34) «"| r = 0, I p T Y + rp" = rVu" + X",

(6.35)

A" + 1 = X" + pn(Vun - p").

(6.36)

The nonlinear system (6.34), (6.35) can be solved by the block-relaxation method of Sec. 4.3, and we observe that if u" and X" are known (or estimated) in (6.35), the computation of p" is an easy task since \p"\ is solution of the singlevariable nonlinear equation ipT1

+r\pn\ = \rVun + Xn\,

(6.37)

which can easily be solved by various methods; once | p" \ is known, we obtain p" by solving a trivial linear equation (in (Ls(Ci))N). Solution of (6.26), (6.27) by ALG 2. We have to replace (6.33) by {p°, k1} e H x H'

(6.38)

and (6.34) by -rAun=f+

V-A" - rV

-p"-\ (6.39)

«"|r = 0.

188

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Remark 6.1 still applies to (6.26), (6.27), and since G is linear, we can take 0 < pn = p < 2r if we are using ALG 2. For more details and comparisons with other methods, see Glowinski and Marrocco [5], [7] and Fortin, Glowinski, and Marrocco [1]. Remark 6.2. ALG 1 and ALG 2 have also been successfully applied to the iterative solution of magneto-static problems (see Glowinski and Marrocco [6]). They have also been applied by Fortin, Glowinski, and Marrocco [1] to the solution of the subsonic flow problem described in Chapter IV, Sec. 3; in the latter case, using ALG 1 and ALG 2, we obtain easy variants of (6.33)(6.36) and (6.38), (6.39), (6.35), and (6.36).

6.4. Application to the solution of mildly nonlinear systems Let A be a N x N symmetric positive-definite matrix, let D be a diagonal positive-semidefinite matrix, and let f e UN. Let : R -> R be a C° and nondecreasing function (we can always suppose that 0(0) = 0). Using the same notation as in Chapter IV, Sec. 2.6, we associate to v = {vu v2, •. •, vN} e UN the vector (v) e UN defined by (
Vi=l,...,N.

(6.40)

Then we consider the nonlinear system Au + D0(u) = f.

(6.41)

In Chapter IV, Sec. 2.6, various methods for solving (6.41) have been given, but in this section we would like to show that (6.41) can also be solved by ALG 1 and ALG 2 once a convenient augmented Lagrangian has been introduced. Remark 6.3. The methods to be described later are easily generalized to the case where A is not symmetric but still positive definite. Let us define

Jo

Since cj) is C° and nondecreasing, we find that $ is C 1 and convex. It then follows from the symmetry of A that solving (6.41) is equivalent to solving the minimization problem J(u) < J(v),

V v e UN, u e UN.

(6.42)

In (6.42) we have

J(y) = \ (Av, v) + £ dMvd - (f, v),

(6.43)

6. Applications

189

where (-, •) denotes the usual inner product of UN and || • || denotes the corresponding norm and where

D =|

d,

From the above properties of A, D, and $, it follows from, e.g., Cea [1], [2], that (6.41), (6.42) has a unique solution. Remark 6.4. In fact (6.41) has a unique solution if A is positive definite and possibly nonsymmetric, the assumptions on and D remaining the same. Problem (6.42) is a particular problem (P) corresponding to V = H = UN, B = I,

(6.44)

G(v)= £ d, <*(»,)-(f,v),

(6.45)

F(q) = F 0 (q) = ^Aq, q) => F 0 (q) = Aq.

(6.46)

From these properties we can solve (6.41), (6.42) by using ALG 1 and ALG 2 (we observe that, unlike in the above examples, G is nonlinear). Remark 6.5. Instead of using G and F defined by (6.44), (6.45), we can use

GOO = £ dMvd, F(q) = i(Aq, q) - (f, q). The augmented Lagrangian to be associated with (6.44)-(6.46) is ^,(v, q, »i) = \ (Aq, q) + £ d,
(6.48)

190

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

then for n > 0, ru" + D
(6.49)

(rl + A)p" = ru" + X",

(6.50)

The nonlinear system (6.49), (6.50) can be solved by the block-relaxation method of Sec. 4.3, and we observe that if p" and X" are known (or estimated) in (6.49), the computation of u" is easy since it is reduced to the solution of N independent single-variable nonlinear equations of the following type: r£ + #( 0).

(6.52)

Since r > 0 and 4> is C° and nondecreasing, (6.52) has a unique solution which can be computed by various standard methods (see, e.g., Householder [1] and Brent [1]). Similarly, if u" and X" are known in (6.50), we obtain p" by solving a linear system whose matrix is rl + A. Since r is independent of n, it is very convenient to prefactorize rl + A (by the Cholesky or Gauss method). Solution of (6.41), (6.42) by ALG 2. We have to replace (6.48) by {p°, X1} eUN xUN

(6.53)

and (6.49) by ru" + D
(6.54)

From Theorem 5.2 it follows that we have convergence of (6.53), (6.54), (6.50), and (6.51) if 0 < Pn = p < (1 + y/S)/2r. Remark 6.6. Suppose that pn = p = r in ALG 2; we then have ru" + D0(u") = f + rp"" 1 - Xn, rp" + Ap" = ru" + X",

(6.55)

Xn+1 = X" + r ( u " - p " ) . From (6.55) it follows that X"+1=Ap".

(6.56)

Then from (6.55), (6.56) we obtain ru" + D(u") + Ap"- J = f + rp"- \ rp" + Ap" + D(u") = f + rp"" 1 .

(6.57) (6.58)

Therefore, if pn = p = r, ALG 2 reduces (with different notation) to the alternating-direction method described in Chapter IV, Sec. 2.6.6. (for more details on the relation existing between alternating-direction methods and

6. Applications

191

augmented Lagrangian methods, we refer to G.L.T. [3, Appendix 2], Gabay [1], Bourgat, Dumay, and Glowinski [1], and Bourgat, Glowinski, and Le Tallec [1]). Remark 6.7. From the numerical experiment performed in Chan and Glowinski [1], ALG 1, combined with the block-relaxation method of Sec. 4.3, is much more robust than ALG 2; this is the case if, for instance, we solve a finite element (or finite difference) approximation of the mildly nonlinear elliptic problem -Aw + M|M| S " 2 = /

onQ,

« | r = 0,

(6.59)

with 1 < s < 2. In Chan and Glowinski, loc. cit., we can find various numerical results and also comparisons with other methods (see also Chan, Fortin, and Glowinski [1]).

6.5. Solution of elliptic variational inequalities on intersections of convex sets 6.5.1. Formulation of the problem Let V be a real Hilbert space and a:V x F->[Rbea bilinear form, continuous, symmetric, and F-elliptic. Let K be a closed convex nonempty subset of V such that N

K= f]Kh

(6.60)

i=l

where, V i = l,...,N,Ktisa. EVI problem

closed convex subset of V. We then consider the

a(u, v - u) > L(v - u),

VveK,

us K,

(6.61)

where L: V —>• U is linear and continuous. Since a(-, •) is symmetric, we know from Chapter I that the unique solution of (6.61) is also the solution of J(u) < J(v),

VceX,

ueK,

(6.62)

where J(v) = \a(v,v) - L(v).

(6.63)

6.5.2. Decomposition of (6.61), (6.62) Let us define (with q = {qu...,

qN})

W = {{v, q} e V x VN, v - qt = 0,

V i = 1 , . . . , JV}

(6.64)

192

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

and Jf = {{v,q}eW,qieKi,

V i = 1,..., N}.

(6.65)

It is clear that (6.62) is equivalent to Min j(v, q),

(6.66)

where

Kv, «) = ^

I «(«*. 9d - UP)-

(6-67)

Remark 6.8. We have to observe that many other decompositions are possible; for instance, W = {{v, q}eV

x VN,v - ql=0,qi+1-

qt = 0,

V i = 1 , . . . , N - 1}

with j and X again defined by (6.67), (6.65). We can also use W = {{v,q}eV

x VN~\ v - qt = 0,

V i = 1,... ,N - 1}

with JtT = {{v,q} eW,ve

Kuqie

Ki+U

Vi = 1.....N - 1}

and 1 1 N~l J(P, ' We suppose that in the sequel we use the decomposition defined by (6.64)(6.67); then (6.66) is a particular problem (P) corresponding to H = VN,

Bv = {v,..., D},

G(v) = -L(i>),

f o(«)•= ^

t

a

^ id,

^i(«) = t IM

(6.68) (6.69)

(6.70) (6.71)

i=l

with IK.: indicator function of Kt. It is easily shown that from the properties of B, G, F we can apply ALG 1 and ALG 2 to solve (6.62), via (6.66), provided that the augmented Lagrangian r N 1N £Cr(v, q, n) = F(q) + G(v) + — J] a(v - qt, v - qt) + — J] Qi{, v - qt) (6.72)

6. Applications

193

has a saddle point over V x VN x VN. Such a saddle point exists if H is finite dimensional, since the constraints v — qi = 0 are linear. 6.5.3. Solution of (6.62) by ALG 1. From (3.2)-(3.4), (6.72) it follows that when applying ALG 1 to (6.62), we obtain: X° then for n >

given;

(6.73)

o, XI, v ) +L(v),

- 1

=r ra(u", v) =

MveV,

u"eV, (6.74)

(1 + r)a(p1,

- pi) > ra(u",

-Pi) + (XI, q<-Pi),

V qt e Kh

pi e Kt (6.75),

/ori=l,2,...,JV; ^ + 1 = ^ + P«(«"-P?)

(6.76)i

/or i = 1 , . . . , AT. The system (6.74), (6.75) is, for X" given, a system of coupled EVI's; a very convenient method for solving it is the block-over-relaxation method with projection described in Chapter V, Sec. 5 and also in Cea and Glowinski [1] and Cea [2]. This method will reduce the solution of (6.62) to a sequence of EVI's on Kh i = 1 , . . . , N. 6.5.4. Solution of (6.62) by ALG 2 From (3.7)-(3.10), (6.72) it follows that to solve (6.62) by ALG 2, we have to use the variant of (6.73)-(6.76) obtained by replacing (6.73), (6.74) with: {p°,X1}eVN

ra(u\ v) = ra(~ |

P1~

x VN given;

\ v\ - l~ | XI, v\ + L(v),

(6.77)

V v e V, u" e V. (6.78)

Remark 6.9. The two algorithms above are well suited for use in multiprocessor computers, since many operations may be done in parallel; this is particularly clear for algorithm (6.77), (6.78), (6.75), (6.76). Remark 6.10. Using different augmented Lagrangians, other than <£r denned by (6.72), we can solve (6.62) by algorithms better suited to sequential computing than to parallel computing. We leave to the reader, as exercises, the task of describing such algorithms.

194

VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications

Remark 6.11. The two algorithms described above can be extended to EVI's where a(-, •) is not symmetric. Moreover, they have the advantage of reducing the solution of (6.62) to the solution of a sequence of simpler EVI's of the same type, to be solved over Kt, i = 1,...,N, instead of K.

7. General Comments As mentioned several times before, the methods described in this chapter may be extended to variational problems which are not equivalent to optimization problems. These methods have been applied by Begis and Glowinski [1] to the solution of fourth-order nonlinear problems in fluid mechanics (see also Begis [2] and G.L.T. [3, Appendix 6]). From a historical point of view, the use of augmented Lagrangians for solving—via ALG 1 and ALG 2—nonlinear variational problems of type (P) (see Sec. 1.1) seems tobedueto Glowinski and Marrocco [3], [4], [5]. For more details and other applications, see Gabay and Mercier [1], Fortin and Glowinski [1], [2], Glowinski and Marrocco, loc. cit., and also Bourgat, Dumay, and Glowinski [1], Glowinski and Le Tallec [1], [2], Le Tallec [1], Bourgat, Glowinski, and Le Tallec [ 1 ], and Glowinski, Le Tallec, and Ruas [ 1 ], where ALG 1 and ALG 2 have been successfully used for solving nonlinear nonconvex variational problems occurring in finite elasticity (particularly in inextensible and/or incompressible finite elasticity). With regard to Sec. 3.2, D. Gabay [1] has recently introduced the following variant of ALG 2: {p°, Xl}eH n

n

x H given; n+1 2

then, {p ~\ X"} known, we define {u , X n

(7.1)

n+1

' , p", X

n

} by

n

G(v) - G(u ) + (X , B(v - «")) + r(Bu" - p ~ \ B(v - if)) > 0, V v e V, uneV, 1/2

y +

F(q) - F(p") - (A"

+1/2

, q-

Xn +

=

n

p(Bun

_

pn-

n

1^

(73) n

p )+ r(p" - Bu , q - p ) > 0, \fqeH,

+J

A"

(7.2)

= X

n+1/2

+ p(Bu" - p"),

p"eH,

(7.4) (7.5)

with p > 0 in (7.3), (7.5). For additional details and convergence properties, see Gabay, loc. cit. (and also Gabay [2]). To conclude this chapter, we have to mention that, using some results due to Opial [1], we have, in fact, in Theorems 4.1 and 5.1 (resp., 4.2 and 5.2) the weak convergence (resp., the convergence) of the whole sequence {X"}n to a A* such that {u, p, X*} is a saddle point of i f (and &r) over V x H x H. We refer to Glowinski, Lions, and Tremolieres [3, Appendix 2] for a proof of the above results in a more general context (see also Gabay [2]).

CHAPTER VII

Least-Squares Solution of Nonlinear Problems: Application to Nonlinear Problems in Fluid Dynamics

1. Introduction: Synopsis In this chapter we would like to discuss the solution of some nonlinear problems in fluid dynamics by a combination of least-squares, conjugate gradient, and finite element methods. In view of introducing the reader to this technical subject, we consider in Sec. 2 the solution of systems of nonlinear equations in UN by least-squares methods; then, in Sec. 3, the solution of a nonlinear Dirichlet model problem; also in Sec. 3 we make some comments about the use of pseudo-arc-length-continuation methods for solving nonlinear problems. In Sec. 4 we discuss the application of the above methods to the solution of the nonlinear equation modelling potential transonic flows of inviscid compressible fluids; finally in Sec. 5 we discuss the solution of the Navier-Stokes equations, for incompressible viscous Newtonian fluids, by similar techniques. This chapter is closely related to Bristeau, Glowinski, Periaux, Perrier, and Pironneau [1] and Bristeau, Glowinski, Periaux, Perrier, Pironneau, and Poirier [1]; other references will be given in the sequel.

2. Least-Squares Solution of Finite-Dimensional Systems of Equations 2.1. Generalities Replacing the solution of finite-dimensional systems of equations by the solution of minimization problems is a very old idea, and many papers dealing with this approach can be found in the literature. Since referring to all those papers would be an almost impossible task, we shall mention just some of them, referring to the bibliographies therein for more references. The methods most widely used have been the least-squares methods in which the solution of F(x) = 0, where F: UN ->• UN with F = {/,,..., fN}, is replaced by:

(2.1)

196

VII Least-Squares Solution of Nonlinear Problems

Find xeUN such that

VyeR",

(2.2)

where in (2.2), || • || denotes some Euclidean norm. If N is not too large, a natural choice for || • || is (if y = {yu ..., yN}) /N

\l/2

nyi = ( w )

•

(23)

Suppose, for example, that F(x) = Ax - b,

(2.4)

where A is an N x N matrix and b e UN. If || • || is denned by (2.3), then the corresponding problem (2.2) is equivalent to the well-known normal equation ArAx = A'b,

(2.5)

where A' is the transpose matrix of A. This simple example shows the main advantage of the method, which is to replace the original problem Ax = b,

(2.6)

whose matrix is possibly nonsymmetric and indefinite, by the problem (2.5) whose matrix is symmetric and positive semidefinite (or equivalently, by the minimization of a quadratic convex functional). This convexification property (which can only be local in nonlinear problems) is fundamental since it will insure the good behavior (locally, at least) of most minimization methods used to solve the least-squares problem (2.2) (once a proper || • || has been chosen; see below). Also, from (2.5) it is clear that a main drawback of the method is the possible deterioration of the conditioning which, for example, may make the solution of (2.2) sensitive to roundoff errors. Actually in many problems this drawback can be easily overcome by the use of a more sophisticated Euclidean norm than (2.3). Indeed, if || • || is defined by llyll = (Sy, y)^ 2

(2.7)

(where S is an N x N positive-definite symmetric matrix and (x, y)RW = Xf= i xiyi)an^ if F is still defined by (2.4), then (2.5) is replaced by A'SAx = A'Sb.

(2.8)

With a proper choice of S we can dramatically improve the conditioning of the matrix in the normal equation (2.8) and make its solution much easier. This matrix S can be viewed as a scaling (or preconditioning) matrix. This idea of preconditioning stiff problems will be systematically used in the sequel. The standard reference for linear least-squares problems is Lawson and Hanson [1]; concerning nonlinear least-squares problems of finite dimension

2 Least-Squares Solution of Finite-Dimensional Systems of Equations

197

and their solution, we shall mention, among many others, Levenberg [1], Marquardt [1], Powell [3], [4], Fletcher [1], Golub and Pereyra [1], Golub and Plemmons [1], Osborne and Watson [1], and More [1] (see also the references therein).

2.2. Conjugate gradient solution of the least-squares problem (2.2) Conjugate gradient methods have been considered in Chapter IV, Sec. 2.6.7; actually they can also be used for solving the least-squares problem (2.2). We suppose that in (2.2) the Euclidean norm ]| • || is defined by (2.7), with S replaced by S~1, and we use the notation /

(x, y) = (x, y)R»

JV

I = X,xf

Let us define J : UN -^ R by (2.9) we clearly have equivalence between (2.2) and the following: Find xeUN such that J(x) < J(y),

VyeR"

(2.10)

In the following we denote by F' and J' the differentials of F and J, respectively, we can identify F' with the (Jacobian) matrix (3/;/<3XJ)I<;,J<;V> a n d we have (J'(y), z) = ( S - ^ ( y ) , F'(y)z),

Vy.zeR"

(2.11)

which implies (2.12) To solve (2.2) (via (2.10)), we can use the following conjugate gradient algorithm in which S is used as a scaling (or preconditioning) matrix (most of the notation is the same as in Chapter IV, Sec. 2.6.7). First algorithm (Fletcher-Reeves) x°eUN given;

(2.13)

g^S-W),

(2.14)

w° = g°.

(2.15)

Then assuming that x" and w" are known, we compute x" + 1 by x" + 1 = x" - pnW,

(2.16)

where pn is the solution of the one-dimensional minimization problem J(x" - pBw") < J(x" - pw"),

V p e R, pneU.

(2.17)

198

VII Least-Squares Solution of Nonlinear Problems

Then g«+i

and compute w

B+1

= S" 1 J'(x" +1 )

(2.18)

by W+1 = g"+1 + /lnw",

(2.19)

where (Sg" +1 g" +1 ) (220) "(Sg«,g") • Second algorithm (Polak-Ribiere). This method is like the first algorithm except that (2.20) is replaced by A

g) (221) • Remarks 2.17, 2.19, and 2.20 of Chapter IV, Sec. 2.6.7 still hold for algorithms (2.13)-(2.20) and (2.13)-(219), (2.21). As a stopping test for the above conjugate gradient algorithms, we may use, for example,either J(x") < £or||g"|| Si e (where eis a "small"positive number), but other tests are possible.

3. Least-Squares Solution of a Nonlinear Dirichlet Model Problem In order to introduce the methods that we shall apply in Sees. 4 and 5 to the solution of fluid dynamics problems, we shall consider the solution of a simple nonlinear Dirichlet problem by least-squares and conjugate gradient methods after briefly describing (in Sec. 3.2) the solution of the model problem introduced in Sec. 3.1 by some more standard interative methods; in Sec. 3.5 we shall briefly discuss the use of pseudo-arc-length-continnation methods for solving nonlinear problems via least-squares and conjugate gradient algorithms. 3.1. Formulation of the model problem Let Q c UN be a bounded domain with a smooth boundary F = dQ; let T be a nonlinear operator from V = Hj(Q) to V* = H " 1 ^ ) {H~l(£l): topological dual space of //J(Q)). We consider the nonlinear Dirichlet problem: Find u e HQ(Q) such that ~Au - T(u) = OinQ,

(3.1)

and we observe that u e HQ(Q) implies u = 0 on T. Here we shall not discuss the existence and uniqueness properties of the solutions of (3.1), since we do not want to be very specific about the operator T.

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

199

3.2. Review of some standard iterative methods for solving the model problem 3.2.1. Gradient methods The simplest algorithm that we can imagine for solving (3.1) is as follows: u° given; +1

then for n > 0, define u"

(3.2)

from u" by -Au"+l

= T(un) in Q, +1

»"

(i

= 0onr.

'

Algorithm (3.2), (3.3) has been extensively used (see, e.g., Norrie and De Vries [1] and Periaux [1]) for the numerical simulation of subsonic potential flows for compressible inviscid fluids like those considered in Chapter IV, Sec. 3. Unfortunately algorithm (3.2), (3.3) usually blows up in the case of transonic flows. Actually (3.2), (3.3) is a particular case of the following algorithm: u° given; +i

then for n > 0, define u"

(3.4)

from u" by _A« B + 1 / 2 = T(u") in fi, M"

+1/2

(3

= 0onr,

u n + 1 = u" + p ( u n + 1 ' 2 - u"),

p > 0 .

-^

(3.6)

If p = 1 in (3.4)-(3.6), we recover (3.2), (3.3). Since (3.5), (3.6) are equivalent to = „" - p ( - A ) - 1 ( - A « " - T(u")), (3.7) with ( — A)" corresponding to Dirichlet boundary conditions, algorithm (3.4)-(3.6) is very close to a gradient method (and is rigorously a gradient algorithm if T is the derivative of some functional). Let us define A: Hj(fi) -> H~ \Q) by M-+i

1

A(v)=

-Av-

T(v).

(3.8)

We can easily prove the following: Proposition 3.1. Suppose that the following properties hold (i) A is Lipschitz continuous on the bounded sets ofHl(Q); (ii) A is strongly elliptic, i.e., there exists a > 0 such that a\\v2-vl\\2Hh(n),

V vu v2 eH^Q)

(3.9)

(where < •, •) denotes the duality between H~ 1(Q) and HQ(Q)). Then problem (3.1) has a unique solution; moreover, for every u° e Hl(Q), there exists pM > 0 (depending upon u° in general) such that 0 < p < pM

(3-10)

implies the strong convergence o/(3.4)-(3.6) to the solution u o/(3.1). PROOF. See, e.g., Brezis and Sibony [2].

•

200

VII Least-Squares Solution of Nonlinear Problems

3.2.2. Newton's methods Assuming that T is differentiable, one may try to solve (3.1) by a Newton's method. For this case, using a prime to denote differentiation, we obtain: u° given; +i

(3.11)

n

then for n > 0, define ii" from u by - AM"+ 1 - T'(u") • un+1 = T(un) - T'(nn) • u" in Q, un+1 =0 on T. (3.12) Algorithm (3.11), (3.12) is the particular case corresponding to p = 1 of the following: u° given;

(3.13)

+1

then for n > 0, define n" from u" by - Aun+1/2 -

T"(M")

• "" + 1 / 2 = T(un) - T'(u") • u" in Q, un+m = 0 on T, (3.14)

n+1

u

n

n+112

=u + p(u

-u"),

p>0.

(3.15)

The various comments made in Chapter IV, Sec. 2.6.4 about Newton's methods still hold for algorithms (3.11), (3.12) and (3.13)-(3.15). 3.2.3. Time-dependent approach A well-known technique is the following: one associates with (3.1) the timedependent problem ~ -

AM - 7X10 = 0 infi,

(3.16)

ot « = 0onr,

(3.17)

u(x, 0) = uo(x) (initial condition). (3.18) Since lim,^ + 00 u(t) are usually solutions of (3.1), a natural method for solving (3.1) is the following: (i) Use a space approximation to replace (3.16)—(3.18) by a system of ordinary differential equations, (ii) Use an efficient method for numerically integrating systems of ordinary differential equations, (iii) Then integrate from 0 to + oo (in practice, to a large value of t). In the case of a stiff problem, it may be necessary to use an implicit method to integrate the initial-value problem (3.16)-(3.18). Therefore each time step will require the solution of a problem like (3.1). If one uses the ordinary backward implicit scheme (see Chapter III, Sec. 3) one obtains u° = u0,

(3.19)

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

201

and for n > 0, w " + 1 - it"

k

- AM"+ 1 - T(un+!) = 0 in Q,

un+1 = 0 on Y

(3.20)

(where k denotes the time-step size). At each step one has to solve

k

+! - Au" +1 - T(u" ) = \ in Q, un+1 = 0 on F, v ' k

(3.21)

which is very close to (3.1) (but usually better conditioned); actually, in practice, instead of (3.21), we solve a finite-dimensional system obtained from (3.16) (3.18) by a space discretization. 3.2.4. Alternating-direction methods These methods have been considered in Chapter IV, Sec. 2.6.6; actually they are also closely related to the time-dependent approach as can be seen in, e.g., Lions and Mercier [1] to which we refer for further results and comments. Two possible algorithms are the following: First Algorithm. This is a nonlinear variant of the Peaceman-Rachford algorithm (see Peaceman and Rachford [1], Varga [1], Kellog [1], Lions and Mercier [1], and Gabay [1]) defined by: u° given;

(3.22) +1/2

then for n > 0, u" being given, we compute u" rnun+1/2

- T(un+1/2)

+1

, u"

from u" by

= rnu" + AM",

rnun+1 - A«" +1 = rnun+m

+ T(u"+'12).

(3.23) (3.24)

Second Algorithm. This is a nonlinear variant of the Douglas-Rachford algorithm (see Douglas and Rachford [1], Lieutaud [1], Varga, Lions and Mercier, and Gabay, loc. cit.) denned by (3.22) and rnun+li2 n+1

rnii

~

T(M"+1/2)

- A«"

+1

= rnu" + Aun,

= rnu" + T(«"

+ 1/2

).

(3.25) (3.26)

In both algorithms {rn}n > 0 is a sequence of positive parameters (usually a cyclic sequence). In the nonlinear case, the determination of optimal sequences {/«}«> o i s a difficult problem. 1 We also have to observe that if in algorithm

1 See, however, Doss and Miller [1] in which alternating-direction methods more sophisticated than (3.22H3.24) and (3.22), (3.25), (3.26) are also discussed and tested.

202

VII Least-Squares Solution of Nonlinear Problems

(3.22)-(3.24) operators T and A play the same role, this is no longer true in (3.22), (3.25), (3.26), and it is usually safer to have A as an "acting" operator in the second step (if we suppose that —A is "more" elliptic than — T).

3.3. Least squares formulations of the model problem (3.1) 3.3.1. Generalities We shall consider least-squares formulations of the model problem (3.1). An obvious least-squares formulation consists of the statement that the required function u minimizes the left-hand side of (3.1) in a L2(Q)-least-squares sense. That is, Min f |Ai> + T(v)\2dx,

(3.27)

•>£!

veV

where V is a space of feasible functions. Let us introduce £ by - A £ = T(t>)inn,

(328)

£ = 0 on F. Then (3.27) is equivalent to Min f |A(t>- O\2dx, veV

(3.29)

Jn

where £ is a (nonlinear) function of v, through (3.28). From Lions [4] and Cea [1], [2], for example, it is clear that (3.28), (3.29) Has the structure of an optimal control problem where (i) (ii) (iii) (iv)

v is the control vector, £ is the state vector, (3.28) is the state equation, and the functional occurring in (3.29) is the cost function.

Another least-squares optimal control formulation is

Min f \ v - £\2 dx, veV

(3.30)

Jn

where £ again satisfies (3.28). This formulation has been used by Cea and Geymonat [1] to solve nonlinear partial differential problems (including the steady Navier-Stokes equations). Actually the two above least-squares formulations may lead to a slow convergence, since the norm occurring in the cost functions is not appropriate for the state equation. An alternate choice, very well suited to nonlinear second-order Dirichlet problems, will be discussed in the next section.

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

203

3.3.2. A H~i-least-squares formulation of (3.1) Let us recall some properties of H~ ^Q), the topological dual space of If L 2 (Q) has been identified with its dual space, then

HQ(Q).

moreover A ( = V2) is an isomorphism from //o(Q) onto H~1(Q). In the sequel the duality pairing < •, • > between H~X(H) and //J(Q) is chosen in such a way that V/eL2(Q),

< / , » > = [fvdx, Jo

VUGHJ(Q).

(3.31)

The topology of H~^Q) is defined by || • ||*, where, V / e i f " \ Q ) , SU

=

P

liTij l\vl\^

(3-32>

•

A convenient2 least-squares formulation for solving the model problem (3.1) seems to be

Min | | A » + TO;) II*.

(3.33)

It is clear that if (3.1) has a solution, then this solution will be a solution of (3.33) for which the cost function will vanish. Let us introduce £ e Hj(Q) by A£ = Av + T(v) in £1, t = 0 on F so that (3.33) reduces to Min ||A£|L,

(3.35)

where £, is a function of v through (3.34). Actually it can be proved that if || • || „. is defined by (3.32) with < •, • > obeying (3.31), then VveHKtt).

(3.36)

From (3.36) it then follows that (3.35) may also be formulated by Min

2

\\Vt;\2dx

(3.37)

Convenient because the space H\,{£i) in (3.33) is also the space in which we want to solve (3.1) (from the properties of A and T).

204

VII Least-Squares Solution of Nonlinear Problems

where ^ is a function of v through (3.34); (3.37) also has the structure of an optimal control problem. Remark 3.1. Nonlinear boundary-value problems have been treated by Lozi [1] using a formulation closely related to (3.34), (3.37).

3.4. Conjugate gradient solution of the least squares problem (3.34), (3.37) Let us define J : i?£(fi) ->• U. by dx,

(3.38)

where ^ is a function of v in accordance with (3.34); then (3.37) may also be written as follows: Find u e HQ(Q) such that J(u) < J(v),

V c e f f J(Q).

(3.39)

To solve (3.39), we shall use a conjugate gradient algorithm. Among the possible conjugate gradient algorithms, we have selected the Polak-Ribiere version (see Polak [1]), since this algorithm produced the best performances in the various experiments that we performed (the good performances of the Polak-Ribiere algorithm are discussed in Powell [2]). Let us denote the differential of J(-) by J'( •); then the Polak-Ribiere version of the conjugate gradient method applied to the solution of (3.39) is Step 0: Initialization u° e HJ(Q) given;

(3.40)

then compute g° e Ho(Q)/rom (

)

g° = 0 on T, and set z° = g°.

(3.42) +1

+1

+i

Then for n > 0, assuming u", g", z" known, compute u" , g" , z"

by:

Step 1: Descent un+1 =un-

Xnz\

(3.43)

where Xn is the solution of the one-dimensional minimization problem Xn e U,

J(un - V ) < J{ii" - Az"),

V X e U.

(3.44)

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

205

Step 2: Construction of the New Descent Direction. Define g"+1 eHj(Q) by -Agn+1

= J'(un+1) in Q,

0" + 1 = O o n r ;

(3.45)

then

z*+1=g»+1+yHz»,

(3.47)

n = n + 1 go to (3.43).

(3.48)

The two nontrivial steps of algorithm (3.40)-(3.48) are: (i) The solution of the single-variable minimization problem (3.44); the corresponding line search can be achieved by dichotomy or Fibonacci methods (see, for example, Polak [1], Wilde and Beightler [1], and Brent [1]). We have to observe that each evaluation of J(v) for a given argument v requires the solution of the linear Dirichlet problem (3.34) to obtain the corresponding £. (ii) The calculation of gn+1 from un+1 which requires the solution of two linear Dirichlet problems (namely (3.34) with v = M" +1 and (3.45)). Calculation ofJ'(u") and g". Due to the importance of step (ii), let us describe the calculation of J'(u") and g" in detail. Let weflJ(Q); then J'{v) may be denned by (3.49) (-•0

(*0

from (3.34), (3.38), (3.49) we obtain <J'(»),w>= [v^-Vndx,

(3.50)

Ja

where t] e Hl(Q) is the solution of AM = Aw + T'(v) • w in Q,

(3.51) n = 0 on F; (3.51) has the following variational formulation: (Vn-Vzdx

= f Vw-Vzdx - ,

Vze#J(Q),

f/eff&fi). (3.52)

206

VII Least-Squares Solution of Nonlinear Problems

Taking z = £ in (3.52), from (3.50) we obtain <J'(v), w> = f V£ • Vw dx - < 7 » • w, O ,

V », w E tf J(fl).

(3.53)

Therefore J'(v) (eH~l(Q.)) may be identified with the linear functional on Hj(Q) defined by -
(3.54)

From (3.45), (3.53), (3.54) it then follows that g" is the solution of the following linear variational problem: Find g" e H£(Q) such that, V w e i/£(fi), f Vg" • Vw dx = f V,

(3.55)

where for w belonging to a basis of the finite dimensional subspace of HQ(Q.) corresponding to the Galerkin or finite element approximation under consideration. 3.1. Extend the above least-squares-conjugate-gradient methodology to the solution in HQ(Q) X H J(Q) of the nonlinear boundary-value system

EXERCISE

-

AMJ

du, dut , . n + ut — - + « 2 ^— = / i m £2>

Ai +u +u LmQ

(156)

- '> ^ >tr ' Uj = u2 = 0 on F,

where Q is a bounded domain of K2 and f1,f2eH~

'(Q).

3.5. A nonlinear least-squares approach to arc-length-continuation methods 3.5.1. Generalities: Synopsis We would like to show that the above least-squares methodology can be (slightly) modified in order to solve nonlinear problems by arc-lengthcontinuation methods, directly inspired by H. B. Keller [1], [2] (where the

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

207

basic iterative methods are Newton's and quasi-Newton's, instead of conjugate gradient). As a test problem we have chosen a variant of the nonlinear Dirichlet problem (3.1); let us consider the following family of nonlinear Dirichlet problems, 3 parametrized by X e U, -A« = XT(u) in Q, u = OonT;

(3.57)

(3.1) corresponds to X = 1. 3.5.2. Solution of(3.57) via arc-length-continuation methods Following H. B. Keller [1], [2] [for which we refer for justification (see also Percell [1])], we associate to (3.57) a " continuation " equation; we have chosen 4 |VM| 2 dx + X2 = 1,

(3.58)

where u = du/ds, X = dXjds, and where the curvilinear abscissa s is defined by Ss = XSX + f VM • VSu dx,

(3.59)

(5s)2 = (5Xf + f VSu • Wdu dx.

(3.60)

or equivalently by

In fact we are considering a path in Hj(Q) x IR whose arc length is defined by (3.58)-(3.60). Then, in order to solve (3.57), we consider the family (parametrized by s) of nonlinear systems (3.57), (3.58). In practice we shall approximate (3.57), (3.58) by the following discrete family of nonlinear systems, where As is an arc-length step, positive or negative (possibly varying with n) and u" ~ u(n As): Take u° = 0, X° = 0 and suppose that 1(0), «(0) are given

(3.61)

(initialization (3.61) is justified by the fact that u = 0 is the unique solution of (3.57) if X = 0); then for n > 0, assuming M"" 1 , X"~l, u", X" known, we obtain {un+1, Xn+1}eHl(Q.) x Uby -Aun+1

=

Xn+1T(un+1)inQ,

un+1 = 0onY,

3 4

T o be solved in j() O t h e r choices are possible

(3.62)

208

VII Least-Squares Solution of Nonlinear Problems

and f Viu1 - u°) • Vu(0) dx + (X1 - A°)l(0) = As ifn = 0,

= 0 or 1, ifn>\.

(3.63)

(3.64)

Obtaining ti(0) and 1(0) is an easy task since we have (from (3.57))

«(0) = 0 on T,

(3.65)

and therefore

X2(0)(l + j \Vu\2 dx\ = I,

(3.66)

where M e //J(Q) is the solution of -Aw = T(0)infi, M=

0 on T

(3.67)

(then we clearly have a(0) = X(0)u). Relations (3.61)-(3.64) look like clearly a discretization scheme for solving the Cauchy problem for first-order ordinary differential equations; from this analogy we can derive many other discretization schemes for the approximation of (3.57), (3.58) (Runge-Kutta, multisteps, etc...) and also methods for the automatic adjustment of As. 3.5.3. Nonlinear least squares and conjugate gradient solution o/(3.62)-(3.64). Without going into details (for which we refer to Reinhart [1] and Glowinski, Keller, and Reinhart [1]) we can solve (3.62)-(3.64) by a variant of (3.40)-(3.48) defined on the Hilbert space Hl(Q) x IR equipped with the metric and innerproduct corresponding to (3.68) Jn it is clear that many other norms than (3.68) are possible, however in all cases the scaling of a conjugate gradient algorithm using a discrete variant of5 -A

0\ I (or similar operators)

will require an efficient solver and the conclusions of Sec. 3.4 still hold. 1

In (3.69), A corresponds to the homogeneous Dirichlet boundary conditions.

(3.69)

3 Least-Squares Solution of a Nonlinear Dirichlet Model Problem

209

Remark 3.4. To initialize the conjugate gradient algorithm solving (3.62)(3.64) we have used {21" - A""1, 2M" - M"" 1 } as initial guess to compute {Xn+1, un+1}; with such a choice we obtain a much faster convergence than by taking {A", «"} as initial guess. 3.5.4. Application 3.5.4.1. Formulation of the problem. We shall apply the methods described in Sees. 3.5.2 and 3.5.3 to the solution of the following classical problem: - A M = Ae"infi, (170) n r u = 0 on F, where Q is a bounded domain of UN. We have to observe that unless N = 1, the mapping T denned by

is not continuous from HQ(O) to V* ( = H~1(Q)). We consider only the case where A > 0, since if X < 0, the operator v -* — Av — Xe" is monotone and therefore the methods of Chapter IV, Sec. 2 can be applied (take (j)(i) = — X(e' — 1 ) , / = A), showing the existence of a unique solution of (3.70) (which is u = 0 if A = 0). With A > 0, problem (3.70) has been considered by many authors (Henri Poincare—with Q = IRN—among them). With regard to recent publications, let us mention, among others, Crandall and Rabinowitz [1], [2], Amann [1], Mignot and Puel [1], and Mignot, Murat, Puel [1]. In particular, in Mignot, Murat, and Puel [1] we may find an interesting discussion showing the relationships between (3.70) and combustion phenomena. From a numerical point of view, problem (3.70) has been investigated by, among others, Kikuchi [1] and Reinhart [1] to which we refer for more details and further references (see also Simpson [1], Moore and Spence [1], Glowinski, Keller, and Reinhart [1], and Chan and Keller [1]). 3.5.4.2. Numerical implementation of the methods of Sees. 3.5.2. and 3.5.3. We have chosen to solve the particular case of (3.70) where Q = ]0, 1[ x ]0, 1[. The practical application of the methods of Sees. 3.5.2 and 3.5.3 requires the reduction of (3.70) to a finite-dimensional problem; to do this we have used the finite element method described in Chapter IV, Sec. 2.5, taking for ^h the triangulation consisting of 512 triangles indicated in Fig. 3.1. The unknowns are the values taken by the approximate solution uh at the interior nodes of 2Th; we have 225 such nodes. Algorithm (3.61)-(3.64) has been applied with As = 0.1, and) = 0 in (3.64); we observe that T(0) = 1 in (3.67); algorithm (3.61)-(3.64) ran "nicely," since an accurate least-squares solution of the nonlinear system (3.62), (3.64) required basically no more than 3 or 4 conjugate gradient iterations, even close to the turning point.

210

VII Least-Squares Solution of Nonlinear Problems Figure 3.1

7

u, (0.5, 0.5) 8.00

I i i i i i i i i i

4.80

Figure 3.2

6.40 8.00 LAMBDA

4 Transonic Flow Calculations by Least-Squares and Finite Element Methods

211

In Fig. 3.2 we show the maximal value (reached at xt = x2 = 0.5) of the computed solution uh as a function of X; the computed turning point is at X = 6.8591... . The initialization of the conjugate gradient algorithm used to solve system (3.62), (3.64), via least squares, was performed according to Remark 3.4. 3.5.5. Further comments

The least-squares conjugate gradient continuation method described in Sees. 3.5.2, 3.5.3, and 3.5.4 has been applied to the solution of nonlinear problems more complicated than (3.70); among them let us mention the Navier-Stokes equations for incompressible viscous fluids at high Reynold's number and also problems involving genuine bifurcation phenomena like the Von Karman equations6 for plates. The details of these calculations can be found in Reinhart [1], [2] and Glowinski, Keller, and Reinhart [1].

4. Transonic Flow Calculations by Least-Squares and Finite Element Methods

4.1. Introduction

In Chapter IV, Sec. 3, we considered the nonlinear elliptic equation modelling the subsonic potential flows of an inviscid compressible fluid. In this section, which closely follows Bristeau, Glowinski, Periaux, Perrier, Pironneau, and Poirier7 [2] (see also G.L.T. [3, Appendix 4]). we would like to show that the least-squares conjugate gradient methods of Sec. 2 and 3 can be applied (via convenient finite element approximations) to the computation of transonic flows for similar fluids. Given the importance and complexity of the above problem, we would like to point out that the following considerations are just an introduction to a rather difficult subject. Many methods, using very different approaches, exist in the specialized literature, and we shall concentrate on a few of them only (see the following references for other methods). We would also like to mention that from a mathematical point of view, the methods to be described in the following sections are widely heuristical. 6 7

For which we refer to the monograph by Ciarlet and Rabier [1] (and the references therein). B.G.4P. in the sequel

212

VII Least-Squares Solution of Nonlinear Problems

4.2. Generalities. The physical problem The theoretical and numerical studies of transonic potential flows for inviscid compressible fluids have always been very important questions. But these problems have become even more important in recent years in relation to the design and development of large subsonic economical aircrafts. From the theoretical point of view, many open questions still remain, with their counterparts in numerical methodology. The difficulties are quite considerable for the following reasons: (1) The equations governing these flows are nonlinear and of changing type (elliptic in the subsonic part of the flow; hyperbolic in the supersonic part). (2) Shocks may exist in these flows corresponding to discontinuities of velocity, pressure, and density. (3) An entropy condition has to be included "somewhere" in order to eliminate rarefaction shocks, since they correspond to nonphysical situations. Concerning the fluids and flows under consideration, we suppose that these fluids are compressible and inviscid (nonviscous) and that their flows are potential (and therefore quasi-isentropic) with weak shocks only; in fact this potential property is no longer true after a shock (cf. Landau and Lifchitz [1]). In the case of flows past bodies, we shall suppose that these bodies are sufficiently thin and parallel to the main flow in order not to create a wake in the outflow. 4.3. Mathematical formulation of the transonic flow problem. References 4.3.1. Governing equations If Q is the region of the flow and T its boundary, it follows from Landau and Lifchitz [1] that the flow is governed by the so-called full potential equation: V • pu = 0 in Q,

(4.1)

where 1 - - [(y

+

i y

^ _

1)]c2

u = \(j>, and (a) (b) (c) (d)

is the velocity potential, p is the density of the fluid, y is the ratio of specific heats (y = 1.4 in air), C* is the critical velocity.

1

,

(4.2) (4.3)

4 Transonic Flow Calculations by Least-Squares and Finite Element Methods

213

Figure 4.1

4.3.2. Boundary conditions

For an airfoil B (see Fig. 4.1), the flow is supposed to be uniform on F^, and tangential at TB. We then have (4.4)

— ^ - = Oonr B .

(4.5)

Since only Neumann boundary conditions are involved in the above case, the potential

V w e //H^),

= 0,VH>£

H 1 ^), which in turn implies

- = 0onT; (4.16) on (4.14), (4.16) imply (4.12) and complete the proof of the lemma. • From Lemma 4.1 it is feasible to take {v, } e Wo in (4.8), and since (from (4.11)) A

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