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 approxi- mate 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 IR 2 , • F = <9Q, • x = {x u x 2 }, a generic point of Q, . V =_{d/8x u d/dx 2 }, • C m (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 C m (Q) | Supp(u) is a compact subset of Q}, • IML, P> n = L«l<m \\D«v\\ LP(n) {or v e C m (Q), where a = {cc 1 ,a 2 };x 1 ,oc 2 aie non-negative integers, |a| = otj + oc 2 and D" = d^/dxf dx 2 2 , 28 II Application of the Finite Element Method • W m ' p (Q): completion of C m (U) in the above norm, • WS'"(Q,): completion of CQ(Q) in the above norm, • H m (Q.)= W m - 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), where a(u, v) = VM • Vv dx, Jn _ du dv du dv • Vv = —- — + dx 2 dx 2 L(v) = </, v) for / e V* = H~\Q) and v e V. Let W e H^O) n C°(Q) and 'PIr < 0. Define K = {u e Hj(Q)|u > i> a.e. on Q}. Then the obstacle problem is a particular (PJ 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 x u x 2 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 displace- ment 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 U N . Then the seminorm on H 1 (Q) \l/2 \Vv\ 2 dx\ i / is a norm on HQ(Q) and it is equivalent to the norm on Hj(Q) induced from H l (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 con- tinuous (i.e., 3 k > 0 such that \f(t) - f(t')\ <k\t - £'|, V t,t' eW) and such that /' has a finite number of points of discontinuity. Then the induced map /* on H 1 (D.) defined by v -> f(v) is a continuous map into H 1 (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 veH 1 (Q.) (respectively, HQ(Q)), then the map v -* {v + , v~} is continuous from H\Q) -> H\Q) x H\Q) (respectively, Hj(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 H l (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 v n -* v strongly in Hj(n), where v n e K and v e HJ(Q). Hence v n —>v strongly in L 2 (Q). Therefore we can extract a subsequence {v n .} such that v n . -* v a.e. on Q. Then v n . > 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)> »F(x)}, Q° = {x\xeQ,u(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 + , (2.3) u = V on Q°, (2.4) u = 0 on T, (2.5) u + \ y = 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 H 1 (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 U 2 with a smooth boundary. If L(v) = f fv dx with f E Z/(Q), 1 < p < + oo (2.7) and *Pe^ 2 '"(Q), (2.8) then the solution of the problem (2.1) is in W 2 ' p (il). 2. An Example of EVI of the First Kind: The Obstacle Problem 31 Remark 2.2. Let QcR* have a smooth boundary. We know that W S '\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 W 2 '"(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 L 2 (Q), (2.10) T-OonQ. (2.11) Before proving that (2.10), (2.11) imply u e H 2 (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 U N with a boundary F sufficiently smooth. Then ||Au|| L 2 (n) defines a norm on H 2 (Q) n Hl(il) which is equivalent to the norm induced by the H 2 (Q)-norm. EXERCISE 2.1. Prove Lemma 2.3 using the following regularity result due to Agmon, Doughs, and Nirenberg [1]: If w e L 2 (Q) and if r is sufficiently smooth, then the Dirichlet problem — Av = w in Q, has a unique solution in HQ(Q) n H 2 (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) = j Q fv dx with feL 2 (Q) then the solution u of the problem (2.1) satisfies ueKn H 2 (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 £ + u t = u in Q, u t \ Y = 0. (2.13) Problem (2.13) has a unique solution in H\{Q), and the smoothness of T implies that u s belongs to H 2 (Ci). Since u > 0 a.e. on Q, by the maximum principle for second-order elliptic differential operators (cf. Necas [1]), we have u e > 0. Hence u c eK. (2.14) 32 II Application of the Finite Element Method From (2.14) and (2.1), we obtain a(u, u c -u)> L{u E - u) = j f(u c - u) dx. (2.15) The F-ellipticity of o(-, •) implies a(u c , u £ — u) = a(u t — u, u z — u) + a(u, u s — u) > a(u, u e — u), so that a(u e , u s -u)> \ f(u s - u) dx. (2.16) By (2.13) and (2.16), we obtain E Vu £ • V(Au £ ) dx > s \ fAu s 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 < II/IIL^,, (2.18) using Schwarz inequality in L 2 (ii). By Lemma 2.3 and relations (2.13), (2.18) we obtain lim u t = u weakly in H 2 (Q), (2.19) (which implies that lim u e = u strongly in H S (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 U 2 . Consider a "classical" triangulation 2T h of Q, i.e. 2T h is a finite set of triangles T such that TcQ VTef», U T = Q, (2.21) ^0^ = 0 V T u T 2 e ^ and T x ^ T 2 . (2.22) 2. An Example of EVI of the First Kind: The Obstacle Problem Figure 2.1 M iT m 2T 33 M lT m 3T Moreover Vr 1 ,r 2 eJ, and Ti # T 2 , exactly one of the following conditions must hold (1) T,nT 2 = 0, (2) Ti and T 2 have only one common vertex, (3) T t and T 2 have only a whole common edge. (2.23) As usual h will be the length of the largest edge of the triangles in the triangula- tion. From now on we restrict ourselves to piecewise linear and piecewise quadratic finite element approximations. 2.5.1. Approximation of V and K • P k : space of polynomials in x t and x 2 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 eZ t . The centroid of the triangle T is denoted by G T . 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), w fc | r = 0 and v h \ T eP k ,\/Te 3T h }, 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\, v h (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(u k h , v h - u\) > L(v h - u k ), Vv h eK k h , u k h sK k h . (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, v h ) - L(v h )l (2.24) v h 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. EXERCISE 2.3. With notations as in Fig. 2.1, prove the following identities for any triangle T: w dx = mea 3 S(r) f w(M iT ), V w e P ls (2.26) wdx = vae ^ r l £ w (m iT ), V w e P 2 . (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 (P k lh )) as h -» 0. Theorem 2.3. Suppose that the angles of the triangles of 2T h are uniformly bounded below by 9 0 > 0 as h -» 0; then for k = 1,2, lim ||u\ - M || H4(n) = 0, (2.28) where u\ and u are the solutions of(P\ h ) and (2.1), respectively- 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 (v h ) h is such that v h e 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 x- Verification 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 <^(G T )^ T , where ^r is the characteristic function 1 of T and G T 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 *P fc such that (2.31) ^(P) = ¥(P), VPeZj. This function *P h satisfies lim «P h = ¥ strongly in L°°(Q). (2.32) h->0 Let us consider a sequence (v h ) h , v h e Kl,V h such that lim v h = v weakly in V. Then lim^ 0 v h = v strongly in L 2 (Q) (cf. Necas [1]) which, using (2.30) and (2.32), implies that lim I {v h - ¥„)(£„ dx = I (v - VW dx, (2.33) lim f (v h - VM>k dx= f (v - (actually, since </>,, -> ^> strongly in L co (fi), the weak convergence of v h in L 2 (Q) is enough to prove (2.33)). We have [ (v h - V h )<f> h dx= X <KG T ) f (»* - Y*) ^. (2.34) From (2.26), (2.27), and from the definition of T ft , we obtain for all Te#~ h , f (t>* - «P») dx = mea ' (r) I K(M, T ) - WM ff )] if /c = 1, (2.35) f (^ - T ft ) dx = ^ (T) E [O^T) - W«ir)] if * = 2. (2.36) l Xi{x) = 1, VxeT, XT(X) = 0 if X <£ T. 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, V cf> € &(Q), 4>>0 •In 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* h veV k h , V v e Hj(fi) n C°(fi), (2-37) V P e SJ for fc = 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 \\r k h v-v\\ v <Ch k \\v\\ Hk , Hn) , Voe©(QX k = 1, 2, with C independent of h and u. This implies that lim \\riv - v\\ v = 0, V t; £ x, k=l,2. On the other hand, it is obvious that r&eKl, VveKnC°(n), so that r k h veK k h , Vvex for fe = 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 C 0 (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. [...]... 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)... 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... (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 ).. .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) ... 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... 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 one= dimensional 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... = (x\ + x|) 1 / 2 , the solution u of (3.1) is given by u{x) = C-(R2-r1) 4 ifc 2/ R, then 2 ii-m < Ch \\u\\W2,nn), ll2 llp with C independent of h and u Then the O(h... 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.), . strongly in H S (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. U 2 . Consider a "classical" triangulation 2T h of Q, i.e. 2T h is a finite set of triangles T such that TcQ VTef», U T = Q, (2. 21) ^0^ = 0 V T u T 2 e ^ and T x ^ T 2 . (2. 22) 2. . in Fig. 2. 1, prove the following identities for any triangle T: w dx = mea 3 S(r) f w(M iT ), V w e P ls (2. 26) wdx = vae ^ r l £ w (m iT ), V w e P 2 . (2. 27) Formula (2. 26) is called