4 Application to the Solution of Elliptic Problems for Partial Differential Operators 387 it follows from (4.301) and (4.297) 4 that the boundary integral in (4.300) reduces to f (K\u)-nvdr, where F, = {x|x = {x 1; x 2 }, 0 < x, < 1, x 2 = 0}, implying, in turn, that (4.300) reduces to f (K\u) -\vdx= f fv dx + f (K\u) • nv dT. (4.302) Jn Jn Jrt Combining (4.297) 5 and (4.302), and using the second relation (4.301), we obtain (after integrating by parts over I\) f (XVu) -\vdx+ f k(x,) ~ (x,, 0) -^- (x,, 0) dx, Jn Jo ox, ox, = [ fvdx+ [ g,v dT. (4.303) Conversely, it can be proved that if (4.303) holds for every vsV, where r = {v\v £ C X (Q), v(0, x 2 ) = v(l, x 2 ) if 0 < x 2 < 1, M. 304^ v = 0 in the neighborhood of F o } where T o = {x|x e {x 1; x 2 }, 0 < x, < 1, x 2 = 1}, then u is a solution of the boundary-value problem (4.297). Relations (4.301), (4.303) suggest the intro- duction of the following subspaces of H 1 (Q): V = {v\v e H^Q), v(0, x 2 ) = v(l, x 2 ) a.e. 0 < x 2 < 1, (d/dx,)v(x,, 0) £ L 2 (0, 1)}, (4.305) V o = {v\v e F, t;(x 1; 1) = 0 a.e. 0 < x, < 1}. (4.306) Suppose that V is endowed with the scalar product (v, w) v = (v, w) HHn) + — v(x,, 0) —- w(x 1; 0) dx, (4.307) Jo Q-X, ax, and the corresponding norm \\v\\ r = (v,v) r 12 . (4.308) We then have the following: Proposition 4.21. The spaces V and V o are Hilbert spaces for the scalar product and norm defined by (4.307) and (4.308), respectively. Moreover, the seminorm l\\Vv\ 2 dx+ C \Jn Jo dv u 0) dx, 1/2 defines a norm equivalent to the V-norm (4.308) over V o . 388 App. I A Brief Introduction to Linear Variational Problems EXERCISE 4.11. Prove Proposition 4.21. We now define a bilinear form a: V x V -* U and a linear functional L:F-+Rby a(v, w) = f (K\v) -\wdx+ f k( Xl ) •/- !<*!, 0) -^- W(JC 1; 0) dx ls J n J o dx t dx t (4.309) L(t>) = f/yrfx+ f s^dT, (4.310) Jn Jr t respectively. We suppose that the following hypotheses concerning A, k, f, 0i hold: /eL 2 (Q), ^EL 2 ^), (4.311) k e L°°(0, 1), k(x x ) > a 0 > 0 a.e. on ]0, 1[, (4.312) X satisfies (4.47). (4.313) From the above hypotheses, we find that a(-, •) is bilinear continuous over V x V and V o -elliptic, and that L(-) is linear continuous over V; we can therefore apply Theorem 2.1 of Sec. 2.3 to prove: Proposition 4.22. If the above hypotheses on A, k,f, g± hold, and if g 0 in (4.297)j satisfies do = 0o | r 0 with So e V, (4.314) then the linear variational problem: Find usV such that u \ To = g 0 and a(u,v) = L(v), \/veV 0 (4.315) has a unique solution; this solution is also the unique solution in V of the boundary- value problem (4.297). PROOF. Define u e V o by u = u — g 0 ; u—if it exists—is clearly a solution of the following linear variational problem in V o : ueV 0 , a(u,v) = L{v)-a(g o ,v). (4.316) Since v —> L(v) — a(g 0 , v) is linear and continuous over V o , and since a{-, •) is F 0 -elliptic, it follows from Theorem 2.1 of Sec. 2.3 that (4.316) has a (unique) solution, in turn implying the existence of u solving problem (4.315). The above u is clearly unique, since if u x and u 2 are two solutions of (4.315), then u 2 — u x e V o and also a(u 2 — u x , u 2 — u t ) = 0; the Po-ellipticity of a{-, •) then implies Ui = u 2 . • Remark 4.23. Suppose that A is symmetric; this in turn implies the symmetry of a(-, •)• From Proposition 2.1 of Sec. 2.4 it then follows that (4.315) is equiva- lent to the minimization problem: 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 389 Find ue V,u = g 0 onT 0 , such that J{u) < J(v), VceF, v = g 0 on F o , where J(v) = - f (X\v) -\vdx + - f k( Xl ) z Ja ^ Jo - f fv dx - f g^ dr. 4.6.3. Finite element approximation of problem (4.297) In this section we consider the approximation of the boundary-value problem (4.297) via the variational formulation (4.315). Actually the finite element approximation discussed in the sequel is closely related to the approximations of the Neumann, Dirichlet, and Fourier problems discussed in Sees. 4.5.4 and 4.5.5. To approximate (4.297), (4.315), we consider a family {3T h } h of triangula- tions of Q satisfying the hypotheses (i)-(iv) of Sec. 4.5.2 and also: (v) If Q = {0, x 2 } is a vertex of ,T h , then Q' = {1, x 2 } is also a vertex of jT h , and conversely (i.e., ST h preserves the periodicity of the functions of the space V (cf. (4.305)). With Hj, still defined by (4.197), we approximate the above spaces V and V o (cf. (4.305), (4.306)) by V h =VnHl = {v h \v h e H{, v h {0, x 2 ) = v h (l, x 2 ) if 0 < x 2 < 1}, (4.317) V Oh =V o n Hi = {v h \v h e V h , v k ( Xl , 1) = 0 if 0 < x x < 1}. (4.318) We suppose that g 0 (in (4.297) 2 , (4.314)) also satisfies g 0 e C°[0, 1] and (withF 0 = {x\x = {x L , l},0 < Xi < 1}) we approximate the problem(4.297), (4.315) by: Find u h e V such that u h (Q) = g o (Q), V Q vertex of^ h located on F o and a(u h , v h ) = L(v h ), Mv h e V Oh , (4.319) where a(•, •) and L are still denned by (4.209), (4.310), respectively. We should easily prove that the approximate problem (4.319) has a unique solution if (4.311)-(4.313) hold. EXERCISE 4.12. Prove that the approximate problem (4.319) has a unique solution if the above hypotheses hold. The convergence of the approximate solutions follows from: 390 App. I A Brief Introduction to Linear Variational Problems Proposition 4.23. Suppose that the above hypotheses on A, k, f g 0 , g 1 and {^~h)h hold. Also suppose that g 0 = <?o|r O ' where g 0 is Lipschitz continuous over Q. // {&~ h } h is a regular family of triangulations o/Q (in the sense of Sec. 4.5.3.4), we then have lim \\u h - u\\ v = 0, (4.320) where u h (resp., u) is the solution of (4.319) (resp., (4.297), (4.315)). PROOF. We sketch the proof in the case where g 0 = 0 on F o , implying that we can take g 0 = 0 over U. Problems (4.315), (4.319) reduce to: Find ueV 0 such that a(u, v) = L(v), V»eF 0 . (4.321) Find u h e V oh such that a(u h ,v h ) = L(v h ), Vv h eV 0h , (4.322) respectively. To apply Theorem 3.2 of Sec. 3.3 (with V h replaced by V oh ), it suffices to find f" and r h obeying (3.1). Define V by (4.304); it has been proved by H. Beresticky and F. Mignot (personal communications) that If v e y and if r h is still defined by (4.255), we have—since condition (v) on {&~ h } h holds— r h v e V oh ; on the other hand, we still have (see the proof of Proposition 4.19 for more details) lim \\r h v - v\\ HHn) = 0, V»£f. (4.323) Since W v = W - Hxi, 0) L 2 (O. 1), 1/2 it follows from (4.323) that to complete the proof of the present proposition, it suffices to prove that lim (r h v-v)( Xl ,0) d L 2 (0,l) = 0, V»ef" (4.324) Let us denote by y^ the function x Y -> v(x u 0); we clearly have Vlfrd" - v ) = s h7yV - JiV, where s h is the interpolation operator defined as follows: s h <t> e C°[0, 1], Vtf>eC°[0, 1], SiXfi) = <f>(£i) for any vertex {£ 1; 0} of $~ h belonging to r i( s h 4>\ [iu(i] e P y for any pair {£ u 0}, {^i, 0} of consecutive vertices of ST h belonging to (we recall that T x = {x\x = {x u 0}, 0 < x, < 1}). 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 391 Since vsY implies that jiV e C°°[0,1], it follows from standard approximation results that lim fz->0 = 0, mo, i dx 1 i.e., (4.324) holds. • EXERCISE 4.13. Prove Proposition 4.23 if g> 0 # 0. Remark 4.24. The approximation by variational methods of problems closely related to (4.297) is considered in Aubin [2]. 4.6.4. Some comments on the practical solution of the approximate problem (4.319) To solve the approximate problem (4.319), we can use a penalty method similar to the one used in Sec. 4.4.4 to approximate the Dirichlet problem by a Fourier one; the main advantage of this formulation is that the discrete space under consideration is still Hi denned by (4.197). A possible penalty approximation of (4.319) is (with e > 0): Find u\ e Hj, such that, V v h e Hi, 1 f 1 a(ul v h ) + - («J(1, x 2 ) - if h (0, x 2 )Xv h (l, x 2 ) - v h (0, x 2 )) dx 2 e Jo If 1 If 1 + - ul(x u \)v h (x u l)dxj = L(v h ) + ~ 0OJI(XIK(XI> !) dx u eJo £ Jo (4.325) where g Oh e C°[0, 1] coincides with g 0 at those vertices of 3~ h located on F o ( = {x\x = {x u 1}, 0 < x l < 1}) and is linear (i.e., belongs to P t ) between two consecutive vertices of 2T h located on F o . Such an approximation is justified by lim u\ = u h , (4.326) E^O where uf, and u h are the solutions of (4.325) and (4.319), respectively. EXERCISE 4.14. Prove (4.326). Another possibility is to work directly with the spaces V h and V Oh , taking into account the periodicity conditions u h (0, x 2 ) = u h {\, x 2 ), v h (0, x 2 ) = v h (l, x 2 ) and also the fact that u h (x u 1) = g Oh (xi), v h (x u 1) = 0, V v h e V Oh . This second approach will require an explicit knowledge of vector bases for 392 App. I A Brief Introduction to Linear Variational Problems V h and V Oh ; obtaining such bases from the basis of Hi defined by (4.198) (see Sec. 4.5.2) is not very difficult and is left to the reader as an exercise. We should again use numerical integration to compute the matrices and right-hand sides of the linear systems equivalent to the approximate problems (4.319) and (4.325). We shall conclude by mentioning that the methods discussed in Sec. 4.5.7 still apply to the solution of the above linear systems. 4.7. On some problems of the mechanics of continuous media and their variational formulations 4.7.1. Synopsis In this section, which very closely follows Mercier [1, Chapter 2] and Ciarlet [2], we briefly discuss some important problems of the mechanics of continuous media which are the three-dimensional linear elasticity equations (Sec. 4.7.2), the plate problem (Sec. 4.7.3), and Stokes problem (Sec. 4.7.4). After describing the partial differential equations modelling the physical phenomena, we discuss the variational formulations and various questions concerning the existence and uniqueness of the solutions of these problems. 4.7.2. Three-dimensional linear elasticity Let Q <= |R 3 be a bounded domain. Let T be the boundary of Q and suppose that r = T o u I\ with r 0 , r\ such that jYonrt dF = 0 (a typical situation is shown in Fig. 4.6). We suppose that Q is occupied by an elastic continuous medium and that the resulting elastic body is fixed along F o . Let f = {/J}f=i be a density of body forces acting in Q and g = {g t }f= x be a density of surface forces acting on F 1 . We denote byu(x) = {Wj(x)}f=i the displacement of the body at x. Figure 4.6 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 393 In linear elasticity the stress-strain relation is ffy(u) = A(V • u)<5 u + 2^», with £i /u) = \{~^ + ^\ (4-327) where a tj and e tj denote the components of the stress and strain tensors, re- spectively; X and JX are positive constants and are known as the Lamme coefficients. The problem is to find the tensor a = (o- y ), the displacement u = {u ; }f = x if Q, f e (L 2 (Q)) 3 and g e (L 2 (r\)) 3 are given. The equilibrium equations are JLa tj + f t = 0 infi, . = 1,2,3, (4.328^ <r t jnj = g t onT h i= 1,2,3, (4.328) 2 u t = 0 onT 0 , i= 1,2,3. (4.328) 3 We have used the summation convention of repeated indices in the above equations. To obtain a variational formulation of the linear elasticity problem (4.327), (4.328), we define a space V, a bilinear form a(-, •) and a linear functional L by V = {v | v e (if H^)) 3 , v = 0 on r 0 }, (4.329) a(u,v)= f ffyOOeyOO dx, (4.330) L(»)= f fli »,.dr+ f/^idx, (4.331) JTI •In respectively. Using (4.327), a(u, v) can be written as a(u, v) = I" {AV • uV • v + 2/iEi/n)e £ /v)} dx, (4.332) from which it is clear that a{-, •) is symmetric. The space V is a Hilbert space for the (H 1 (n)) 3 -norm, and the functionals a(-, •) and L are clearly continuous over V. Proving the F-ellipticity of a(-, •) is nontrivial; actually this ellipticity property follows from the so-called Korn inequality for which we refer to Duvaut and Lions [1] (see also Ciarlet [3] and Nitsche [1]). Now consider the variational problem associated with V, a, and L, i.e.: Find u e V such that a(u,v) = L(v), VveF; (4.333) from the above properties of V,a(-, •), L, the variational problem (4.333) has a unique solution (from Theorem 2.1 of Sec. 2.3). Applying the Green- Ostrogradsky formula, (4.33) shows that the boundary-value problem corresponding to (4.333) is precisely (4.328). 394 App. I A Brief Introduction to Linear Variational Problems The finite element solution of (4.328), via (4.333), is discussed in great detail in e.g., Ciarlet [l]-[3], Zienkiewicz [1], and Bathe and Wilson [1]. Remark 4.25. The term a(u, v) can be interpreted as the work of the internal elastic forces and L(v) as the work of the external (body and surface) forces. Thus, the equation a(u, v) = L(v), V v e V is a reformulation of the virtual work theorem. 4.7.3. A thin plate problem We follow the presentation of Ciarlet [3, Chapter 1]. Let Q be a bounded domain of R 2 and consider V,a(-, •), L defined by V = Hl(il) = Wii) HHa) = \v\ve H 2 (Q), v = ~ = 0 on rk (4.334) a{u, v)= [ \AU\V + (1 - <T)(2 d U 8 V dx x dx 2 dx y dx 2 8 2 u 8 2 v d 2 u 3 2 v\} ox 1 ox 2 dx 2 ox\)) 'dh^d 2 ^ dx 2 dx 2 ' dx\ dx = ioAuAv + (1 - ff) Jo I. \ 1 ox 2 ox l ox 2 L(v) = \ fvdx, feL 2 (Q). (4.336) The associated variational problem: Find ue V such that a(u, v) = L(v), \/veV, (4.337) corresponds to the variational formulation of the clamped plate problem, which concerns the equilibrium position of a plate of constant (and very small) thickness under the action of a transverse force whose density is proportional to/. The constant a is the Poisson coefficient of the plate material (0 < a < j). If f = 0, the plate is in the plane of coordinates {x 1; x 2 }. The condition u e HQ(Q) takes into account the fact that the plate is clamped. The derivation of (4.337) from (4.333) is discussed in Ciarlet and Destuynder [1]. The Poisson coefficient a satisfies 0 < a < ^; the bilinear form a(-, •) is Ho(Q)-elliptic, since we have a(v, v) = <7||Ac||£, (n) + (1 - <r)|!>|! >n (4.338) (where | -| 2 ,n nas been defined in Sec. 4.5.3.2). 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 395 Actually, using the fact that Hl(Q) = @(Q) H2(n) , we can easily prove that a(v,w)= [\oAwdx, Vv,weH 2 0 (Q). (4.339) From (4.339) it follows that the solution u of the variational problem (4.337) is also the unique solution in H 2 (Q) of the biharmonic problem A 2 u = / in Q, u = ^ = 0, (4.340) on and conversely (here A 2 = AA). Problem (4.340) also plays an important role in the analysis of incompressible fluid flows (see, e.g., Girault and Raviart [1], Glowinski and Pironneau [1], and Glowinski, Keller, and Reinhart [1] for further details). For the finite element solution of (4.337), (4.340), see Strang and Fix [1], Ciarlet and Raviart [1], Ciarlet [3], and Brezzi [1], and the references therein. 4.7.4. The Stokes problem As mentioned in Chapter VI, Sec. 5.2, the motion of an incompressible viscous fluid is modelled by the Navier-Stokes equations; if we neglect the nonlinear term (u • V)u, the steady case reduces to the steady Stokes problem -Au + Vp = finQ, V-u = 0inQ, u| r = g with g-ndF = 0; (4.341) in (4.341), Q is the flow domain (Q c M N , N = 2 or 3 in practical applications), F is its boundary, u = {M;}f =1 is the flow velocity, p is the pressure, and f is the density of external forces. For simplicity we suppose that g = 0 on F (for the case g # 0, see Chapter VII, Sec. 5, and also Appendix III) and also that Q is bounded. There are many possible variational formulations of the Stokes problem (4.341), and some of them are described in Chapter VII, Sec. 5; we shall concentrate on one of them, obtained as follows: Let • ={&}?=! e (0W (=>c|»| r = 0); taking the [R^-scalar product of <J> with both sides of the first equation (4.341) and integrating over Q, we obtain - f Au • <|> dx + \\p-^dx- f f • <|> dx (4.342) JQ Jn Jn 396 App. I A Brief Introduction to Linear Variational Problems Using the Green-Ostrogradsky formula (4.33), it follows from (4.342) that (\u-\$dx- f ~-($>dr- (p\-$dx + [p Jn Jron Jn Jr (4.343) (with Vu V(|) = YA=I VM< • V^i). Since <$> = 0 on T, the above relation (4.343) reduces to f Vu • V<|> dx = f f • * dx + f pV • <> dx. (4.344) •Jn Jn Jn Now suppose that §e"V, where TT = {<|>|c|>G(^(Q)r, V-ct> = 0}; from (4.344) it follows that f Vu • V<|) dx = f f • <J> rfx, V <|> e "T. (4.345) Relation (4.345) suggests the introduction of F, a(•, -),L denned by V = {v | v € (tf J(fi))", V • v = 0}, (4.346) a(v, w) = f Vv • Vw dx, V v, w e (H\Q)) N , (4.347) •la L(v)= ff-vdx, (4.348) respectively, and then, in turn, the following variational problem: Find u e V such that a(u, v) = L(v), VveF. (4.349) Since the mapping v -> V • v is linear and continuous from (Hl(Q)) N into L 2 (Q), F is a closed subspace of (HQ(Q)) N and, since Q is bounded is therefore a Hilbert space for the scalar product {v, w} -» j n Vv • Vw dx. The bilinear form a( •, •) is clearly continuous and (Ho(n)) iV -elliptic, and the linear functional L is continuous over F if f E (L 2 (Q)) iV . From the above properties of V,a( •), L, it follows from Theorem 2.1 of Sec. 2.3 that the variational problem (4.349) has a unique solution. Since we have (cf. Ladyshenskaya [1]) ^(Hi(«)) w _ y it follows from (4.345), (4.349) that if {u, p} is a solution of the Stokes problem (4.341), then u is also the solution of (4.349). Actually the reciprocal property is true, but proving it is nontrivial, particularly obtaining a pressure p e L 2 (Q) from the variational formulation (4.349); for the reciprocal property we refer [...]... and Girault and Raviart [1] We refer to Chapter VII, Sec 5 for finite element approximations of the Stokes problem (4.341) (see also the references therein) and also to Appendix III for some complements 5 Further Comments: Conclusion Variational methods provide powerful and flexible tools for solving a large variety of boundary-value problems for partial differential operators The various examples discussed... method, founded on the application of the Green-Ostrogradsky formula, can also be applied to the computation of fluxes through lines (or surfaces if Q < R3) = inside Q; this is done, for example, in some solution methods for partial differential equations using domain decomposition APPENDIX II A Finite Element Method with Upwinding for Second-Order Problems with Large First-Order Terms 1 Introduction Upwinding... Sec 5, we observe that we do not require the incompressibility condition V • u = 0 to be satisfied for the nonlinear steps In order to improve the well-posedness properties of the elliptic problems to solve at these nonlinear steps (and also to simplify convergence proofs), one may replace the original nonlinear term B(u) by = (u • V)u + iu(V • u), following Temam [1] It is clear that B(n) = B(u) if... the scope of this book, and therefore will not be discussed here (indeed these existence and equivalence results hold if we suppose that gx e (L2(F1))IV, for example) However, proving the 420 App Ill Some Complements on the Navier-Stokes Equations existence and uniqueness of solutions for the corresponding Stokes problem (i.e., for the problem obtained by neglecting the nonlinear term (u • V)u) is not... Element Method with Upwinding for Second-Order Problems Figure 6.7 Triangulation 3~l (s = 10" 3, Q = 0) Figure 6.8 Triangulation 9~\ (s = 10~3, d = 0) 6 Numerical Experiments 409 Figure 6.9 Triangulation ,Tt {& = 10 " 3 , 9 = 0) Figure 6.10 Triangulation $~l{e= 10" 3 , 0 = re/3) 410 App II A Finite Element Method with Upwinding for Second-Order Problems Figure 6 .11 Triangulation 0 is continuous (but not C 1 ), and therefore the numerical process, is less sensitive to phase distorsion (but exhibits... more complicated problems are quite obvious 2 The Model Problem Let Q be a bounded domain of U2 and Y = dQ We consider the problem (with e > 0) -eAu + p • Su = / i n Q, u = 0, on Y, (2.1) where p = {cos 9, sin 8} We are mainly interested in solving (2.1) for small values of e; in the following we shall suppose that / e L2(Q), and we shall use the notation Problem (2.1) has as variational formulation (see . moment, we are testing more sophisticated methods like the Lanczos- type methods described in, e.g., Widlund [1] (see also Strikwerda [1] for iterative methods for solving finite difference approximations. Green-Ostrogradsky formula, can also be applied to the computation of fluxes through lines (or surfaces if Q <= R 3 ) inside Q; this is done, for example, in some solution methods for partial differential. work of the internal elastic forces and L(v) as the work of the external (body and surface) forces. Thus, the equation a(u, v) = L(v), V v e V is a reformulation of the virtual