Numerical_Methods for Nonlinear Variable Problems Episode 10 potx

4 Application to the Solution of Elliptic Problems for Partial Differential Operators 347 then (4.29) has a unique solution in H 1 (Q), which is also the unique solution of the variational problem: Find u e H^Q) such that I (XVM) • \v dx + I a o uv dx = f fv dx + j gv dT, VHE H\Q). Jn Jn Jn Jr (4.95) PROOF. It suffices to prove that the bilinear form occurring in (4.95) is H 1 (Q)-elliptic. This follows directly from Lemma 4.1 and from the fact that if v = constant = C, then a o (x) dx > 0 •Jn and a o (x)v 2 dx = C 2 a o (x) dx = 0 •'n •'n imply C = 0, i.e., v = 0. • Proposition 4.7. We consider the Neumann problem (4.29) with Q bounded and A still obeying (4.47); if we suppose that a 0 = 0, then (4.29) has a unique solution u in H^nyU 20 if and only if L ; (4.96) (4.97) f fdx+ LdT = 0; Jn JT u is also the unique solution in H 1 (Q)/U of the variational problem: Find uef/'P such that f (SVu) -\vdx= f fv dx + f gv dT, Vce H\£l). Jn Jn Jr PROOF. For clarity we divide the proof into several steps. Step 1. Suppose that a 0 = 0; if u is a solution of (4.29) and if C is a constant, it is clear from V(« + C) = \u that u + C is also a solution of (4.29). If u is a solution of (4.29), we can show, as in Sec. 4.2.2, that (4.97) holds; taking v = 1 in (4.97), we obtain (4.96). Step 2. Consider the bilinear form over H l (ii) x H 1 (Q.) defined by r I r \/r \ a(v, w)= \ \v • Vw dx + (\ v dx w dx , Vt,w6 H^O); (4.98) 1 This means that u is determined in H l (Q) only to within an arbitrary constant. 348 App. I A Brief Introduction to Linear Variational Problems the bilinear form a(-, •) is clearly continuous, and from Lemma 4.1, it is if'(fi)-elliptic (it suffices to observe that if v = constant = C, then 0= [ \ vdx] = C 2 (meas(Q)) 2 => C = v = 0). Vn / From these properties, v -> {a(v, v)) 112 defines, over if l (Q), a norm equivalent to the usual H^nynorm defined by (4.40). Step 3. We now consider the space Fj = ivlveH^O), Vi being the kernel of the linear continuous functional i; -> v(x) dx is a closed subspace of H i (il). If we suppose that H l (Q) has been equipped with the scalar product defined by a( •, •) (see (4.98)), it follows from Step 2, and from the definition of V lt that over V lt v 1/2 defines a norm equivalent to the H '(Sll-norm (4.49); henceforth we shall endow V x with the following scalar product: 7' •In • Vw dx. From these properties of V u and from (4.47), the variational problem: Find ueV^ such that f (KVu) • \v dx = [fvdx+ [gvdT, VueFj, (4.99) has a unique solution. Step 4. Returning to H l (Q) equipped with its usual product, let us introduce V o a H 1 (Q) defined by V o = {v\v = constant over fi}; (4.100) if v e VQ , we have r t r = c \ v dx, VceR, 0 = \ \v-Vcdx+ \ vc dx = c I ; which shows that Kg = V t . We then have if \Q) =V 0 ® V u and for any v e if J (f2), we have a unique decomposition t> = v 0 + v u v t eVi, V i = 0, 1. 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 349 Step 5. From (4.96), (4.99) it follows that we also have f (AVu) • V(v + c)dx= f f(v + c)dx + \g(v + c)dT, V v e V u V c e R, u € V x . (4.101) From the results of Step 4, relation (4.101) implies that u is a solution of (4.97) (but the only one belonging to FJ; actually if we consider a second solution of (4.97), say u*, we clearly have (from (4.47)) a f \V(u* - u)\ 2 dx < f (XV(«* - «)) • V(«* - u) dx = 0. (4.102) From (4.102) it follows that u* — u = const; this completes the proof of the proposition. • Remark 4.10. In many cases where a 0 = 0 in (4.29), one is more interested in Su than in u itself [this is the case, for example, in fluid mechanics (resp., electrostatics), where u would be a velocity potential (resp., an electrical potential) and Vu (resp., — Vu) the corresponding velocity (resp., electrical field)]; in such cases, the fact that u is determined only to within an arbitrary constant does not matter, since V(« + c) = Vu, VceR. 4.3. Solution of Dirichlet problems for second-order elliptic partial differential operators We shall now discuss the formulation and the solution via variational methods of Dirichlet problems for linear second-order elliptic partial differential operators. The finite element approximation of these problems will be dis- cussed in Sec. 4.5. 4.3.1. The classical formulation. With Q, A, a 0 , f, g, and the notation as in Sec. 4.2.1, we consider the following Dirichlet problem: - V • (XVu) + V • (PM) + a o u = / in Q, u = g on F, (4.103) where P is a given vector function denned over Q and taking its values in U N . Remark 4.4 of Sec. 4.2.1 still holds for the Dirichlet problem (4.103). Remark 4.11. If X = I, a 0 = 0, p = 0, the Dirichlet problem (4.103) reduces to -Au = / in Q, u = g on T, (4.104) which is the classical Dirichlet problem for the Laplace operator A. 350 App. I A Brief Introduction to Linear Variational Problems 4.3.2. A variational formulation of the Dirichlet problem (4.103) Let v e 9{Q) (where ®(Q) is still denned by (4.38)); we then have c = 0onr. (4.105) Multiplying the first relation (4.103) by v, we obtain (still using the Green- Ostrogradsky formula, and taking (4.105) into account) [ (SVu) • \v dx - tufl-Vvdx+ [a o uvdx = [ fv dx, Vi>eS>(Q). Jn . J(i Jn Jn (4.106) Conversely it can be proved that if (4.106) holds, then u satisfies the second- order partial differential equation in (4.103) (at least in a distribution sense). Let us now introduce the Sobolev space HQ(Q) defined by H l 0 (Q) = S>(Q) H ' (n) ; (4.107) if F = dQ is sufficiently smooth, we also have Hj(fi) = {v\veHHQ), y o ^ = 0}, (4.108) where y 0 is the trace operator introduced in Sec. 4.2.2. From (4.107), (4.108), HQ(Q) is a closed subspace of H J (Q). An important property of HQ(Q) is the following: Suppose that Q. is bounded in at least one direction of U N ; then \l/2 \Vv\ 2 dx) (4.109) Ja ) defines a norm over Hj(Q) equivalent to the H 1 (Q)-norm. Property (4.109) holds, for example, for Q = ]a, /?[ x R^" 1 with a, jS e U, a < )3, but does not hold for Returning to the Dirichlet problem (4.103), and to (4.106), we suppose that the following hypotheses on A, a 0 , P, /, and g hold: feL 2 (Q), lgeH\Q) such that g = y o g, (4.110) a 0 e L^fi), a o (x) > a 0 > 0 a.e. on Q, (4.111) X satisfies (4.47), (4.112) p e (L OO (Q)) /V , V • p = 0 (in the distribution sense). (4.113) 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 351 We now define a: H\Q) x H\SX) -> U and L: H l (ii) -^Uby a(v,w)= \ (K\v) • Vw dx - \v$-\wdx+ \ a Q vw dx, Vu.wer/'P, Jn Jn Jfi (4.114) L(v) = f /t> dx, Voe/f 1 (Q), (4.115) respectively; a(-, •) (resp., L) is clearly bilinear continuous (resp., linear continuous). Before discussing the variational formulation of the Dirichlet problem (4.103), we shall prove the following useful lemma: Lemma 4.2. Suppose that p satisfies (4.113); we then have |\p- \wdx = - f wP- Vvdx, Vu,wG/fJ(Q) (4.116) Jn Jn (i.e., the bilinear form {v, w} -+ up • Vw dx Jn is skew symmetric over HQ(Q,) X Hl(Q)). PROOF. Let v, w e 3){Q); we have I up • Vw dx = I P • \(vw) dx- I wp • \v dx. (4.117) Jn Jn •'n Since vw e @(£l) and V • p = 0, we also have f p • \(vw) dx = <P, V(w)> = - <V • P, vw} = 0 (4.118) (where < •, • > denotes the duality between £)'(Q.) and From (4.117), (4.118), we then have -\wdx= - \ wfl-\vdx, V»,tf6 9(fi). (4.119) Jn From the density of @(£l) in tf J(J2), (4.119) implies (4.116). • Lemma 4.2 implies: Proposition 4.8. Suppose that (4.Ill)—(4.113) hold; then the bilinear form a{-, •) defined by (4.114) fs Hl(Q.)-elliptic. PROOF. From (4.111), (4.112), (4.114) we have a(v, v) > Min(a, a o )|M|£i (n) - (" up • Vu rfx, VUG Hj(fl); (4.120) 352 App. I A Brief Introduction to Linear Variational Problems since p satisfies (4.113), from Lemma 4.2 we have yp • Vv dx = 0, V v e H&Q). (4.121) [2 Combining (4.120) and (4.121), we have a(v, v) > Min(a, a o )||y|||i ( jj), V»e Hj(fl), i.e., the HJ(Q)-ellipticity of a{ •, •). • Using the above results we now prove: Proposition 4.9. Suppose that (4.110)—(4.115) hold; then the variational problem: Find u e H 1 (Q) such that y o u — g and a(u,v) = L(v), Vcer/JfQ) (4.122) has a unique solution. This solution is also the unique solution in H 1 (D.) of the Dirichlet problem (4.103). PROOF. (1) Uniqueness. Suppose that (4.122) has two solutions u t and u 2 ; we then have a(u!, v) = L(v), V v e HQ(Q), a(u 2 ,v) = L(v), Vt)eHj(Q), and by substraction, a(u 2 -u 1 ,v) = 0, V v e Hj(Q). (4.123) Since u u u 2 e H 1 (Q) with y o u l = y o u 2 (=g), we have u 2 — u l eH 1 ^), y o (u 2 — u y ) = 0, i.e., « 2 — "i e HQ(Q). Taking v = u 2 — u t in (4.123) and using Proposition 4.8, we have 0 < Min(a, a o )||« 2 - "IIIHUD) ^ 0, i.e., u 2 = «!• (2) Existence. We have (from (4.110)) u = y o 0> where g e H 1 (Q); this leads to the introduction of u e HQ(Q) such that u = u-g {&u = u + g). (4.124) There is clearly equivalence between (4.122) and the linear variational problem in Hl(il) below: Find u e Hj(fi) such that a(u, v) = L(v) - a(g, v), V v e H^(£i). (4.125) From Proposition 4.8, a{-, •) is bilinear, continuous over Hj(Q), and i/J(fl)-elliptic; moreover, the linear functional v -* L(v) - a(g, v) (4.126) 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 353 is clearly continuous over Ho(ii). From the properties of HQ(Q), a(-, •), and of the linear functional (4.126), we can apply Theorem 2.1 of Sec. 2.3 to prove that (4.125) has a unique solution in HQ(Q); this, in turn, implies (taking part (1) into account) that (4.122) has a unique solution as well. (3) The Solution u of (4.122) Satisfies (4.103) and Conversely. Taking v e &(Q) in (4.122), we find that u satisfies (4.103) in the sense of distributions. Conversely, if u e H 1 (Q.), with y o u = g, satisfies (4.103), we can easily prove that a(u, v) = L(v), VD£ ®(Q), (4.127) and using the density of &(Q) in tfj(fi), we find that (4.127) also holds for all the v in Hj(fl). • Remark 4.12. Suppose that A is symmetric and that P = 0; this implies the symmetry of the bilinear form a(-, •). There is then equivalence between (4.122) and the minimization problem: Find u e H g such that J(u)<J(v), ~iveH g , (4.128) where J( V ) = I f (K\v) -\vdx + l f a 0 v 2 dx - f fv dx (4.129) 2 Jn 2 J n J n and H g = {v\veH 1 (Q),y o v = g}. (4.130) 4.3.3. Further remarks and comments Remarks 4.7 and 4.8 of Sec. 4.2.3.1 still hold for the Dirichlet problem (4.103) in that we can replace L defined by (4.115) by more complicated linear con- tinuous functionals like the one in (4.63), or L(v) = f fv dx + [hv Jn Jy dy with h and y as in (4.68). Concerning the case where a 0 = 0 in (4.103), (4.114), we have the following: Proposition 4.10. Suppose that Q is bounded in at least one direction of U N ; also suppose that a 0 = 0, the hypotheses on A, P, /, g remaining the same. Then the variational problem (4.122) still has a unique solution which is also the unique solution in H J (Q) of the Dirichlet problem (4.103). PROOF. It suffices to prove that the bilinear functional a: H^Sl) x H^il) -> R defined by a(v, w) = (A\v) -\wdx - \ vfl • \w dx 354 App. I A Brief Introduction to Linear Variational Problems is Ho(£l)- elliptic. This follows from (4.112) (and (4.47)) and from Lemma 4.2 which implies that (v, v) > a f a(v, v)>a\ \Vv\ 2 dx, V v e tf£(Q), and from (4.109) which implies that v -> (J n | Vu | 2 dx) 1/2 is a norm over HJ(Q) equivalent to the H 1 (Q)-norm. D 4.3.4. Some comments on mixed boundary-value problems (Neumann-Dirichlet problems) We briefly discuss the solution (via variational methods) of elliptic problems for linear second-order partial differential operators combining the boundary conditions of Sec. 4.2 (Neumann's boundary conditions) and Sees. 4.3.1., 4.3.2, and 4.3.3 (Dirichlet's boundary conditions). With fi as in the above sections, we suppose that F (= d£l) is the union of F o , Fj such that F o u Fi = F, r o nr, = 0; (4.131) such a situation is described in Fig. 4.1 of Chapter II, Sec. 4.1. We now consider the mixed boundary-value problem — V • (AVu) + a o u = /in D., u = g 0 onF 0 , (AVM) • n = g l on r 1( (4.132) where A, a o ,/are as in Sec. 4.3.1 and where g 0 , g x are given functions defined over F o , T u respectively. Taking a test function v "sufficiently smooth" and such that v\r o =0, (4.133) we obtain (still using the Green-Ostrogradsky formula (4.33)) that any solution of (4.132) satisfies f (SVw) • \v dx + f a o uv dx = f fv dx + f g r v dY. (4.134) Jn Jn Ja Jr t To solve (4.132) by variational methods, we first introduce a(-, •) and L(-) defined by a(v,w)= I (KVv) • Vw dx + \ a o vw dx, V v, w e H\Q.), (4.135) Jn Jn L(v) = [ fv dx + f gi v dT, VceH'p. (4.136) We suppose that a 0 and A still satisfy (4.46), (4.47) and that /eL 2 (Q), g.eViT,). (4.137) 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 355 These hypotheses on A,a o ,f,g 1 imply that a(-, •) is bilinear continuous over H*(Q) x H^Q) and H 1 (Q)-elliptic and that L is a linear continuous functional over H^Q). We now introduce (motivated by (4.133)) the following subspace V o of H\ny. V o = {v|v e H^Q), y o v = O a.e. on F o }; (4.138) actually V o is a closed subspace of H 1 (Q). Using a variant of the proof of Proposition 4.9 in Sec. 4.3.2, we should prove the following: Proposition 4.11. Suppose that there exists g 0 e H 1 (Q) such that 0o = y O <7olr o ; (4-139) also suppose that the above hypotheses on A, a 0 , f, g 1 hold. Then the variational problem: Find u e H 1 (Q) such that y o u = g 0 on F o and f (K\u) • \v dx + \a o uvdx= \ fv dx + j g x vdT, V v e V o Jn Jn J " JFl (4.140) has a unique solution, which is also the unique solution in H\Q) of the mixed boundary-value problem (4.132). For a proof see, e.g., Necas [1]. Most remarks of Sees. 4.2.3 and 4.3.3 still hold for (4.132), (4.140); in particular, if D. is bounded and if | ro dT > 0, then we can suppose that a 0 = 0 [this follows from the fact that if Q, is bounded, then \\v\ 2 dx V ' 2 defines a norm over V o which is equivalent to the H^Q^norm (Lemma 4.1 can be used to prove this equivalence property)]. 4.4. Solution of second-order elliptic problems with Fourier boundary conditions 4.4.1. Synopsis In this section we shall discuss the solution—via variational methods—of the so-called Fourier problem for linear second-order elliptic partial differential operators; our interest in this problem is twofold: (i) The Fourier problem occurs in the modelling of several heat-transfer phenomana. 356 App. I A Brief Introduction to Linear Variational Problems (ii) The Neumann and Dirichlet problems discussed in Sees. 4.2 and 4.3, respectively, can be considered, in fact, to be particular cases 21 of the Fourier problem; this claim will be justified in the sequel. 4.4.2. Classical formulations of the Fourier problem With Q, A, a 0 , f, g as in Sec. 4.2, we consider the following boundary-value problem: Find u such that - V • (AVM) + a 0 u= f in Q, (X\u) • n + ku = g onT, (4.141) where k is a given function denned over F. 4.4.3. A variational formulation of the Fourier problem (4.141) Let v be a smooth function defined over Q; multiplying the first relation (4.141) by v and integrating over Q, we obtain (using the Green-Ostrogradsky formula (4.33)) f = f f C = (A\u)-\vdx + a o uvdx= fvdx+ (AVw) • nv dT; (4.142) Jn Jn Jn Jr combining (4.142) and the boundary condition in (4.141), we finally obtain | (X\u) • \v dx + \ a 0 uv dx + \ kuv dT = \ fv dx + \ gv dT. (4.143) Jn Jn Jr Jn Jr Conversely, it can be proved that if (4.143) holds, Vuef, where V is still defined by (4.35), then u is in some sense a solution of the Fourier problem (4.141). We now define a: H\Q) x H l (Q) -» R and L: H^O) -> U by f = f a(v, w) = [(AVu) • Vw + a o i;w] dx + kvw dT, Vtt,we H 1 (Q), Jn Jr (4.144) L(v) = f fv dx + f gy o vdT, V v e H ! (Q), (4.145) Jn Jr respectively. We suppose that the following hypotheses concerning X, a 0 , k, f, g hold: /eL 2 (Q), geL 2 (T), (4.146) a 0 e L°°(fi), a o (x) > a 0 > 0 a.e. on Q, (4.147) k G L CO (F), k(x) > 0 a.e. on F, (4.148) X satisfies (4.47). (4.149) Similarly for the mixed boundary-value problem of Sec. 4.3.4. [...]... solution of Dirichlet and mixed Neumann-Dirichlet problems Since (see Sec 4.3.4 for more details) the Dirichlet problem appears as a particular case of the other problem, we shall consider the second one only (actually the extension to the mixed FourierDirichlet problem is straightforward) 4.5.5.1 Formulation of the continuous and discrete Neumann-Dirichlet problems We suppose that F ( = dQ) is the union... problems compute the solution of the (discrete) Dirichlet, Neumann, Neumann-Dirichlet, etc problems via discrete Fourier formulations (derived from (4.157), (4.175)) with e > 0 "very small." In the finite element code, MODULEF, for example, one uses e = 10~ 30 (see also Perronnet [1] for more details about MODULEF-and further references) From the practical interest of these approximations by penalization... Dirichlet, and Fourier problems 4.5.1 Synopsis We now consider the finite element approximation of the second-order elliptic problems discussed in Sees 4.2, 4.3, and 4.4; actually the finite element techniques to be described in this section will also be used to approximate the second-order elliptic problem discussed in Sec 4.6 (and have been used all along in this book to solve nonlinear problems much more... used all along in this book to solve nonlinear problems much more complicated than those linear problems discussed in this appendix) 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 365 The following subsections cover only a small part of this very important subject; for more details (theoretical and practical) concerning finite element approximations, see Aubin... having Qi as a common vertex Remark 4.19 In this appendix we consider piecewise linear approximations on triangles only; for more sophisticated methods making use of higherdegree polynomials, quadrilateral elements, curved elements, and also for finite element methods for three-dimensional problems, see the references given in Sec 4.5.1 4.5.3 Some fundamental results on finite element approximations 4.5.3.1... element approximation of important problems (like the Stokes and Navier-Stokes problems discussed in Chapter VII, Sec 5).] 4.5.3.5 Some approximation properties of C2 functions Let T be a triangle of R2 whose vertices are denoted by Au A2, A3 (with At = {aa, ai2}) With nT as in (4.212), we shall estimate, by elementary methods, for v e C2{T) and m = 0, 1 \KTV - u|m>00>T For more results in this direction,... Variational Problems The proof of the convergence of the solution u£ of (4.175) to the solution u of (4.132), (4.140) is left to the reader as an exercise The above approximation properties are, in fact, of practical interest since many modern finite element codes for solving elliptic partial differential equation problems compute the solution of the (discrete) Dirichlet, Neumann, Neumann-Dirichlet, etc problems. .. [2], [3] and Mercier [1]), we shall give some results about the 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 367 approximation properties of the particular finite element method introduced in Sec 4.5.2 We shall be particularly concerned with providing estimates for the finite element approximation errors; we shall use Sobolev norms to express these estimates 4.5.3.2... independent of T such that llph l\v\2^^, V ve W2-P(t) (4.215) PT 25 Q can be a polyhedral d o m a i n , for example In fact, it is compact In the sequel, we shall use the symbol c? to denote the inclusion, with continuity of the injection 26 27 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 369 Remark 4.20 In the particular case where q = p, (4.215) holds if m... Application to the Solution of Elliptic Problems for Partial Differential Operators 359 Find ne eif'ffl) such that J£ut) < J£v), V v e H\Q), (4.158) where Jlv) = \ f [_(X\v) • \v + a0v21 dx - L(v) + i - f (v - gf dT z Jn ze J r (4.159) Thus (4.157), (4.158) is obtained from (4.156) by using a penalty procedure to handle the boundary condition you = g (see Chapter I, Sec 7, for more details on penalty procedures) . of Dirichlet problems for second-order elliptic partial differential operators We shall now discuss the formulation and the solution via variational methods of Dirichlet problems for linear second-order. triangles only; for more sophisticated methods making use of higher- degree polynomials, quadrilateral elements, curved elements, and also for finite element methods for three-dimensional problems, . this book to solve nonlinear problems much more complicated than those linear problems discussed in this appendix). 4 Application to the Solution of Elliptic Problems for Partial Differential