GALERKIN APPROXIMATION IN BANACH AND HILBERT SPACES Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
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Wolfgang Arendt, Isabelle Chalendar, Robert Eymard. GALERKIN APPROXIMATION IN BA- NACH AND HILBERT SPACES. 2019. hal-02264895v2
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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GALERKIN APPROXIMATION IN BANACH AND HILBERT SPACES
W. ARENDT, I. CHALENDAR, AND R. EYMARD
Abstract. In this paper we study the conforming Galerkin ap- proximation of the problem: find u such that a(u, v)= L, v for all v , where and are Hilbert∈U or Banach spaces, ha is ai continuous∈ V bilinear orU sesquilinearV form and L ′ a given data. The approximate solution is sought in a finite dimensional∈ V subspace of , and test functions are taken in a finite dimensional subspace of U. We provide a necessary and sufficient condition on the form a forV convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that has a finite dimensional Schauder decomposition in terms of propertiesU related to the Galerkin approximation. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the uni- versal Galerkin property) are the essentially coercive forms. In this case, a generalization of the Aubin-Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side L, as shown by several applications.
1. Introduction Due to its practical importance, the approximation of elliptic prob- lems in Banach or Hilbert spaces has been the object of numerous works. In Hilbert spaces, a crucial result is the simultaneous use of the Lax-Milgram theorem and of C´ea’s Lemma to conclude the con- vergence of conforming Galerkin methods in the case that the elliptic problem is resulting from a coercive bilinear or sesquilinear form.
But the coercivity property is lost in many practical situations: for example, consider the Laplace operator perturbed by a convection term or a reaction term (see the example in Section 7.2), and the approxi- mation of non-coercive forms must be studied as well. For particular
2010 Mathematics Subject Classification. 65N30,47A07,47A52,46B20. Key words and phrases. Galerkin approximation, sesquilinear coercive forms, approximation properties in Banach spaces, essential coercivity, universal Galerkin convergence. 1 2 W.ARENDT,I.CHALENDAR,ANDR.EYMARD bilinear or sesquilinear forms, the Fredholm alternative provides an existence result in the case where the problem is well-posed in the Hadamard sense. Such results have been extended by Banach, Neˇcas, Babuˇska and Brezzi in the case of bilinear forms on Banach spaces. The conforming approximation of such problems enters into the framework of the so-called Petrov–Galerkin methods, for which sufficient condi- tions for the convergence are classical (see for example the references [2, 4, 8, 21] which also include the case of non-conforming approxima- tions). Nevertheless, these sufficient conditions do not guarantee that for a given problem, there exists a converging Galerkin approximation. Moreover, they do not answer the following question, which is impor- tant in practice: under which conditions does the Galerkin approxima- tion exist and converge to the solution of the continuous problem for any sufficiently fine approximation (for example, letting the degree of an approximating polynomial or the number of modes in a Fourier ap- proximation be high enough, or letting the size of the mesh for a finite element method be small enough, and, in the case of Hilbert spaces, using the Galerkin method and not the Petrov–Galerkin method)? The aim of this paper is precisely to address such questions for not necessarily coercive bilinear or sesquilinear forms defined on some Ba- nach or Hilbert spaces (we treat the real and complex cases simulta- neously). We shall restrict this study to conforming approximations, in the sense that the approximation will be sought in subspaces of the underlying space, using the continuous bilinear or sesquilinear form. In the first part we consider the Banach space framework. Given a continuous bilinear form a : R where and are reflexive, separable Banach spaces, oneU×V→ is interested in theU existenceV and the convergence of the Galerkin approximation to u, where u is the solution of the following problem:
(1.1) Find u such that a(u, v)= L, v , for all v , ∈ U h i ∈V where L ′ is given (the existence and uniqueness of u are obtained under the∈V Banach-Neˇcas-Babuˇska conditions, see for example [8, The- orem 2.6]). For approximating sequences ( n)n N∗ , ( n)n N∗ (see Sec- tion 2 for the definition), the Galerkin approximationU ∈ V of (1.1)∈ is given by the sequence (un)n N∗ such that, for any n N∗, un is the solution of the following finite∈ dimensional linear problem:∈
(1.2) Find u such that a(u , χ)= L, χ , for all χ . n ∈ Un n h i ∈Vn GALERKIN APPROXIMATION 3
It is known that, if dim n = dim n, the uniform Banach-Neˇcas- Babuˇska condition (BNB)U given in SectionV 2 is sufficient for these exis- tence and convergence properties (see for example [8, Theorem 2.24]). We show here that this condition is also necessary and, surprisingly, that the convergence of the Galerkin approximation of (1.1) is equiva- lent to that of the Galerkin approximation of the dual problem. These two results seem to be new and are presented in Section 2. In Section 3, we ask the following: given a form a such that (1.1) is well-posed, do there always exist approximating sequences in and such that the Galerkin approximation converges? Surprisingly,U the answerV is negative (even though the spaces and are supposed to be reflexive and separable). In fact, such approximatingU V sequences exist if and only if the Banach space has a finite dimensional Schauder decomposition, a property whichU is strictly more general than having a Schauder basis. In the remainder of the paper, merely Hilbert spaces are considered and moreover we assume that = and n = n for all n N∗. Given is a continuous bilinear formU aV: U RV, where is a∈ sep- arable Hilbert space. Assuming that (1.1)V×V→ is well-posed, weV show that the convergence of the Galerkin approximation for all approximating sequences in (which we call here the universal Galerkin property) is equivalent toV a being essentially coercive, which means that a com- pact perturbation of a is coercive. This notion of essential coercivity can also be characterized by a certain weak-strong inverse continuity of a, which, in fact, we take as definition of essential coercivity (Defi- nition 4.1). We then derive improved a priori error estimates by generalizing the Aubin–Nitsche argument to non-symmetric forms and also allowing the given right hand side L of (1.1) to belong to arbitrary interpolation spaces in between and ′. These generalizations are finally applied to two cases: the approximationV V of selfadjoint positive operators with compact resolvent (in this case, it is seen that our a priori error estimate is optimal, with the fastest speed of convergence for L in , the slowest V for L ′) and the finite element approximation of a non-selfadjoint elliptic∈ differential V operator, including convection and reaction terms which is indeed essentially coercive. To avoid any ambiguity, in the sequel, we let N = 0, 1, 2, and { ···} N∗ = N 0 . \{ } The paper is organized as follows: 4 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
Contents 1. Introduction 1 2. Petrov–Galerkin approximation 4 3. Existence of a converging Galerkin approximation 10 4. Essentially coercive forms 13 5. Characterization of the universal Galerkin property 18 6. The Aubin-Nitsche trick revisited 21 7. Applications 23 7.1. Selfadjoint positive operators with compact resolvent 23 7.2. Finite elements for the Poisson problem 26 References 31
2. Petrov–Galerkin approximation In this section we give a characterization of the convergence of Petrov– Galerkin methods, that, for short, we call Galerkin convergence. A basic definition is the following. Definition 2.1 (Approximating sequences of Banach spaces). Let be a separable Banach space. An approximating sequence of isV a V sequence ( n)n N∗ of finite dimensional subspaces of such that V ∈ V dist(v, ) 0 as n Vn → →∞ for all v , where dist(u, ) := inf u χ : χ . ∈V Vn {k − k ∈Vn} Now let and be two separable, reflexive Banach spaces over K = R or CUand a aV continuous sesquilinear form such that a(u, v) M u v for all u , v | | ≤ k kU k kV ∈ U ∈V where M > 0 is a constant. We assume that and are infinite di- U V mensional and that ( n)n N∗ and ( n)n N∗ are approximating sequences of and respectively.U ∈ We also assumeV ∈ throughout that U V 0 = dim = dim for all n N∗. 6 Un Vn ∈ Given L ′ we search a solution u of the problem: ∈V (2.1) find u such that a(u, v)= L, v , for all v . ∈ U h i ∈V Moreover we want to approximate such a solution by un, the solution of the problem: (2.2) find u such that a(u , χ)= L, χ , for all χ . n ∈ Un n h i ∈Vn GALERKIN APPROXIMATION 5
Note that, given n N∗, there exists a unique un n satisfying (2.2) if and only if ∈ ∈ U (2.3) for all u , a(u, χ) = 0 for all χ u =0, ∈ Un ∈Vn ⇒ since, by assumption, n and n have the same finite dimension. Let us briefly recallU the originV of the Banach-Neˇcas-Babuˇska condi- tions for the well-posedness of (2.1) as stated for example in [8, 21, 2] (equivalent conditions are proposed in [4] in the case of Hilbert spaces). Let us consider the associated operator : ′ defined by A U→V u, v = a(u, v) (u , v ). hA i ∈ U ∈V Then is linear, bounded with M. By the Inverse Mapping Theorem,A has closed range andkAk is injective ≤ if and only if there exists β > 0 suchA that
(2.4) u ′ β u for all u . kA kV ≥ k kU ∈ U By the definition of the norm of ′, this can be reformulated by V (2.5) sup a(u, v) β u for all u . v V 1 | | ≥ k kU ∈ U k k ≤ Recall that is invertible if and only if is injective and has a closed and dense range.A By the Hahn-BanachA theorem, has dense range A if and only if no non-zero continuous functional on ′ annihilates the range of . By reflexivity, this is equivalent to the followingV uniqueness property:A
(2.6) for all v , a(u, v) = 0 for all u v =0. ∈V ∈ U ⇒ Thus (2.1) is well-posed (i.e. for all L ′ there exists a unique u satisfying (2.1)) if and only if (2.5) and∈V (2.6) are satisfied. ∈ U In order to obtain a result of convergence of the approximate solu- tions we consider the following uniform Banach-Neˇcas-Babuˇska condi- tion (called Ladyzenskaia-Babuˇska-Brezzi condition in the framework of the mixed formulations), which is the estimate (2.5) for a n n uni- |U ×V formly in n N∗, namely ∈ (BNB) β > 0; n N∗, u n, sup a(u, v) β u . ∃ ∀ ∈ ∀ ∈ U v n, v V =1 | | ≥ k kU ∈V k k We recall that (BNB) implies that the approximate solutions converge to the solution if the problem is well-posed (see for example [8, 21, 2]). Here we will show that (BNB) is actually equivalent to Galerkin- convergence, and surprisingly also to Galerkin-convergence for the dual problem. 6 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
Definition 2.2 (Convergence of Galerkin approximation). We say that the Galerkin-approximation converges if (2.1) as well as (2.2) are well- posed for all n N∗ and if, in addition, there exists a constant γ > 0 ∈ such that for each L ′, ∈V (2.7) u un γ dist(u, n), k − kU ≤ U where u is the solution of (2.1) and u the solution of (2.2) for n N∗. n ∈ In particular, limn un = u in . →∞ U We may also consider the dual problem of (2.1) where a is replaced by the adjoint form a∗ : K given by V×U→ a∗(v,u)= a(u, v) (u , v ). ∈ U ∈V If in Definition 2.2 the form a is replaced by a∗, then we say that the dual Galerkin approximation converges. Similarly we note the following dual uniform Banach-Neˇcas-Babuˇska condition
(BNB∗) β∗ > 0; n N∗, sup a∗(u, v) β∗ v (v n). ∃ ∀ ∈ u n, u U =1 | | ≥ k kV ∈V ∈U k k Then the following theorem holds. Theorem 2.3. The following assertions are equivalent: (i) the Galerkin approximation converges; (ii) (BNB) holds; (iii) (BNB∗) holds; (iv) the dual Galerkin approximation converges.
It is surprising that (BNB) and (BNB∗) are equivalent even though the corresponding condition (2.5) is obviously not equivalent to its dual form. In fact, it can well happen that is injective and has closed range (so that there exists β > 0 satisfyingA (2.5)) but the range of A is a proper subspace of ′ so that there exists v such that v = 0 and a(u, v) = 0 for all uV ; in particular the dual∈ V form of (2.5) does6 ∈ U not hold for any β∗ > 0. We will give the proof of Theorem 2.3 in several steps which give partly even stronger results. At first we show that (ii) implies (i), where γ can even be expressed in terms of β and M. Although the proof of this result is classical (see for example [21, 8]), we provide it for the convenience of the reader, but also to establish the well- posedness of (2.1) which we did not assume. This will be important for the proof of Theorem 2.3 and for the main result in Section 5.
Proposition 2.4. Let β > 0. Assume that for all n N∗, ∈ (2.8) sup a(u, v) β u (u n). v n, v V =1 | | ≥ k kU ∈ U ∈V k k GALERKIN APPROXIMATION 7
Then the Galerkin-approximation converges and (2.7) holds with γ = M 1+ β .
Proof. Let L ′. Note that (2.8) implies (2.3). Thus, for each n N∗ ∈V ∈ there exists a unique solution un of (2.2). By (2.8), 1 1 (2.9) un sup L, v L ′ . k kU ≤ β v n, v V 1 |h i| ≤ β k kV ∈V k k ≤ Since is reflexive, we find u such that a subsequence of (un)n, say, (uU ) , converges weakly to∈u U. Let v . By assumption we find nk k ∈V vk nk such that limk v vnk = 0. It follows that ∈V →∞ k − kV a(u, v) = lim a(unk , vk) = lim L, vk = L, v . k k →∞ →∞h i h i Thus we find a solution u of (2.1). But so far we do not know its uniqueness. This will be a consequence of (2.7) which we prove now. Indeed, observe that (2.10) a(u, χ)= L, χ = a(u , χ) for all χ . h i n ∈Vn It follows that a(u, χ) = a(un, χ) for all χ n (Galerkin orthogonal- ity). Using this, for all χ , ∈ V ∈ Un u un u χ + χ un k − kU ≤ k − kU k − kU 1 u χ + sup a(χ un, v) ≤ k − kU β v n, v V =1 | − | ∈V k k 1 = u χ + sup a(χ u, v) k − kU β v n, v V =1 | − | ∈V k k M 1+ χ u . ≤ β k − kU Taking the infimum over all χ we obtain (2.7). In particular ∈ Un limn u un = 0 which shows uniqueness. →∞ k − kU The following result is due to Xu and Zikatanov [21, Theorem 2] (see also [2, Satz 9.41]). We nevertheless provide its proof for the sake of completeness. Proposition 2.5. Assume that is a Hilbert space and that β > 0 is such that (2.8) holds. Then the Galerkin-approximationU converges and M (2.7) holds with γ = β . Proof. Note that (2.8) implies (2.3). Consequently for each w there exists a unique Q w such that ∈ U n ∈ Un a(Q w, χ)= a(w, χ) (χ ). n ∈Vn 8 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
Then Q is a projection from onto . Moreover, n U Un β Qnw sup a(Qnw, χ) k kU ≤ χ n, χ V =1 | | ∈V k k = sup a(w, χ) χ n, χ V =1 | | ∈V k k M w . ≤ k kU Thus Q M . k nk ≤ β Since 0 = n = , one has Qn = 0, Id. It follows from a result due to Kato [12,6 U Lemma6 U 4] that Q 6 = Id Q . k nk k − nk Now let L ′ and u the solution of (2.3), un the solution of (2.2). Then for any∈Vχ , ∈ Un u u = (Id Q )u = (Id Q )(u χ). − n − n − n − Hence M u un Id Qn u χ = Qn u χ u χ . k − kU ≤k − kk − kU k kk − kU ≤ β k − kU This implies that M u un dist(u, n). k − kU ≤ β U
Remark 2.6. Also in certain Banach spaces an improvement of the M constant 1+ β is possible, see Stern [19]. Next we show that even a weaker assumption than the convergence of the Galerkin-approximation implies (BNB∗).
Proposition 2.7. Assume (2.3) for all n N∗ and that ∈ sup un < n N∗ k kU ∞ ∈ whenever L ′ and u is the solution of (2.2). Then (BNB∗) holds. ∈V n Proof. Since the spaces n and n have the same finite dimension, our assumption (2.3) impliesV also dualU uniqueness, i.e. a(χ, v) = 0 for all χ implies v = 0 whenever v , and this for all n N∗. Thus ∈ Un ∈Vn ∈
v n := sup a(u, v) k kV u n, u U =1 | | ∈U k k defines a norm on . Moreover, Vn a(u, v) u v n for all u n, v n. | |≤k kU k kV ∈ U ∈V GALERKIN APPROXIMATION 9
We show that the set v := : n N∗, v n, v =0 B v n ∈ ∈V 6 k kV is bounded. For that purpose, let L ′. By assumption there exist c> 0 and u such that ∈ V n ∈ Un a(u , v)= L, v for all v n h i ∈Vn v and u c for all n N∗. Now, for , n v Vn k kU ≤ ∈ k k ∈ B v 1 L, = a(un, v) un c. h v n i | | v n ≤k kU ≤ V V k k k k This shows that is weakly bounded and thus, owing to the Banach– B Steinhaus theorem, norm-bounded. Therefore there exists β∗ > 0 such 1 that v ∗ v n , i.e. k kV ≤ β k kV β∗ v sup a(u, v) for all v n, n N∗. k kV ≤ u n, u U =1 | | ∈V ∈ ∈U k k This is (BNB∗). Proof of Theorem 2.3. (ii) (i) and (iii) (iv) via Proposition 2.4, whereas (i) (iii) and (iv)⇒ (ii) follows⇒ from Proposition 2.7. ⇒ ⇒
Remark: The hypothesis on and to be reflexive is not needed in Proposition 2.4. U V
Finally we mention that the best lower bounds β for (BNB) and β∗ for (BNB∗) are the same if and are Hilbert spaces. U V Proposition 2.8. Let β > 0. Then the two conditions (2.11) and (2.12) are equivalent: (2.11) sup a(u, v) β u for all u n and for all n N∗; v V 1,v n | | ≥ k kU ∈ U ∈ k k ≤ ∈V (2.12) sup a(u, v) β u for all v n and for all n N∗; u U 1,u n | | ≥ k kU ∈V ∈ k k ≤ ∈U
Proof. Let n N∗ and A : be given by ∈ n Un →Vn Anu, v = a(u, v). h iV Then
An∗ v,u = a∗(v,u)= a(u, v), h iU 10 W.ARENDT,I.CHALENDAR,ANDR.EYMARD where An∗ is the adjoint of A. Moreover, since An is invertible, sup a(u, v) β u v V =1,v n | | ≥ k kU k k ∈V 1 1 1 1 for all u n if and only if An− β . Since (An∗ )− = (An− )∗, it ∈ U 1 1k k ≤ 1 1 follows that (A∗ )− = (A− )∗ = A− and hence k n k k n k k n k ≤ β sup a∗(v,u) β v for all v n. u U 1,u n | | ≥ k kV ∈V k k ≤ ∈U
3. Existence of a converging Galerkin approximation In this section, we again let and be separable reflexive real Banach spaces and let a : U R be aV continuous sesquilinear form U×V → such that the problem (2.1) is well-posed; i.e. for all L ′ there exists a unique u satisfying (2.1). Since and are∈V separable, there ∈ U U V always exist approximating sequences ( n)n N∗ of and ( n)n N∗ of . Our question is whether there is a choiceU ∈ of theseU sequencesV ∈ which isV adapted to the problem (2.1); i.e. such that the associated Galerkin approximation converges. We will show that the answer is related to the approximation property. In fact, different versions of this property play a role; we recall them in the next definition. Definition 3.1 (Approximation property and Schauder decomposi- tion). Let be a separable Banach space. X a) The space has the approximation property (AP) if, for every compact subsetX K of and every ε > 0, there exists a finite rank operator R ( X) such that ∈L X Rx x <ε for all x K. k − k ∈ b) The space has the bounded approximation property (BAP) X if there exists a sequence (Pn)n N∗ of finite rank operators in such that ∈ X
for all x , lim Pnx = x. n ∈ X →∞ c) The space has the bounded projection approximation prop- X erty (BPAP) if each Pn in b) can be chosen as a projection (i.e. 2 such that Pn = Pn). d) The space possesses a finite dimensional Schauder decompo- X sition if one finds (Pn)n N∗ as in c) with the additional property ∈ (3.1) P P = P P = P for all n m. m n n m m ≥ GALERKIN APPROXIMATION 11
e) The space has a Schauder basis if d) holds with dim P = 1 X n for all n N∗. ∈ It is known that (BAP) is equivalent to (AP) if is reflexive. The first counterexample of a Banach space without (AP)X has been given by Enflo [7]. He constructed a space which is even separable and reflexive. Obviously the properties a)–e) have decreasing generality. It was Read [16] who showed that (BAP) does not imply (BPAP), even if re- flexive and separable spaces are considered. Szarek [20] constructed a reflexive, separable Banach space having a finite dimensional Schauder decompositon but not a Schauder basis. Finally, it seems to be un- known whether (BPAP) implies the existence of a finite dimensional Schauder decomposition (see [15, Sec. 5.7.4.6] and [3, Problem 6.2]). However, if is reflexive and separable, then these two properties are equivalent byX [3, Theorem 6.4 (3)]). Concerning the notion of finite dimensional Schauder decomposition, there is an equivalent formulation, namely the existence of finite dimen- sional subspaces of such that for each x there exist unique Xn X ∈ X xn n such that x = n N∗ xn This explains the name. We refer ∈ X ∈ to [14, Chapter I] , [3] forP more information and to [15, Sec. 5.7.4] for the history of the approximation property. In the following theo- rem, by the hypothesis of well-posedness, the two Banach spaces and are isomorphic. For this reason they have the same BanachU space properties.V Theorem 3.2. Let and be separable reflexive Banach spaces and let a : K beU a continuousV sesquilinear form such that (2.1) is well-posed.U×V→ Then the following assertions are equivalent.
(i) There exist approximating sequences ( n)n N∗ of and ( n)n N∗ of such that the associated GalerkinU approximation∈ U convergesV ∈ . (ii) TheV space has the (BPAP). (iii) The space U has a finite dimensional Schauder decomposition. U Here convergence of the associated Galerkin approximation is under- stood in the sense of Definition 2.2. Proof of Theorem 3.2. (i) (ii) Let u . Then L, v := a(u, v) ⇒ ∈ V h i defines an element L ′. By Definition 2.2, for each n N∗, there exists a unique P u ∈ Vsuch that ∈ n ∈Vn a(P u, χ)= a(u, χ) for all χ . n ∈Vn Moreover, P u u γ dist( ,u) for all n N∗ and some γ > 0. k n − k ≤ Un ∈ In particular, limn Pnu = u. It follows from the definition that P 2 = P . Since P →∞ , each P has finite rank. We have shown n n nU ⊂ Un n 12 W.ARENDT,I.CHALENDAR,ANDR.EYMARD that the space has the (BPAP). (ii) (iii) SeeU [3, Theorem 6.4 (3)]. ⇒ (iii) (i) Let : ′ be the operator defined by u, v = a(u, v⇒). Then Ais invertible.U →V By hypothesis there exist finitehA ranki A projections (Pn)n N∗ such that limn Pnu = u for all u . Let 1 ∈ →∞ ∈ U f ′, u := − f be the solution of (2.1). Then ∈V A 1 (3.2) u := P − f u in as n . n nA → U →∞ We show that un is obtained as a Galerkin approximation. In fact, fix n N∗. There exist b1, , bm , ϕ1, ,ϕm ′ such that ϕ , b ∈= δ and ··· ∈ U ··· ∈ U h i ji i,j m (3.3) P x = ϕ , x b n h k i k Xk=1 for all x . Since is reflexive there exist v such that ∈ U V k ∈V 1 (3.4) ϕ , − g = g, v h k A i h ki for all g ′ and k = 1, , m. Define = Span v , , v and ∈ V ··· Vn { 1 ··· m} n = Span b1, , bm . Now consider the given f ′. Let w = U m λ b { ···. Then} ∈ V k=1 k k ∈ Un (3.5)P a(w, χ)= f, χ for all χ h i ∈Vn if and only if (3.6) a(w, v )= f, v for j =1, , m. j h ji ··· By (3.4), m m m a(w, v )= λ a(b , v )= λ b , v = λ ϕ , b = λ . j k k j khA k ji kh j ki j Xk=1 Xk=1 Xk=1 Therefore w = m f, v b is the unique solution of (3.5). Again, by k=1h ki k (3.4), P m m 1 1 u = P − f = ϕ , − f b = f, v b = w, n nA h k A i k h ki k Xk=1 Xk=1 and it follows from (3.2) that limn un = u. This also implies that →∞ dist( n,u) 0 as n . Thus the sequence ( n)n N∗ is approximat- ing. U → →∞ U ∈ It remains to show that the sequence ( n)n N∗ is approximating in . For this we need the the additional propertyV ∈ (3.1). Consider the V adjoint P ′ ( ′) of P . Then P ′ ϕ weakly converges to ϕ as n n ∈L U n n →∞ for all ϕ ′. Thus ∈ U ∗ := n N Pn′ ′ W ∪ ∈ U GALERKIN APPROXIMATION 13 is weakly dense in ′. But, because of (3.1), is a subspace of ′. U W U Thus, by Mazur’s Theorem, is dense in ′. If ψ , then there W U ∈ W exist m N∗,ϕ ′ such that ψ = P ′ ϕ. Thus ∈ ∈ U m Pn′ ψ = Pn′ Pm′ ϕ = Pm′ ϕ = ψ, N for all n ∗ by (3.1), and then limn Pn′ ψ = ψ for all ψ . Since ∈ →∞ ∈ W supn N∗ Pn′ < , it follows that limn Pn′ ϕ = ϕ for all ϕ ′. ∈ →∞ k k ∞ ∗ ∈ U This implies that the sequence (Pn′ ′)n N is approximating in ′. It 1 U ∈ U follows from (3.4) that n ( − )′Pn′ ′. In fact, fix n and consider Pn V ⊃ A U 1 ∗ as in (3.3). Then (3.4) says that vk =( − )′ϕk. Since (Pn′ ′)n N is an A1 U ∈ approximating sequence in ′ and ( − )′ is an isomorphism from ′ U A U to , it follows that ( n)n N∗ is an approximating sequence in . V V ∈ V 4. Essentially coercive forms Let be a separable Hilbert space over K = C or R and a : K be aV sesquilinear form satisfying V×V → a(u, v) M u v for all u, v | | ≤ k kV k kV ∈V for some M > 0. Then we may associate with a the operator A ∈ ( , ′) defined by L V V u, v = a(u, v). hA i If a is coercive, i.e. if a(u,u) α u 2 (u ) | | ≥ k kV ∈V for some α > 0, then is invertible. This consequence is the well- known Lax-Milgram lemma.A Our aim is to find weaker assumptions than coercivity which help to decide whether the operator is invertible. Note that a is coercive if andA only if
lim a(un,un) = 0 implies that lim un =0. n n V →∞ →∞ k k We weaken this property in the following way. Definition 4.1 (Essential coercivity). The continuous sesquilinear form a (or the operator ) is called essentially coercive if for each sequence A (un)n N∗ in weakly converging to 0 and such that limn a(un,un)= ∈ V →∞ 0, one has limn un =0. →∞ k kV The following is a characterization of this new property. Theorem 4.2. The following assertions are equivalent: (i) the form a is essentially coercive; 14 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
(ii) there exist an orthogonal projection P ( ) of finite rank and α> 0 such that ∈L V a(u,u) + Pu 2 α u 2 for all u ; | | k kV ≥ k kV ∈V (iii) there exist a Hilbert space , a compact operator J : and α> 0 such that H V → H a(u,u) + Ju 2 α u 2 (u ); | | k kH ≥ k kV ∈V (iv) there exist a compact operator ( , ′) and α > 0 such that K∈L V V a(u,u) + u,u α u 2 (u ). | | |hK i| ≥ k kV ∈V Proof. (i) (ii): Let (en)n N∗ be an orthonormal basis of and con- ⇒ ∈ V sider the orthogonal projections Pn given by n
Pnv := v, ek ek. h iV Xk=1
Assume that (ii) is false for every Pn. Then there exists a sequence (un)n N∗ such that un = 1 and ∈ ⊂V k kV 2 1 a(un,un) + Pnun < . | | k kV n Note that, since Id P is a self-adjoint operator, − n (Id Pn)un, v = un, (Id Pn)v (Id Pn)v , |h − iV| |h − iV |≤k − kV with limn (Id Pn)v = 0 for all v . This implies that →∞ k − kV ∈ V (Id Pn)un converges weakly to 0. Since limn Pnun = 0, it fol- − →∞ k kV lows that un converges weakly to 0. Moreover limn a(un,un) 1 →∞ | | ≤ limn n = 0. Therefore a is not essentially coercive. (ii) →∞(iii): Choose = and J = P . ⇒ H V (iii) (iv): There exists a unique operator J ∗ : ′ such that ⇒ H→V J ∗u, v = u,Jv h i h iH for all v . Choose = J ∗J. ∈V K (iv) (i): Let (un)n N∗ that tends weakly to 0 and such that a(u ,u⇒ ) = u ,u ∈ tends⊂ V to 0 as n . Since is compact, n n hA n ni → ∞ K un 0 as n . Hence un,un 0 as n . By kKassumptionkV → there exists→ ∞β > 0 such|hK that iV | → → ∞ 2 un,un + un,un β un . |hA i| |hK i| ≥ k kV It follows that un 0 as n . k kV → →∞ GALERKIN APPROXIMATION 15
Next we want to justify the notion ”essentially coercive”. We recall that by the Toeplitz–Hausdorff theorem [10], the numerical range of a, W (a) := a(u,u): u , u =1 , { ∈V k kV } is a convex set. Hence also W (a) is convex. For α> 0, a(u,u) α u 2 (u ) | | ≥ k kV ∈V if and only if W (A) D = , ∩ α ∅ where Dα = ( α,α) in the real case and Dα = w C : w < α if K = C. This observation− leads to the following more{ precise∈ | description| } of coerciveness. Lemma 4.3. The form a is coercive if and only if there exist α > 0 and λ K with λ =1 such that ∈ | | Re(λz) α for all z W (a). ≥ ∈ Proof. We give the proof for K = C. Assume that a is coercive. There exists a maximal α > 0 such that W (a) Da = . Then there exists iθ∩ ∅ z0 W (a) of modulus α; i.e. z0 = e α for some θ R. The set ∈ iθ ∈ C := e− W (a) is convex and closed. Moreover α C and Dα C = . This implies that Re(z) α for all z C. Indeed,∈ let z C such∩ that∅ Re(z) <α. Then the segment≥ [α, z] has∈ a non-empty intersection∈ with Dα. Since C is convex it follows that z C. Conversely, clearly, if there exists α >6∈ 0 such that Re(λz) α for all z W (a), then a is coercive. ≥ ∈ Theorem 4.4. Let ( , ′). The following assertions are equiv- alent: A∈L V V (i) the operator is essentially coercive; A (ii) there exists a finite rank operator : ′ such that + is coercive; K V→V A K (iii) there exists a compact operator : ′ such that + is coercive. K V→V A K Proof. (i) (ii): Choose the orthogonal finite rank projection P on ⇒ V and α > 0 as in Theorem 4.2 (ii). Let 1 = ker V and 2 = range P . 2V V Then dim 2 < and a(u,u) α u for all u 1. Let j : ′ be the RieszV isomorphism∞ | given| ≥ by k kV ∈V V→V j(u), v = u, v . h i h iV 16 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
1 Let A = j− ( ). Then a(u, v) = Au, v for all u, v . Moreover A has◦A∈L a matrixV decomposition h iV ∈ V A A A = 11 12 A21 A22 according to the decomposition = 1 2 of . Since P is orthogonal, A is coercive. Thus, by LemmaV 4.3,V ⊕V thereV exists z C such that 11 0 ∈ z0 = 1 and | | 2 Re z0 A11u,u α u h i ≥ k kV for all u 1. Since dim 2 < , there exists a finite rank operator K ( ∈) Vsuch that V ∞ 1 ∈L V A 0 A + K = 11 . 1 0 0
Choose a further finite rank perturbation K2 such that
A11 0 B := A + K1 + K2 = . 0 αz0 Id 2 V Since P is orthogonal, for Q = Id P , we get − Bu,u = A11Qu, Qu + αz0 Pu,Pu . h iV h iV h iV Hence 2 2 2 Re z0Bu,u α Qu + α Pu = α u . h iV ≥ k kV k kV k kV Now let = j (K1 + K2). Then + is coercive. (ii) (iiiK) is obvious.◦ A K (iii)⇒ (i): Condition (iii) implies clearly Condition (iv) of Theo- rem 4.2;⇒ thus the claim (i) follows from that theorem. Corollary 4.5. Let a be a continuous essentially coercive sesquilinear form. The following assertions are equivalent:
(i) for all L ′ there exists a unique u such that ∈V ∈V a(u, v)= L, v for all v ; h i ∈V (ii) a(u, v)=0 for all v implies that u =0 ( uniqueness); ∈V (iii) for all L ′ there exists u such that a(u, v) = L, v for all v ∈( existence V ). ∈ V h i ∈V Proof. The assertion (i) means that is invertible, the assertion (ii) means that is injective and the assertionA (iii) means that is sur- A A jective. By Theorem 4.4, there exists a compact operator ( , ′) such that + =: is invertible. K∈L V V (ii) (i):A AssumeK thatB is injective. Write ⇒ A 1 = = (Id − ). A B−K B −B K GALERKIN APPROXIMATION 17
1 1 Then also (Id − ) is injective. Since − is compact, it follows −B K B K 1 from the classical Fredholm alternative that (Id − ) is invertible. Consequently also is invertible. −B K A 1 (iii) (i): If is surjective, write = (Id − ) to conclude ⇒ 1A A −KB B 1 that (Id − ) is surjective. Again we deduce that (Id − ) is invertible−KB and so is . −KB A Moreover, we deduce from Theorem 4.4 the following properties of essential coercivity. Corollary 4.6. (a) The set of all essentially coercive operators on is open in ( , ′). V L V V (b) If ( , ′) is essentially coercive and ( , ′) is compact,A∈L thenV V + is essentially coercive. K ∈L V V A K (c) If ( , ′) is essentially coercive, then is a Fredholm operatorA∈L ofV indexV 0. A The following example shows that the invertibility of does not imply the essential coercivity of a. A
2 Example 4.7. Let = ℓ (N∗), K = R and V ∞ a(u, v)= ( 1)nu v . − n n Xn=0 Let j be the Riesz isomorphism introduced in the proof of Theorem 4.4. 1 Then A := j− is a diagonal operator with merely 1 and 1 in ◦A − the diagonal. Thus A and obviously are clearly invertible. Let fn = (0, , 1, 1, 0, ) where the 1 is aA coordinate for k = 2n and k = ··· ··· 2n +1. Then f = √2 and (f ) tends weakly to 0 as n . k nk n n → ∞ Moreover a(fn, fn)=0 for all n, which shows that a is not essentially coercive. Remark 4.8. Let K = C. In [1] a continuous sesquilinear form a is called compactly elliptic if there exists a compact operator J : , where is some Hilbert space and there exists α> 0 such thatV → H H Re a(u,u)+ Ju 2 α u 2 . k kH ≥ k kV In view of Theorem 4.2, each compactly elliptic form is essentially co- ercive. In fact the following holds: the form a is essentially coercive if and only if there exists λ C 0 such that λa is compactly elliptic. ∈ \{ } Proof. If λa is compactly elliptic, then λa is essentially coercive and hence also a is essentially coercive. Conversely, let a be essentially 18 W.ARENDT,I.CHALENDAR,ANDR.EYMARD coercive. By Theorem 4.4, there exists a compact operator : ′ such that the form b defined by K V→V b(u, v)= a(u, v)+ u, v hK i is coercive. By Lemma 4.3 there exist λ C of modulus one and α> 0 2 ∈ such that Re(λb(u,u)) α u for all u . Now let j : ′ ≥ k kV 1∈ V V →V be the Riesz isomorphism. Then J := j− : is compact. Choosing = we see that λb is compactly◦K elliptic.V →V It follows from [1, PropositionH V 4.4 (b)] that λa is compactly elliptic.
5. Characterization of the universal Galerkin property In this section we want to characterize those forms on a Hilbert space for which every Galerkin approximation converges, whatever be the choice of the approximating sequence. Let be a separable, infinite dimensional separable Hilbert space over KV= R or C, and let a : K be a continuous sesquilinear V×V→ form. Given L ′ we again consider solutions of the problem: ∈V (5.1) Find u , a(u, v)= L, v for all v . ∈V h i ∈V We say that the form a satisfies uniqueness if for u , ∈V a(u, v) = 0 for all v implies u =0. ∈V We say that (5.1) is well-posed if for all L ′ there exists a unique solution u . ∈ V ∈V Definition 5.1 (Universal Galerkin property). The sesquilinear and continuous form a has the universal Galerkin property if (5.1) is well- posed and the following holds. Let ( n)n N∗ be an arbitrary approxi- V ∈ mating sequence of . Then there exist n N∗ and γ > 0 such that V 0 ∈ for each L ′ and each n n , there exists a unique u solving ∈V ≥ 0 n ∈Vn a(u , χ)= L, χ for all χ , n h i ∈Vn and u un γ dist(u, n) for all n n0, k − kV ≤ V ≥ where u is the solution of (5.1). As recalled in the introduction and in the preceding section, the Lax-Milgram Theorem and C´ea’s Lemma imply the universal Galerkin property if a is coercive. We now show that the weaker notion of essential coercivity also provides a sufficient condition for ensuring the universal Galerkin property, and moreover that it is necessary. Theorem 5.2. The following assertions are equivalent. (i) The form a is essentially coercive and satisfies uniqueness. GALERKIN APPROXIMATION 19
(ii) The form a has the universal Galerkin property.
Proof. (i) (ii): let ( n)n N∗ be an approximating sequence in . By ⇒ V ∈ V Theorem 2.3 it suffices to show that there exist β > 0 and n0 N∗ such that ∈
(5.2) sup a(u, v) β u for all u n, n n0. v n, v V =1 | | ≥ k kV ∈V ≥ ∈V k k Assume that (5.2) is false. We then find a subsequence (nk)k N∗ and ∈ unk nk such that unk = 1 and ∈V k kV 1 N sup a(unk , v) < for all k ∗. v n , v V =1 | | k ∈ ∈V k k k
We may assume that (unk )k converges weakly to u taking a further subsequence otherwise. Let v . Then there exist v such ∈ V k ∈ Vnk that limk v vk = 0. Thus →∞ k − kV a(u, v) = lim a(unk , vk)=0. k →∞
It follows from the uniqueness assumption that u = 0. Thus (unk )k converges weakly to 0, limk a(unk ,unk ) = 0, but unk = 1 for all k. Therefore the form a is not→∞ essentially coercive. k kV (ii) (i): the uniqueness condition is part of (ii). It remains to ⇒ show that a is essentially coercive. Let (en)n N∗ be an orthonormal basis of and := Span e , , e . By our∈ assumption, there exist V Vn { 1 ··· n} 2 n N∗ and for all n n an operator Q : such that ≤ 0 ∈ ≥ 0 n V→Vn a(Q u, χ)= a(u, χ) for all χ (n n ). n ∈Vn ≥ 0 Denote by P : the orthogonal projection. Define the operator n V→Vn J : n V→V×V by J u =(P u, Q u), n n . n n n ≥ 0 Now assume that a is not essentially coercive. Then it follows from Theorem 4.2 that for all n n0 we find un such that un = 1 and ≥ ∈ V k kV 2 2 1 a(un,un) + Pnun + Qnun < . | | k kV k kV (n + 2)2 1 In particular Pnun < 2 . This implies that un n. Let ˜n = k kV (n+2) 6∈ V V ˜ Span n un . Then ( n)n n0 and ( n)n n0 are both approximating {V ∪{ }} V ≥ V ≥ sequences. Let n n0 and let v ˜n be arbitrary with unit norm. There exist a unique≥ w and λ∈ VK such that 1 ∈Vn ∈ v = w + λu = w + λ(u P u ), 1 n n − n n 20 W.ARENDT,I.CHALENDAR,ANDR.EYMARD where w := w + λP u . Thus 1 n n ∈Vn 2 2 2 2 1= v = w + λ un Pnun . k kV k kV | | k − kV 2 1 Consequently w 1 and, since Pnun < 2 , it follows that k kV ≤ k kV 1 un Pnun , k − kV ≥ 2 which implies that λ 2 4, i.e. λ 2. | | ≤ | | ≤ Observe that the definition of Qn implies that a(un Qnun,w) = 0. Hence − a(u , v) = a(u ,w)+ λa(u ,u P u ) | n | | n n n − n n | = a(u Q u ,w)+ a(Q u ,w)+ λa(u ,u P u ) | n − n n n n n n − n n | a(Q u ,w) +2 a(u ,u ) +2 a(u ,P u ) ≤ | n n | | n n | | n n n | M 2 2M + + . ≤ n +2 (n + 2)2 (n + 2)2 Consequently
lim sup a(un, v) =0. n →∞ v V˜n, v V =1 | | ∈ k k ˜ Thus (BNB) is violated for the approximating sequence ( n)n n0 . But then (ii) does not hold by Theorem 2.3, which shows thatV the≥ assump- tion that a is not essentially coercive is false.
It is obvious that a form a is essentially coercive if and only if its adjoint a∗ is essentially coercive. However, a surprising consequence of Theorem 5.2 is that, for an essentially coercive form, uniqueness for the form and uniqueness for its adjoint are equivalent, as the following corollary shows. Corollary 5.3. Let be a separable Hilbert space on K and a : K be a continuous essentiallyV coercive form. The following asV×Vsertions → are equivalent: (i) for all u , a(u, v)=0 for all v implies u =0; (ii) for all v ∈V, a(u, v)=0 for all u ∈V implies v =0; ∈V ∈V (iii) for all L ′ there exists u in such that a(u, v)= L, v , for all v ;∈V V h i ∈V (iv) for all L ′ there exists v in such that a(u, v)= L, u , for all u .∈V V h i ∈V Proof. (i) (ii): this follows from Theorem 5.2 and Theorem 2.3. The other⇐⇒ equivalences follow from Corollary 4.5. GALERKIN APPROXIMATION 21
6. The Aubin-Nitsche trick revisited In this section we want to prove that on suitable Hilbert spaces containing the space continuously the approximation speed in the Galerkin approximationV can be improved. We refer also to [18] for related, but different results in this direction. Let be a separable Hilbert space over K = R or C, and a : K a sesquilinearV form satisfying V×V → a(u, v) M u v . | | ≤ k kV k kV Let ( n)n N∗ be an approximating sequence of . We assume that (BNB)V holds;∈ i.e. there exists β > 0 such that V
(6.1) For all n N∗, sup a(u, v) β u for all u n. ∈ v n, v V =1 | | ≥ k kV ∈V ∈V k k Given L ′ and n N∗, let u be the solution of ∈V ∈ n ∈Vn (6.2) a(u , χ)= L, χ for all χ , n h i ∈Vn and u the solution of ∈V (6.3) a(u, v)= L, v for all v . h i ∈V Note that, by subtracting (6.3) and (6.2), we obtain the following Galerkin orthogonality: (6.4) a(u u , v) = 0 for all v . − n ∈Vn We know from Proposition 2.4 and Proposition 2.5 that M (6.5) u un dist(u, n) k − kV ≤ β V for all n N∗. We want to improve this estimate if the given data ∈ L ′ is in a suitable subspace of ′. ∈V V Let ֒ ′; i.e. is a Banach space such that ′ and X →V X X ⊂V f c f ′ k kX ≤ X k kV for all f and some c > 0. We define for n N∗ ∈ X X ∈
1 (6.6) γn( ) := sup dist( − f, n), X f , f X =1 A V ∈X k k where the distance is taken in . Thus V 1 (6.7) dist(w, n) γn( ) w for all w − , V ≤ X kA kX ∈A X where : ′ is the isomorphism given by A V→V u, v = a(u, v). hA i 22 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
Thus, if u is the solution of (6.3) and un the approximate solution of (6.2), then, if L , we have the estimate ∈ X M (6.8) u un γn( ) L , k − kV ≤ β X k kX which has the advantage of being uniform for L in the unit ball of . X Remark 6.1. Let Qn : , L un be the solution operator for (6.2). Then (6.8) says thatX →V 7→
1 M − Qn ( , ) γn( ). kA − kL X V ≤ β X We can characterize when γ ( ) 0 as n . n X → →∞ Proposition 6.2. One has
.lim γn( )=0 if and only if ֒ ′ is compact n →∞ X X →V Proof. Denote by Pn : n the orthogonal projection onto n. Then V →V V 1 1 1 1 γn( ) = sup − f Pn − f = − j Pn − j ( , ), X f , f X =1 kA − A kV kA ◦ − A ◦ kL X V ∈X k k where j : ′ is the canonical injection. If j is compact, then 1 X →V K := − j(B ), where B is the unit ball of , is relatively compact A ◦ X X X in . Now, Pn converges strongly to the identity of . Since Pn 1, thisV convergence is uniform on compact subsets of V. This showsk k ≤that X γn( ) 0 as n . X → →∞ 1 Conversely, if γn( ) 0, then − j is compact as limit of finite rank operators. ThenX also→ j is compact.A ◦ Similarly, we define 1 γn∗( ) := sup dist( ∗− f, n), X f , f X =1 A V ∈X k k where ∗ : ∗ is given by A V→V ∗u, v = a∗(u, v) := a(u, v). hA i As before we have γn∗( ) defined as γn( ) but with a replaced by the X X 1 adjoint form a∗ of a. Thus we have for all w ∗− , ∈A X (6.9) dist(w, ) γ∗( ) ∗w . Vn ≤ n X kA k Now we apply the Aubin–Nitsche trick in the following proof. In con- trast to the literature [8] we allow non-selfadjoint forms and also let L where ֒ ′ is arbitrary. However, as usual, we fix a Hilbert space∈ X such thatX →V֒ with dense range. Thus we have the Gelfand triple H V → H GALERKIN APPROXIMATION 23
.′ ֒ ֒ V → H →V Now we let ֒ ′ be another Banach space in which we choose the given data LX, whereas→ V our error estimate is done with respect to the norm of . H Theorem 6.3. Let L and let u be the solution of (6.3), un the solution of (6.2). Then∈ X M 2 (6.10) u un γn( )γn∗( ) L , k − kH ≤ β X H k kX for all n N∗. ∈ Proof. Let n N∗. Then, on the footsteps of Aubin–Nitsche, we con- sider the solution∈ w of ∈V (6.11) a∗(w, v)= u un, v (v ). h − iH ∈V Then, by (6.11), for any χ , ∈Vn 2 u un = u un,u un = a∗(w,u un)= a(u un,w) k − kH h − − iH − − = a(u un,w χ) M u un w χ − − ≤ k − kV k − kV where in the last identity we used the Galerkin orthogonality (6.4). Since χ is arbitrary, this implies that ∈Vn 2 u un M u un dist(w, n). k − kH ≤ k − kV V Now we use (6.8) and (6.9) to deduce
2 M u un M. γn( ) L γn∗( ) u un . k − kH ≤ β X k kX H k − kH Consequently, we obtain M 2 u un γn( )γn∗( ) L . k − kH ≤ β X H k kX
7. Applications 7.1. Selfadjoint positive operators with compact resolvent. As an illustration, we apply Theorem 6.3 to selfadjoint positive operators with compact resolvent. Let , be infinite dimensional, separable Hilbert spaces over K = R or VC Hsuch that is compactly injected in and dense in . Thus we have the GelfandV triple H H .′ ֒ ֒ V → H →V 24 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
Let a : K be continuous, symmetric and coercive. Then the V×V→ operator : ′ given by A V→V u, v = a(u, v) hA i is invertible. Moreover, there exist an orthonormal basis (en)n 0 of and λ R such that ≥ H n ∈ 0 <λ0 λ1 , lim λn = n ≤ ≤··· →∞ ∞ and ∞ 2 = u : λn u, en < V { ∈ H |h iH| ∞} Xn=0 (see e.g. [2, Satz 4.49]) and
∞ a(u, v)= λn u, en en, v . h iHh iH Xn=0 Passing to an equivalent scalar product we may and will assume that u, v = a(u, v) (u, v V ). h iV ∈
Thus a(u, v) u v and sup v V =1 a(u, v) = u ; i.e. we have M = β| = 1 in|≤k the abovekVk k estimates.V k k | | k kV Consider n = Span e0, , en 1 , n = 1, 2, . Then ( n)n N∗ is an approximatingV sequence{ ··· of . We− } define for ···s [ 1, 1] V ∈ V ∈ − s 2 s := f ′ : λn f, en < , V { ∈V |h iH| ∞} Xn 0 ≥ which is a Hilbert space for the norm
2 s 2 f s = λn f, en . k kV |h iH| Xn 0 ≥ Then it is easy to see that 1 = ′, 0 = , 1 = with identity of the norms. Morever, for s V−(0, 1),V V H V V ∈ =( , ) Vs V0 V1 s (the complex interpolation space) and for s ( 1, 0), ∈ − s =( 0, 1) s. V V V− − Lemma 7.1. One has for s [ 1, 1], ∈ − (1+s)/2 γ ( )= λ − (n =1, 2, ). n Vs | n| ··· In particular, 1/2 γ ( )= λ − . n H | n| GALERKIN APPROXIMATION 25
1 Proof. Let en = en. Then (en)n 0 is an orthonormal basis of . For √λn ≥ V u , ∈V b b u, ek = λn u, en en, ek = λk u, ek h iV h iHh iH h iH Xn 0 p ≥ Thus b b n 1 n 1 − − Pnu = u, ek ek = u, ek ek h iV h iH Xk=0 Xk=0 defines the orthogonal projectionb ofb onto . Moreover, in one has V Vn V 2 2 2 dist(u, n) = u Pnu = λk u, ek . V k − kV |h iH| Xk n ≥ 1 Let f , u = − f. Then ∈Vs A f, ek = u, ek = λk u, ek . h iH hA iH h iH Thus 2 2 2 u Pnu k n λk u, ek γ ( ) = sup k − kV = sup ≥ |h iH| n s 2 P s 2 V u=f f s f s, u=f k 0 λk f, ek A k kV ∈V A ≥ |h iH| 1 2 P 1 s s 2 k n λk− f, ek k n λ−k − λk f, ek ≥ |h iH| ≥ |h iH| = sup s 2 = sup s 2 f s P k 0 λk f, ek f s P k 0 λk f, ek ∈V ≥ |h iH| ∈V ≥ |h iH| 1 s λ− − P P ≤ n since (λk)k 0 is increasing. ≥ −1 2 λn s 1 Taking f = en, one sees that γn( s) s = λ− − . V ≥ λn n 1 Now let f s, where 1 s 1 and let u = − f. Let un such that ∈ V − ≤ ≤ A ∈ V a(u , χ)= f, χ (χ ), n h i ∈Vn i.e. un is the approximate solution. Then by Theorem 6.3 (1+s)/2 1/2 u un γn( )γn( ) f s = λn − λn − f s . k − kH ≤ X H k kV | | | | k kV Thus we obtain the following error estimate
1 s/2 (7.1) u un λ − − f s . k − kH ≤| | k kV
Remark 7.2. In this special case one can compute the error directly. 1 n 1 In fact u = ∞ f, ek ek and un = − f, ek ek. Thus k=0 λk h iH k=0h iH P P ∞ ∞ 2 1 2 2 s s 2 2 s 2 u un = 2 f, ek = λ−k − λk f, ek λ−n − f s , k − kH λ |h iH| |h iH| ≤ k kV Xk=n k Xk=n 26 W.ARENDT,I.CHALENDAR,ANDR.EYMARD which is exactly the estimate (7.1). This means that Theorem 6.3 gives the best possible estimate of the error. Let us provide an example of application of (7.1). Let K = C, 2 2 1 2π 2 = L (0, 2π) with norm u = 2π 0 u(t) dt. Let = u H k kH | | V { ∈ H1(0, 2π): u(0) = u(2π) with norm R } 2π 2π 2 1 2 1 2 u = u′(t) dt + u(t) dt. k kV 2π Z0 | | 2π Z0 | | Then the injection ֒ is compact. Let a : C be given by V → H V×V → 1 2π 1 2π a(u, v)= u′(t)v′(t)dt + u(t)v(t)dt. 2π Z0 2π Z0 Let f L2(0, 2π). Then there exists a unique u H2(0, 2π) such that ∈ ∈ u u′′ = f, u(0) = u(2π), u′(0) = u′(2π). − In fact, u is the unique element of such that a(u, v) = f, v for all v . V h i ∈V 1 2π ikt For u , let u(k) = u(t)e− dt be the k-th Fourier coeffi- ∈ H 2π 0 cient. Then R b a(u, v)= (1 + k2)u(k)v(k). Xk Z ∈ ikt b b Let ek(t)= e , t (0, 2π). Then (ek)k Z is an orthonormal basis of ∈ ∈ H and u(k) = u, ek . Let n = Span ek : k < n and let un be the approximateh solutioniH i.e. V { | | } b a(un, χ)= f, χ (χ n). h iH ∈V Then our estimate shows that 1 u u 2 f 2 . k n − kL ≤ (1 + n2)1/2 k kL
2 2 s 2 Let 0
1, for all x Ω, where α > 0. Moreover, let b ,c W ∞(Ω) for j = ∈ j j ∈ 1, ,d and b L∞(Ω). We consider the operator A given by ··· 0 ∈ d d d Au := D (a D u)+ b D u D (c u)+b u (u H2(Ω)). − i ij j j j − j j 0 ∈ i,jX=1 Xj=1 Xj=1 Note that A : H2(Ω) L2(Ω) is linear and continuous. Our aim is to study→ the Poisson equation (7.3) Au = f
2 1 2 where f L (Ω) is given and a solution u H0 (Ω) H (Ω) is to be determined∈ and calculated by approximation.∈ We will∩ impose the uniqueness condition (7.4) For all u H1(Ω) H2(Ω), Au = 0 implies u =0. ∈ 0 ∩ We use the continuous, coercive form a : H1(Ω) H1(Ω) R 0 0 × 0 → given by d
a0(u, v)= aijDjuDiv Z i,jX=1 Ω and also the perturbed form a given by
d
a(u, v)= a0(u, v)+ (bjDjuv + cjuDjv)+ b0uv. Z Z Xj=1 Ω Ω
Note that the adjoint form a∗ defined by a∗(u, v)= a(v,u) has the same form as a. This is the reason why we also consider the coefficients cj. Then the following well posedness result holds. Theorem 7.3. i) The form a is essentially coercive. ii) Assume (7.4). Then for each f L2(Ω) there exists a unique solution u H1(Ω) H2(Ω) of (7.3).∈ ∈ 0 ∩ 28 W.ARENDT,I.CHALENDAR,ANDR.EYMARD
2 1 2 Proof. a) We first show H -regularity. Let u H0 (Ω), f L (Ω) such that a(u, v)= fv for all v H1(Ω). Then∈u H2(Ω)∈ and Au = f. Ω ∈ 0 ∈ In fact, let R d g := f b u (b D u D (c u)). − 0 − j j − j j Xj=1 Then g L2(Ω) and a (u, v)= gv for all v H1(Ω). Now it follows ∈ 0 Ω ∈ 0 from the classical H2-result ofR Kadlec [11] (see [9, Theorem 3.2.1.2]) that u H2(Ω). It clearly follows that Au = f. ∈ b) We show that a is essentially coercive. Let un ⇀ 0 as n in 1 → ∞ H0 (Ω) and a(un,un) 0 as n . Then Djun ⇀ 0 as n in 2 → →1 ∞ 2 → ∞ L (Ω). Since the embedding of H0 (Ω) in L (Ω) is compact, it follows that u 0 in L2(Ω). Consequently n →
bjDjun.un 0, cjDjun.un 0 and b0un.un 0 as n . ZΩ → ZΩ → ZΩ → →∞ Thus also a (u ,u ) 0 as n . Since a is coercive this implies 0 n n → → ∞ 0 un H1 0 as n . k k → →∞ 1 c) The form a satisfies uniqueness. In fact, let u H0 (Ω) such that 1 2 ∈ a(u, v) = 0 for all v H0 (Ω). Then u H (Ω) by part a) of the proof. Hence u = 0 by our∈ assumption (7.4).∈ d) Let f L2(Ω). It follows from Corollary 4.5 that there exists a ∈ 1 2 1 unique u H0 (Ω) such that a(u, v)= f, v L for all v H0 (Ω). Now a) implies∈ that u H2(Ω) and Au = fh. i ∈ ∈
Concerning the uniqueness property, we make the following remark. Remark 7.4 (Eigenvalues and uniqueness). Replace the operator A by Aλ := A λ Id (i.e. b0 by b0 λ) where λ R. Then there exists a finite or countable− infinite set− such that ∈
λ : (7.4) is violated for A = λ : n N∗,n
(7.7) a(uh, χ)= fχ for all χ Vh. ZΩ ∈
The finite elements (uh)0 (7.8) u u 1 c h f 2 k − hkH (Ω) ≤ 1 k kL (Ω) and 2 (7.9) u u 2 c h f 2 k − hkL (Ω) ≤ 2 k kL (Ω) where u is the solution of (7.3). Proof. Applying the closed graph theorem in the situation of Theo- rem 7.3, we find a constant c3 > 0 such that (7.10) u 2 c f 2 k kH (Ω) ≤ 3k kL (Ω) 30 W.ARENDT,I.CHALENDAR,ANDR.EYMARD whenever f L2(Ω) and u solves (7.3). ∈ By Theorem 5.2, there exist γ > 0, 0 < h1 h0, both independent of f, such that ≤ u uh H1(Ω) γ inf χ u H1(Ω) χ Vh k − k ≤ ∈ k − k for all 0 < h h . Thus (7.5) implies that for 0 < h h , ≤ 1 ≤ 1 u u 1 chγ u 2 . k h − kH (Ω) ≤ | |H (Ω) Now (7.8) follows from (7.10). Next we establish the L2-estimate (7.9). For that we compute using (7.5), dist(w, h) ch w H2(Ω) γh( ) = sup V sup | | . H w H1(Ω) H2(Ω) Aw L2(Ω) ≤ w H1(Ω) H2(Ω) Aw L2(Ω) ∈ 0 ∩ k k ∈ 0 ∩ k k Since w H2(Ω) w H2(Ω), it follows from (7.10) that γh( ) cc3h for all h>| 0.| ≤k k H ≤ The same estimate is true for γh∗( ). Now assume that (7.9) is false. Then there exists a sequence hH 0 as n such that (7.9) n ↓ → ∞ does not hold for all h = hn and any constant c2. This contradicts Theorem 6.3. Remark 7.7. There are other methods to approximate the solution of a non-coercive advection-diffusion equation as (7.3). In fact, Le Bris, Legoll and Madiot [13] use the Banach-Neˇcas-Babuska lemma (instead of essential coercivity as we do) and a special measure to construct an approximation. The advantage is that no initial mesh h1 has to be considered; on the other hand there seems to be no such precise error estimate as our quadratic convergence obtained in Theorem 7.6 even though numerical examples are given in [13]. Still, another approach (based on Fredholm perturbation) is presented by Christensen [5], which also involves the Babuska inf-sup condition. 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Wolfgang Arendt, Institute of Applied Analysis, University of Ulm. Helmholtzstr. 18, D-89069 Ulm (Germany) E-mail address: [email protected] Isabelle Chalendar, Universite´ Paris-Est, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Valle (France) E-mail address: [email protected] Robert Eymard, Universite´ Paris-Est, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Valle (France) E-mail address: [email protected]