Ana Alonso 1 and Anahí Dello Russo 2

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1 Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher SPECTRAL APPROXIMATION OF VARIATIONALLY FORMULATED EIGENVALUE PROBLEMS ON CURVED DOMAINS Ana Alonso 1 and Anahí Dello Russo 2 Abstract. This paper deals with the spectral approximation of variationally formulated eigenvalue problems posed on curved domains. As an example of the present theory, convergence and optimal error estimates are proved for the piecewise linear finite element approximation of the eigenvalues and eigenfunctions of a scalar second order elliptic differential operator on a general curved threedimensional domain. Résumé. Dans cet article nous traitons l approximation spectrale de problèmes de valeurs propres formulés sous la forme variationnelle et posés dans des domaines au bord curbe. Á titre d exemple de cette théorie, on démontre la convergence pour les approximations par des éléments finis linéaires des valeurs propes et des fonctions propres d un opérateur différentiel scalaire elliptique de second ordre posé dans un domaine général tridimensionnel curbe par morceaux. On obtient de même des estimations optimales de l erreur pour ces approximations Mathematics Subject Classification. 65N15, 65N25, 65N Introduction In this paper we present an extension of the spectral approximation theory for non compact operators in Hilbert spaces. In particular, we deal with the numerical approximation of the eigenvalues and eigenvectors of variationally formulated problems posed over general curved domains. There are not many references about error estimates for this kind of problems. In particular, the finite element approximations of the spectrum of the Laplace operator on non convex domains with curved boundaries have been studied only in a few papers. The first proof of the convergence for Laplace eigenproblem, for simple eigenvalues and Dirichlet boundary conditions, was given by Vanmaele and Ženíšek [15] by using the min-max characterization (see [14]. The same authors generalized their previous results to include multiple eigenvalues [16] and numerical integration effects [17]. Keywords and phrases: Spectral approximation; eigenvalue problems; curved domains 1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172, 1900 La Plata, Argentina. 2 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172, 1900 La Plata, Argentina. Member of CIC, Provincia de Buenos Aires, Argentina. c EDP Sciences, SMAI 1999

2 2 TITLE WILL BE SET BY THE PUBLISHER Almost at the same time, Lebaud [11] analyzed a similar problem in the framework of the classical spectral approximation theory (see [1]. To do that, she construct a good approximation of the boundary by using isoparametric finite element method of Lagrangian type. She also considered simple eigenvalues and Dirichlet boundary conditions but assuming exact integration. More recently, Hernández and Rodríguez [8] considered the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on curved non convex domains. By using the abstract spectral approximation theory, they proved optimal order error estimates for the eigenfunctions and a double order for eigenvalues. Later, the same authors [9] proved convergence results and error estimates for the Raviart-Thomas approximations of the spectral acoustic problem on a general curved two-dimensional domain. The goal of this paper is to prove some abstract results on spectral approximation that can be applied to a variety of eigenvalue problems defined over curved domains. Moreover, our analysis is suitable for studying numerical approximations for eigenvalue problems on bounded three-dimensional domains with piecewise curved boundaries. Let us remark that none of the theories mentioned above seems to admit a direct extension to cover this situation. The remainder of the paper is organized as follows. Section 2 is devoted to notations. In Section 3 we give a precise statement of the eigenvalue problems and approximation methods we will consider. In Section 4 we prove the abstract results. These results are obtained by introducing suitable modifications in the theory developed by Descloux, Nassif and Rappaz (see [4, 5]. Finally in Section 5, as an application of our results, we establish error estimates for the piecewise linear finite element approximation of the eigenvalues and eigenfunctions of a symmetric uniformly elliptic second order differential operator on a general curved three-dimensional domain with boundary conditions of the Dirichlet type. To the best of the authors knowledge, these estimations have not been proved yet. 2. Notation Throughout this paper Ω denotes a bounded open domain in R n, n = 2 or 3, in general non convex, with a Lipschitz continuous boundary Ω. Let W (R n be a complex Hilbert function space with norm R n. Given an open set O R n, let W (O denote a generic complex Hilbert space of functions defined in O and O its norm. First, we define the restriction operator Š by Š : W (R n W (O f f O. (2.1 We restrict our attention to Hilbert spaces such that the norm R n satisfies 2 R n = 2 O + 2 R n \O. Then, as an immediate consequence of this assumption, we get that Š is a bounded operator. We will need to provide extensions for functions on O to R n. So, if u W (O, we extend it by zero from its original domain to R n and we denote this extended function by ū. Now, let W 0 (O be the space of all the function in W (O defined in such a way that the extension operator Ŝ, given by Ŝ : W 0 (O W (R n u ū, (2.2

3 TITLE WILL BE SET BY THE PUBLISHER 3 is well defined and bounded. Finally, we can define the function space W (R n := Ŝ(W 0(O endowed with the norm R n. In what follows, to simplify notation, we will write R n =. 3. Statement of the eigenvalue problem Let X(Ω be a complex Hilbert function space with norm Ω. Let V (Ω be a closed subspace of X(Ω, with norm Ω, such that the inclusion V (Ω X(Ω is continuous. We denote by V 0 (Ω the subspace of V (Ω defined as in (2.2. Consider the eigenvalue problem: Find µ C, u 0, u V 0 (Ω, such that a(u, v = µb(u, v, v V 0 (Ω (3.1 where a : V (Ω V (Ω C is a continuous and coercive sesquilinear form and b : X(Ω X(Ω C is a continuous sesquilinear form. Let T be the linear operator defined by where u V 0 (Ω is the solution of T : X(Ω V 0 (Ω X(Ω x u, a(u, y = b(x, y, y V 0 (Ω. (3.2 Since a is elliptic, b is continuous and V (Ω X(Ω, Lax-Milgram s Lemma allows us to conclude that T is a bounded linear operator. It is simple to show that µ is an eigenvalue of (3.1 if and only if λ = 1/µ is an eigenvalue of the operator T and the corresponding associated eigenfunctions u coincide. Now, we define the linear operator A by A : X(R n Ṽ (Rn x ū = ŜTŠx. It is clear that ū Ω = u, where u V 0 (Ω is the solution of problem (3.2. The curved domain Ω is approximated by a family of domains Ω h, h > 0, with polygonal boundary Ω h. Let T h be a standard partition of Ω h into n-simplices such that each vertex of Ω h is also an element of Ω. The index h denotes, as usual, the mesh size of T h. We assume that the family {T h } is regular in the sense of the minimal angle condition, i.e., there is a constant C independent of the choice of T h such that vol(t Cdiam n (T for all T T h, where vol(t denotes the n-dimensional volume of T (see [2], for instance. Let V h (Ω h be a finite dimensional space on Ω h such that V h (Ω h V (Ω h, for all h. We denote by V 0h (Ω h the space of all the functions in V h (Ω h defined as in (2.2. Then, we consider the following discretization of eigenvalue problem (3.1: Find µ h C, u h 0, u h V 0h (Ω h, such that a h (u h, v = µ h b h (u h, v, v V 0h (Ω h. (3.3 In what follows we shall assume that the approximate sesquilinear forms a h and b h are continuous on V (Ω h uniformly on h and that a h is coercive on V (Ω h uniformly on h. We remark that, since V 0h (Ω h / V 0 (Ω, (3.3 represents a nonconforming approximation to (3.1.

4 4 TITLE WILL BE SET BY THE PUBLISHER Let us now define the function space Ṽh(R n := Ŝ(V 0h(Ω h. Then, the discrete analogue of the operator A can be define as follows: A h : X(R n Ṽh(R n x ū h : ū h Ωh = u h, where u h V 0h (Ω h is the solution of a h (u h, y = b h (x, y, y V 0h (Ω h. (3.4 Once again, because of Lax-Milgram s Lemma, the operator A h is bounded uniformly on h. As in the continuous case, it is simple to show that µ h is an eigenvalue of problem (3.3 if and only if λ h = 1/µ h is an eigenvalue of the operator A h, and the corresponding associated eigenfunctions are related by u h = ū h Ωh. We end this section by making other assumptions over the sesquilinear forms a and a h. We assume that the form a(x, y can be expressed as a(x, y = a 1 (x Ω Ωh, y Ω Ωh + a 2 (x Ω\Ωh, y Ω\Ωh, (3.5 where a 1 and a 2 are continuous bilinear forms on V (Ω Ω h and V (Ω \ Ω h, respectively. We also assume that a h (x h, y h = a 1h (x h Ω Ωh, y h Ω Ωh + a 2h (x h Ωh \Ω, y h Ωh \Ω. (3.6 Finally, if x, y V (R n, we assume that holds. a 1 (x Ω Ωh, y Ω Ωh = a 1h (x Ω Ωh, y Ω Ωh ( Spectral approximation In this section we present several abstract results on the approximation of eigenvalues and eigenvectors of non compact operators defined over curved domains. These results are obtained by suitable modifications of the theory presented in [4] and [5]. As a consequence of these modifications, consistency terms arise in the error estimates. First, we introduce some notation that will be used in the sequel. For further information on eigenvalue problem we refer the reader to [1]. We denote by ρ(a the resolvent set of A and by σ(a the spectrum of A. For any z ρ(a, R z (A = (z A 1 defines the resolvent operator. Let λ be a nonzero isolated eigenvalue of A with algebraic multiplicities m. Let Γ be a circle in the complex plane centered at λ which lies in ρ(a and which encloses no other points of σ(a. The continuous spectral projector, E : V (R n Ṽ (Rn, relative to λ is defined by E = 1 R z (A dz. 2πi Γ We assume the following properties to be satisfied: P1: lim h 0 (A A h Ṽh (R n = 0. P2: For each function x of E(V (R n, lim h 0 x Ω\Ω h = 0.

5 TITLE WILL BE SET BY THE PUBLISHER 5 P3: For each function x of E(V (R n, P4: ( lim h 0 inf x h Ṽh(R n x x h = 0. lim (A A h E(V (R h 0 n = 0. We are going to give an extension of the theory developed in [4] to deal with curved domains. Most of the proof of the results stated below are slight modifications of those in [4]. From now and on, C denotes a constant not necessary the same at each occurrence but always independent on h. Lemma 4.1. Let G be a closed subset of ρ(a. Under assumption P1, there exist positive constants C and h 0, independent of h, such that (z A h Ṽh (R n 1 C, z G, h < h 0. Proof. The proof is identical to that of Lemma 1 in [4]. Theorem 4.2. Let O C be a compact set not intersecting σ(a. There exist h 0 > 0 such that, if h < h 0, then O does not intersect σ(a h Ṽh (R n. Proof. The proof is a direct consequence of assumption P1, as it is shown in Theorem 1 in [4]. So, in virtue of the previous theorem, if h is small enough, Γ ρ(a h Ṽh (R n and the discrete spectral projector, E h : V (R n Ṽh(R n, can be defined by E h = 1 R z (A h Ṽh 2πi (R n dz. Γ Let us recall the definition of the gap δ between two closed subspaces, Y and Z, of V (R n. We define where δ(y, Z := max{δ(y, Z, δ(z, Y }, δ(y, Z := y Y ( inf y z. z Z The following theorem implies uniform convergence of E h to E as h goes to 0. Theorem 4.3. Under assumption P1, lim (E E h Ṽh h 0 (R n = 0. Proof. It follows combining Lemma 4.1 with assumption P1 and is essentially identical to that of Lemma 2 in [4]. Theorem 4.4. Under the assumption P1, for all x E h (V (R n there holds lim δ(x, E(V h 0 (Rn = 0.

6 6 TITLE WILL BE SET BY THE PUBLISHER Proof. It is a direct consequence of Theorem 4.3. Theorem 4.5. Under the assumptions P1 and P3, for all x E(V (R n holds lim δ(x, E h(v (R n = 0. h 0 Proof. The proof is identical to that of Theorem 3 in [4]. Theorem 4.6. Under the assumptions P1 and P3, lim δ(e(v (R n, E h (V (R n = 0. h 0 Proof. It is direct consequence of Theorem 4.4 and Theorem 4.5. As a consequence of the previous theorems, isolated parts of the spectrum of A are approximated by isolated parts of the spectrum of A h (see [10] and [4]. More precisely, for any eigenvalue λ of A of finite multiplicity m, there exist exactly m eigenvalues λ 1h,, λ mh of A h, repeated according to their respective multiplicities, converge to λ as h goes to zero. Next we are going to give estimates which show how the eigenvalues of T are approximated by those of T h. To attain this goal, we extend the theory developed in [5] so that it can be applied to more general situations where the original and the discrete domains do not coincide. By so doing, consistency terms arise in the error estimates. These consistency terms are associated with the variational crime committed by approximating the curved boundary with a polyhedral one. We shall give general expressions for these additional consistency terms. We begin considering the bounded operator A defined by A : X(R n Ṽ (Rn x ū : ū Ω = u, where u V 0 (Ω is the solution of a(y, u = b(y, x, y V 0 (Ω. (4.1 It is known that λ is an eigenvalue of A with the same multiplicity m as that of λ. We also consider the bounded operator A h defined by where u h V 0h (Ω h is the solution of A h : X(R n Ṽh(R n x ū h : ū h Ωh = u h, a h (y, u h = b h (y, u h, y V 0h (Ω h. (4.2 Here, λ 1h,, λ mh are the eigenvalues of A h which converge to λ as h goes to zero. Let E be the spectral projector of A relative to λ. We also assume the following properties for A and A h : P5: lim h 0 (A A h Ṽh (R n = 0. P6: For each function x of E (V (R n, lim h 0 x Ω\Ω h = 0.

7 TITLE WILL BE SET BY THE PUBLISHER 7 P7: For each function x of E (V (R n, P8: ( lim h 0 We now need to introduce other operators. inf x h Ṽh(R n x x h = 0. lim (A A h E(V (R h 0 n = 0. Let Π h : V (R n V (R n be the projector defined by the relations a h (x Π h x, y = 0, y V 0h (Ω h (Π h x R n \Ω h = 0. (4.3 Because V 0h (Ω h is a closed subset of V (Ω h, (Π h x Ωh V 0h (Ω h. Hence, we have Π h x Ṽh(R n. Analogously, we define the projector Π h : V (R n V (R n with range Ṽh(R n by the relations: a h (y, x Π h x = 0, y V 0h (Ω h (Π h x R n \Ω h = 0. (4.4 Since a h is continuous and coercive on V (Ω h, both uniformly on h, the operators Π h and Π h are bounded uniformly on h. Let B h := A h Π h : V (R n V (R n. Notice that σ(a h = σ(b h and that, for any non null eigenvalue, the corresponding invariant subspaces coincide. Let F h : V (R n V (R n be the spectral projector of B h relative to its eigenvalues λ 1h,, λ mh. It can be proved that R z (B h is bounded uniformly on h for z Γ (see Lemma 1 in [5]. Consequently, the spectral projectors F h are bounded uniformly on h. Finally, let B h := A h Π h : V (R n V (R n and let F h be the spectral projector of B h relative to λ 1h,, λ mh. It is easy to show that B h is the actual adjoint of B h with respect to a h. In fact, for all x and y V (R n, we have a h (B h x, y = a h (A h Π h x, y = a h (A h Π h x, Π h y = b h (Π h x, Π h y. Similarly, we get a h (x, B h y = b h (Π h x, Π h y. Therefore, the spectral projector F h is also the adjoint of F h with respect to a h. Let γ h := δ(e(v (R n, Ṽh(R n + y E(V (R n Properties P2 and P3 imply that γ h 0 as h 0. Analogously, let y Ω\Ωh. γ h := δ(e (V (R n, Ṽh(R n + y E (V (R n y Ω\Ωh. Here, because P6 and P7, γ h 0 as h 0. Lemma 4.7. (I Π h E(V (R n Cγ h, (I Π h E (V (R n Cγ h.

8 8 TITLE WILL BE SET BY THE PUBLISHER Proof. For a x E(V (R n, we have Using that a h is coercive on V (Ω h uniformly on h we have (I Π h x 2 = (I Π h x 2 Ω h + x 2 Ω\Ω h. (4.5 (I Π h x 2 Ω h Ca h ((I Π h x, (I Π h x = Ca h ((I Π h x, x y h, y h V 0h (Ω h, where the last equality yields from the definition of Π h. Now, taking into account that a h is continuous on V (Ω h uniformly on h, we obtain (I Π h x Ωh C inf x y h, y h Ṽh(R n which together (4.5 allows us to conclude the proof of the first estimation. Analogous proof is valid for the second one. Lemma 4.8. (E F h E(V (R n C (A B h E(V (R n, (E F h E (V (R n C (A B h E (V (R n. Proof. The proof is identical to that of Lemma 3 in [5]. Let δ h := γ h + (A A h E(V (R n. From properties P2, P3 and P4 it is easily seen that δ h 0 as h 0. Analogously, let Since P6, P7 and P8 δ h 0 as h 0. Lemma 4.9. Proof. Let x E(V (R n with x = 1. We have δ h := γ h + (A A h E(V (R n. (A B h E(V (R n Cδ h, (A B h E(V (R n Cδ h. (A B h x (A A h x + A h (I Π h x (A A h E(V (R n + A h (I Π h E(V (R n ( (A A h E(V (R n + γ h, where the last inequality follows from Lemma 4.7 and the fact that A h is uniformly bounded with respect to h. Analogous proof is valid for the second estimate of the Lemma. Let Λ h := F h E(V (R n : E(V (R n F h (V (R n. Lemma For h small enough, Λ h is a bijection and Λ 1 h is bounded uniformly on h. Proof. See the proof of Theorem 1 in [5]. Theorem δ(f h (V (R n, E(V (R n Cδ h.

9 TITLE WILL BE SET BY THE PUBLISHER 9 Proof. The proof is identical to that of Theorem 1 in [5]. Let us now define the operators  := A E(V (R n : E(V (R n E(V (R n and ˆB h := Λ 1 h B hλ h : E(V (R n E(V (R n. From these definitions, it follows that  has a unique eigenvalue λ of algebraic multiplicity m and that ˆB h has the eigenvalues λ 1h,, λ mh. Let us consider the following consistency terms: M h = x E(V (R n x = 1 y E (V (R n a h (Ax, Π h y y b h (x, Π h y y, M h = x E(V (R n x = 1 y E (V (R n a h (Π h x x, A y b h (Π h x x, y, N h = x E(V (R n x = 1 y E (V (R n a h (Ax, y b h (x, y. Theorem  ˆB h C (δ h δ h + M h + M h + N h. Proof. We have  ˆB h = x E(V (R n x = 1 C x E(V (R n x = 1 C x E(V (R n x = 1 ( ˆB h x = x E(V (R n x = 1 ( ˆB h x Ω a((â ˆB h x, y y Ṽ (Rn y E (V (R n a((â ˆB h x, y. (4.6 Since ( ˆB h x, y Ṽ (Rn, we can use (3.5 and (3.7 to get a((â ˆB h x, y = a 1h (( ˆB h x Ω Ωh, y Ω Ωh + a 2 (( ˆB h x Ω\Ωh, y Ω\Ωh = a h (( ˆB h x, y + a 2 (( ˆB h x Ω\Ωh, y Ω\Ωh. (4.7 Now, using that (Λ 1 h F h IA E(V (R n = 0 and that B h commutes with its spectral projector F h, we obtain  ˆB h = (A B h E(V (R n + (Λ 1 h F h I(A B h E(V (R n. (4.8 Let x E(V (R n and y E (V (R n, with x =. Since F h (Λ 1 h F h I = 0 and F h is the adjoint of F h with respect to a h, we have a h ((Λ 1 h F h I(A B h x, y = a h ((Λ 1 h F h I(A B h x, (I F h y C Λ 1 h F h I (A B h E(V (R n (I F h E (V (R n Cδ h δ h. (4.9 The last inequality in (4.9 follows from Lemmas 4.8, 4.9 and 4.10, the fact that a h is continuous on V (Ω h independently of h and that F h is bounded uniformly on h.

10 10 TITLE WILL BE SET BY THE PUBLISHER On the other hand, a h ((A B h x, y = a h ((A B h x, Π h y + a h ((A B h x, (I Π h y. (4.10 To bound the second term in the right hand side of this equation, we use Lemmas 4.7 and 4.9. We thus obtain a h ((A B h x, (I Π h y C (A B h E(V (R n (I Π h E (V (R n C δ h γ h. (4.11 For the first term, we have a h ((A B h x, Π h y = a h ((A A h x, Π h y + a h ((A h B h x, Π h y. (4.12 Now, a h ((A A h x, Π h y = a h (Ax, Π h y b h (x, Π h y M h + N h, (4.13 and a h ((A h B h x, Π h y = a h (A h (I Π h x, Π h y = b h ((I Π h x, Π h y (4.14 = b h ((I Π h x, y b h ((I Π h x, (I Π h y. The first term in the right hand side of (4.14 can be written as b h ((I Π h x, y = [a h ((Π h Ix, A y b h ((Π h Ix, y] a h ((Π h Ix, (I Π h A y. (4.15 Now, the last term of the right hand side above can be easily bounded by a h ((Π h Ix, (I Π h A y C (Π h I E(V (R n (I Π h E (V (R n A. (4.16 Then, Lemma 4.7, (4.15 and (4.16 immediately yield b h ((I Π h x, y C(M h + γ h γ h. (4.17 Finally, we estimate the last term in (4.7. Using that (Λ 1 h F h I is bounded uniformly on h, we obtain from (4.8 and Lemma 4.9 a 2 ((Â ˆB h x Ω\Ωh, y Ω\Ωh C (A B h E(V (R n y Ω\Ωh Cδ h y E (V (R n y Ω\Ωh Cδ h γ h. (4.18 Now, the theorem is a consequence of formulae (4.6 to (4.18. By using the previous theorem, we deduce the following result about the approximation of the eigenvalue λ: Theorem i ii λ 1 m m i=1 where α is the ascent of the eigenvalue λ of Â. λ ih C ( δ h δ h + M h + M h + N h 1/α max λ λ ih C (δ h δ h + M h + M h + N h i=1,,m

11 TITLE WILL BE SET BY THE PUBLISHER 11 Proof. Taking into account that σ(â = λ and that λ 1h,, λ mh are the eigenvalues of ˆB h, we have tr(â = mλ and tr( ˆB h = m i=1 λ ih. Then, from the continuity of the traces λ 1 m On the other hand, it is know that, m i=1 λ ih = 1 m tr(â tr( ˆB h C Â ˆB h. λ λ ih α C Â ˆB h, for any 1 i m. Therefore, we can conclude i and ii directly from Theorem Remark In many applications, the operator A is selfadjoint. In this case, if µ is a nonzero eigenvalue of A, the ascent α of (µ A is one. So, the space of generalized eigenvectors E(R n coincide with the space of the actual eigenvectors corresponding to µ (see [1]. 5. Example Let Ω be a bounded three-dimensional domain with a Lipschitz continuous boundary Ω. We assume that Ω is piecewise smooth, more precisely, is piecewise of class C 2. To avoid additional technical difficulties, we will assume that the set of points where the condition of C 2 - smoothness of Ω is not satisfied consists of a finite number of straight lines and single points. Let (, be the scalar product in L 2 (Ω and let denote the corresponding L 2 norm. Further, H s (Ω denotes the standard Sobolev spaces with the usual norms s and H 1 0 (Ω denotes the subspace of functions in H 1 (Ω satisfying Dirichlet boundary conditions. We consider the spectral problem: Find λ R and u 0 such that { Lu = λu in Ω, u = 0 on Ω, where the operator L is defined on C 2 ( Ω by (5.1 Lu = ( k ij (x u. x j x i We assume that the coefficients functions k ij are defined on Ω (the domain Ω Ω and will be specified below and have the following properties: k ij C 1 ( Ω i, j = 1, 3 (5.2 with k 0 a positive constant. k i,j (xξ i ξ j k 0 k ij (x = k ji (x x Ω (5.3 Ω ξi 2, x Ω, ξ R 3, (5.4 i=1 Let X(Ω := L 2 (Ω, V (Ω := H 1 (Ω and V 0 (Ω := H0 1 (Ω. Let a 0 and b be the symmetric bilinear forms defined by a 0 (u, v := k ij (x u v dx, u, v V (Ω, x i x j

12 12 TITLE WILL BE SET BY THE PUBLISHER b(u, v := Ω uv dx, u, v X(Ω. The bilinear form a 0 is coercive on V 0 (Ω but is not coercive on V (Ω. However, a := a 0 + b can be used in our problem and it turns out to be coercive on V (Ω. Furthermore, since k i,j are bounded in Ω, a is continuous on V (Ω. The variational formulation of problem (5.1 associated to a is given by: Find λ R and u V 0 (Ω, u 0, such that a(u, v = (λ + 1 b(u, v, v V 0 (Ω. (5.5 It is well know that problem (5.5 attains a sequence of finite multiplicity eigenvalues λ n > 0, n N, diverging to +, with corresponding L 2 (Ω-orthonormal eigenfunctions u n belonging to V 0 (Ω. Now, as in section 2, we consider the bounded linear operator T : X(Ω X(Ω defined by Tf = u V 0 (Ω and By virtue of Lax-Milgram Lemma, we have a(u, y = b(f, y, y V 0 (Ω. (5.6 u Ω C f 0,Ω. As a consequence of the classical a priori estimates, for any f X(Ω, u = Tf is known to satisfy some further regularity. In fact, u H 1+r (Ω for r (1/2, 1] (see [3] and there holds u 1+r,Ω C f 0,Ω. (5.7 Now, we consider the bounded linear operator A : X(R 3 Ṽ (R3 defined by Af = ŜTŠf, where Ŝ and Š are the extension and the restriction operators, respectively, defined in section 2. Since a y b are symmetric, T is self-adjoint with respect to a. Clearly, A is also self-adjoint with respect to a. Notice that (λ, u is a solution of problem (5.5 if and only if ( 1 1 λ+1, u is an eigenpair of T which, in its turn, is equivalent to ( λ+1, ū being an eigenpair of A, where ū = Ŝ(u. Let the curved domain Ω be approximated by a polyhedron Ω h with vertices on Ω. Let T h be a partition of Ω h, i.e., a set of a finite number of closed tetrahedra T, which has the following properties: each vertex of Ω h is a vertex of a T T h, each T T h has a least one vertex in the interior of Ω h, any two tetrahedra, T, T T h share at most a vertex, a whole side or a whole face. Let N h and E h denote the set of all vertices and the set of all edges in T h, respectively. We assume that N h Ω N h Ω h Ω E h contains all the points where the boundary Ω is not C 2 for all T T h, at most one face of T lies on Ω h. We also assume that the family {T h } is regular. In what follows we will use some notation and definitions introduced in [6]. Consider a T T h which has a face Sh T Ω h, called a boundary tetrahedra. We enumerate the vertices of T such that the vertices of Sh T are numbered first and we denote them by P1 T, P2 T, P3 T and P4 T, in local notation. Let Σ T h be the part of Ω which is approximated by the face Sh T. We denote by T id the closed tetrahedra with three plane slides, having P 4 as a common vertex, and with one curved slide, coinciding with Σ T h, and we call it the ideal tetrahedra associated with T T h. For the sake of simplicity, we assume that the partitions T h are such that, for each boundary

13 TITLE WILL BE SET BY THE PUBLISHER 13 tetrahedra T, either T T id or T T id. If we replace all boundary tetrahedra in T h by their associated ideal tetrahedra T id, we obtain the so called ideal partition Th id of the domain Ω. and Finally, for a family {T h }, we denote Ω a closed domain in R 3 satisfying Ω Ω Ω h, regardless of h. With the triangulation T h we consider the finite element spaces where V (Ω h is the functional space H 1 (Ω h. V h (Ω h := {v h V (Ω h : v h T P 1 (T T T h }, V 0h (Ω h := {v h V h (Ω h : v h Ωh = 0}, Let a h and b h be the symmetric bilinear forms defined by a h (u, v := Ω h k ij (x u x i v dx + uv dx, u, v V (Ω h, x j Ω h b h (u, v := uv dx, Ω h u, v X(Ω h, where X(Ω h is the functional space L 2 (Ω h. The bilinear form a h is coercive on V (Ω h uniformly on h. Once again, since k i,j are bounded on Ω, a h is continuous on V (Ω h uniformly on h. Then the discretization of the spectral problem (5.5 is given by Find λ h R and u h V 0h (Ω h, u h 0, such that a h (u h, v h = (λ h + 1 b h (u h, v h, v h V 0h (Ω h. (5.8 We consider the bounded linear operator A h : X(R 3 ṼhR 3 defined by A h f Ṽh(R 3 and a h (A h f, v h = b h (f, v h, v h V 0h (Ω h. (5.9 Consider the bilinear form a ω (u, v := ω k ij (x u x i v dx + uv dx, x j ω u, v H 1 (ω, with ω being a closed subset of Ω Ω h. Observe that a ω is continuous on H 1 (ω uniformly on h. Then, we can take a 1 (u Ω Ωh, v Ω Ωh = a 1h (u Ω Ωh, v Ω Ωh = a Ω Ωh (u, v, a 2 (u Ω\Ωh, v Ω\Ωh = a Ω\Ωh (u, v, a 2h (u Ωh \Ω, v Ωh \Ω = a Ωh \Ω(u, v. In order to prove properties P1, P2, P3 and P4 for this problem, we establish the following lemmas and definitions.

14 14 TITLE WILL BE SET BY THE PUBLISHER Lemma 5.1. There exists a positive constant C such that: v 0,Ω\ Ωh Ch s v s,ω v H s (Ω, 0 s 1, v 0,Ωh \ Ω Ch s v s,ωh v H s (Ω h, 0 s 1. Proof. By adapting the arguments used in the proof of Lemma in [6] for the three dimensional case, the inequalities can be proved for s = 1. Since the two inequalities are clearly true for s = 0, they follow for 0 < s < 1 from standard results on interpolation in Sobolev spaces. Definition 5.2. Let w h V 0h (Ω h. A function ŵ V 0 (Ω is called to be associated with w h if it has the following properties: ŵ C 0 ( Ω ŵ(p i = w h (P i, P i N h ŵ is linear on each tetrahedra T T h Th id if T T id, ŵ = 0 on T id \ T and ŵ = w h on T if T id T, ŵ T id Ω = 0. The definition above is due to Feistauer and Ženíšek (see [6]. The construction of such a function follows basically from the interpolation theory developed to Zlámal [18] for two dimensional curved finite elements. The extension of his ideas to the three dimensional case is relatively straightforward so we do not include the details here. Lemma 5.3. Let ŵ V 0 (Ω be associated with w h V 0h (Ω h. Let T id Th id approximation. If T id T, then ŵ w h T id C h w h T, where C is a constant independent of h. lie along Ω and let T T h be its Proof. The proof is a consequence of Definition 5.2 and a suitable extension of Theorem 2 in [18]. In what follows, we will use smooth extensions of functions originally defined in Ω. We denote by ϕ e an extension of ϕ from H s (Ω, s > 0, into H s (R 3 satisfying ϕ e H s (R 3 and (see [7], for instance. Let f Ṽh(R 3 and define ū := Af and ū h := A h f. Lemma 5.4. There exists a positive constant C such that Proof. We have ( ū ū h C inf v h u e Ωh + v h V 0h (Ω h ϕ e s,r 3 C ϕ s,ω ( u e Ωh \Ω w h V 0h (Ω h. a h (u e u h, w h w h Ωh ū ū h 2 = ū ū h 2 Ω Ω h = u u h 2 Ω Ω h + u 2 Ω\Ω h + u h 2 Ω h \Ω. Now, let v h be an arbitrary element in the space V 0h (Ω h. We can write u u h 2 Ω Ω h 2 ( u v h 2 Ω Ω h + v h u h 2 Ω Ω h, + u Ω\Ωh

15 TITLE WILL BE SET BY THE PUBLISHER 15 and u h Ωh \Ω v h u h Ωh \Ω + v h Ωh \Ω. By using the uniform coerciveness and continuity of the bilinear form a h, we obtain v h u h 2 Ω h from which we deduce On the other hand, and ( α k ij (x (v h u h (v h u h dx + (v h u h 2 dx Ω h x i x j Ω h ( α k ij (x (v h u e (v h u h dx Ω h x i x j + k ij (x (ue u h (v h u h dx Ω h x i x j + (v h u e (v h u h dx + (u e u h (v h u h dx ( Ω h Ω h C v h u e Ωh v h u h Ωh + a h ((u e u h, (v h u h, v h u h Ωh C ( v h u e Ωh + Combining the above inequalities, we conclude the proof. w h V 0h (Ω h a h (u e u h, w h. w h Ωh u v h Ω Ωh = u e v h Ω Ωh, (5.11 v h Ωh \Ω v h u e Ωh \Ω + u e Ωh \Ω. (5.12 We are going to estimate the terms appearing in the right hand side of the inequality in Lemma 5.4. In the sequel, we shall assume that r is the constant appearing in equation (5.7. Lemma 5.5. There exists a positive constant C such that inf v h V 0h (Ω h ue v h Ωh C h r u 1+r,Ω. Proof. Since u e H 1+r (R 3, u e C 0 (R 3. So, Lu e, the Lagrange linear interpolant of u e Ωh, is well defined (see [2], for instance. By using standard interpolation results, we have u e Lu e Ωh Ch r u e 1+r,Ωh. Observe that Lu e V 0h (Ω h although u e Ωh 0. Then, using the estimate (5.10, we conclude the proof. Lemma 5.6. There exists a positive constant C such that a h (u e u h, w h C h r f. w h V 0h (Ω h w h Ωh

16 16 TITLE WILL BE SET BY THE PUBLISHER Proof. For any function w h V 0h (Ω h, we have a h (u e u h, w h = = = k ij (x (ue u h Ω h x i k ij (x Ω Ωh ue + Ω x i w h x j dx + k ij (x u x i w h x j dx + Ω h \Ω w h dx + (u e u h w h dx x j Ω h Ω k ij (x ue x i w h x j dx + u e w h dx fw h dx Ω Ω h Ω h u w h dx fw h dx Ω h u e w h dx. Ω h \Ω The last two terms can be easily bounded. In fact, by using Cauchy-Schwarz inequality, Lemma 5.1 and estimate (5.10, we obtain Ω h \Ω k ij (x ue w h dx + u e w h dx C u e Ωh \Ω w h Ωh \Ω x i x j Ω h \Ω C h r u e 1+r,R 3 w h Ωh C h r u 1+r,Ω w h Ωh. (5.13 We are going to estimate the remainder three terms. To this end, we need to introduce some notation. We denote by ω T the domain bounded by Σ T h and ST h, with ΣT h Ω being the curved side of an ideal tetrahedra and with Sh T Ω h being the corresponding side of the associated tetrahedra T T h. Now, we consider a function ŵ as in Lemma 5.3. Since ŵ V 0 (Ω, we may take it as a test function in (5.6. Then, we can obtain = = Ω ( fŵ dx T T id h T Ω ( T u(ŵ w h dx T T id h T Ω T [ C T ( k ij (x u (ŵ w h dx + u(ŵ w h dx fw h dx Ω x i x j Ω Ω h fŵ dx k ij (x u (ŵ w h dx x i x j T u(ŵ w h dx T T id h T Ω T T T h T Ω h f(ŵ w h dx T T h T Ω h T fw h dx T fw h dx ω T k ij (x u (ŵ w h dx x i x j ( f 0,T ŵ w h 0,T + u T ŵ w h T + f 0,Ωh \Ω w h 0,Ωh \Ω ]

17 [ C TITLE WILL BE SET BY THE PUBLISHER 17 (h f 0,T w h T + h u T w h T + h r f 0,Ωh \Ω w h r,ωh ] [ C h T T h T Ω h ( f + u Ω ] + h r f w h Ωh, where we have used Lemmas 5.1 and 5.3. We conclude the proof combining the previous inequality with (5.13 and the estimate (5.7. Theorem 5.7. (P1 There exists a positive constant C such that (A A h f C h r f f Ṽh(R 3. Proof. It is an immediate consequence of the previous lemmas. Theorem 5.8. (P2 For each eigenfunction ū of A associated to λ, there exists a strictly positive constant C such that ū Ω\Ωh = u Ω\Ωh C h r u 1+r,Ω. Proof. It is an immediate consequence of the regularity of the eigenfunctions u of the operator T and Lemma 5.1. Theorem 5.9. (P3 For each eigenfunction ū of A associated to λ, there exists a strictly positive constant C such that inf v h V 0h (Ω h ū v h C h r u 1+r,Ω. Proof. We have ū v h 2 = ū v h 2 Ω h + u 2 Ω\Ω h ū u e 2 Ω h + u e v h 2 Ω h + u 2 Ω\Ω h u e 2 Ω h \Ω + ue v h 2 Ω h + u 2 Ω\Ω h h 2r u 2 1+r,Ω, which follows because the regularity of the eigenfunctions u of the operator T, estimate 5.10 and Lemma 5.5. Then, taking the infimum with respect to v h V 0h (Ω h, we can conclude the proof. By virtue of the previous theorems, the spectrum of A h furnishes the approximations of the spectrum of A as we stated in section 3. Theorem (P4 There exists a positive constant C such that (A A h E(V (R 3 C h r. Proof. For x E(V (R 3, ū = Ax H 1+r (R 3. Then, the proof runs identically to that of Theorem 5.7. Observe that, since A and A h are self-adjoint, properties P5, P6, P7 and P8 are equally valid. Now, we are going to estimate the consistency terms appearing in Theorem Notice that M h and M h also coincide because of the symmetry of a h and b h. Lemma There exists a positive constant C such that M h = x E(V (R 3 x = 1 y E(V (R 3 a h (Ax, Π h y y b h (x, Π h y y C h 2r, with Π h being the projection onto Ṽh(R 3 with respect to a h, defined by equation (4.3.

18 18 TITLE WILL BE SET BY THE PUBLISHER Proof. Let x E(V (R 3, with x = 1, and put w = ŠAx. w = ŠŜTŠx = TŠx. From (5.7 we know that w H1+r (Ω and According to the definition of A, we have w 1+r,Ω C x = C. Now, taking v D(Ω as test functions in (5.6, it can be shown that w is the solution of the following strong problem { Lw + w = x in Ω, w = 0 on Ω. Let us denote by w the extension of w by zero from Ω to R 3. v h = Π h y y. Integrating by parts, we obtain Let y E(V (R 3 with, and take a h ( w, v h b h (x, v h = k ij (x w (v h dx + wv h dx + xv h dx Ω\Ω h x i x j Ω\Ω h Ω\Ω h C ( w 1,Ω\Ωh v h 1,Ω\Ωh + w 0,Ω\Ωh v h 0,Ω\Ωh + x 0,Ω\Ωh v h 0,Ω\Ωh C (h 2r w 1+r,Ω y 1+r,Ω + h 2r w r,ω y r,ω + h 2r x r,ω y r,ω. In the last inequality, we have used that v h Ω\Ωh = y Ω\Ωh and the estimate in Lemma 5.1. Finally, we can conclude the proof using estimate (5.7. Lemma There exists a positive constant C such that N h = x E(V (R 3 x = 1 y E(V (R 3 a h (Ax, y b h (x, y C h 2r. Proof. It is identical to that of the previous lemma by substituting v h by y. Theorem There exists a positive constant Csuch that max λ λ ih Ch 2r. i=1,,m Proof. It is a immediate consequence of the properties P2, P3, P4 and the previous lemmas. The authors wish to thank Rodolfo Rodríguez for many valuable comments on this paper. References 1. I. Babǔska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions, eds., North Holland, P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions, eds., North Holland, M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, Lecture Notes in Mathematics, 1341, Springer, Berlin, J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence, R.A.I.R.O. Anal. Numer., 12 ( J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 2: Error estimates for the Galerkin methods, R.A.I.R.O. Anal. Numer., 12 ( M. Feistauer and A. Ženíšek, Finite element solution of nonlinear elliptic problems, Numer. Math., 50 ( P. Grisvard, Elliptic Problems for Non-smooth Domains, Pitman, Boston, 1985.

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