- Email: [email protected]
Coupling wavelets and finite elements by the Mortar method
C. R. Acad. Sci. Paris, t. 332, Série I, p. 845–850, 2001 Analyse numérique/Numerical Analysis
Coupling wavelets and finite elements by the Mortar method Silvia BERTOLUZZA a , Valérie PERRIER b a b
IAN-CNR, v. Ferrata 1, 27100 Pavia, Italy Laboratoire de modélisation et calcul de l’IMAG, B.P. 53, 38041 Grenoble cedex 9, France E-mail: [email protected]; [email protected]
(Reçu le 24 janvier 2001, accepté le 12 février 2001)
Abstract.
We propose and analyse a mortar method with approximate constraint which we show to be particularly well suited for application in the framework of the wavelet/FEM coupling. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Couplage ondelettes/éléments finis par la méthode des joints Résumé.
Nous proposons et nous étudions une méthode de joints avec contrainte approchée. Nous montrons que cette méthode est particulièrement bien adaptée pour définir un couplage ondelettes/éléments finis. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Considérons le problème modèle suivant (Ω ⊂ R2 domaine polygonal, f ∈ L2 (Ω)) : −∇ · (a∇u) = f
dans Ω,
u = 0 sur Γ = ∂Ω,
où a est une matrice supposée constante, symétrique définie positive. Pour l’exemple simplifié d’une décomposition de domaine géométriquement conforme : Ω = Ω+ ∪ Ω− avec γ = ∂Ω+ ∩ ∂Ω− , nous introduisons des espaces d’approximation Vh+ et Vh− dans chaque sous-domaine (formules (2)), ainsi qu’un espace de multiplicateurs Mh ⊂ H−1/2 (γ), construit à partir de l’espace de trace Vh− |γ . La méthode des joints [1] reformulée dans le cadre d’un espace de multiplicateurs assez général [2], consiste à chercher une solution uh du problème vérifiant : uh |Ω+ ∈ Vh+ , uh |Ω− ∈ Vh− , et la approchée + − relation de continuité faible : γ (uh − uh )λ = 0, pour tout λ ∈ Mh . Nous montrons alors que cette contrainte peut être relaxée en une contrainte approchée, pour permettre + − le calcul en pratique des intégrales γ u+ h λ si les espaces de discrétisations Vh et Vh ne sont pas du même type. Pour cela nous introduisons deux espaces de même dimension, Uδ− ⊂ L2 (γ) et Uδ+ ⊂ L2 (γ), convenablement couplés (relation (4)) et dépendant d’un paramètre δ. Nous supposons que ces deux Note présentée par Philippe G. C IARLET. S0764-4442(01)01905-X/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
845
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espaces sont choisis de façon à permettre le calcul pratique des intégrales de la forme : γ ζ η si ζ ∈ Vh+ |γ et η ∈ Uδ+ , ou si ζ ∈ Mh et η ∈ Uδ− . Pour tout ζ ∈ L2 (γ) nous considérons P − (ζ) l’unique élément de Uδ− vérifiant : γ P − (ζ) η = γ ζ η, pour tout η ∈ Uδ+ , et nous proposons la contrainte approchée : − (P − (u+ h ) − uh ) λ = 0, pour tout λ ∈ Mh . γ Cette contrainte permet de définir un nouvel espace d’approximation X∗h explicité en (5), dans la formulation du problème avec joints (6). Nous sommes alors en mesure de démontrer le résultat [3] : T HÉORÈME 1. – Si l’espace de multiplicateur Mh vérifie de bonnes hypothèses (A1)–(A2) et si les espaces auxiliaires Uδ+ et Uδ− sont choisis de façon à vérifier des inégalités de type Jackson (7), alors si uh est la solution du problème avec joints et contrainte approchée (6), et si u, solution exacte de (1) vérifie u ∈ Hs (Ω) pour s donné, l’estimation d’erreur suivante est vérifiée : u − uh 1,∗ δ s−1 u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf u − vh H1 (Ω+ ) + inf u − vh H1 (Ω− ) , vh ∈Vh−
vh ∈Vh+
λ∈Mh
∂νa désignant la trace sur γ de la dérivée conormale extérieure au sous-domaine Ω+ , et · 1,∗ = · H1 (Ω+ ) + · H1 (Ω− ) la norme H1 par morceaux. Le couplage ondelettes/éléments finis s’effectue alors de la manière suivante : l’espace Vh+ est un espace éléments finis associé à une grille non uniforme et non structurée, tandis que l’espace Vh− est un espace de type ondelettes (cas EF maître/ondelettes esclave), ou inversement (cas ondelettes maître/EF esclave). Le point-clé de la méthode est le choix des espaces auxiliaires Uδ+ et Uδ− . Celui-ci s’obtient à partir d’un couple d’analyses multirésolution bi-orthogonales {Vj }jj0 et {Vj }jj0 de L2 (γ) dans lequel Vj ⊂ H1 (γ) est choisi comme un espace d’éléments finis P1 sur la grille uniforme de γ de pas 2−j , et tel que son espace dual bi-orthogonal Vj contient les polynômes de degré M − 1 (Vj ne contenant que les polynômes de degré 1) [5]. Il suffit alors de choisir Uδ+ = Vj ∩ H10 (γ) et de construire Uδ− comme un espace bi-orthogonal à Uδ+ [2] (cas EF maître), ou bien de choisir Uδ+ = Vj ∩ H10 (γ) et de construire Uδ− , comme un espace bi-orthogonal à Uδ+ (cas ondelettes maître). Dans les deux cas, le calcul de la contrainte approchée est effectué par des intégrales ne faisant intervenir que des fonctions du même type. L’application du théorème 1 permet d’estimer l’effet de la contrainte approchée sur l’estimation d’erreur finale : si u ∈ Hs (Ω) avec 2 s T , alors u − uh 1,∗ 2−j(s−1) u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf + u − vh H1 (Ω+ ) λ∈Mh
vh ∈Vh
+ inf u − vh H1 (Ω− ) , vh ∈Vh−
où la limite T de l’ordre d’approximation dépend du choix du côté maître et vaut respectivement T = min{7/2, M + 1/2} dans le cas « EF maître », et T = min{5/2, M + 3/2} dans le cas « ondelettes maître ». L’effet de la contrainte approchée est contrôlé par le premier terme à droite de l’inégalité, et pour un bon choix de j, peut être du même ordre de grandeur que les trois autres termes.
1. Introduction One of the advantages of nonconforming domain decomposition methods, is that they allow in principle to couple discretizations of different type. This feature is particularly attractive when considering discretizations whose applicability is, by nature, limited to tensor product domains. This is the case of wavelet type methods, which perform in a very promising way on academic examples, but whose
846
Coupling wavelets and finite elements by the Mortar method
application to real life problems is seriously limited by the issue of geometry, among other things. Coupling (in a domain decomposition approach) with finite elements allows in principle to overcome this limitation. In such methods, and in particular in the mortar method [1] which we are considering here, continuity across the interface is relaxed to a weak continuity constraint. In order to couple discretizations of different type the need arises eventually of applying the constraint operator, which implies the need of computing integrals of products of functions belonging to the two different discretizations. In particular, in the wavelet/FEM coupling the problem arises of computing the integral of a wavelet type function times a piecewise polynomial defined on an unstructured grid. Unfortunately, due to the particular nature of wavelets, which are not known in closed form, there is up to date no way of computing such an integral exactly, and it is therefore necessary to approximate it somehow. The problem of the effect of using quadrature formulae for the constraint within the mortar method has already been considered by [4]. In this paper we propose a different way of approximating such integrals which is particularly well suited for the coupling of wavelet and finite elements. 2. The mortar method with approximate constraint Let Ω ⊂ R2 be a bounded polygonal domain and consider the model problem: given f ∈ L2 (Ω) find u : Ω → R such that −∇ · (a∇u) = f
in Ω,
u=0
on Γ = ∂Ω,
(1)
where for simplicity we assume that the matrix a is constant symmetric positive definite. We consider here a very simple example of domain decomposition. More precisely, consider a geometrically conforming splitting of Ω in two subdomains as Ω = Ω+ ∪ Ω− , with γ = ∂Ω+ ∩ ∂Ω− . Denote by Vh+ and Vh− Vh+ ⊂ H1Γ (Ω+ ) = u ∈ H1 (Ω+ ) : u = 0 on Γ ∩ Ω+ , (2) Vh− ⊂ H1Γ (Ω− ) = u ∈ H1 (Ω− ) : u = 0 on Γ ∩ Ω− the two discrete spaces chosen for approximating u in Ω+ and Ω− respectively, and let Mh ⊂ H−1/2 (γ), with dim(Mh ) = dim(Vh− |γ ) be a suitable multiplier space—which in the mortar method is obtained from a subspace of Vh− |γ with suitable modifications at the vertices of γ [1], or which coincides, in a more general formulation, with a suitable “dual space” of Vh− |γ [2]. In the classical formulation of the mortar method, the approximation of the solution of (1) is sought in the constrained space Xh defined as: Xh = u : u|Ω+ ∈ Vh+ , u|Ω− ∈ Vh− , [u]λ = 0, ∀λ ∈ Mh , γ
where introducing the notation u+ = u|Ω+ and u− = u|Ω− , [u] = u+ |γ − u− |γ denotes the jump of the function u across the interface γ. The solution u to problem (1) is approximated by looking for uh in Xh such that for all vh ∈ Xh it holds a∇uh ∇vh + a∇uh ∇vh = f vh . Ω+
Ω−
Ω
In the solution of the linear system resulting from such problem the need arises eventually of computing the integrals appearing in the constraint − (u+ ∀λ ∈ Mh . (3) h − uh )λ = 0, γ
Since in the mortar method the multiplier space Mh is strongly related to the “slave” space Vh− , it is reasonable to assume that the integrals of the products u− h λ are computable in practice. This is not
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S. Bertoluzza, V. Perrier
necessarily the case of the products u+ h λ where functions originating from totally unrelated spaces are involved. We will concentrate here on approximating this term in the constraint. In order to do that, let us introduce two auxiliary spaces Uδ− ⊂ L2 (γ) and Uδ+ ⊂ L2 (γ) depending on a parameter δ, which we assume to have the same finite dimension and to satisfy ζη γ α > 0. (4) inf− sup ζ∈Uδ η∈U + ζ H1/2 (γ) η H−1/2 (γ) δ 00 Assume that the two auxiliary spaces are chosen in such a way that the integrals of the form γ ζ η are computable provided either ζ ∈ Vh+ |γ and η ∈ Uδ+ or ζ ∈ Mh and η ∈ Uδ− . For all ζ ∈ L2 (γ) let P − (ζ) ∈ Uδ− be the unique element in Uδ− such that − P (ζ) η = ζ η, ∀η ∈ Uδ+ . γ
γ
− + We propose here to approximate the integral of the product u+ h λ with the integral of P (uh )λ (where, by + + abuse of notation we will write uh instead of uh |γ ). The constraint (3) is then replaced by the approximated constraint − + P (uh ) − u− ∀λ ∈ Mh , h λ = 0, γ
which corresponds to defining a new constrained space as − + + − ∗ − P (u ) − u λ = 0, ∀λ ∈ Mh , Xh = u : u|Ω+ ∈ Vh , u|Ω− ∈ Vh ,
(5)
γ
and approximating the solution to (1) by the solution of the following discrete problem: find uh ∈ X∗h such that for all vh ∈ X∗h it holds a∇uh ∇vh + a∇uh ∇vh = f vh . (6) Ω+
Ω−
Ω
Denoting by · 1,∗ = · H1 (Ω+ ) + · H1 (Ω− ) the broken H norms, we can prove the following bound [3] 1
T HEOREM 1. – Let the multiplier space Mh be chosen in such a way that the following assumptions are satisfied: 1/2 (A1) there exists a bounded projection π : L2 (γ) → Vh− |γ , such that, for all η ∈ H00 (γ) and for all λ ∈ Mh , it holds that η − π(η) λ = 0 and πη H1/2 (γ) η H1/2 (γ) ; 00
γ
00
(A2) there exists a discrete lifting Rh : Vh− |γ → Vh− such that, for all η ∈ Vh− |γ , Rh η H1 (Ω− ) η H1/2 (γ) . 00
Moreover, let the two auxiliary spaces Uδ+ and Uδ− be chosen in such a way that the following Jackson R 1/2: for all r, 1/2 r R (resp. for all r˜, −1/2 r˜ R), type inequality holds for some R, ∀ η ∈ Hr0 (γ), ∀η ∈ Hr˜(γ),
848
inf
ηδ ∈Uδ−
η − ηδ H1/2 (γ) δ r−1/2 η Hr (γ) ,
inf + η − ηδ H−1/2 (γ) δ r˜+1/2 η Hr˜(γ) .
ηδ ∈Uδ
(7)
Coupling wavelets and finite elements by the Mortar method
Then, if uh is the solution of problem (6), and if the solution u of problem (1) verifies u ∈ Hs (Ω) for some + 3/2, R + 1/2}, the following error estimate holds: s, 2 s min{R u−uh 1,∗ δ s−1 u Hs(Ω) + inf ∂νa u−λ H−1/2 (γ) + inf + u−vh H1 (Ω+ ) + inf − u−vh H1 (Ω− ) , vh ∈Vh
λ∈Mh
vh ∈Vh
where ∂νa denotes the trace on γ of outer co-normal derivative to the subdomain Ω+ . Remark 1. – The extremely simple configuration considered (only two subdomains), hides some of the issues related to the analysis of the mortar method in more general configurations—namely the treatment of cross points. However, the approach used and the results obtained in this paper carry over to more complex cases (with the presence of cross-points), with, in the worse case, a loss of a logarithmic factor in the error estimate. 3. Wavelet/FEM coupling Let us now consider the case of wavelet/FEM coupling. In order to get two suitable auxiliary spaces, we will in such a case need a couple of biorthogonal multiresolution analyses {Vj }jj0 and {Vj }jj0 of L2 (γ) with the following characteristics [5]. – Vj ⊂ H1 (γ) is the subspace of P1 finite elements on the uniform grid Gj obtained by splitting γ into 2j equal segments; – Vj ⊂ H1 (γ) is a subspace having the same dimension as Vj , which is biorthogonal to Vj in the following sense: denoting by ej,k (k = 0, . . . , 2j ) the nodal basis function in Vj corresponding to space Vj has a Riesz’s basis {˜ ej,k , k = 0, . . . , 2j } which satisfies the k-th point in the grid Gj , the j ˜j,k = δkk , for all k, k = 0, . . . , 2 ; γ ej,k e – the functions e˜j,k can be obtained as linear combination of the restriction to γ (identified through a suitable mapping with the interval (0, 1)), of the translates and contracted (with a contraction factor 2j ) of a compactly supported function
N e˜, which we assume to be refinable, i.e., to verify, for suitable values of the coefficients hk , e˜(s) = k=0 hk e˜(2s − k); – Vj satisfies a Strang–Fix condition of order M , that is it contains polynomials up to degree M − 1, while, of course, Vj contains polynomials of order 1. Let Vj0 = Vj ∩ H10 (γ) and Vj0 = Vj ∩ H10 (γ), it is possible (see [2]) to construct two subspaces Vj∗ ⊂ Vj and Vj∗ ⊂ Vj satisfying dim(Vj∗ ) = dim Vj0 , dim(Vj∗ ) = dim Vj0 , and γηζ γηζ inf 0 sup α1 , inf sup α2 , ∗ ζ H−1/2 (γ) ζ H−1/2 (γ) η∈Vj j0 ζ∈Vj η H1/2 η∈V j∗ η H1/2 ζ∈V 00 (γ) 00 (γ) in such a way that they satisfy a Strang–Fix condition with the same order as Vj and Vj , respectively. Moreover, it is possible to construct Riesz’s bases e∗j,k and e˜∗j,k for Vj∗ and Vj∗ respectively in such a way that the two following biorthogonality relations hold: ej,k e˜∗j,k = δk,k , e˜j,k e∗j,k = δk,k , ∀k, k = 1, . . . , 2j − 1. (8) γ
γ
Since the function e˜ is refinable, it is well known that it is possible to compute integrals of the product of a wavelet type function times any function in Vj (and therefore in Vj0 and in Vj∗ ), while the product of a function in Vj , Vj0 and Vj∗ with a finite element type function can be computed by standard techniques, already implemented in the mortar method for finite elements with non-matching grids. For using such spaces for coupling wavelets and finite elements in the mortar method we distinguish two cases.
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S. Bertoluzza, V. Perrier
Case 1. FEM master/wavelet slave: in this case we set Vh+ to be a finite element space on an unstructured, non-uniform grid while Vh− and Mh are two wavelet type spaces. The approximate integration is done by setting Uδ+ = Vj0 and Uδ− = Vj∗ . Case 2. Wavelet master/FEM slave: in this case we set Vh+ to be a wavelet type space, while Vh− and Mh are two finite element spaces defined on unstructured, non-uniform grid, the grid for Mh being the trace on γ of the grid for Vh− . The approximate integration is this time performed by setting Uδ+ = Vj0 and Uδ− = Vj∗ . Once P − (vh+ ) is known, the space Uδ− is chosen in both cases in such a way that the integrals of the product λh P − (vh+ ) can be computed. We then only need to compute the P − (vh+ ). This can be done by taking advantage of the biorthogonality property (8). In fact it is not difficult to see that, depending on which of the two cases we are in, we have j j
2 −1 2 −1 − ∗ − ∗ Case 1: P u = u ej,k e˜j,k , Case 2: P u = u e˜j,k ej,k . k=1
γ
k=1
γ
Again, in both cases the two auxiliary spaces have been chosen in such a way that the two integrals defining the projectors are computable. Moreover, biorthogonality implies that no linear system has to be solved in order to compute the auxiliary projector. By applying Theorem 1 we can finally estimate the effect of using the approximate integration technique proposed in the previous section. In both cases we get the following bound: if u ∈ Hs (Ω) with 2 s T it holds u − uh 1,∗ 2−j(s−1) u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf u − vh H1 (Ω+ ) λ∈Mh
vh ∈Vh+
+ inf − u − vh H1 (Ω− ) , vh ∈Vh
where, depending on the choice of the master and slave, the limit T in the bound is respectively T = min{7/2, M + 1/2} for the ‘FEM master’ case and T = min{5/2, M + 3/2} for the ‘wavelet master’ case. The effect of approximating the constraint is contained in the first term on the r.h.s. which, by suitably choosing j can be tuned up in such a way it is comparable to the other three terms. Acknowledgements. This work is partially supported by the “European project TMR ERB-FMRX-CT98-0184” and by “la région Rhone-Alpes”.
References [1] Bernardi C., Maday Y., Patera A.T., A new nonconforming approach to domain decomposition: the mortar element method, in: Nonlinear Partial Differential Equations and their Applications, Notes Math. Ser., Vol. 299, 1994, pp. 13–51. [2] Bertoluzza S., Perrier V., The mortar method in the wavelet context, Technical Report 99-17, LAGA, Université Paris-13, 1999. [3] Bertoluzza S., Perrier V., Coupling wavelets and finite elements by the mortar method, Technical Report 1191, IAN, 2000. Available at: http://www.ian.pv.cnr.it/~aivlis/pubbl.html. [4] Cazabeau L., Lacour C., Maday Y., Numerical quadratures and mortar methods, in: Bristeau et al. (Eds.), Computational Sciences for the 21st Century, Wiley and Sons, 1997, pp. 119–128. [5] Cohen A., Numerical analysis of wavelet methods, in: Ciarlet P.G., Lions J.-L. (Eds.), Handbook in Numerical Analysis, Vol. VII, Elsevier Science Publishers, North-Holland, 2000.
850
Coupling wavelets and finite elements by the Mortar method Silvia BERTOLUZZA a , Valérie PERRIER b a b
IAN-CNR, v. Ferrata 1, 27100 Pavia, Italy Laboratoire de modélisation et calcul de l’IMAG, B.P. 53, 38041 Grenoble cedex 9, France E-mail: [email protected]; [email protected]
(Reçu le 24 janvier 2001, accepté le 12 février 2001)
Abstract.
We propose and analyse a mortar method with approximate constraint which we show to be particularly well suited for application in the framework of the wavelet/FEM coupling. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Couplage ondelettes/éléments finis par la méthode des joints Résumé.
Nous proposons et nous étudions une méthode de joints avec contrainte approchée. Nous montrons que cette méthode est particulièrement bien adaptée pour définir un couplage ondelettes/éléments finis. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Considérons le problème modèle suivant (Ω ⊂ R2 domaine polygonal, f ∈ L2 (Ω)) : −∇ · (a∇u) = f
dans Ω,
u = 0 sur Γ = ∂Ω,
où a est une matrice supposée constante, symétrique définie positive. Pour l’exemple simplifié d’une décomposition de domaine géométriquement conforme : Ω = Ω+ ∪ Ω− avec γ = ∂Ω+ ∩ ∂Ω− , nous introduisons des espaces d’approximation Vh+ et Vh− dans chaque sous-domaine (formules (2)), ainsi qu’un espace de multiplicateurs Mh ⊂ H−1/2 (γ), construit à partir de l’espace de trace Vh− |γ . La méthode des joints [1] reformulée dans le cadre d’un espace de multiplicateurs assez général [2], consiste à chercher une solution uh du problème vérifiant : uh |Ω+ ∈ Vh+ , uh |Ω− ∈ Vh− , et la approchée + − relation de continuité faible : γ (uh − uh )λ = 0, pour tout λ ∈ Mh . Nous montrons alors que cette contrainte peut être relaxée en une contrainte approchée, pour permettre + − le calcul en pratique des intégrales γ u+ h λ si les espaces de discrétisations Vh et Vh ne sont pas du même type. Pour cela nous introduisons deux espaces de même dimension, Uδ− ⊂ L2 (γ) et Uδ+ ⊂ L2 (γ), convenablement couplés (relation (4)) et dépendant d’un paramètre δ. Nous supposons que ces deux Note présentée par Philippe G. C IARLET. S0764-4442(01)01905-X/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
845
S. Bertoluzza, V. Perrier
espaces sont choisis de façon à permettre le calcul pratique des intégrales de la forme : γ ζ η si ζ ∈ Vh+ |γ et η ∈ Uδ+ , ou si ζ ∈ Mh et η ∈ Uδ− . Pour tout ζ ∈ L2 (γ) nous considérons P − (ζ) l’unique élément de Uδ− vérifiant : γ P − (ζ) η = γ ζ η, pour tout η ∈ Uδ+ , et nous proposons la contrainte approchée : − (P − (u+ h ) − uh ) λ = 0, pour tout λ ∈ Mh . γ Cette contrainte permet de définir un nouvel espace d’approximation X∗h explicité en (5), dans la formulation du problème avec joints (6). Nous sommes alors en mesure de démontrer le résultat [3] : T HÉORÈME 1. – Si l’espace de multiplicateur Mh vérifie de bonnes hypothèses (A1)–(A2) et si les espaces auxiliaires Uδ+ et Uδ− sont choisis de façon à vérifier des inégalités de type Jackson (7), alors si uh est la solution du problème avec joints et contrainte approchée (6), et si u, solution exacte de (1) vérifie u ∈ Hs (Ω) pour s donné, l’estimation d’erreur suivante est vérifiée : u − uh 1,∗ δ s−1 u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf u − vh H1 (Ω+ ) + inf u − vh H1 (Ω− ) , vh ∈Vh−
vh ∈Vh+
λ∈Mh
∂νa désignant la trace sur γ de la dérivée conormale extérieure au sous-domaine Ω+ , et · 1,∗ = · H1 (Ω+ ) + · H1 (Ω− ) la norme H1 par morceaux. Le couplage ondelettes/éléments finis s’effectue alors de la manière suivante : l’espace Vh+ est un espace éléments finis associé à une grille non uniforme et non structurée, tandis que l’espace Vh− est un espace de type ondelettes (cas EF maître/ondelettes esclave), ou inversement (cas ondelettes maître/EF esclave). Le point-clé de la méthode est le choix des espaces auxiliaires Uδ+ et Uδ− . Celui-ci s’obtient à partir d’un couple d’analyses multirésolution bi-orthogonales {Vj }jj0 et {Vj }jj0 de L2 (γ) dans lequel Vj ⊂ H1 (γ) est choisi comme un espace d’éléments finis P1 sur la grille uniforme de γ de pas 2−j , et tel que son espace dual bi-orthogonal Vj contient les polynômes de degré M − 1 (Vj ne contenant que les polynômes de degré 1) [5]. Il suffit alors de choisir Uδ+ = Vj ∩ H10 (γ) et de construire Uδ− comme un espace bi-orthogonal à Uδ+ [2] (cas EF maître), ou bien de choisir Uδ+ = Vj ∩ H10 (γ) et de construire Uδ− , comme un espace bi-orthogonal à Uδ+ (cas ondelettes maître). Dans les deux cas, le calcul de la contrainte approchée est effectué par des intégrales ne faisant intervenir que des fonctions du même type. L’application du théorème 1 permet d’estimer l’effet de la contrainte approchée sur l’estimation d’erreur finale : si u ∈ Hs (Ω) avec 2 s T , alors u − uh 1,∗ 2−j(s−1) u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf + u − vh H1 (Ω+ ) λ∈Mh
vh ∈Vh
+ inf u − vh H1 (Ω− ) , vh ∈Vh−
où la limite T de l’ordre d’approximation dépend du choix du côté maître et vaut respectivement T = min{7/2, M + 1/2} dans le cas « EF maître », et T = min{5/2, M + 3/2} dans le cas « ondelettes maître ». L’effet de la contrainte approchée est contrôlé par le premier terme à droite de l’inégalité, et pour un bon choix de j, peut être du même ordre de grandeur que les trois autres termes.
1. Introduction One of the advantages of nonconforming domain decomposition methods, is that they allow in principle to couple discretizations of different type. This feature is particularly attractive when considering discretizations whose applicability is, by nature, limited to tensor product domains. This is the case of wavelet type methods, which perform in a very promising way on academic examples, but whose
846
Coupling wavelets and finite elements by the Mortar method
application to real life problems is seriously limited by the issue of geometry, among other things. Coupling (in a domain decomposition approach) with finite elements allows in principle to overcome this limitation. In such methods, and in particular in the mortar method [1] which we are considering here, continuity across the interface is relaxed to a weak continuity constraint. In order to couple discretizations of different type the need arises eventually of applying the constraint operator, which implies the need of computing integrals of products of functions belonging to the two different discretizations. In particular, in the wavelet/FEM coupling the problem arises of computing the integral of a wavelet type function times a piecewise polynomial defined on an unstructured grid. Unfortunately, due to the particular nature of wavelets, which are not known in closed form, there is up to date no way of computing such an integral exactly, and it is therefore necessary to approximate it somehow. The problem of the effect of using quadrature formulae for the constraint within the mortar method has already been considered by [4]. In this paper we propose a different way of approximating such integrals which is particularly well suited for the coupling of wavelet and finite elements. 2. The mortar method with approximate constraint Let Ω ⊂ R2 be a bounded polygonal domain and consider the model problem: given f ∈ L2 (Ω) find u : Ω → R such that −∇ · (a∇u) = f
in Ω,
u=0
on Γ = ∂Ω,
(1)
where for simplicity we assume that the matrix a is constant symmetric positive definite. We consider here a very simple example of domain decomposition. More precisely, consider a geometrically conforming splitting of Ω in two subdomains as Ω = Ω+ ∪ Ω− , with γ = ∂Ω+ ∩ ∂Ω− . Denote by Vh+ and Vh− Vh+ ⊂ H1Γ (Ω+ ) = u ∈ H1 (Ω+ ) : u = 0 on Γ ∩ Ω+ , (2) Vh− ⊂ H1Γ (Ω− ) = u ∈ H1 (Ω− ) : u = 0 on Γ ∩ Ω− the two discrete spaces chosen for approximating u in Ω+ and Ω− respectively, and let Mh ⊂ H−1/2 (γ), with dim(Mh ) = dim(Vh− |γ ) be a suitable multiplier space—which in the mortar method is obtained from a subspace of Vh− |γ with suitable modifications at the vertices of γ [1], or which coincides, in a more general formulation, with a suitable “dual space” of Vh− |γ [2]. In the classical formulation of the mortar method, the approximation of the solution of (1) is sought in the constrained space Xh defined as: Xh = u : u|Ω+ ∈ Vh+ , u|Ω− ∈ Vh− , [u]λ = 0, ∀λ ∈ Mh , γ
where introducing the notation u+ = u|Ω+ and u− = u|Ω− , [u] = u+ |γ − u− |γ denotes the jump of the function u across the interface γ. The solution u to problem (1) is approximated by looking for uh in Xh such that for all vh ∈ Xh it holds a∇uh ∇vh + a∇uh ∇vh = f vh . Ω+
Ω−
Ω
In the solution of the linear system resulting from such problem the need arises eventually of computing the integrals appearing in the constraint − (u+ ∀λ ∈ Mh . (3) h − uh )λ = 0, γ
Since in the mortar method the multiplier space Mh is strongly related to the “slave” space Vh− , it is reasonable to assume that the integrals of the products u− h λ are computable in practice. This is not
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necessarily the case of the products u+ h λ where functions originating from totally unrelated spaces are involved. We will concentrate here on approximating this term in the constraint. In order to do that, let us introduce two auxiliary spaces Uδ− ⊂ L2 (γ) and Uδ+ ⊂ L2 (γ) depending on a parameter δ, which we assume to have the same finite dimension and to satisfy ζη γ α > 0. (4) inf− sup ζ∈Uδ η∈U + ζ H1/2 (γ) η H−1/2 (γ) δ 00 Assume that the two auxiliary spaces are chosen in such a way that the integrals of the form γ ζ η are computable provided either ζ ∈ Vh+ |γ and η ∈ Uδ+ or ζ ∈ Mh and η ∈ Uδ− . For all ζ ∈ L2 (γ) let P − (ζ) ∈ Uδ− be the unique element in Uδ− such that − P (ζ) η = ζ η, ∀η ∈ Uδ+ . γ
γ
− + We propose here to approximate the integral of the product u+ h λ with the integral of P (uh )λ (where, by + + abuse of notation we will write uh instead of uh |γ ). The constraint (3) is then replaced by the approximated constraint − + P (uh ) − u− ∀λ ∈ Mh , h λ = 0, γ
which corresponds to defining a new constrained space as − + + − ∗ − P (u ) − u λ = 0, ∀λ ∈ Mh , Xh = u : u|Ω+ ∈ Vh , u|Ω− ∈ Vh ,
(5)
γ
and approximating the solution to (1) by the solution of the following discrete problem: find uh ∈ X∗h such that for all vh ∈ X∗h it holds a∇uh ∇vh + a∇uh ∇vh = f vh . (6) Ω+
Ω−
Ω
Denoting by · 1,∗ = · H1 (Ω+ ) + · H1 (Ω− ) the broken H norms, we can prove the following bound [3] 1
T HEOREM 1. – Let the multiplier space Mh be chosen in such a way that the following assumptions are satisfied: 1/2 (A1) there exists a bounded projection π : L2 (γ) → Vh− |γ , such that, for all η ∈ H00 (γ) and for all λ ∈ Mh , it holds that η − π(η) λ = 0 and πη H1/2 (γ) η H1/2 (γ) ; 00
γ
00
(A2) there exists a discrete lifting Rh : Vh− |γ → Vh− such that, for all η ∈ Vh− |γ , Rh η H1 (Ω− ) η H1/2 (γ) . 00
Moreover, let the two auxiliary spaces Uδ+ and Uδ− be chosen in such a way that the following Jackson R 1/2: for all r, 1/2 r R (resp. for all r˜, −1/2 r˜ R), type inequality holds for some R, ∀ η ∈ Hr0 (γ), ∀η ∈ Hr˜(γ),
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inf
ηδ ∈Uδ−
η − ηδ H1/2 (γ) δ r−1/2 η Hr (γ) ,
inf + η − ηδ H−1/2 (γ) δ r˜+1/2 η Hr˜(γ) .
ηδ ∈Uδ
(7)
Coupling wavelets and finite elements by the Mortar method
Then, if uh is the solution of problem (6), and if the solution u of problem (1) verifies u ∈ Hs (Ω) for some + 3/2, R + 1/2}, the following error estimate holds: s, 2 s min{R u−uh 1,∗ δ s−1 u Hs(Ω) + inf ∂νa u−λ H−1/2 (γ) + inf + u−vh H1 (Ω+ ) + inf − u−vh H1 (Ω− ) , vh ∈Vh
λ∈Mh
vh ∈Vh
where ∂νa denotes the trace on γ of outer co-normal derivative to the subdomain Ω+ . Remark 1. – The extremely simple configuration considered (only two subdomains), hides some of the issues related to the analysis of the mortar method in more general configurations—namely the treatment of cross points. However, the approach used and the results obtained in this paper carry over to more complex cases (with the presence of cross-points), with, in the worse case, a loss of a logarithmic factor in the error estimate. 3. Wavelet/FEM coupling Let us now consider the case of wavelet/FEM coupling. In order to get two suitable auxiliary spaces, we will in such a case need a couple of biorthogonal multiresolution analyses {Vj }jj0 and {Vj }jj0 of L2 (γ) with the following characteristics [5]. – Vj ⊂ H1 (γ) is the subspace of P1 finite elements on the uniform grid Gj obtained by splitting γ into 2j equal segments; – Vj ⊂ H1 (γ) is a subspace having the same dimension as Vj , which is biorthogonal to Vj in the following sense: denoting by ej,k (k = 0, . . . , 2j ) the nodal basis function in Vj corresponding to space Vj has a Riesz’s basis {˜ ej,k , k = 0, . . . , 2j } which satisfies the k-th point in the grid Gj , the j ˜j,k = δkk , for all k, k = 0, . . . , 2 ; γ ej,k e – the functions e˜j,k can be obtained as linear combination of the restriction to γ (identified through a suitable mapping with the interval (0, 1)), of the translates and contracted (with a contraction factor 2j ) of a compactly supported function
N e˜, which we assume to be refinable, i.e., to verify, for suitable values of the coefficients hk , e˜(s) = k=0 hk e˜(2s − k); – Vj satisfies a Strang–Fix condition of order M , that is it contains polynomials up to degree M − 1, while, of course, Vj contains polynomials of order 1. Let Vj0 = Vj ∩ H10 (γ) and Vj0 = Vj ∩ H10 (γ), it is possible (see [2]) to construct two subspaces Vj∗ ⊂ Vj and Vj∗ ⊂ Vj satisfying dim(Vj∗ ) = dim Vj0 , dim(Vj∗ ) = dim Vj0 , and γηζ γηζ inf 0 sup α1 , inf sup α2 , ∗ ζ H−1/2 (γ) ζ H−1/2 (γ) η∈Vj j0 ζ∈Vj η H1/2 η∈V j∗ η H1/2 ζ∈V 00 (γ) 00 (γ) in such a way that they satisfy a Strang–Fix condition with the same order as Vj and Vj , respectively. Moreover, it is possible to construct Riesz’s bases e∗j,k and e˜∗j,k for Vj∗ and Vj∗ respectively in such a way that the two following biorthogonality relations hold: ej,k e˜∗j,k = δk,k , e˜j,k e∗j,k = δk,k , ∀k, k = 1, . . . , 2j − 1. (8) γ
γ
Since the function e˜ is refinable, it is well known that it is possible to compute integrals of the product of a wavelet type function times any function in Vj (and therefore in Vj0 and in Vj∗ ), while the product of a function in Vj , Vj0 and Vj∗ with a finite element type function can be computed by standard techniques, already implemented in the mortar method for finite elements with non-matching grids. For using such spaces for coupling wavelets and finite elements in the mortar method we distinguish two cases.
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Case 1. FEM master/wavelet slave: in this case we set Vh+ to be a finite element space on an unstructured, non-uniform grid while Vh− and Mh are two wavelet type spaces. The approximate integration is done by setting Uδ+ = Vj0 and Uδ− = Vj∗ . Case 2. Wavelet master/FEM slave: in this case we set Vh+ to be a wavelet type space, while Vh− and Mh are two finite element spaces defined on unstructured, non-uniform grid, the grid for Mh being the trace on γ of the grid for Vh− . The approximate integration is this time performed by setting Uδ+ = Vj0 and Uδ− = Vj∗ . Once P − (vh+ ) is known, the space Uδ− is chosen in both cases in such a way that the integrals of the product λh P − (vh+ ) can be computed. We then only need to compute the P − (vh+ ). This can be done by taking advantage of the biorthogonality property (8). In fact it is not difficult to see that, depending on which of the two cases we are in, we have j j
2 −1 2 −1 − ∗ − ∗ Case 1: P u = u ej,k e˜j,k , Case 2: P u = u e˜j,k ej,k . k=1
γ
k=1
γ
Again, in both cases the two auxiliary spaces have been chosen in such a way that the two integrals defining the projectors are computable. Moreover, biorthogonality implies that no linear system has to be solved in order to compute the auxiliary projector. By applying Theorem 1 we can finally estimate the effect of using the approximate integration technique proposed in the previous section. In both cases we get the following bound: if u ∈ Hs (Ω) with 2 s T it holds u − uh 1,∗ 2−j(s−1) u Hs(Ω) + inf ∂νa u − λ H−1/2 (γ) + inf u − vh H1 (Ω+ ) λ∈Mh
vh ∈Vh+
+ inf − u − vh H1 (Ω− ) , vh ∈Vh
where, depending on the choice of the master and slave, the limit T in the bound is respectively T = min{7/2, M + 1/2} for the ‘FEM master’ case and T = min{5/2, M + 3/2} for the ‘wavelet master’ case. The effect of approximating the constraint is contained in the first term on the r.h.s. which, by suitably choosing j can be tuned up in such a way it is comparable to the other three terms. Acknowledgements. This work is partially supported by the “European project TMR ERB-FMRX-CT98-0184” and by “la région Rhone-Alpes”.
References [1] Bernardi C., Maday Y., Patera A.T., A new nonconforming approach to domain decomposition: the mortar element method, in: Nonlinear Partial Differential Equations and their Applications, Notes Math. Ser., Vol. 299, 1994, pp. 13–51. [2] Bertoluzza S., Perrier V., The mortar method in the wavelet context, Technical Report 99-17, LAGA, Université Paris-13, 1999. [3] Bertoluzza S., Perrier V., Coupling wavelets and finite elements by the mortar method, Technical Report 1191, IAN, 2000. Available at: http://www.ian.pv.cnr.it/~aivlis/pubbl.html. [4] Cazabeau L., Lacour C., Maday Y., Numerical quadratures and mortar methods, in: Bristeau et al. (Eds.), Computational Sciences for the 21st Century, Wiley and Sons, 1997, pp. 119–128. [5] Cohen A., Numerical analysis of wavelet methods, in: Ciarlet P.G., Lions J.-L. (Eds.), Handbook in Numerical Analysis, Vol. VII, Elsevier Science Publishers, North-Holland, 2000.
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