The Mortar method in the wavelet context
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 647-673.

This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method, such multiplier spaces are not a subset of the space of traces of interior functions, but rather of their duals. For the resulting method, we provide with an error estimate, which is optimal in the geometrically conforming case.

Classification : 65N55, 42C40, 65N30, 65N15
Mots-clés : domain decomposition, mortar method, wavelet approximation 
@article{M2AN_2001__35_4_647_0,
     author = {Bertoluzza, Silvia and Perrier, Val\'erie},
     title = {The {Mortar} method in the wavelet context},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {647--673},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     mrnumber = {1862873},
     zbl = {0995.65131},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_647_0/}
}
TY  - JOUR
AU  - Bertoluzza, Silvia
AU  - Perrier, Valérie
TI  - The Mortar method in the wavelet context
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2001
SP  - 647
EP  - 673
VL  - 35
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/M2AN_2001__35_4_647_0/
LA  - en
ID  - M2AN_2001__35_4_647_0
ER  - 
%0 Journal Article
%A Bertoluzza, Silvia
%A Perrier, Valérie
%T The Mortar method in the wavelet context
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2001
%P 647-673
%V 35
%N 4
%I EDP-Sciences
%U http://www.numdam.org/item/M2AN_2001__35_4_647_0/
%G en
%F M2AN_2001__35_4_647_0
Bertoluzza, Silvia; Perrier, Valérie. The Mortar method in the wavelet context. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 647-673. http://www.numdam.org/item/M2AN_2001__35_4_647_0/

[1] Y. Achdou, G. Abdoulaev, Y. Kutznetsov and C. Prud'Homme, On the parallel inplementation of the mortar element method. ESAIM: M2AN 33 (1999) 245-259. | Numdam | Zbl

[2] L. Anderson, N. Hall, B. Jawerth and G. Peters, Wavelets on closed subsets on the real line, in Topics in the theory and applications of wavelets, L.L. Schumaker and G. Webb, Eds., Academic Press, Boston (1993) 1-61. | Zbl

[3] F. Ben Belgacem, The mortar finite element method with Lagrange multiplier. Numer. Math. 84 (1999) 173-197. | Zbl

[4] F. Ben Belgacem, A. Buffa and Y. Maday, The mortar element method for 3D Maxwell's equations. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 903-908. | Zbl

[5] F. Ben Belgacem and Y. Maday, Non conforming spectral method for second order elliptic problems in 3D. East-West J. Numer. Math. 4 (1994) 235-251. | Zbl

[6] C. Bernardi, Y. Maday, C. Mavripilis and A.T. Patera, The mortar element method applied to spectral discretizations, in Finite element analysis in fluids. Proc. of the seventh international conference on finite element methods in flow problems, T. Chung and G. Karr, Eds., UAH Press (1989). | Zbl

[7] C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar element method, in Asymptotic and numerical methods for partial differential equations with critical parameters, H.G. Kaper and M. Garbey, Eds., N.A.T.O. ASI Ser. C 384 . | Zbl

[8] C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear partial differential equations and their applications, Collège de France Seminar XI, H. Brezis and J.L.Lions, Eds. (1994) 13-51. | Zbl

[9] S. Bertoluzza, An adaptive wavelet collocation method based on interpolating wavelets, in Multiscale wavelet methods for partial differential equations. W. Dahmen, A.J. Kurdila and P. Oswald, Eds., Academic Press 6 (1997) 109-135.

[10] S. Bertoluzza and V. Perrier, The mortar method in the wavelet context. Technical Report 99-17, LAGA, Université Paris 13 (1999).

[11] S. Bertoluzza and P. Pietra, Space frequency adaptive approximation for quantum hydrodynamic models. Transport Theory Statist. Phys. 28 (2000) 375-395. | Zbl

[12] D. Braess and W. Dahmen, Stability estimate of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6 (1998) 249-264. | Zbl

[13] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl

[14] C. Canuto and A. Tabacco, Multilevel decomposition of functional spaces. J. Fourier Anal. Appl. 3 (1997) 715-742. | Zbl

[15] C. Canuto, A. Tabacco and K. Urban, The wavelet element method. Part I: Construction and analysis. Appl. Comput. Harmon. Anal. ACHA 6 (1999) 1-52. | Zbl

[16] L. Cazabeau, C. Lacour and Y. Maday, Numerical quadratures and mortar methods1997) 119-128. | Zbl

[17] P. Charton and V. Perrier, A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equation. Comput. Appl. Math. 15 (1996) 139-160. | Zbl

[18] G. Chiavassa and J. Liandrat, On the effective construction of compactly supported wavelets satisfying homogeneous boundary conditions on the interval. Appl. Comput. Harmon. Anal. ACHA 4 (1997) 62-73. | Zbl

[19] A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. ACHA 1 (1993) 54-81. | Zbl

[20] A. Cohen and R. Masson, Wavelet methods for second order elliptic problems, preconditioning and adaptivity. SIAM J. Sci. Comput. 21 (1999) 1006-1026. | Zbl

[21] A. Cohen and R. Masson, Wavelet adaptive method for second order elliptic problems. boundary conditions and domain decomposition. Numer. Math. 86 (1999) 193-238. | Zbl

[22] S. Dahlke, W. Dahmen ans R. Hochmut and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems. Appl. Numer. Math. 23 (1997) 21-48. | Zbl

[23] W. Dahmen, Stability of multiscale transformations. J. Fourier Anal. Appl. 2 (1996) 341-361. | Zbl

[24] W. Dahmen and A. Kunoth, Multilevel preconditioning. Numer. Math. 63 (1992) 315-344. | Zbl

[25] W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline-wavelets on the interval - stability and moment condition. Appl. Comput. Harmon. Anal. ACHA 6 (1999) 132-196. | Zbl

[26] W. Dahmen and R. Schneider, Composite wavelet bases for operator equations. Math. Comp. 68 (1999) 1533-1567. | Zbl

[27] I. Daubechies, Ten lectures on wavelets, in CBMS-NSF Regional Conference Series in Applied Mathematics 61. SIAM, Philadelphia (1992). | MR | Zbl

[28] S. Jaffard, Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29 (1992) 965-986. | Zbl

[29] Y. Maday, V. Perrier and J.C. Ravel, Adaptivité dynamique sur bases d'ondelettes pour l'approximation d'équations aux dérivées partielles. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) 405-410. | Zbl

[30] R. Masson, Biorthogonal spline wavelets on the interval for the resolution of boundary problems 6 (1996) 749-791. | Zbl

[31] Y. Meyer, Ondelettes et opérateurs. Hermann, Paris (1990). | MR | Zbl

[32] P. Monasse and V. Perrier, Orthonormal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal. 29 (1998) 1040-1065. | Zbl

[33] C. Prud'Homme, A strategy for the resolution of the tridimensional incompressible Navier-Stokes equations, in Méthodes itératives de décomposition de domaines et communications en calcul parallèle. Calcul. Parallèles Réseaux Syst. Répartis 10 Hermès (1998) 371-380.

[34] S. Grivet Talocia and A. Tabacco, Wavelets on the interval with optimal localization 10 (2000) 441-462. | Zbl

[35] H. Triebel, Interpolation theory, function spaces, differential operators. North Holland-Elsevier Science Publishers, Amsterdam (1978). | MR | Zbl

[36] B. Wohlmut, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012. | Zbl