Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T11:07:20.377Z Has data issue: false hasContentIssue false

The Mortar Method in the Wavelet Context

Published online by Cambridge University Press:  15 April 2002

Silvia Bertoluzza
Affiliation:
I.A.N.-C.N.R., v. Ferrata 1, 27100, Pavia, Italy. ([email protected])
Valérie Perrier
Affiliation:
Laboratoire de Modélisation et Calcul de l'IMAG, BP 53, 38041 Grenoble Cedex 9, France. ([email protected])
Get access

Abstract

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 fornon conforming domain decomposition, and point out some basic assumptionsunder which stability and convergence of such method can be proven. We studythe application of the mortar method in the biorthogonal wavelet framework.In particular we define suitable multiplier spaces for imposing weakcontinuity. Unlike in the classical mortar method, such multiplier spaces arenot a subset of the space of traces of interior functions, but rather oftheir duals.For the resulting method, we provide with an error estimate, which is optimal in thegeometrically conforming case.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y., Abdoulaev, G., Kutznetsov, Y. and Prud'homme, C., On the parallel inplementation of the mortar element method. ESAIM: M2AN 33 (1999) 245-259.
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.
Ben Belgacem, F., The mortar finite element method with Lagrange multiplier. Numer. Math. 84 (1999) 173-197. CrossRef
Ben Belgacem, F., Buffa, A. and Maday, Y., The mortar element method for 3D Maxwell's equations. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 903-908. CrossRef
Ben Belgacem, F. and Maday, Y., Non conforming spectral method for second order elliptic problems in 3D. East-West J. Numer. Math. 4 (1994) 235-251.
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).
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 .
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.
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.
S. Bertoluzza and V. Perrier, The mortar method in the wavelet context. Technical Report 99-17, LAGA, Université Paris 13 (1999).
Bertoluzza, S. and Pietra, P., Space frequency adaptive approximation for quantum hydrodynamic models. Transport Theory Statist. Phys. 28 (2000) 375-395. CrossRef
Braess, D. and Dahmen, W., Stability estimate of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6 (1998) 249-264.
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).
Canuto, C. and Tabacco, A., Multilevel decomposition of functional spaces. J. Fourier Anal. Appl. 3 (1997) 715-742. CrossRef
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. CrossRef
L. Cazabeau, C. Lacour and Y. Maday, Numerical quadratures and mortar methods, in Computational Sciences for the 21st Century, Bristeau et al., Eds., John Wiley & Sons, New York (1997) 119-128.
Charton, P. and Perrier, V., A pseudo-wavelet scheme for the two-dimensional Navier-Stokes equation. Comput. Appl. Math. 15 (1996) 139-160.
Chiavassa, G. and Liandrat, J., On the effective construction of compactly supported wavelets satisfying homogeneous boundary conditions on the interval. Appl. Comput. Harmon. Anal. ACHA 4 (1997) 62-73. CrossRef
Cohen, A., Daubechies, I. and Vial, P., Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. ACHA 1 (1993) 54-81. CrossRef
Cohen, A. and Masson, R., Wavelet methods for second order elliptic problems, preconditioning and adaptivity. SIAM J. Sci. Comput. 21 (1999) 1006-1026. CrossRef
Cohen, A. and Masson, R., Wavelet adaptive method for second order elliptic problems. boundary conditions and domain decomposition. Numer. Math. 86 (1999) 193-238. CrossRef
Dahlke, S., Dahmen, W. ans R. Hochmut and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems. Appl. Numer. Math. 23 (1997) 21-48. CrossRef
Dahmen, W., Stability of multiscale transformations. J. Fourier Anal. Appl. 2 (1996) 341-361.
Dahmen, W. and Kunoth, A., Multilevel preconditioning. Numer. Math. 63 (1992) 315-344. CrossRef
Dahmen, W., Kunoth, A. and Urban, K., Biorthogonal spline-wavelets on the interval - stability and moment condition. Appl. Comput. Harmon. Anal. ACHA 6 (1999) 132-196. CrossRef
Dahmen, W. and Schneider, R., Composite wavelet bases for operator equations. Math. Comp. 68 (1999) 1533-1567. CrossRef
I. Daubechies, Ten lectures on wavelets, in CBMS-NSF Regional Conference Series in Applied Mathematics 61. SIAM, Philadelphia (1992).
Jaffard, S., Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29 (1992) 965-986. CrossRef
Maday, Y., Perrier, V. and Ravel, J.C., 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.
R. Masson, Biorthogonal spline wavelets on the interval for the resolution of boundary problems. M 3 AS (Math. Models Methods Appl. Sci.) 6 (1996) 749-791.
Y. Meyer, Ondelettes et opérateurs. Hermann, Paris (1990).
Monasse, P. and Perrier, V., Orthonormal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal. 29 (1998) 1040-1065. CrossRef
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.
S. Grivet Talocia and A. Tabacco, Wavelets on the interval with optimal localization. M 3 AS (Math. Models Methods Appl. Sci.) 10 (2000) 441-462.
H. Triebel, Interpolation theory, function spaces, differential operators. North Holland-Elsevier Science Publishers, Amsterdam (1978).
B. Wohlmut, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012.