Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 4, pp. 801-824.

This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

DOI : 10.1051/m2an:2007035
Classification : 65N35, 65N55
Mots-clés : Mortar method, spectral elements, Laplace equation, Darcy equation 
@article{M2AN_2007__41_4_801_0,
     author = {Belhachmi, Zakaria and Bernardi, Christine and Karageorghis, Andreas},
     title = {Mortar spectral element discretization of the {Laplace} and {Darcy} equations with discontinuous coefficients},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {801--824},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {4},
     year = {2007},
     doi = {10.1051/m2an:2007035},
     mrnumber = {2362915},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007035/}
}
TY  - JOUR
AU  - Belhachmi, Zakaria
AU  - Bernardi, Christine
AU  - Karageorghis, Andreas
TI  - Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2007
SP  - 801
EP  - 824
VL  - 41
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007035/
DO  - 10.1051/m2an:2007035
LA  - en
ID  - M2AN_2007__41_4_801_0
ER  - 
%0 Journal Article
%A Belhachmi, Zakaria
%A Bernardi, Christine
%A Karageorghis, Andreas
%T Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2007
%P 801-824
%V 41
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007035/
%R 10.1051/m2an:2007035
%G en
%F M2AN_2007__41_4_801_0
Belhachmi, Zakaria; Bernardi, Christine; Karageorghis, Andreas. Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 4, pp. 801-824. doi : 10.1051/m2an:2007035. http://www.numdam.org/articles/10.1051/m2an:2007035/

[1] Y. Achdou and C. Bernardi, Un schéma de volumes ou éléments finis adaptatif pour les équations de Darcy à perméabilité variable. C.R. Acad. Sci. Paris Série I 333 (2001) 693-698. | Zbl

[2] Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17-42. | Zbl

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

[4] C. Bernardi and N. Chorfi, Mortar spectral element methods for elliptic equations with discontinuous coefficients. Math. Models Methods Appl. Sci. 12 (2002) 497-524. | Zbl

[5] C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical Analysis V, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209-485.

[6] C. Bernardi and Y. Maday, Spectral element discretizations of the Poisson equation with mixed boundary conditions. Appl. Math. Inform. 6 (2001) 1-29. | Zbl

[7] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579-608. | Zbl

[8] C. Bernardi, M. Dauge and Y. Maday, Relèvements de traces préservant les polynômes. C.R. Acad. Sci. Paris Série I 315 (1992) 333-338. | Zbl

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

[10] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques et Applications 45. Springer-Verlag (2004). | MR | Zbl

[11] C. Bernardi, Y. Maday and F. Rapetti, Basics and some applications of the mortar element method. GAMM - Gesellschaft für Angewandte Mathematik und Mechanik 28 (2005) 97-123.

[12] S. Bertoluzza and V. Perrier, The mortar method in the wavelet context. ESAIM: M2AN 35 (2001) 647-673. | Numdam | Zbl

[13] S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. RAIRO Modél. Math. Anal. Numér. 31 (1997) 845-870. | Numdam | Zbl

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | Zbl

[15] Y. Maday and E.M. Rønquist, Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries. Comput. Methods Appl. Mech. Engrg. 80 (1990) 91-115. | Zbl

[16] N.G. Meyers, An L p -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | Zbl

[17] NAG Library Mark 21, The Numerical Algorithms Group Ltd, Oxford (2004).

Cité par Sources :