Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 4, p. 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 : https://doi.org/10.1051/m2an:2007035
Classification:  65N35,  65N55
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {4},
year = {2007},
pages = {801-824},
doi = {10.1051/m2an:2007035},
zbl = {pre05289497},
mrnumber = {2362915},
language = {en},
url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 4, pp. 801-824. doi : 10.1051/m2an:2007035. http://www.numdam.org/item/M2AN_2007__41_4_801_0/

[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 0996.65123

[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 1050.76035

[3] F. Ben Belgacem, The Mortar finite element method with Lagrangian multiplier. Numer. Math. 84 (1999) 173-197. | Zbl 0944.65114

[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 1027.65156

[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 1004.65119

[7] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579-608. | Zbl 0962.65096

[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 0755.65103

[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 0797.65094

[10] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques et Applications 45. Springer-Verlag (2004). | MR 2068204 | Zbl 1063.65119

[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 0995.65131

[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 0894.35035

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | Zbl 0585.65077

[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 0728.65078

[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 0127.31904

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