Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T19:17:27.604Z Has data issue: false hasContentIssue false
  • English
  • Français

Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients

Published online by Cambridge University Press:  04 October 2007

Zakaria Belhachmi ,
Christine Bernardi  and
Andreas Karageorghis
Zakaria Belhachmi
Affiliation:
L.M.A.M. (UMR 7122), Université Paul Verlaine-Metz, Ile de Saulcy, 57045 Metz Cedex 01, France.
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr
Andreas Karageorghis
Affiliation:
Dept. of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus.
Get access

Abstract

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.

Keywords

Mortar methodspectral elementsLaplace equationDarcy equation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 4 , July 2007 , pp. 801 - 824
Copyright
© EDP Sciences, SMAI, 2007

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

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.
Achdou, Y., Bernardi, C. and Coquel, F., A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17–42. CrossRef
Ben Belgacem, F., The Mortar finite element method with Lagrangian multiplier. Numer. Math. 84 (1999) 173–197. CrossRef
Bernardi, C. and Chorfi, N., Mortar spectral element methods for elliptic equations with discontinuous coefficients. Math. Models Methods Appl. Sci. 12 (2002) 497–524. CrossRef
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.
Bernardi, C. and Maday, Y., Spectral element discretizations of the Poisson equation with mixed boundary conditions. Appl. Math. Inform. 6 (2001) 1–29.
Bernardi, C. and Verfürth, R., Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579–608. CrossRef
C. Bernardi, M. Dauge and Y. Maday, Relèvements de traces préservant les polynômes. C.R. Acad. Sci. Paris Série I 315 (1992) 333–338.
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.
C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques et Applications 45. Springer-Verlag (2004).
Bernardi, C., Maday, Y. and Rapetti, F., Basics and some applications of the mortar element method. GAMM – Gesellschaft für Angewandte Mathematik und Mechanik 28 (2005) 97–123.
Bertoluzza, S. and Perrier, V., The mortar method in the wavelet context. ESAIM: M2AN 35 (2001) 647–673. CrossRef
Clain, S. and Touzani, R., Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. RAIRO Modél. Math. Anal. Numér. 31 (1997) 845–870. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms . Springer-Verlag (1986).
Maday, Y. and Rønquist, E.M., Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries. Comput. Methods Appl. Mech. Engrg. 80 (1990) 91–115. CrossRef
Meyers, N.G., An Lp -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189–206.
NAG Library Mark 21, The Numerical Algorithms Group Ltd, Oxford (2004).