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

Zakaria Belhachmi; Christine Bernardi; Andreas Karageorghis

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 4, page 801-824
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Belhachmi, Zakaria, Bernardi, Christine, and Karageorghis, Andreas. "Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 41.4 (2007): 801-824. <http://eudml.org/doc/250046>.

@article{Belhachmi2007,
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. },
author = {Belhachmi, Zakaria, Bernardi, Christine, Karageorghis, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mortar method; spectral elements; Laplace equation; Darcy equation.; mortar method; Darcy equation; optimal error estimates; numerical experiments},
language = {eng},
month = {10},
number = {4},
pages = {801-824},
publisher = {EDP Sciences},
title = {Mortar spectral element discretization of the Laplace and Darcy equations with discontinuous coefficients},
url = {http://eudml.org/doc/250046},
volume = {41},
year = {2007},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 801
EP - 824
AB - 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.
LA - eng
KW - Mortar method; spectral elements; Laplace equation; Darcy equation.; mortar method; Darcy equation; optimal error estimates; numerical experiments
UR - http://eudml.org/doc/250046
ER -

References

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  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 I333 (2001) 693–698.  Zbl0996.65123
  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.  Zbl1050.76035
  3. F. Ben Belgacem, The Mortar finite element method with Lagrangian multiplier. Numer. Math.84 (1999) 173–197.  Zbl0944.65114
  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.  Zbl1027.65156
  5. C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical AnalysisV, 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.  Zbl1004.65119
  7. C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math.85 (2000) 579–608.  Zbl0962.65096
  8. C. Bernardi, M. Dauge and Y. Maday, Relèvements de traces préservant les polynômes. C.R. Acad. Sci. Paris Série I315 (1992) 333–338.  
  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.  Zbl0797.65094
  10. C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques et Applications45. Springer-Verlag (2004).  
  11. C. Bernardi, Y. Maday and F. Rapetti, Basics and some applications of the mortar element method. GAMM – Gesellschaft für Angewandte Mathematik und Mechanik28 (2005) 97–123.  Zbl1177.65178
  12. S. Bertoluzza and V. Perrier, The mortar method in the wavelet context. ESAIM: M2AN35 (2001) 647–673.  Zbl0995.65131
  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.  Zbl0894.35035
  14. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms . Springer-Verlag (1986).  Zbl0585.65077
  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.  Zbl0728.65078
  16. N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa17 (1963) 189–206.  Zbl0127.31904
  17. NAG Library Mark 21, The Numerical Algorithms Group Ltd, Oxford (2004).  

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