Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

Duality-based asymptotic-preserving method for highly anisotropic diffusion equations

Pages: 1 – 31

DOI: https://dx.doi.org/10.4310/CMS.2012.v10.n1.a2

Authors

Pierre Degond (Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France)

Fabrice Deluzet (Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France)

Alexei Lozinski (Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France)

Jacek Narski (Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France)

Claudia Negulescu (CMI/LATP, Université de Provence, Marseille, France)

Abstract

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [P. Degond, F. Deluzet, and C. Negulescu, Multiscale Model. Simul., 8(2), 645–666, 2009/10] to the case of an arbitrary anisotropy direction field.

Keywords

anisotropic diffusion, asymptotic preserving scheme, finite element method

2010 Mathematics Subject Classification

65N30

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Published 14 October 2011