An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition

TL;DR

This paper introduces an efficient Asymptotic Preserving scheme for highly anisotropic elliptic equations that maintains accuracy across different anisotropy levels and handles complex anisotropy fields on Cartesian grids.

Contribution

The work presents a novel asymptotic-preserving method based on micro-macro decomposition, improving upon previous schemes by handling non-uniform and non-aligned anisotropy fields effectively.

Findings

Uniform convergence with respect to the anisotropy parameter <

Applicable to non-uniform and non-aligned anisotropy fields on Cartesian grids

Simple extension to variable anisotropy intensity 1/

Abstract

The concern of the present work is the introduction of a very efficient Asymptotic Preserving scheme for the resolution of highly anisotropic diffusion equations. The characteristic features of this scheme are the uniform convergence with respect to the anisotropy parameter $0<\eps <<1$, the applicability (on cartesian grids) to cases of non-uniform and non-aligned anisotropy fields $b$ and the simple extension to the case of a non-constant anisotropy intensity $1/\eps$. The mathematical approach and the numerical scheme are different from those presented in the previous work [Degond et al. (2010), arXiv:1008.3405v1] and its considerable advantages are pointed out.