Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations

TL;DR

This paper presents a novel asymptotic-preserving numerical scheme for highly anisotropic elliptic equations with variable anisotropy directions, achieving accurate solutions regardless of anisotropy strength without specialized meshes.

Contribution

It extends previous methods to handle arbitrary anisotropy directions using a reformulation that maintains accuracy and efficiency, despite increased system size.

Findings

Accurately solves highly anisotropic equations independently of anisotropy strength.

Generalizes previous methods to variable anisotropy directions.

Maintains efficiency without mesh adaptation.

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 [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.