Anisotropic finite elements with high aspect ratio for an Asymptotic Preserving method for highly anisotropic elliptic equation

TL;DR

This paper extends an Asymptotic Preserving method for highly anisotropic elliptic equations by incorporating anisotropic finite elements with high aspect ratios, stabilization, and adaptive mesh refinement to improve efficiency and accuracy.

Contribution

It introduces a stabilized Asymptotic Preserving method using anisotropic finite elements with high aspect ratios and adaptive mesh algorithms for better handling of anisotropic elliptic equations.

Findings

Stabilization improves method robustness.

Anisotropic meshes with aspect ratio > 500 are effective.

Mesh adaptation reduces computational cost.

Abstract

The concern of this work is the generalization of an Asymptotic Preserving method for the highly anisotropic elliptic equations presented in [P. Degond, A. Lozinski, J. Narski, and C. Negulescu. An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition. J. Comput. Phys., 231(7):2724{2740, 2012]. The limitations of the method introduced there in are omitted by the introduction of a stabilization term in the Asymptotic Reformulation. Furthermore, anisotropic error indicators and mesh adaptation algorithms are proposed and tested allowing to reduce considerably the number of mesh points required to achieve prescribed precision. Reported meshes have maximum aspect ratio greater than 500.