Numerical analysis of an asymptotic-preserving scheme for anisotropic elliptic equations

TL;DR

This paper analyzes asymptotic-preserving numerical schemes for highly anisotropic elliptic equations, demonstrating their robustness and convergence independent of the anisotropy parameter.

Contribution

It provides rigorous convergence analysis for AP-schemes, ensuring their effectiveness across a wide range of anisotropy levels in elliptic problems.

Findings

Schemes are effective for a broad range of epsilon values

Proved epsilon-independent convergence results

Schemes capture macroscopic properties as epsilon approaches zero

Abstract

The main purpose of the present paper is to study from a numerical analysis point of view some robust methods designed to cope with stiff (highly anisotropic) elliptic problems. The so-called asymptotic-preserving schemes studied in this paper are very efficient in dealing with a wide range of $\varepsilon$-values, where $0 < \varepsilon \ll 1$ is the stiffness parameter, responsible for the high anisotropy of the problem. In particular, these schemes are even able to capture the macroscopic properties of the system, as $\varepsilon$ tends towards zero, while the discretization parameters remain fixed. The objective of this work shall be to prove some $\varepsilon$-independent convergence results for these numerical schemes and put hence some more rigor in the construction of such AP-methods.