A hybrid method for anisotropic elliptic problems based on the coupling of an Asymptotic-Preserving method with the Asymptotic-Limit model

TL;DR

This paper introduces a hybrid numerical approach combining asymptotic-preserving and limit models to efficiently solve highly anisotropic elliptic problems with variable anisotropy strength across the domain.

Contribution

The paper develops a novel hybrid method that couples asymptotic-preserving and asymptotic limit models for efficient computation in anisotropic elliptic problems.

Findings

Reduces computational time in regions with small anisotropy parameter

Effectively couples two models with appropriate boundary conditions

Demonstrates efficiency in large-scale anisotropic problems

Abstract

This paper presents a hybrid numerical method to solve efficiently a class of highly anisotropic elliptic problems. The anisotropy is aligned with one coordinate-axis and its strength is described by a parameter $\eps \in (0,1]$, which can largely vary in the study domain. Our hybrid model is based on asymptotic techniques and couples (spatially) an Asymptotic-Preserving model with its asymptotic Limit model, the latter being used in regions where the anisotropy parameter $\eps$ is small. Adequate coupling conditions link the two models. Aim of this hybrid procedure is to reduce the computational time for problems where the region of small $\eps$-values extends over a significant part of the domain, and this due to the reduced complexity of the limit model.