A Hybrid Method for Anisotropic Elliptic Problems Based on the Coupling of an Asymptotic-Preserving Method with the Asymptotic Limit Model

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 ε ∈ (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 ε is small. Adequate coupling conditions link the two models. The aim of this hybrid procedure is to reduce the computational time for problems where the region of small ε-values extends over a significant part of the domain, and this is due to the reduced complexity of the limit model. Introduction.The present work is a contribution to the numerical resolution of highly anisotropic elliptic equations, the anisotropy being aligned with one coordinate axis and described by a perturbation parameter ε ∈ (0, 1], varying considerably in the study domain. The approach presented here is based on a coupling strategy, solving an asymptotic-preserving (AP) reformulation of the elliptic problem, where ε is nonnegligible, and solving the corresponding asymptotic limit model, where ε is quasi-vanishing. The strategy we propose is particularly well suited for physical systems in which the anisotropy parameter ε is very small in a large part of the study domain.Such types of directionally anisotropic diffusion systems are common in physical applications, such as plasma physics [4,13,14,16,25,26].The application which was at the origin of the present work comes from strongly magnetized ionospheric plasmas [1,15,21]. The problem we shall study here is extracted from the dynamo model and represents an elliptic equation for the computation of the electric potential in two-dimensions, i.e.,

Asymptotic-Preserving methods and multiscale models for plasma physics

“…In contrast, for Problem (44)(45)(46)(47)(48) the non zeros elements remain 2.3 times larger that of Problem (40) whatever the mesh size. The efficiency is further improved by implementing a hybrid method coupling the AP reformulation (44)(45)(46)(47)(48) and the limit problem in the region where the asymptotic parameter is small [50,37]. Indeed this last model furnishes a solution that does not depend on the z coordinate, the discretization of this model gives rise to a smaller system matrix.…”

“…), number of rows (# rows), number of non zero elements stored in the system matrix (Nnz Mat.) and precision of the numerical approximation (L 2 -error norm of the relative error), for Ω z = Ω 1 z ∪Ω 2 z , where mes(Ω 2 z ) = 7 /10 mes(Ω z ), Ω 2 z being the sub-domain of the limit problem [37] of the AP method compared to the numerical resolution of the singular perturbation problem, with respect to the memory requirements is roughly 2.3 for the most refined meshes. The fill-in of the factorized matrix does not scale as badly, with an increase of the number of non zero elements stored from 50% to 80%.…”

“…The existence of such a region supposes that the approximation error of the numerical method dominates the modelling error produced by the use of the limit model for non vanishing asymptotic parameter values. For refined numerical parameters this requirement may not be met (an example of such a situation is discussed in [37]), or at the price of important computational efforts.…”