Block preconditioning methods for asymptotic preserving scheme arising in anisotropic elliptic problems

TL;DR

This paper introduces block preconditioning methods for asymptotic-preserving schemes in anisotropic elliptic problems, improving solver robustness and efficiency across different anisotropic strengths and grid refinements.

Contribution

The paper develops new block preconditioners based on approximate Schur complements for MMAP schemes, enhancing solver stability for small anisotropic parameters.

Findings

Preconditioned GMRES effectively solves anisotropic elliptic systems.

Block preconditioners are robust against grid refinement.

Improved Schur complement approximation enhances stability for small anisotropic parameters.

Abstract

Efficient and robust iterative solvers for strong anisotropic elliptic equations are very challenging. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro-macro asymptotic-preserving (MMAP) scheme. MMAP method was developed by Degond {\it et al.} in 2012 where the discrete matrix has a $2\times2$ block structure. By the approximate Schur complement a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original non-AP discretization. However it tends to be singular for very small anisotropic parameters. We then improve it by using more suitable approximation for boundary rows of the exact Schur complement. With these block preconditioners, preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a robust strategy with respect to grid refinement and the anisotropic strengths.