Optimized Schwarz preconditioning for SEM based magnetohydrodynamics

TL;DR

This paper adapts an algebraic optimized Schwarz algorithm to a finite element preconditioning strategy for spectral element discretizations of magnetohydrodynamics equations, improving solver efficiency.

Contribution

It demonstrates how to modify a FEM-based Schwarz preconditioner to its optimized form for magnetohydrodynamics spectral element problems.

Findings

Enhanced preconditioning efficiency for MHD spectral element discretizations

Successful adaptation of algebraic Schwarz optimization to FEM preconditioners

Potential for faster convergence in MHD simulations

Abstract

A recent theoretical result on optimized Schwarz algorithms demonstrated at the algebraic level enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how to modify a bilinear FEM based Schwarz preconditioning strategy originally presented in [Fischer, JCP 133:84 1997] to its optimized version. The latter is employed to precondition the pseudo--Laplacian operator arising from the spectral element discretization of the magnetohydrodynamic equations in Elsasser form.