Preconditioned HSS Method for Finite Element Approximations of Convection-Diffusion Equations

TL;DR

This paper introduces a preconditioned Hermitian/Skew-Hermitian splitting method for efficiently solving nonsymmetric linear systems from finite element discretizations of convection-diffusion equations, with theoretical analysis and numerical validation.

Contribution

It develops a superlinear preconditioning approach with spectral analysis for FE discretizations, demonstrating optimality and effectiveness on structured and unstructured meshes.

Findings

Strong spectral clustering at unity for preconditioned matrices

Proven optimality of the PHSS method under certain conditions

Numerical experiments confirm theoretical spectral and convergence results

Abstract

A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the PHSS method is proved and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the some convergence behavior.