A Combined Preconditioning Strategy for Nonsymmetric Systems

TL;DR

This paper introduces a new combined preconditioning strategy for nonsymmetric systems, enhancing GMRES efficiency by integrating additive Schwarz methods with weighted least-squares approaches, supported by theoretical analysis and numerical validation.

Contribution

It proposes a novel combined preconditioning approach for nonsymmetric matrices, with theoretical convergence analysis and practical implementation insights.

Findings

The combined preconditioner converges for nonsymmetric problems.

Numerical results demonstrate improved GMRES performance.

The approach effectively handles finite element discretizations.

Abstract

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.