A Combined Preconditioning Strategy for Nonsymmetric Systems
For practitioners solving nonsymmetric linear systems from finite element discretizations, this provides a practical and theoretically grounded preconditioning approach.
This work introduces a class of nonsymmetric preconditioners in a normal matrix form for GMRES, combining them with a weighted least-squares version to ensure convergence. Numerical results demonstrate the effectiveness of the proposed strategy for nonsymmetric finite element problems.
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.