Preconditioned Hermitian and skew-Hermitian splitting method for finite element approximations of convection-diffusion equations

A two-step preconditioned iterative method based on the Hermitian and skew- Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the finite element approximation of diffusion-dominated convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to finite element 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 preconditioned Hermitian and skew-Hermitian splitting (PHSS) method is proved, and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the same convergence behavior. © 2009 Society for Industrial and Applied Mathematics.