Overlapping Schwarz Preconditioners for Spectral Element Discretizations of Convection-Diffusion Problems

The classical overlapping Schwarz algorithm is here extended to stabilized spectral element discretizations of convection-diffusion problems. The algorithm solves iteratively the resulting non-symmetric system of linear equations by a preconditioned GMRES method. The preconditioner is built from local convection-diffusion solvers on overlapping subdomains and from a coarse convection-diffusion solver on a coarse mesh defined by the subdomain boundaries. Several numerical experiments on test problems in the plane indicate that this algorithm retains the fast convergence rate and optimal scalability properties of classical overlapping methods for diffusion dominated problems. Fast convergence is also obtained for convection dominated problems without closed streamlines and with a moderate number of subdomains.