Numerical Study of the High-Contrast Stokes Equation and Its Robust Preconditioning

Abstract

We numerically study the Stokes equation with high-contrast viscosity coefficients. The high-contrast viscosity values create complications in the convergence of the underlying solver methods. To address this complication, we construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 11:319-331, 2008). We examine the performance of our preconditioner against multigrid and provide a comparative study reflecting the effect of the underlying discretization and the aspect ratio of the mesh by utilizing the preconditioned inexact Uzawa and Minres solvers. Our preconditioner turns out to be most effective when used as a preconditioner to the inexact p-Uzawa solver and we observe contrast size and mesh size robustness simultaneously. As the contrast size grows asymptotically, we numerically demonstrate that the inexact p-Uzawa solver converges to the exact one. We also observe that our preconditioner is contrast size and mesh size robust under p-Minres when the Schur complement solve is accurate enough. In this case, the multigrid preconditioner loses both contrast size and mesh size robustness. © Springer Science+Business Media New York 2013.