Robust preconditioners for the high-contrast Stokes equation

Abstract

We consider the Stokes equation with high-contrast viscosity coefficients. 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. (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. We address the rigorous justification of the solver methods, p-Uzawa and p-Minres, used in Aksoylu and Unlu (2013), and compare the results with additional solver method, Schur complement reduction (SCR). We observe that our preconditioner is only contrast size robust under the p-SCR solver. The inexact p-Uzawa solver remains to be the best choice for the most effective performance of our preconditioner as we observe contrast size and mesh size robustness simultaneously in this case. As the contrast size grows asymptotically, we prove and numerically demonstrate that the inexact p-Uzawa solver converges to the exact one. Finally, we show that our preconditioner is contrast size and mesh size robust under p-Minres when the Schur complement solve is accurate enough.