Preconditioning for the time-dependent Stokes equations: stage-weighted Radau IIA and weighted discontinuous Galerkin time-stepping

Abstract

We determine field-of-values equivalent preconditioners for the time-dependent Stokes equations discretized in time by stage-weighted Radau IIA Runge--Kutta time-stepping and discontinuous Galerkin time-stepping methods. By weighing each stage of the Radau IIA Runge--Kutta method differently we obtain preconditioners that are field-of-values equivalent, with constants independent of viscosity, timestep, and mesh size, to the system matrix. Similarly, by using weighted inner products in time to define the discontinuous Galerkin time-stepping method, we again obtain field-of-values equivalent preconditioners for the resulting system matrix. Numerical experiments are performed in two and three-dimensions which demonstrate robustness of the proposed preconditioners.