Fast multigrid solution of high-order accurate multiphase Stokes problems

Abstract

A fast multigrid solver is presented for high-order accurate Stokes problems discretized by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an elementwise block Gauss–Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilization parameter — provided the parameter is suitably chosen, numerical experiment shows that (i) for steady-state Stokes problems, the convergence rate of the multigrid solver can match that of classical geometric multigrid methods for Poisson problems and (ii) for unsteady Stokes problems, the convergence rate further accelerates as the effective Reynolds number is increased. An extensive range of two- and three-dimensional test problems demonstrates the solver performance as well as high-order accuracy — these include cases with periodic, Dirichlet, and stress boundary conditions; variable-viscosity and multiphase embedded interface problems containing density and viscosity discontinuities several orders in magnitude; and test cases with curved geometries using semiunstructured meshes.