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.
Keywords
Stokes equations, multigrid, high order, multiphase,
discontinuous Galerkin methods
