A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations

We consider multigrid methods for discontinuous Galerkin (DG) H(div, Ω)-conforming discretizations of the Stokes equation. We first describe a simple Uzawa iteration for the solution of the Stokes problem, which requires a solution of a nearly incompressible linear elasticity problem on every iteration. Then, based on special subspace decompositions of H(div, Ω), as introduced in [J. Schöberl. Multigrid methods for a parameter dependent problem in primal variables. Numerische Mathematik, 84(1):97–119, 1999], we analyze variable V-cycle and W-cycle multigrid methods with nonnested bilinear forms. We prove that these algorithms are robust, and their convergence rates are independent of the material parameters such as Poisson ratio and of the mesh size.