Uniformly Stable Discontinuous Galerkin Discretization and Robust Iterative Solution Methods for the Brinkman Problem

Abstract

We consider robust iterative methods for discontinuous Galerkin (DG) $H(div,\Omega)$-conforming discretizations of the Brinkman equations. We describe a simple Uzawa iteration for the solution of this problem, which requires the solution of a nearly incompressible linear elasticity type equation with mass term on every iteration. We prove the uniform stability of the DG discretization for both problems. Then, 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 parameters in the Brinkman problem and of the mesh size. The theoretical analysis is confirmed by numerical results.