Navier--Stokes Equations in Rotation Form: A Robust Multigrid Solver for the Velocity Problem

Abstract

The topic of this paper is motivated by the Navier--Stokes equations in rotation form. Linearization and application of an implicit time stepping scheme results in a linear stationary problem of Oseen type. In well-known solution techniques for this problem such as the Uzawa (or Schur complement) method, a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion-reaction type must be solved to update the velocity vector field. In this paper we analyze a standard finite element method for the discretization of this coupled system, and we introduce and analyze a multigrid solver for the discrete problem. Both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed. We prove stability results and discretization error bounds for the Galerkin finite element method. We present a convergence analysis of the multigrid method which shows the robustness of the solver. Results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver.