Box-relaxation based multigrid solvers for the variable viscosity Stokes problem

This paper concerns the numerical study of a class of box-relaxation (coupled) smoothers for multigrid solvers applied to the variable viscosity Stokes problem. In particular, we address the case of multi-phase flows of non miscible fluids with a high viscosity contrast, which are particularly challenging to solve using iterative methods and need robust preconditioning strategies. Three variants of box-relaxation are presented. These are defined based on the algebraic structure of the discrete operators only and therefore can be be compatible, in principle, with most of discretization techniques. Several tests are carried out to assess the robustness of the proposed approach against the varying viscosity contrast (up to 107), mesh size and mesh stretching factor. Finally, as a practical application, the box-relaxation smoother is used in the framework of flow simulation for multi-layer coextrusion of polymeric fluids.