Multigrid relaxation methods for systems of saddle point type

In this paper, we give an overview of multigrid methods for two systems of equations, namely the Stokes equations and the incompressible poroelasticity equations. We emphasize the saddle point type aspect in these two systems and discuss their discretization on staggered and collocated grids. The basic problem is that of smoothing a system of equations that has a zero (or almost zero) block in the matrix for one of the unknowns. In particular, we discuss the coupled relaxation approach, with its “box-wise” and “line-wise” versions and distributive relaxation, that gives a decoupled system of equations for smoothing. For general systems of equations it is a challenge to design an efficient distributive relaxation scheme. This paper may help in finding one.