A new algebraic multigrid approach for Stokes problems

Standard discretizations of Stokes problems lead to linear systems of equations in saddle point form, making difficult the application of algebraic multigrid methods. In this paper, a new approach is proposed. It consists in first transforming the system by pre- and post-multiplication with simple, algebraic, sparse block triangular matrices. This is a form of pre-conditioning in the literal sense, designed to make sure that the transformed matrix is well adapted to multigrid. In particular, after transformation, all the diagonal blocks are symmetric and positive definite, and correspond to, or resemble, a discrete Laplace operator. Then, to each of these diagonal blocks is associated a prolongation that works well for it, using any relevant algebraic or geometric multigrid method. Next, a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a Galerkin coarse grid matrix. For this approach combined with damped Jacobi-smoothing, a uniform two-grid convergence bound is derived for the global system under the assumption that the two-grid schemes for the different diagonal blocks are themselves uniformly convergent. This result is illustrated by a few examples, showing further that time-dependent problems and variable viscosity can be handled in a natural way, without requiring parameter adjustment. A numerical comparison also shows that the new approach can be more effective than state-of-the-art block preconditioning techniques.