Algebraic Multigrid for Stokes Equations

Abstract

A method is investigated for solving stationary or time-dependent discrete Stokes equations. It uses one of the standard flavors of algebraic multigrid for coupled partial differential equations, which, however, is not applied directly to the linear system stemming from discretization, but to an equivalent system obtained with a simple algebraic transformation (which may be seen as a form of preconditioning in the literal sense). A two-grid analysis is provided, showing that the eigenvalues of the preconditioned matrix are within a region of the complex plane that is both bounded and away from the origin, independently of the mesh or grid size, as well as of other main problem parameters. On the other hand, whereas the approach can in principle be combined with any type of algebraic multigrid scheme, an investigation of the properties of the coarse grid matrices reveals that plain aggregation has to be preferred to maintain nice two-grid convergence at coarser levels. Eventually, numerical experiments are reported showing that the resulting method is both robust and cost effective, being significantly faster than a state-of-the-art competitor which combines MINRES with optimal block diagonal preconditioning.