An Analytic Basis for Multigrid Methods for Stabilized Finite Element Methods for the Stokes Problem

Zhiqiang Cai and Jim Douglas, Jr.

Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette, IN 47907-1395


Abstract

In recent years, so-called ``stabilization" techniques have been used extensively to stabilize unstable numerical methods for partial differential equations. While existing results indicate that such methods have great promise, a fast solver for the resulting algebraic equations has been missing for many such methods, possibly because of a too simple treatment of the perturbation term and the lack of symmetry of the schemes. The effect of the former is that the bilinear form is either not elliptic or not continuous with respect to norms separating velocity and pressure. The effect of the latter is that existing iterative methods cannot be applied directly. In this paper, we first describe a new absolutely stabilized finite element method for the Stokes problem, with the method being a modification of the approach of Douglas and Wang. In it, a weighted $L^2$-inner product is replaced by a discrete $H^{-1}$-inner product. Our bilinear form is then elliptic and continuous with respect to the $H^1$-norm for the velocity and the $L^2$-norm for the pressure, and an error estimate of the finite element approximation follows immediately. Then, we introduce a symmetrized form of the method which retains ellipticity and continuity with respect to the same norm. Hence, we can use any effective elliptic preconditioner associated with velocity, including one of multigrid or domain-decomposition type, along with a simple preconditioner associated with pressure, such as one of diagonal matrix type. The condition number of the preconditioned problem is then uniform in the number of unknowns.