Uniform stability analysis of Austin, Manteuffel and McCormick finite elements and fast and robust iterative methods for the Stokes‐like equations

Abstract

AbstractIn this paper, we discuss uniformly stable discretizations and fast solvers for the steady and semi‐discrete time‐dependent Stokes equations, which will be called the Stokes‐like equations. We study two pairs of conforming finite elements that are uniformly accurate with respect to all relevant parameters in the equations, one of which is the Scott–Vogelius (RAIRO Modél. Math. Anal. Numér. 1985; 19:111–143) element and the other of which is the finite element pair by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140). We prove that the finite element pair by Austin, Manteuffel and McCormick is uniformly accurate and we design and analyze the fast and robust solution techniques for the linear algebraic systems arising from the discretizations of the Stokes‐like equations as well. Our method is based on the augmented Lagrangian Uzawa iterative methods (Augmented Lagrangian Methods. North‐Holland: Amsterdam, 1983) and the multigrid methods designed by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140) for the linear algebraic equations from the discretizations of the differential operator ρ2I−κ2Δ−µ2∇div. We prove our methods are robust with respect to all relevant parameters that appear in the equations and also the mesh size, thereby, achieve the fast and robust methods for the solutions to the Stokes‐like equations. In particular, the convergence analysis for the multigrid methods designed by Austin et al., has been posed as an open question (Numer. Linear Algebra Appl. 2004; 11:115–140), which is resolved in this paper. Copyright © 2009 John Wiley & Sons, Ltd.