First-Order System Least Squares for the Stokes
Equations, with Application to Linear Elasticity

Z. Cai
Department of Mathematics
University of Southern California
1042 W. 36th Place
DRB-155
Los Angeles, CA 90089-1113

T. A. Manteuffel and S. F. McCormick
Program in Applied Mathematics
Campus Box 526
University of Colorado at Boulder
Boulder, CO 80309-0526

Abstract

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for m ultiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, with estimates that are uniform in the Lamè constants.


Contributed April 24, 1995.