Equations, with Application to Linear Elasticity

Department of Mathematics

University of Southern California

1042 W. 36th Place

DRB-155

Los Angeles, CA 90089-1113

Program in Applied Mathematics

Campus Box 526

University of Colorado at Boulder

Boulder, CO 80309-0526

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 H^{1}
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.