FIRST-ORDER SYSTEM LEAST SQUARES FOR THE STOKES
EQUATIONS, WITH APPLICATION TO LINEAR ELASTICITY
Z. Cai (USC), T. A. Manteuffel (UC-Boulder), and S. F. McCormick (UC-Boulder)
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 asso ciated 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 multiplicative 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 Lam{\'e} constants.