FIRST-ORDER SYSTEM LEAST-SQUARES FOR THE STOKES EQUATIONS, WITH APPLICATION TO LINEAR ELASTICITY

Following our earlier work on general second-order scalar equations, h ere we develop a least-squares functional for the two-and three-dimens ional Stokes equations, generalized slightly by allowing a pressure te rm in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ell ipticity in an H-1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimate s for finite element spaces in this norm and optimal algebraic converg ence estimates for multiplicative and additive multigrid methods appli ed to the resulting discrete systems. Both estimates are naturally uni form in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous res ult for the Dirichlet problem for linear elasticity, where we obtain t he more substantive result that the estimates are uniform in the Poiss on ratio.