First-Order System Least Squares for the Stokes and Linear Elasticity Equations: Further Results

Abstract

First-order system least squares (FOSLS) was developed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741; SIAM J. Numer. Anal., 35 (1998), pp. 320--335] for Stokes and elasticity equations. Several new results for these methods are obtained here. First, the inverse-norm FOSLS scheme that was introduced but not analyzed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741] is shown to be continuous and coercive in the L2 norm. This result is shown to hold for pure displacement or pure traction boundary conditions in two or three dimensions, and for mixed boundary conditions in two dimensions. Next, the FOSLS schemes developed in [ SIAM J. Numer. Anal., 35 (1998), pp. 320--335] are applied to the pure displacement problem in planar and spatial linear elasticity byeliminating the pressure variable in the FOSLS formulations of [SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741]. The idea of two-dimensional variable rotation is then extended to three dimensions to make the intervariable coupling subdominant (uniformly so in the Poisson ratio for elasticity). This decoupling ensures optimal (uniform) performance of finite element discretization and multigrid solution methods. It also allows special treatment of the new trace variable, which corresponds to the divergence of velocity in the case of Stokes, so that conservation can be easily imposed, for example. Numerical results for various boundary value problems of planar linear elasticity are studied in a companion paper [SIAM J. Sci. Comput., 21 (2000), pp. 1706--1727].