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

Zhiqiang Cai

Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette, IN 47907-1395 Tom Manteuffel and Steve McCormick Program in Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526


In this paper, we study the two-stage first-order system least-squares (FOSLS) approaches developed in [1] and [2]. First, we establish the ellipticity of the FOSLS functional at the first-stage based on $H^{-1}$ norms in [1]. Next, we consider the same approaches in [2] for the pure displacement problem in planar and spatial linear elasticity by eliminating the pressure variable in FOSLS formulations of [1]. We then extend the idea of two dimensional variable rotation to three dimensions in order to decouple the dependence of intervariables and to obtain higher-order approximation of the trace (which is corresponding to the divergence of velocity for the Stokes equations). Finally, we study a different algorithm at the second-stage as an alternative. [1] Z. Cai, T. Manteuffel, and S. McCormick, First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal, 34 (1997), pp.~1727-1741. [2] Z. Cai, T. Manteuffel, S. McCormick, and S. Parter, First-order system least squares (FOSLS) for planar linear elasticity: pure traction, SIAM J. Numer. Anal, 35 (1998), pp.~320--335.