First-Order System Least Squares for Second-Order
Partial Differential Equations: Part II

Z. Cai
Department of Mathematics
University of Southern California
1042 W. 36th Place
Los Angeles, CA 90089-1113

T. A. Manteuffel and S. F. McCormick
Program in Applied Mathematics
Campus Box 526
University of Colorado at Boulder
Boulder, CO 80309-0526


This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in n = 2 or 3 dimensions as a system of first-order equations. In part I [11] a similar functional was developed and shown to be elliptic in the H(div)xH1 norm and to yield optimal convergence for finite element subspaces of H(div)xH1. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the (H1)n+1 norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of (H1)n+1. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation based approaches, the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.

Contributed April 24, 1995.