Partial Differential Equations: Part II

Department of Mathematics

University of Southern California

1042 W. 36th Place

DRB-155

Los Angeles, CA 90089-1113

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)xH^{1} norm and to yield optimal convergence for finite element
subspaces of H(div)xH^{1}. 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 (H^{1})^{n+1} norm. This immediately implies optimal
error estimates for finite element approximation by standard subspaces of
(H^{1})^{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.