First-Order System Least Squares for Second-Order Partial Differential Equations: Part II
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 [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the $H(÷v) × H^1$ norm and to yield optimal convergence for finite element subspaces of $H(÷v) × H^1$. 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.