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 [11] a similar functional was developed and shown to be elliptic in the H(div) \Theta H 1 norm and to yield optimal convergence for finite element subspaces of H(div) \Theta 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 ap..