This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower-order terms) that are reformulated as a least-squares problem for an equivalent first-order system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers of the discrete systems.

Abstract. 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(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div)×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.Key words. least-squares discretization, multigrid, second-order elliptic problems, iterative methods AMS subject classifications. 65F10, 65F30PII. S00361429942660661. Introduction. The object of study of this paper, and its earlier companion [11], is the solution of elliptic equations (including convection-diffusion and Helmholtz equations) by way of a least-squares formulation for an equivalent first-order system. Such formulations have been considered by several researchers over the last few decades (see the historical discussion in [11]), motivated in part by the possibility of a well-posed variational principle for a general class of problems. In [11] a similar functional was developed and shown to be elliptic in the H(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div) × H 1 . In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. It is shown that the resulting functional is elliptic in the (H 1 ) n+1 norm. Direct consequences of this result are optimal approximation error estimates for standard finite element subspaces of (H 1 ) n+1 and optimal convergence of multiplicative and additive multigrid algorithms applied to the resulting discrete functionals. As an alternative to perturbation-based approaches (cf. [1,3,9,10,25,34,35]), the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits an efficient multilevel solver, historically a missing ingredient in least-squares methodology.

Following our earlier work on general second-order scalar equations, here we develop a leastsquares functional for the two-and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity ux variable and associated curl and trace equations, we are able to establish ellipticity in an H 1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for nite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, with estimates that are uniform in the Lam e constants.

We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading to the success of our proposed algorithm are the use of an adaptive projection onto coarse grids that preserves the near null space of the system matrix together with a simplified form of the correction based on the so-called γ5-Hermitian symmetry of the Dirac operator. We demonstrate that the algorithm nearly eliminates critical slowing down in the chiral limit and that it has weak dependence on the lattice volume.