Abstract. We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

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.

Summary. The finite volume element method (FVE) is a discretization techniquefor partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. This paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establish O(h) estimates of the error in a discrete H 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved to O(h2). Results on the effects of numerical integration are also included.