In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken H 1 and L 2 norms are O(h + ε δ ) and O(h 2 + h ε δ ), respectively, and for the dual variable is O(h + ε δ ) in the H(div; ·) norm, where 0 < δ ≤ 1/2 (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions. Flows in porous media, which commonly exhibit multiple scale structures, are usually modeled by a second order elliptic problem (Darcy equation) with rough discontinuous coefficients. Such a model arises when we consider the simulation of oil reservoirs in a highly heterogenous and/or fractured media. Multiscale problems necessarily require the use of very fine meshes, which makes their numerical approximation extremely expensive. Since the pioneering work of Babuska and Osborn [9] and its extension to higher dimensions by Hou and Wu [23], multiscale numerical methods have emerged as an attractive "divide and conquer" option to handle heterogeneous problems (see [15,14,36], just to cite a few). Overall, the idea relies on basis functions specially designed to upscale submesh oscillations to an overlying coarse mesh. As a result, such numerical method becomes precise on coarse meshes. Also interesting, the multiscale basis functions can be locally computed through completely independent problems. This makes the resulting numerical algorithm particularly attractive for use in parallel computing environments.

Abstract. We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation, which can be viewed as the gradient flow of a convex but non-smooth energy with respect to the H −1 per inner product. Our numerical scheme uses implicit discretization in time and a mixed finite-element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method. We study the convergence of this scheme, both theoretically and numerically.

SUMMARYWe develop a numerical method to solvewhere the matrix a(y) = (a ij (y)) is symmetric positive deÿnite, whose entries are periodic functions of y with periodic cell Y . More speciÿcally we assume a ij ∈ C 1; ÿ ( 2 ); ÿ ¿ 0. It is also assumed that there exists positive constants a and ÿa such that a 2 6 a ij (y) i j 6 ÿa 2 for all ∈ 2 and y ∈ Y . The major goal in this paper is to develop a numerical approximation scheme on a mesh size h ¿ (or h ) with quasi-optimal approximation on L 2 and broken H 1 norms. The new method is based on asymptotic analysis and a careful treatment of the boundary corrector term. This kind of equation has applications in areas such as the study of ow through porous media and composite materials.

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