This work presents a priori and a posteriori error analyses of a new multiscale hybridmixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable, which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence and uniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the natural norms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of both variables in the natural norms. Several numerical tests assess the theoretical results.

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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.

This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and k + 1, k ≥ 0, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree k + 1 posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order k + 1 and k + 2 in the broken H 1 and L 2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.