On the robustness of multiscale hybrid-mixed methods

Abstract

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.

The multiscale hybrid mixed method in general polygonal meshes

“…The global problem needs for its construction the solution of local problems that fulfill the role of upscaling the under-mesh structures. Introduced and analysed in [24,2,31] for the Laplace (Darcy) equation, the MHM method has been further extended to other elliptic problems in [25,23] as well as to mixed and hyperbolic models in [3] and [29], respectively. See also [26] for an abstract setting for the MHM method.…”

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

“…This strategy makes the resulting numerical algorithm particularly attractive for use in parallel computing environments. The method has been analyzed in [3,4,5], and extended to reactive-advective dominated problems [6], linear elasticity [7], Stokes problems [8], and Maxwell equations [9].…”

A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems

“…We consider the periodic test-case studied in [39] (and also in [43]). We let d " 2, and Ω be the unit square.…”

“…On general polytopal meshes, the literature on multiscale methods is more scarce. For constructions in the spirit of the msFEM, one can cite the msFEM à la Crouzeix-Raviart of [39,40], the so-called Multiscale Hybrid-Mixed (MHM) [4,43] approach, and the (polynomial-based) method of [26] in the HDG context. Each one of these methods has its proper design, but they all share the same construction principles: they are based, more or less directly, on oscillatory basis functions that solve local Neumann problems with polynomial boundary data, and result in global systems (posed on the coarse mesh) that can be expressed in terms of face unknowns only.…”