A Combined Finite Element and Multiscale Finite Element Method for the Multiscale Elliptic Problems

Abstract

The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inefficient in some portions of the computational domain, e.g., near long narrow channels inside the domain due to the fact that the high-conductivity features cannot be localized within a coarse-grid block, or in a near-well region since the solution behaves like the Green function there. In this paper we develop a combined finite element and multiscale finite element method (FE-MsFEM), which deals with such portions by using the standard finite element method on a fine mesh and the other portions by the oversampling MsFEM. The transmission conditions across the FE-MSFE interface is treated by the penalty technique. To illustrate this idea, a rigorous error analysis for this FE-MsFEM is given under the assumption that the diffusion coefficient is periodic. Numerical experiments are carried out for the elliptic equations with periodic and random highly oscillating coefficients, as well as multiscale problems with high contrast channels or well singularities, to demonstrate the accuracy and efficiency of the proposed method.