Coarse-fine Mesh Multiscale Finite Element Methods For The Multiscale Elliptic Problems

In this paper we develop a coarse-fine mesh multiscale finite element method (CFM-MsFEM) for solving a class of second order elliptic multi-scale problems. In the literary, the oversampling technique is introduced into multiscale finite element method (MsFEM) to alleviate the resonance error which exists generally in the mul-tiscale methods. In order to construct the multiscale base, the oversamping MsFEM assumes that there is enough information available outside the research domain, which is not applicable in practice. To handle this problem, we separate the research area into two sub-domains, in which inside the domain we use the oversampling multiscale basis functions to capture the information, while, in the sub-domain which adjacent to the boundary we use the fine mesh to resolve the multiscale information. Error estimates for this special CFM-MsFEM are given under the assumption that the diffusion coeffi-cient is periodic. Numerical experiments are carried out for the elliptic equations with periodic oscillating coefficients to demonstrate the accuracy of the proposed method. Numerical experiments which deal with the multiscale problems with high contrast channels are performed to demonstrate the proposed method.