A combined finite element and oversampling multiscale Petrov-Galerkin method for the multiscale elliptic problems with singularities

In this paper, we construct a combined finite element and oversampling multiscale Petrov-Galerkin method (FE-OMsPGM) to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of subsurface flow, singularities lie in the porous media with channelized features, or in near-well regions since the solution behaves like the Green function. The basic idea of FE-OMsPGM is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the Petrov-Galerkin version of oversampling multiscale finite element method (OMsPGM) on a coarse mesh of the other part. The transmission condition across the FE-OMsPG interface is treated by the penalty technique. The FE-OMsPGM takes advantages of the FEM and OMsPGM, which uses much less DOFs than the standard FEM and may be more accurate than the OMsPGM for problems with singularities. Although the error analysis is carried out under the assumption that the oscillating coefficients are periodic, our method is not restrict to the periodic case. Numerical examples with periodic and random highly oscillating coefficients, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method.