  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 inapplicable or inefficient
in some portions of the computational domain, e.g., near the domain boundary or
near long narrow channels inside the domain due to the lack of permeability
information outside of the domain or the fact that the high-conductivity
features cannot be localized within a coarse-grid block. 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. A rigorous convergence analysis for this special 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, to demonstrate the accuracy and efficiency of the proposed
method.
