A Combined Multiscale Finite Element Method Based On Rectangular Mesh Generation

The mathematical modeling and efficient computation of the multiscale problem is a hot research direction in the field of Applied Mathematics and scientific comput-ing, and it has important theoretical significance and application prospects. In this paper, under the framework of the FE-MsFEM [1], we propose a combined multiscale method based on rectangular mesh generation for the efficient numerical simulation of the multiscale problems with singularities. The main idea of the FE-MsFEM is to use the standard finite element method on a fine mesh of the problematic part of the domain and using the MsFEM on a coarse mesh of the other part. The transmission condition across the FE-MsFEM interface is treated by the penalty technique. The FE-MsFEM has overcome the difficult that the MsFEM can not solve the multiscale problems with singularities effectively. However, the error generated by the mismatch between the triangulation and the period of the coefficient still exists. A direct method to remove this error is to utilize the rectangular mesh for the domain. This article presents the FE-MsFEM for rectangular mesh to match the period of the coefficient can remove the kind of error easily. Error analysis is given under the assumption that the oscillating coefficient is periodic. Compared to the triangular mesh generation, numerical experi-ments for rectangular mesh with periodic highly oscillating coefficient and multiscale problems with high contrast channels are presented to demonstrate the efficiency and accuracy of the proposed method.