A Backward FE-MsFEM Method For Multiscale Parabolic Problems

Lots of problems in science and engineering have many scales. For these prob-lems, in order to get the required accuracy, we must ensure the size of mesh h is less than ε, which makes the calculations cost prohibitive. The multiscale finite element method has been proposed to reduce the calculations and to capture large scale charac-teristic of multiscale problems, which is accomplished by constructing the multiscale base functions from the corresponding problems. However, the multiscale finite el-ement method leads to the resonant error between the grid size and the small scale. The oversampling technique which constructs base functions in larger domain than the element of the grid can reduce the resonant error.In this thesis, we handle the multiscale parabolic problems based on a combined finite element and multiscale finite element method for the multicale elliptic problems. The emphasis of the combined method proposed is to attain the effective approxima-tion for the multiscale parabolic problems. The main idea is to integrate the merit of traditional finite element method and multiscale finite element method, as well as uti-lizing the penalty technique in the interface between different parts. In order to extend the method to apply to the parabolic problems, we firstly discrete the time by using the backward Euler method, which can transform a parabolic problem into an elliptic problem. Then we solve the problem by the combined method proposed before, we give a detailed error analysis based on some mild assumptions and demonstrate the performance of our proposed method through numerical experiments.