A Multiscale Finite Element Method for Pdes with Oscillatory Coefficients1

We present a multiscale finite element method (MFEM) for solving partial differential equations with highly oscillatory coefficients. MFEM provides a general and systematic framework for effectively capturing the large scale solutions without directly resolving the small scale details. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. The construction of the base functions is fully decoupled from element to element; thus the method is naturally adapted to parallel computing. We briefly overview the analysis of the method, emphasizing on the “scale resonance” effect and its resolution—an over-sampling technique. Then, we demonstrate the accuracy and efficiency of the method through numerical experiments.