Discontinuous Petrov-Galerkin Finite Element Method For The Multi-scale Elliptic Problems

In this thesis we consider the elliptic problems with oscillating coefficients. Such kind of multi-scale problems have variant applications in the research fields of science and engineering. As the equations contain multi-scale characteristics, one is forced to handle the micro-scale information when using classical finite element method to solve these problems, which will cost enormous calculations to match the desired computa-tional accuracy.We introduce a new multi-scale finite element method in this paper. With the main idea of Petrov-Galerkin method, we choose an oversampling technique to obtain the base functions and employ a penalty technique to reduce the computational error produced by the discontinuity of these base functions. We name this new method by discontinuous Petrov-Galerkin multi-scale finite element method (DPG-MsFEM). Actually, our method could be also treated as a generalized discontinuous Galerkin method, or generalized Petrov-Galerkin method, for solving such multi-scale problems.We present a detailed derivation of error analysis and also provide several numer-ical examples to validate the theoretical results, which demonstrate the efficiency of DPG-MsFEM.