Interior Penalty Discontinuous Galerkin Methods For The Multiscale Elliptic Problems

Many problems in science and engineering take on multiscale characteristic. For these problems, when the small scale ε is small, in order to get the required accura-cy, we must ensure the size of mesh h is less than ε which makes the computational costs prohibitive. The multiscale finite element method has been proposed to cap-ture the large scale solutions of multiscale elliptic equations. This is accomplished by constructing the multiscale base functions from the local solutions of elliptic operator. However this leads to the resonant error. We introduce an oversampling technique to reduce the resonant error which constructs base function in larger domain than mesh grid. A consequence of the oversampling method is that the resulting finite element method is no longer conforming.In this paper we introduce interior penalty discontinuous Galerkin method for solving elliptic problems with multiple scales. The proposed method may be seen as a generalization of the multiscale finite element method. in fact, this method is derived by combining multiscale finite element method and dincontinuous element method, by using the approximation spaces for the multiscale FE method and relaxing the continu-ity constraints at the inter-element interfaces. In this paper we give a detailed deriva-tion and analysis of the error. Finally we demonstrate the performance of our proposed method through numerical experiment.