An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.