An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media (Communications in Computational Physics, 2018)
Offline computation is an essential component in most multiscale modelreduction techniques. However, there are multiscale problems in which offline procedureis insufficient to give accurate representations of solutions, due to the fact thatoffline computations are typically performed locally and global information is missingin these offline information. To tackle this difficulty, we develop an online localadaptivity technique for local multiscale model reduction problems. We design newonline basis functions within Discontinuous Galerkin method based on local residualsand some optimally estimates. The resulting basis functions are able to capturethe solution efficiently and accurately, and are added to the approximation iteratively.Moreover, we show that the iterative procedure is convergent with a rate independentof physical scales if the initial space is chosen carefully. Our analysis also gives a guidelineon how to choose the initial space. We present some numerical examples to showthe performance of the proposed method.