# Advanced Numerical Methods in the Mathematical Sciences

## Eric Chung, Chinese University of Hong Kong Adaptive Multiscale Discontinuous Galerkin Methods

### Abstract

In this talk, we present an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. The proposed error indicators are $$L_2$$-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.