An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media

TL;DR

This paper introduces an online adaptive multiscale discontinuous Galerkin method that iteratively enhances solution accuracy in heterogeneous media by adding basis functions based on local residuals, ensuring convergence independent of physical scales.

Contribution

The paper develops a novel online adaptive basis function approach within the DG framework for multiscale problems, improving accuracy and convergence over traditional offline methods.

Findings

The method achieves accurate solutions with fewer offline computations.

Iterative basis enrichment converges independently of physical scales.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

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.