Online adaptive algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method

TL;DR

This paper introduces an online adaptive enrichment algorithm for the CEM-GMsDGM, which adaptively constructs basis functions using residuals and oversampling to rapidly improve multiscale solution accuracy.

Contribution

It develops a novel online basis enrichment strategy within CEM-GMsDGM, enabling fast convergence and effective error reduction through adaptive basis construction.

Findings

Fast convergence from multiscale to fine-scale solutions

Error reduction depends on oversampling region size

Numerical results confirm efficiency and effectiveness

Abstract

In this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method (CEM-GMsDGM). Combining the technique of oversampling, one makes use of the information of the current residuals to adaptively construct basis functions in the online stage to reduce the error of multiscale approximation. A complete analysis of the method is presented, which shows the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling regions and the number of oversampling layers. Further, the convergence rate of the enrichment algorithm depends on a factor of exponential decay regarding the number of oversampling layers and a user-defined parameter. Numerical results are provided to demonstrate the effectiveness and efficiency of the proposed online adaptive algorithm.