Adaptive generalized multiscale finite element methods for H(curl)-elliptic problems with heterogeneous coefficients

TL;DR

This paper introduces an adaptive multiscale finite element method for efficiently solving H(curl)-elliptic problems in media with highly variable properties, ensuring robustness and accuracy through adaptive basis selection.

Contribution

It develops a novel adaptive generalized multiscale finite element method with an offline-online strategy and rigorous error analysis for H(curl)-elliptic problems in heterogeneous media.

Findings

Method achieves robustness against media heterogeneities.

Adaptive basis selection improves computational efficiency.

Numerical results confirm theoretical convergence and accuracy.

Abstract

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on the generalized multiscale finite element method. We will first construct a suitable snapshot space, and a dimensional reduction procedure to identify important modes of the solution. We next develop and analyze an a posteriori error indicator, and the corresponding adaptive algorithm. In addition, we will construct a coupled offline-online adaptive algorithm, which provides an adaptive strategy to the selection of offline and online basis functions. Our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media. We present several numerical results to illustrate the performance of our method.