Online Adaptive Local Multiscale Model Reduction for Heterogeneous Problems in Perforated Domains

TL;DR

This paper introduces an adaptive multiscale modeling approach for heterogeneous perforated domains, combining offline and online strategies to efficiently capture local features and ensure rapid convergence in complex multiscale problems.

Contribution

The paper presents a novel adaptive multiscale method with rigorous analysis, online procedures, and strategies for fast convergence in heterogeneous perforated domain problems.

Findings

Online procedure achieves rapid convergence with few iterations.

Adaptive strategies effectively improve the offline solution.

Method performs well for both small and large perforations.

Abstract

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. The main contributions of this paper are (1) the rigorous analysis of the offline approach (2) the development of the online procedures and their analysis (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.