An analysis of a class of variational multiscale methods based on subspace decomposition

TL;DR

This paper analyzes a class of variational multiscale methods for numerical homogenization of elliptic PDEs with oscillating coefficients, emphasizing their relation to subspace decomposition and iterative methods without explicit scale separation.

Contribution

It introduces a simplified analysis framework for these methods using variational multiscale reformulation and subspace decomposition theory, extending prior approaches.

Findings

Methods do not require explicit scale separation

Analysis based on additive Schwarz and subspace decomposition

Provides a unified framework for understanding these multiscale methods

Abstract

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of M{\aa}lqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of M{\aa}lqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.