Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

TL;DR

This paper introduces a variational multiscale stabilization method for finite element discretizations of multiscale PDEs, leveraging exponentially decaying correctors to improve accuracy and eliminate pre-asymptotic effects.

Contribution

It rigorously justifies the exponential decay of fine-scale correctors and develops a localized Petrov-Galerkin stabilization approach for multiscale PDEs.

Findings

Exponential decay of fine-scale correctors is rigorously proven.

Localization of correctors to local cell problems is effective.

Stabilization improves approximation quality for oscillatory problems.

Abstract

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.