Variational multiscale stabilization and the exponential decay of fine-scale correctors

Abstract

This paper reviews 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 <em>L²</em> approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.