Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
Provides a theoretical foundation for improving finite element discretizations of multiscale PDEs, addressing accuracy issues in homogenization and wave propagation.
The paper rigorously justifies the exponential decay of fine-scale correctors in variational multiscale stabilization for finite element methods, enabling localization to local cell problems. This eliminates scale-dependent pre-asymptotic effects such as poor L^2 approximation in homogenization and pollution in high-frequency scattering.
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.