Variational Multiscale Stabilization and the Exponential Decay of Fine Scale Correctors
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 L2 approximation in homogenization problems or the pollution effect in high frequency acoustic scattering.