Oversampling for the Multiscale Finite Element Method

This paper reviews standard oversampling strategies as performed in the multiscale finite element method (MsFEM). Common to those approaches is that the oversampling is performed in the full space restricted to a patch including coarse finite element functions. We suggest, by contrast, performing local computations with the additional constraint that trial and test functions be linear independent from coarse finite element functions. This approach reinterprets the variational multiscale method in the context of computational homogenization. This connection gives rise to a general fully discrete error analysis for the proposed multiscale method with constrained oversampling without any resonance effects. In particular, we are able to give the first rigorous proof of convergence for an MsFEM with oversampling.