  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 but
including coarse finite element functions. We suggest, by contrast, to perform
local computations with the additional constraint that trial and test functions
are linear independent from coarse finite element functions. This approach
re-interprets 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 a MsFEM with oversampling.
