  We consider a variant of the conventional MsFEM approach with enrichments
based on Legendre polynomials, both in the bulk of mesh elements and on their
interfaces. A convergence analysis of the approach is presented. Residue-type a
posteriori error estimates are also established. Numerical experiments show a
significant reduction in the error at a limited additional off-line cost. In
particular, the approach developed here is less prone to resonance errors in
the regime where the coarse mesh size $H$ is of the order of the small scale
$\varepsilon$ of the oscillations.
