  We study the numerical approximation of advection-diffusion equations with
highly oscillatory coefficients and possibly dominant advection terms by means
of the Multiscale Finite Element Method. The latter method is a now classical,
finite element type method that performs a Galerkin approximation on a
problem-dependent basis set, itself pre-computed in an offline stage. The
approach is implemented here using basis functions that locally resolve both
the diffusion and the advection terms. Variants with additional bubble
functions and possibly weak inter-element continuity are proposed. Some
theoretical arguments and a comprehensive set of numerical experiments allow to
investigate and compare the stability and the accuracy of the approaches. The
best approach constructed is shown to be adequate for both the diffusion- and
advection-dominated regimes, and does not rely on an auxiliary stabilization
parameter that would have to be properly adjusted.
