  We consider an advection-diffusion equation that is both non-coercive and
advection-dominated. We present a possible numerical approach, to our best
knowledge new, and based on the invariant measure associated to the original
equation. The approach has been summarized in [C. Le Bris, F. Legoll and F.
Madiot, C. R. Acad. Sci. Paris, Serie I, vol. 354, 799-803 (2016)]. We show
that the approach allows for an unconditionally well-posed finite element
approximation. We provide a numerical analysis and a set of comprehensive
numerical tests showing that the approach can be stable, as accurate as, and
more robust than a classical stabilization approach.
