Stabilisation de problèmes non coercifs via une méthode numérique utilisant la mesure invariante

We study 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 with the original equation. We show that the approach allows for an unconditionally well-posed finite-element approximation. Two variants of the approach are studied. One of them is stable, and as accurate as a classical stabilization approach. This suggests a possible general strategy, applicable to a large class of non-coercive problems.