Stable approximation of the advection-diffusion equation using the invariant measure

TL;DR

This paper introduces a novel numerical method for advection-diffusion equations, leveraging the invariant measure to achieve stable, accurate, and robust finite element approximations in challenging advection-dominated regimes.

Contribution

The paper proposes a new approach based on the invariant measure for stable finite element approximation of non-coercive, advection-dominated equations, with comprehensive analysis and testing.

Findings

Unconditionally well-posed finite element approximation.

Comparable or better accuracy than classical stabilization methods.

Enhanced robustness in advection-dominated scenarios.

Abstract

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.