Hypocoercivity-compatible finite element methods for the long-time computation of Kolmogorov's equation

TL;DR

This paper introduces a new family of finite element methods for Kolmogorov's equation that preserve decay properties over long times, enabling robust error analysis for degenerate, kinetic-type equations.

Contribution

The paper develops hypocoercivity-compatible finite element methods that maintain decay properties and provide time-independent error bounds for Kolmogorov's equation.

Findings

Methods exhibit decay properties similar to continuous solutions.

Error bounds are independent of final time, ensuring long-time stability.

Extension to three spatial dimensions is feasible.

Abstract

This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov's equation. Kolmogorov's equation serves as a sufficiently rich, for our purposes, model problem for kinetic-type equations and is characterised by diffusion in one of the two (or three) spatial directions only. Nonetheless, its solution admits typically decay properties to some long time equilibrium, depending on closure by suitable boundary/decay-at-infinity conditions. A key attribute of the proposed family of methods is that they also admit similar decay properties at the (semi)discrete level for very general families of triangulations. The method construction uses ideas by the general theory of hypocoercivity developed by Villani [23], along with judicious choice of numerical flux functions. These developments turn out to be sufficient to imply that the proposed finite element methods admit a priori error bounds with constants independent of the final time, despite Kolmogorov equation's degenerate diffusion nature. Thus, the new methods provably allow for robust error analysis for final times tending to infinity. The extension to three spatial dimensions is also briefly discussed.