A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

TL;DR

This paper introduces a novel stabilised finite element method for the Kolmogorov equation that leverages hypocoercivity to ensure robust long-term stability despite degenerate diffusion, supported by theoretical proofs and numerical validation.

Contribution

The paper develops a stabilised finite element approach that exploits hypocoercivity, enabling spectral gap analysis for degenerate diffusion equations like Kolmogorov.

Findings

The method exhibits spectral gap properties in a stronger stabilisation norm.

Stability and a priori error bounds are proven for both spatially discrete and fully discrete schemes.

Numerical experiments confirm the theoretical stability and accuracy results.

Abstract

We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.