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A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

Zhaonan Dong, Emmanuil H. Georgoulis, Philip J. Herbert

TL;DR

The paper addresses the challenge of long-time stability for Kolmogorov-type equations with degenerate diffusion by designing a hypocoercivity–exploiting stabilised finite element method. It combines an $A$-weighted weak formulation with streamline-upwinded Petrov–Galerkin stabilization to achieve a spectral gap in a strengthened norm, enabling robust decay to equilibrium. The authors present both spatially discrete and fully discrete (discontinuous Galerkin in time) analyses, proving stability and a priori error estimates, and they validate the theory with numerical experiments that exhibit numerical hypocoercivity and exponential convergence. The work reduces the effective computational cost relative to higher-order, fourth-order-like approaches while delivering provable long-time behavior and optimal convergence rates, making it practically relevant for kinetic-type simulations with degenerate diffusion.

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.

A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

TL;DR

The paper addresses the challenge of long-time stability for Kolmogorov-type equations with degenerate diffusion by designing a hypocoercivity–exploiting stabilised finite element method. It combines an -weighted weak formulation with streamline-upwinded Petrov–Galerkin stabilization to achieve a spectral gap in a strengthened norm, enabling robust decay to equilibrium. The authors present both spatially discrete and fully discrete (discontinuous Galerkin in time) analyses, proving stability and a priori error estimates, and they validate the theory with numerical experiments that exhibit numerical hypocoercivity and exponential convergence. The work reduces the effective computational cost relative to higher-order, fourth-order-like approaches while delivering provable long-time behavior and optimal convergence rates, making it practically relevant for kinetic-type simulations with degenerate diffusion.

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.
Paper Structure (16 sections, 5 theorems, 110 equations, 3 figures)

This paper contains 16 sections, 5 theorems, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A special weak formulation
  3. The hypocoercive structure of a
  4. A streamline-upwinded Petrov-Galerkin formulation
  5. Coercivity in SUPG-norm
  6. Error estimate
  7. A hypocoercivity-exploiting SUPG method
  8. Numerical hypocoercivity
  9. Error estimate
  10. A fully discrete hypocoercivity-exploiting method
  11. Numerical hypocoercivity
  12. Error estimate
  13. Numerical experiments
  14. Semi-discrete error
  15. Fully-discrete error
  16. ...and 1 more sections

Key Result

Lemma 4.2

\newlabellem:coercivity0 For all $w\in V_h$, we have when $\alpha = (8\delta)^{-1}$, $\beta =(24\delta^2)^{-1}$, $\gamma=(64\delta^3)^{-1}$, whereby for each $T\in\mathcal{T}$, where $C_{inv}>0$ the trace-inverse inequality constant depending only on the shape-regularity of the mesh, viz., $\|{v}\|_{{\partial T}}\le C_{inv}ph_T^{-\frac{1}{2}}\|{v}\|_{{T}}$, valid for $v\in \mathbb{P}_p(T)$, $T\

Figures (3)

  • Figure 1: Experiment for the semi-discrete error.
  • Figure 2: Experiment for the fully-discrete error.
  • Figure 3: Convergence to equilibrium for different values of the discretization parameters.

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 3.1: $\tau\equiv 0$
  • Remark 4.1
  • Lemma 4.2: (Hypo)coercivity
  • Proof 1
  • Remark 4.3
  • Lemma 4.4
  • Proof 2
  • Theorem 4.5
  • Proof 3
  • ...and 9 more