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.

翻译：我们针对经典Kolmogorov方程提出了一种新的稳定化有限元方法。该方程是各类动力学型方程的基本模型问题，其关键特征在于退化的扩散项。所构建的稳定化方法能够使离散格式具备与偏微分方程问题相对应的"数值超松弛性"性质。具体而言，通过构造稳定化项，使得在Kolmogorov方程扩散退化的情况下，所得的"强于能量"的稳定化范数中仍能存在谱间隙，从而该方法在时间变量趋于无穷时具有可证明的稳健性。我们分别考虑了稳定化有限元方法的空间离散版本以及采用间断Galerkin时间步进的全离散版本，并在所有情形下证明了稳定性和先验误差界。数值实验验证了理论分析结果。