A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

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 thetime'' 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.