A Hypocoercivity-Exploiting Stabilized Finite Element Method for Kolmogorov Equation

We propose a new stabilized 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 characterized by degenerate diffusion. The stabilization is constructed so that the resulting method admits a numerical hypocoercivity property, analogous to the corresponding property of the PDE problem. More specifically, the stabilization is constructed so that a spectral gap is possible in the resulting “stronger-than-energy” stabilization norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behavior as the “time” variable goes to infinity. We consider both a spatially discrete version of the stabilized finite element method and a fully discrete version, with the time discretization realized by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.