Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations

Abstract

We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the (inhomogeneous) kinetic Fokker--Planck (kFP) equations, a class of evolution PDEs with degenerate diffusion. The proposed methods mimic Villani's framework of enhanced quadratic forms [23], yielding a coercive bilinear form in an exponentially weighted norm that admits a spectral gap/Poincaré inequality despite the degeneracy. The problem is formulated as a fourth-order-in-space evolution PDE on the whole space $\mathbb{R}^{d}\times\mathbb{R}^d$. The spatial discretisation employs continuous piecewise polynomial finite element spaces on simplicial and/or box-type meshes comprising both finite and ``infinite'' elements, while nonconformity is handled by numerical fluxes in the spirit of $C^0$ interior penalty ($C^0$-IP) methods. The analysis requires new polynomial inverse trace inequalities in exponentially weighted norms for simplicial, box-type, and semi-infinite prismatic elements, which are proved for a broad class of exponential weights and are of independent interest. Coercivity of the Galerkin method then leads to exponential convergence to equilibrium via an exponentially weighted Poincaré inequality. We further develop a fully discrete scheme by coupling the spatial discretisation with an $hp$-version discontinuous Galerkin time-stepping method of arbitrary order and establish the same exponential convergence. The proposed methods preserve the total mass and exhibit \emph{provably} exponential convergence to equilibrium, making them well suited for long-time kFP simulations. Numerical experiments validate the theoretical results and demonstrate the convergence behaviour of the proposed methods.