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/Poincar\'{e} 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 Poincar\'{e} 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.