Year: 2026
Type: preprint
Abstract: We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the kinetic Fokker--Planck (kFP) equations, a class of evolution partial differential equations (PDE) with \emph{degenerate} diffusion. The key idea in the method design is to mimic the framework of enhanced quadratic forms of Villani \cite{villani} to obtain a coercive bilinear form in a sufficiently strong exponentially weighted norm admitting a spectral gap/Poincaré inequality, despite the degeneracy. The problem is posed as a fourth-order evolution PDE on the whole space $\mathbb{R}^{d}\times\mathbb{R}^d$. The spatial discretization employs continuous piecewise polynomial approximation spaces on simplicial and/or box-type meshes comprising finite and infinite elements. Nonconformity is treated by constructing appropriate numerical fluxes in the spirit of $C^0$-interior penalty ($C^0$-IP) methods for fourth-order elliptic problems. The proof of coercivity requires new trace inverse inequalities for exponentially weighted norms on $d$-dimensional simplicial and box-type domains, and prismatic domains with infinite length. We prove these inequalities for a variety of exponential weights. Once coercivity is established, exponential convergence to equilibrium follows from an exponentially weighted Poincaré inequality. A fully discrete extension is obtained by coupling the spatial discretization with an $hp$-version discontinuous Galerkin time-stepping scheme. Exponential convergence to equilibrium is also proven for the fully discrete scheme. The proposed methods preserve total mass and exhibit \emph{provably} exponential convergence to equilibrium, a manifestation of `numerical hypocoercivity,' making them well suited for long-time kFP simulations. Numerical experiments validate the theoretical results and investigate the convergence behaviour of the proposed method.
Source: arXiv (Cornell University)
Authors: Dong, Zhaonan, Georgoulis, Emmanuil H.
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166696710