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 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} and, thus, to arrive at a coercive bilinear form in a sufficiently strong exponentially weighted norm, admitting spectral gap/Poincar\'e inequality, despite the degeneracy. The problem is posed as a fourth order evolution PDE on the whole space ℝd×ℝd. The spatial discretization employs continuous piecewise polynomial approximation spaces, subordinate to simplicial and/or box-type meshes comprising finite, as well as `infinite' elements. The non-conformity is treated by constructing appropriate numerical fluxes in the spirit of C0-interior penalty (C0-IP) methods for fourth order elliptic problems. The proof of coercivity of the method requires new trace inverse inequalities for exponentially weighted norms on d-dimensional simplicial and box-type domains, and also for prismatic domains with infinite length. We prove weighted trace inverse inequalities in all cases and for a variety of exponential weights. Once coercivity is established for the Galerkin method, the proof of exponential convergence to the equilibrium state follows by employing an exponentially weighted Poincar\'e inequality. A fully discrete extension of the method is also considered by coupling the non-standard spatial discretization with an hp-version discontinuous Galerkin time-stepping scheme. The exponential convergence to the equilibrium state for the fully discrete scheme is also proven. The proposed methods preserve the total mass and exhibit \emph{provably} exponential convergence to the equilibrium state, a manifestation of `numerical hypocoercivity,' making them well suited for long-time kFP simulations. A series of numerical experiments is presented to validate the theoretical results and investigate the convergence behaviour of the proposed method.
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BibTeX
@misc{dong2026hypocoercivitypreserving,
title = {{Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations}},
author = {Zhaonan Dong and Emmanuil H Georgoulis},
howpublished = {HAL},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.29621},
eprint = {2606.29621},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05672281v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations
AU - Zhaonan Dong
AU - Emmanuil H Georgoulis
PY - 2026
DA - 2026/06/28
PB - HAL
DO - 10.48550/arxiv.2606.29621
UR - https://inria.hal.science/hal-05672281v2
AB - 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 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} and, thus, to arrive at a coercive bilinear form in a sufficiently strong exponentially weighted norm, admitting spectral gap/Poincar\'e inequality, despite the degeneracy. The problem is posed as a fourth order evolution PDE on the whole space ℝ^d×ℝ^d. The spatial discretization employs continuous piecewise polynomial approximation spaces, subordinate to simplicial and/or box-type meshes comprising finite, as well as `infinite' elements. The non-conformity 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 of the method requires new trace inverse inequalities for exponentially weighted norms on d-dimensional simplicial and box-type domains, and also for prismatic domains with infinite length. We prove weighted trace inverse inequalities in all cases and for a variety of exponential weights. Once coercivity is established for the Galerkin method, the proof of exponential convergence to the equilibrium state follows by employing an exponentially weighted Poincar\'e inequality. A fully discrete extension of the method is also considered by coupling the non-standard spatial discretization with an hp-version discontinuous Galerkin time-stepping scheme. The exponential convergence to the equilibrium state for the fully discrete scheme is also proven. The proposed methods preserve the total mass and exhibit \emph{provably} exponential convergence to the equilibrium state, a manifestation of `numerical hypocoercivity,' making them well suited for long-time kFP simulations. A series of numerical experiments is presented to validate the theoretical results and investigate the convergence behaviour of the proposed method.
ER -