In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, we introduce new tools of discrete functional analysis, such as discrete nonlocal Poincar\'e and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.

翻译：在本文中,我们引入并分析同质和动能L\'evy-Fokker-Planck等方程式的数值计划。离散化设计旨在维护持续模型的主要特征,如保护质量、重尾平衡和(合金)协调性特性等。我们对数值计划进行透彻的分析,并显示这个计划的指数稳定性和趋同性。与此同时,我们引入了离散功能分析的新工具,如离散的非本地波因卡尔,以及适应分数扩散的内插不平等。我们用数字模拟来说明和补充我们的理论结论。