SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 173-194, January 2021.
This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov equation. Kolmogorov's equation serves as a sufficiently rich, for our purposes, model problem for kinetic-type equations and is characterized by diffusion in one of the two (or three) spatial directions only. Nonetheless, its solution typically admits decay properties to some long-time equilibrium, depending on closure by suitable boundary/decay-at-infinity conditions. A key attribute of the proposed family of methods is that they also admit similar decay properties at the (semi)discrete level for very general families of triangulations. The method construction uses ideas by the general theory of hypocoercivity developed by Villani, along with a judicious choice of numerical flux functions. These developments turn out to be sufficient to imply that the proposed finite element methods admit a priori error bounds with constants independent of the final time, despite the Kolmogorov equation's degenerate diffusion nature. Thus, the new methods provably allow for robust error analysis for final times tending to infinity. The extension to three spatial dimensions is also briefly discussed.

中文翻译：

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长期求解Kolmogorov方程的低矫顽力兼容有限元方法

SIAM数值分析学报，第59卷，第1期，第173-194页，2021年1月。
这项工作与经典Kolmogorov方程的Galerkin有限元方法族的发展有关。就我们的目的而言，Kolmogorov方程可为动力学类型方程提供足够丰富的模型问题，其特征仅在于在两个（或三个）空间方向之一上的扩散。尽管如此，它的解决方案通常允许衰减特性达到某种长时间的平衡，这取决于通过合适的边界/无穷大衰减条件的封闭。所提出的方法系列的一个关键特性是，它们也允许非常普通的三角测量族在（半）离散水平上具有相似的衰减特性。该方法的构造使用了Villani提出的一般低矫顽力理论的思想以及对数值通量函数的明智选择。这些发展足以表明，尽管Kolmogorov方程具有简并的扩散性质，但所提出的有限元方法允许常数具有与最终时间无关的先验误差范围。因此，新方法可证明地允许对趋于无穷大的最终时间进行可靠的错误分析。还简要讨论了对三个空间维度的扩展。