HYPOCOERCIVITY-COMPATIBLE FINITE ELEMENT METHODS FOR THE LONG-TIME COMPUTATION OF KOLMOGOROV'S EQUATION

摘要

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.