Stable and efficient Petrov-Galerkin methods for a kinetic Fokker-Planck equation

TL;DR

This paper introduces a stable Petrov-Galerkin discretization for the kinetic Fokker-Planck equation that guarantees uniform stability and well-posedness, with efficient computation and confirmed effectiveness through numerical tests.

Contribution

The paper develops a novel Petrov-Galerkin method ensuring uniform inf-sup stability directly from the variational formulation for the kinetic Fokker-Planck equation.

Findings

The method achieves a discrete inf-sup constant equal to the continuous one, independent of mesh size.

Stable basis functions can be computed via low-dimensional elliptic problems.

Numerical experiments confirm the method's practicality and performance.

Abstract

We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck equation constructed in such a way that uniform inf-sup stability can be inferred directly from the variational formulation. Inspired by well-posedness results for parabolic equations, we derive a lower bound for the dual inf-sup constant of the Fokker-Planck bilinear form by means of stable pairs of trial and test functions. The trial function of such a pair is constructed by applying the kinetic transport operator and the inverse velocity Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin projection we choose an arbitrary discrete test space and then define the discrete trial space using the same application of transport and inverse Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs of the continuous level and we obtain a well-posed numerical method with a discrete inf-sup constant identical to the inf-sup constant of the continuous problem independently of the mesh size. We show how the specific basis functions can be efficiently computed by low-dimensional elliptic problems, and confirm the practicability and performance of the method with numerical experiments.