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

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 for a numerical example.

Pencil-Beam Approximation of Stationary Fokker--Planck

“…The identity (2.5) is obtained as in [8]. More recent proofs of similar density results are presented in [2,12]. The density used in this paper is easier to establish as we do not deal with boundaries.…”