STABLE AND EFFICIENT PETROV―GALERKIN METHODS FOR A KINETIC FOKKER―PLANCK EQUATION

Julia Brunken, Kathrin Smetana

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.

Original languageEnglish
Pages (from-to)157-179
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume60
Issue number1
DOIs
StatePublished - 2022

Keywords

  • Petrov-Galerkin method
  • inf-sup stability
  • kinetic Fokker-Planck equation
  • well-posedness

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