Numerical method for the Fokker-Planck equation of Brownian motion subordinated by inverse tempered stable subordinator with drift

TL;DR

This paper develops a numerical scheme for the Fokker-Planck equation of subordinated Brownian motion with drift, providing regularity analysis, convergence proof, and supporting numerical experiments.

Contribution

It introduces a generalized finite element method with backward Euler convolution quadrature for this class of equations, extending to more general diffusion equations.

Findings

Optimal-order convergence of the numerical scheme

Regularity analysis including maximal L^p regularity

Numerical experiments confirming theoretical results

Abstract

In this work, based on the complete Bernstein function, we propose a generalized regularity analysis including maximal $\mathrm{L}^p$ regularity for the Fokker--Planck equation, which governs the subordinated Brownian motion with the inverse tempered stable subordinator that has a drift. We derive a generalized time--stepping finite element scheme based on the backward Euler convolution quadrature, and the optimal-order convergence of the numerical solutions is established using the proven solution regularity. Further, the analysis is generalized to more general diffusion equations. Numerical experiments are provided to support the theoretical results.