Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion

TL;DR

This paper develops a numerical scheme for the fractional Fokker-Planck equation with two-scale diffusion, combining L1 time discretization and finite element methods, validated through numerical experiments.

Contribution

It introduces the first derivation of the two-scale diffusion fractional Fokker-Planck equation and provides an optimal error analysis for its numerical approximation.

Findings

The scheme achieves optimal spatial and temporal error estimates.

Numerical experiments confirm the scheme's effectiveness.

The derivation extends fractional Fokker-Planck models to two-scale diffusion contexts.

Abstract

Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker-Planck equation with two-scale diffusion from the L\'evy process framework, and then the fully discrete scheme is built by using the $L_{1}$ scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.