# A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data

## TL;DR

This paper develops a new stability and convergence analysis for a finite element method applied to a time-fractional Fokker-Planck equation with nonsmooth initial data, supported by numerical experiments.

## Contribution

It introduces a novel energy-based analysis for spatial discretization of the equation with low-regularity initial data, enhancing understanding of numerical stability and convergence.

## Key findings

- The analysis confirms stability and convergence for the spatial discretization.
- Numerical experiments align with theoretical convergence predictions.
- The method effectively handles nonsmooth initial data.

## Abstract

We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker--Planck equation in a convex polyhedral domain, using continuous, piecewise-linear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.