Uniform stability for a spatially-discrete, subdiffusive Fokker-Planck equation

TL;DR

This paper establishes uniform stability estimates for a spatially discrete Galerkin solution of a fractional Fokker-Planck equation, including inhomogeneous terms and gradient bounds, simplifying error analysis.

Contribution

It provides the first uniform stability bounds in the fractional diffusion exponent and extends stability results to inhomogeneous terms and gradient estimates.

Findings

Stability constants are bounded uniformly in the fractional exponent .

Inhomogeneous terms are incorporated into the stability estimates.

Simplified proofs for finite element error bounds are achieved.

Abstract

We prove stability estimates for the spatially discrete, Galerkin solution of a fractional Fokker-Planck equation, improving on previous results in several respects. Our main goal is to establish that the stability constants are bounded uniformly in the fractional diffusion exponent $\alpha\in(0,1]$. In addition, we account for the presence of an inhomogeneous term and show a stability estimate for the gradient of the Galerkin solution. As a by-product, the proofs of error bounds for a standard finite element approximation are simplified.