On the finite element approximation for fractional fast diffusion equations

TL;DR

This paper develops a finite element method for fractional fast diffusion equations on bounded domains, providing convergence analysis and extending previous results to a broader class of parabolic integral equations.

Contribution

It introduces a fully Galerkin finite element approximation with convergence rates for fractional fast diffusion equations, extending prior work to more general parabolic integral equations.

Findings

Proved a priori estimates for the approximation

Established convergence rates of the finite element method

Extended analysis to a broader class of parabolic integral equations

Abstract

Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler discretization in time, a priori estimates and the rates of convergence for the approximate solutions are proved, which extends the results of \emph{Carsten Ebmeyer and Wen Bin Liu, SIAM J. Numer. Anal., 46(2008), pp. 2393--2410}. We also generalize the a priori estimates and the rates of convergence to a parabolic integral equation under the framework of \emph{Qiang Du, Max Gunzburger, Richaed B. Lehoucq and Kun Zhou, SIAM Rev., 54 (2012), no. 4, pp. 667--696.}