Optimal error analysis of a FEM for fractional diffusion problems by energy arguments

TL;DR

This paper develops optimal error bounds for a finite element method solving time-fractional diffusion equations, overcoming challenges posed by low regularity of solutions through advanced energy analysis and integral operators.

Contribution

It introduces a novel energy analysis approach using integral operators and fractional Leibniz formulas to achieve optimal error estimates for FEM in fractional diffusion problems.

Findings

Optimal error bounds in L2 and H1 norms

Quasi-optimal bounds in L-infinity norm

Numerical experiments confirming theoretical results

Abstract

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, {\it a priori} optimal error bounds in $L^2(\Omega)$-, $H^1(\Omega)$-norms, and a quasi-optimal bound in $L^{\infty}(\Omega)$-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $t^m$ type of weights to take care of the singular behavior of the continuous solution at $t=0.$ The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.