Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

TL;DR

This paper develops an error analysis framework for L1 discretizations of fractional-derivative problems, accounting for initial singularities, and validates findings with numerical experiments.

Contribution

It introduces a simple, unified error analysis framework for L1 methods on graded and uniform meshes for fractional PDEs in multiple dimensions.

Findings

Error bounds for L1 discretizations on graded and uniform meshes

Validation of theoretical error estimates through numerical experiments

Applicability to both finite difference and finite element methods

Abstract

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.