Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

TL;DR

This paper analyzes the error of an L2-type numerical method on graded meshes for solving semilinear subdiffusion equations with fractional derivatives, providing sharp error bounds despite initial singularities.

Contribution

It offers the first sharp pointwise-in-time error bounds for an L2-type method on graded meshes applied to semilinear subdiffusion equations with fractional derivatives.

Findings

Sharp error bounds established for graded meshes

Error estimates accommodate initial singularity behavior

Method achieves high accuracy for fractional subdiffusion problems

Abstract

A semilinear 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 an L2-type discretization of order $3-\alpha$, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.