Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes

TL;DR

This paper derives sharp pointwise-in-time error bounds for numerical schemes solving semilinear and quasilinear fractional subdiffusion equations with initial singularities, using graded meshes and various discretizations.

Contribution

It introduces new error bounds for time-fractional equations on graded meshes, accommodating initial singularities and general discretizations, with comprehensive theoretical and numerical analysis.

Findings

Sharp error bounds on graded meshes for fractional equations.

Validation of theoretical results through numerical experiments.

Applicability to both semi-discrete and fully discrete schemes.

Abstract

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^\sigma$ for any fixed $\sigma \in (0,1) \cup (1,2)$. The L1 scheme in time is combined with a general class of discretizations for the semilinear term. For such discretizations, we obtain sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading. Both semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretcal findings are illustrated by numerical experiments.