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arXiv · 2506.12954

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

Natalia Kopteva·Sean Kelly
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Abstract

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^σ$ for any fixed $σ\in (0,1) \cup (1,2)$. 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.

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Natalia Kopteva, Sean Kelly. 2026-01-23. Pointwise-in-time error bounds for semilinear and quasilinear fractional subdiffusion equations on graded meshes. https://arxiv.org/abs/2506.12954

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