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

Kopteva, Natalia

doi:10.1090/mcom/3410

Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions
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Math. Comp. 88 (2019), 2135-2155 Request permission

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.

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