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


Author: Natalia Kopteva
Journal: Math. Comp. 88 (2019), 2135-2155
MSC (2010): Primary 65M06, 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3410
Published electronically: January 23, 2019
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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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Additional Information

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
Email: natalia.kopteva@ul.ie

DOI: https://doi.org/10.1090/mcom/3410
Keywords: Fractional-order parabolic equation, L1 scheme, graded mesh
Received by editor(s): September 23, 2017
Received by editor(s) in revised form: September 25, 2017, October 18, 2017, May 19, 2018, and October 16, 2018
Published electronically: January 23, 2019
Additional Notes: The author acknowledges financial support from Science Foundation Ireland Grant SFI/12/IA/1683.
Article copyright: © Copyright 2019 American Mathematical Society