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Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem


Author: Natalia Kopteva
Journal: Math. Comp. 90 (2021), 19-40
MSC (2000): Primary 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3552
Published electronically: July 14, 2020
MathSciNet review: 4166451
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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. An L2-type discrete fractional-derivative operator of order $ 3-\alpha $ is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-monotonicity of this operator are established, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The 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
MR Author ID: 610720
ORCID: 0000-0001-7477-6926
Email: natalia.kopteva@ul.ie

DOI: https://doi.org/10.1090/mcom/3552
Keywords: Fractional-order parabolic equation, L2 scheme, graded temporal mesh, arbitrary degree of grading, pointwise-in-time error bounds
Received by editor(s): May 22, 2019
Received by editor(s) in revised form: March 28, 2020, and April 3, 2020
Published electronically: July 14, 2020
Additional Notes: This research was supported by Science Foundation Ireland Grant SFI/12/IA/1683.
Article copyright: © Copyright 2020 American Mathematical Society