Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem
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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.References
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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
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 19-40
- MSC (2000): Primary 65M15, 65M60
- DOI: https://doi.org/10.1090/mcom/3552
- MathSciNet review: 4166451