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 | References | Similar Articles | Additional Information
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
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