## 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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Natalia Kopteva
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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.## 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): 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.
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 2135-2155 - MSC (2010): Primary 65M06, 65M15, 65M60
- DOI: https://doi.org/10.1090/mcom/3410
- MathSciNet review: 3957889