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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels


Author: Kassem Mustapha
Journal: Math. Comp. 82 (2013), 1987-2005
MSC (2010): Primary 65-XX
Published electronically: April 18, 2013
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Abstract: We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use an $ h$-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-uniformly graded meshes. Our theoretical results are numerically validated in a sample of test problems.


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Additional Information

Kassem Mustapha
Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
Email: kassem@kfupm.edu.sa

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02689-0
PII: S 0025-5718(2013)02689-0
Keywords: Integro-differential equation, weakly singular kernel, smooth kernel, DG time-stepping, error analysis, variable time steps
Received by editor(s): January 22, 2011
Received by editor(s) in revised form: November 18, 2011, and January 31, 2012
Published electronically: April 18, 2013
Additional Notes: Support of the KFUPM through the project SB101020 is gratefully acknowledged.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.