An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
Mustapha, Kassem and Brunner, Hermann and Mustapha, Hussein and Schoetzau, D (2011) An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type. SIAM Journal on Numerical Analysis, 49 (4). pp. 1369-1396. ISSN 0036-1429 (https://doi.org/10.1137/100797114)
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Abstract
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an $hp$-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal $hp$-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near $t=0$ caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the $h$-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems.
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Item type: Article ID code: 46548 Dates: DateEvent2011PublishedSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 23 Jan 2014 10:02 Last modified: 04 Oct 2024 00:20 URI: https://strathprints.strath.ac.uk/id/eprint/46548