An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels
Year: 2021
Author: Shuangshuang Li, Lina Wang, Lijun Yi
East Asian Journal on Applied Mathematics, Vol. 11 (2021), Iss. 1 : pp. 20–42
Abstract
An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels is studied. In contrast to the methods reducing second-order problems to first-order systems, here the CG and DG methodologies are combined to directly discretise the second-order derivative. An a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical results are illustrated by numerical examples.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.020520.120620
East Asian Journal on Applied Mathematics, Vol. 11 (2021), Iss. 1 : pp. 20–42
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: $hp$-version second-order Volterra integro-differential equation weakly singular kernel continuous Petrov-Galerkin method exponential convergence.