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Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

An hp-version discontinuous galerkin method for integro-differential equations of parabolic type

Language

English

Abstract

We study the numerical solution of a class of pa rabolic 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. © 2011 Society for Industrial and Applied Mathematics.

Keywords

Exponential convergence, Finite element method, Fully discrete scheme, Hp-version DG time-stepping, Parabolic volterra integro-differential equation, Weakly singular kernel

Publication Date

2011

Source Publication Title

SIAM Journal on Numerical Analysis

Volume

49

Issue

4

Start Page

1369

End Page

1396

Publisher

Society for Industrial and Applied Mathematics

ISSN (print)

00361429

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