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IMA Journal of Numerical Analysis Advance Access originally published online on July 16, 2008
IMA Journal of Numerical Analysis 2009 29(4):856-881; doi:10.1093/imanum/drn037
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Hermann Brunner{dagger}

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, Canada A1C 5S7

Penny J. Davies{ddagger}

Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK

Dugald B. Duncan§

Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

{dagger} Email: hermann{at}math.mun.ca

{ddagger} Corresponding author. Email: penny{at}maths.strath.ac.uk

§ Email: d.b.duncan{at}ma.hw.ac.uk

Received on 16 April 2007. Revised on 5 May 2008.


  Abstract

Motivated by the problem of developing accurate and stable time-stepping methods for the single-layer potential equation for acoustic scattering from a surface, we present new convergence results for piecewise polynomial discontinuous Galerkin (DG) approximations of a first-kind Volterra integral equation of convolution kernel type, where the kernel K is smooth and satisfies K(0) != 0. We show that an mth degree DG approximation exhibits global convergence of order m when m is odd and order m + 1 when m is even. There is local superconvergence of one order higher (i.e. order m + 1 when m is odd and m + 2 when m is even), but in the even order case, there is superconvergence only if the exact solution u of the equation satisfies u(m + 1)(0) = 0. We also present numerical test results which show that these theoretical convergence rates are optimal.

Key Words: Volterra integral equations of the first kind; discontinuous Galerkin approximations; global convergence; local superconvergence


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