Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Hermann Brunner, Penny J. Davies, Dugald B. Duncan

Research output: Contribution to journalArticle

21 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)856-881
Number of pages26
JournalIMA Journal of Numerical Analysis
Volume29
Issue number4
DOIs
Publication statusPublished - Oct 2009

Keywords

  • Discontinuous Galerkin approximations
  • Global convergence
  • Local superconvergence
  • Volterra integral equations of the first kind

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