Global convergence and local superconvergence of first-kind Volterra integral equation approximations

Hermann BRUNNER, Penny J. Davies*, Dugald B. Duncan

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

3 Citations (Scopus)


We present a comprehensive convergence analysis for discontinuous piecewise polynomial approximations of a first-kind Volterra integral equation with smooth convolution kernel, examining the attainable order of (super-) convergence in collocation, quadrature discontinuous Galerkin (QDG) and full discontinuous Galerkin (DG) methods. We introduce new polynomial basis functions with properties that greatly simplify the convergence analysis for collocation methods. This also enables us to determine explicit formulae for the location of superconvergence points (i.e., discrete points at which the convergence order is one higher than the global bound) for all convergent collocation schemes. We show that a QDG method, which is based on piecewise polynomials of degree m and uses exactly m + 1 quadrature points and nonzero quadrature weights, is equivalent to a collocation scheme, and so its convergence properties are fully determined by the previous collocation analysis and they depend only on the quadrature point location (in particular, they are completely independent of the accuracy of the quadrature rule). We also give a complete analysis for QDG with more than m + 1 quadrature points when the degree of precision (d.o.p.) is at least 2m + 1. The behaviour (but not the approximation) is the same as that for a DG scheme when the d.o.p. is at least 2m + 2. Numerical test results confirm that the theoretical convergence rates are optimal.

Original languageEnglish
Pages (from-to)1117-1146
Number of pages30
JournalIMA Journal of Numerical Analysis
Issue number3
Publication statusPublished - Jul 2012

Scopus Subject Areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • collocation
  • discontinuous Galerkin
  • global convergence
  • local superconvergence
  • quadrature Galerkin
  • Volterra integral equations of the first kind


Dive into the research topics of 'Global convergence and local superconvergence of first-kind Volterra integral equation approximations'. Together they form a unique fingerprint.

Cite this