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

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Abstract

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 (DC), 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 \textbf{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 (dop) is at least $2m+1$. The behaviour (but not the approximation)
is the same as for a DG scheme when the dop 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
Volume32
Issue number3
Early online date4 Nov 2011
DOIs
Publication statusPublished - 2012

Keywords

  • volterra integral equations of the first kind
  • discontinuous galerkin
  • quadrature galerkin
  • collocation
  • global convergence
  • local superconvergence

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