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

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.
LanguageEnglish
Pages1117-1146
Number of pages30
JournalIMA Journal of Numerical Analysis
Volume32
Issue number3
Early online date4 Nov 2011
DOIs
Publication statusPublished - 2012

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Superconvergence
Volterra Integral Equations
Galerkin methods
Global Convergence
Quadrature
Integral equations
Collocation
Discontinuous Galerkin
Approximation
Convolution
Discontinuous Galerkin Method
Polynomials
Convergence Analysis
Point Location
Polynomial Basis
Convergence Order
Quadrature Rules
Polynomial function
Collocation Method
Convergence Properties

Keywords

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

Cite this

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title = "Global convergence and local superconvergence of first-kind Volterra integral equation approximations",
abstract = "We present a comprehensive convergence analysis for discontinuous piecewisepolynomial approximations of a first-kind Volterra integral equation withsmooth convolution kernel, examining the attainable order of (super-) convergencein collocation (DC), quadrature discontinuous Galerkin (QDG) andfull discontinuous Galerkin (DG) methods.We introduce new polynomial basis functions with properties that greatlysimplify the convergence analysis for collocation methods. This also enables us todetermine explicit formulae for the location of superconvergence points (i.e.\ discrete pointsat 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 piecewisepolynomials of degree $m$ and uses exactly $m+1$ quadrature points and nonzero quadratureweights is equivalent to a collocation scheme, and so its convergence properties arefully determined by the previous collocation analysis andthey depend only on the quadrature point location (in particular, they are completely independent ofthe accuracy of the quadrature rule). We also give acomplete analysis for QDG with more than $m+1$ quadrature points whenthe 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 resultsconfirm that the theoretical convergence rates are optimal.",
keywords = "volterra integral equations of the first kind, discontinuous galerkin, quadrature galerkin, collocation, global convergence, local superconvergence",
author = "Hermann Brunner and P.J. Davies and D.B. Duncan",
year = "2012",
doi = "10.1093/imanum/DRR029",
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Global convergence and local superconvergence of first-kind Volterra integral equation approximations. / Brunner, Hermann; Davies, P.J.; Duncan, D.B.

In: IMA Journal of Numerical Analysis, Vol. 32, No. 3, 2012, p. 1117-1146.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Global convergence and local superconvergence of first-kind Volterra integral equation approximations

AU - Brunner, Hermann

AU - Davies, P.J.

AU - Duncan, D.B.

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N2 - We present a comprehensive convergence analysis for discontinuous piecewisepolynomial approximations of a first-kind Volterra integral equation withsmooth convolution kernel, examining the attainable order of (super-) convergencein collocation (DC), quadrature discontinuous Galerkin (QDG) andfull discontinuous Galerkin (DG) methods.We introduce new polynomial basis functions with properties that greatlysimplify the convergence analysis for collocation methods. This also enables us todetermine explicit formulae for the location of superconvergence points (i.e.\ discrete pointsat 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 piecewisepolynomials of degree $m$ and uses exactly $m+1$ quadrature points and nonzero quadratureweights is equivalent to a collocation scheme, and so its convergence properties arefully determined by the previous collocation analysis andthey depend only on the quadrature point location (in particular, they are completely independent ofthe accuracy of the quadrature rule). We also give acomplete analysis for QDG with more than $m+1$ quadrature points whenthe 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 resultsconfirm that the theoretical convergence rates are optimal.

AB - We present a comprehensive convergence analysis for discontinuous piecewisepolynomial approximations of a first-kind Volterra integral equation withsmooth convolution kernel, examining the attainable order of (super-) convergencein collocation (DC), quadrature discontinuous Galerkin (QDG) andfull discontinuous Galerkin (DG) methods.We introduce new polynomial basis functions with properties that greatlysimplify the convergence analysis for collocation methods. This also enables us todetermine explicit formulae for the location of superconvergence points (i.e.\ discrete pointsat 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 piecewisepolynomials of degree $m$ and uses exactly $m+1$ quadrature points and nonzero quadratureweights is equivalent to a collocation scheme, and so its convergence properties arefully determined by the previous collocation analysis andthey depend only on the quadrature point location (in particular, they are completely independent ofthe accuracy of the quadrature rule). We also give acomplete analysis for QDG with more than $m+1$ quadrature points whenthe 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 resultsconfirm that the theoretical convergence rates are optimal.

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