### Abstract

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.

Language | English |
---|---|

Pages | 1117-1146 |

Number of pages | 30 |

Journal | IMA Journal of Numerical Analysis |

Volume | 32 |

Issue number | 3 |

Early online date | 4 Nov 2011 |

DOIs | |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

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

### Cite this

}

*IMA Journal of Numerical Analysis*, vol. 32, no. 3, pp. 1117-1146. https://doi.org/10.1093/imanum/DRR029

**Global convergence and local superconvergence of first-kind Volterra integral equation approximations.** / Brunner, Hermann; Davies, P.J.; Duncan, D.B.

Research output: Contribution to journal › Article

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.

PY - 2012

Y1 - 2012

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.

KW - volterra integral equations of the first kind

KW - discontinuous galerkin

KW - quadrature galerkin

KW - collocation

KW - global convergence

KW - local superconvergence

U2 - 10.1093/imanum/DRR029

DO - 10.1093/imanum/DRR029

M3 - Article

VL - 32

SP - 1117

EP - 1146

JO - IMA Journal of Numerical Analysis

T2 - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 3

ER -