### 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.

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 language | English |
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Pages (from-to) | 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 |

### Keywords

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