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

Brunner, Hermann and Davies, P.J. and Duncan, D.B. (2012) Global convergence and local superconvergence of first-kind Volterra integral equation approximations. IMA Journal of Numerical Analysis, 32 (3). pp. 1117-1146. ISSN 0272-4979 (https://doi.org/10.1093/imanum/DRR029)

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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.

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