The superconvergence of composite Newton-Cotes rules for Hadamard finite-part integral on a circle (Q841138)

The authors consider the numerical approximation of a finite-part integral with second-order singularity at the point \(t\) in the periodic case. For this purpose they suggest to use a compound quadrature formula based on a standard Newton-Cotes rule of order \(k\), weighted with the singular kernel in question. They prove that the error at the point \(t\) may be bounded from above by \(C h^{k+\alpha}\) if the \(k\)th derivative of the integrand satisfies a Hölder condition of order \(\alpha\) and if \(t\) is a zero of a certain function. Here \(h\) is the mesh size of the quadrature formula. This is slightly better than what one can expect uniformly from a suitable quadrature formula. The two key problems with this result are: (1) The condition on the point \(t\) depends on the mesh size \(h\), and thus a point \(t^*\), say, that allows such an estimate for some \(h\) is unlikely to allow the estimate if \(h\) is changed. (2) The approximation method underlying the construction of the quadrature produces a piecewise continuous but not differentiable approximant. Thus, the finite-part integral of this approximant (and hence the value of the quadrature formula itself) is likely to be unbounded if \(t\) approaches a point of the basic grid of size \(h\). Thus, even though the quadrature formula provides a rather good approximation for some values of \(t\), it will be very poor for others.