A Quadrature-Based Approach to Improving the Collocation Method for Splines of Even Degree

Abstract

The "qualocation" method, a recently proposed quadrature-based extension of the collocation method, is here applied to a class of boundary integral equations, using an even degree spline trial space on a uniform partition. The problems handled are of the form $(L+K)u=f$ where $L$ is a convolutional operator with even symbol, and $K$ is an operator with a greater smoothing effect than $L$. For a trial space of dimension $n$, it is shown that a certain $2n$-point quadrature rule, which is a generalization of the repeated 2-point Gauss rule, gives a stable qualocation method, and yields an order of convergence, in suitable negative norms, two powers of $h$ higher than achieved by the mid-point collocation method in the recent analysis of Saranen. The treatment of the smooth perturbation covers also the earlier analysis of the odd degree spline case by Sloan.