Zeitschrift für Analysis und ihre Anwendungen
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Published online: 1989-08-31
A Quadrature-Based Approach to Improving the Collocation Method for Splines of Even DegreeIan H. Sloan and Wolfgang L. Wendland (1) University of New South Wales, Sydney, Australia
(2) Universität Stuttgart, Germany
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.
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Sloan Ian, Wendland Wolfgang: A Quadrature-Based Approach to Improving the Collocation Method for Splines of Even Degree. Z. Anal. Anwend. 8 (1989), 361-376. doi: 10.4171/ZAA/359