Abstract
It has been shown that if a Volterra integral equation of the first kind with continuous kernel is solved numerically in a given intervalI by collocation in the space of piecewise polynomials of degreem≧0 and possessing finite discontinuities at their knotsZ N then a careful choice of the collocation points yields convergence of orderp=m+2 on a certain finite subset ofI (while the global convergence order ism+1; this subset does not contain the knotsZ N . In this note it will be shown that superconvergence onZ N can be attained only if some of the collocation points coalesce (Hermite-type collocation).
Zusammenfassung
Wird eine Volterrasche Integralgleichung erster Art mit stetigem Kern in einem gegebenen IntervallI numerisch mittels Kollokation im Raum der stückweisen Polynome vom Gradm≧0, welche endliche Sprungstellen an den KnotenZ N besitzen, gelöst, so liefert eine spezielle Wahl der Kollokationspunkte die Konvergenzordnungp=m+2 auf einer gewissen endlichen Teilmenge vonI (aufI hat das Verfahren die globale Ordnungm+1); die KnotenZ N sind in dieser Teilmenge nicht enthalten. In der vorliegenden Arbeit wird gezeigt, daß Konvergenz der Ordnungp=m+2 aufZ N nur dann erreicht werden kann, wenn gewisse Kollokationspunkte zusammenfallen (Hermitesche Kollokation).
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This research was supported by the National Research Council of Canada (Grant No. A-4805).
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Brunner, H. A note on collocation methods for Volterra integral equations of the first kind. Computing 23, 179–187 (1979). https://doi.org/10.1007/BF02252096
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DOI: https://doi.org/10.1007/BF02252096