Abstract
A class of variable order composite quadrature formulas for the numerical integration of functions with a singularity in or near to the region of integration is introduced. Exponential convergence of the method is shown for all integrands in the countably normed spaceB β. Numerical examples are presented which demonstrate that the asymptotic exponential convergence rates obtained here are sharp and already observed for a small number of quadrature points.
Zusammenfassung
Wir stellen eine Klasse von zusammengesetzten Quadraturformeln variabler Ordnung vor, die sich zur numerischen Integration von Funktionen mit einer Singularität im Inneren oder in der Nähe des Integrationsbereichs eignen. Für alle Integranden in dem abzählbar normierten RaumB β wird eine exponentielle Konvergenz des Verfahrens bewiesen. Numerische Beispiele zeigen, daß die ermittelten asymptotischen exponentiellen Konvergenzraten scharf sind und schon bei einer kleinen Zahl von Quadraturknoten erreicht werden.
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This research was supported in part by the AFOSR under grant No. F49620-J-0100.
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Schwab, C. Variable order composite quadrature of singular and nearly singular integrals. Computing 53, 173–194 (1994). https://doi.org/10.1007/BF02252988
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DOI: https://doi.org/10.1007/BF02252988