Abstract
The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered. Two fast algorithms are combined: an iterative Block Lanczos method and a capacitance matrix method. The capacitance matrix is generated and factored only once for a given problem. In each iteration of the Block Lanczos method, a discrete Helmholtz equation is solved twice on a rectangle at a cost of the order ofn 2 log2 n operations wheren is the number of mesh points across the rectangle in which the region is imbedded.
Zusammenfassung
Es wird über das Auffinden mehrerer Eigenfunktionen und Eigenwerte des inneren Dirichlet-Problems für die Laplace-Gleichung mit willkürlich begrenzten ebenen Gebieten berichtet. Zwei schnelle Algorithmen werden miteinander kombiniert. Eine iterative Block-Lanczos-Methode und eine Kapazitäts-Matrizen-Methode. Die Kapazitäts-Matrix wird berechnet und nur einmal für ein gegebenes Problem faktorisiert. Bei jedem Iterationsschritt der Block-Lanczos-Methode wird eine diskrete Helmholtz-Gleichung zweimal auf einem Rechteck mit einer zu n2log2 n proportionalen Anzahl von Operationen gelöst, wobein die Zahl der Netzpunkte zu dem Rechteck ist, in das das Gebiet eingebettet ist.
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This work was done with support from the U.S. Energy Research and Development Administration.
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Proskurowski, W. On the numerical solution of the eigenvalue problem of the laplace operator by a capacitance matrix method. Computing 20, 139–151 (1978). https://doi.org/10.1007/BF02252343
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DOI: https://doi.org/10.1007/BF02252343