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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bramble, J. H., Hubbard, B. E.: Effects of boundary regularity on the discretization error in the fixed membrane eigenvalue problem. SIAM J. Numer. Anal.5, 835–863 (1968).
Bus, J. C. P., Dekker, T. J.: Two efficient algorithms with guaranteed convergence for finding a zero of a function. Report NW 13/74, Amsterdam: Mathematisch Centrum 1974.
Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 1. Interscience 1953.
Cullum, J., Donath, W. E.: A block generalization of the symmetric S-step Lanczos algorithm. Report No. RC 4845 (NO. 21570), IBM Thomas. J. Watson Research Center, Yorktown Heights, New York, May 1974.
Forsythe, G. E., Wasow, W. R.: Finite-Difference Methods for Partial Differential Equations. New York: Wiley 1960.
Garabedian, P. R.: Partial Differential Equations. New York: Wiley 1964.
Golub, G. H.: Some uses of the Lanczos algorithm in numerical linear algebra, in: Topics in Numerical Analysis (Miller, T., ed). Academic Press 1972.
Golub, G. H., Jenning, L., Yang, W. H.: Waves in periodically structure media. J. Comp. Phys.17, 349–357 (1975).
Golub, G. H., Underwood, R., Wilkinson, J. H.: The Lanczos algorithm for the symmetricAx=λBx problem. Stanford Report CS-72-270, March. 1972.
Kaniel, S.: Estimates for some computational techniques in linear algebra. Math. Comp.20, 369–378 (1966).
Kuttler, J. R.: Direct methods for computing eigenvalues of the finite-difference Laplacian. SIAM J. Numer. Anal.11, 732–740 (1974).
Kuttler, J. R.: Finite-difference approximations for eigenvalues of uniformly elliptic operators. SIAM J. Numer. Anal.7, 206–232 (1970).
Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Standards45, 255–283 (1950).
Moler, C. B.: Finite difference methods for the eigenvalues of Laplace's operator, Stanford Report CS-22, 1965.
Paige, C. C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Ph. D. dissertation, The University of London, 1971.
Paige, C. C.: Computational variants of the Lanczos method for the eigenproblem. J. Inst. Maths. Applics.10, 373–381 (1972).
Paige, C. C.: Error analysis of the Lanczos algorithm for tridiagonalization of a symmetric matrix. J. Inst. Math. Appl.18, 341–349 (1976).
Paige, C. C., Saunders, M. A.: Solutions of sparse indefinite systems of equations. SIAM J. Numer. Anal.12, 617–629 (1975).
Pereyra, V., Proskurowski, W., Widlund, O.: High order fast Laplace solver for the Dirichlet problem on general regions. Math. Comp.31, 1–16 (1977).
Proskurowski, W., Widlund, O.: On the numerical solution of Helmholtz's equation by the capacitance matrix method. Math. Comp.30, 433–468 (1976). Appeared also as ERDA Report COO-3077-99, New York University, November 1975.
Rutishauser, H.: Simultaneous iteration method for symmetric matrices. Numer. Math.16, 205–223 (1970).
Shieh, A.: Fast Poisson solver on nonrectangular domains. Ph. D. thesis, New York University, June 1976.
Stewart, G. W.: The numerical treatment of large eigenvalue problems. IFIP74, 666–672 (1974).
Underwood, R.: An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems. Stanford Report CS-496, May 1975.
Watson, G. N.: A treatise on the theory of Bessel functions. Cambridge Univ. Press 1952.
Wilkinson, J. H., Reinsch, C.: Handbook for Automatic Computation, Vol. II, Linear Algebra, Part 2. New York-Heidelberg-Berlin, Springer 1971.
Author information
Authors and Affiliations
Additional information
This work was done with support from the U.S. Energy Research and Development Administration.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02252343