A fourth-order finite-difference approximation for the fixed membrane eigenproblem

Author:
J. R. Kuttler

Journal:
Math. Comp. **25** (1971), 237-256

MSC:
Primary 65N25

DOI:
https://doi.org/10.1090/S0025-5718-1971-0301955-6

MathSciNet review:
0301955

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The fixed membrane problem in on , for a bounded region of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type, convergence is shown for the approximating eigenvalues and eigenfunctions, where *h* is the mesh width. An application to an approximation of the forced vibration problem in in , is also given.

**[1]**J. H. Bramble, "Error estimates for difference methods in forced vibration problems,"*SIAM J. Numer. Anal.*, v. 3, 1966, no. 1, pp. 1-12. MR**34**#969. MR**0201084 (34:969)****[2]**J. H. Bramble & B. E. Hubbard, "New monotone type approximations for elliptic problems,"*Math. Comp.*, v. 18, 1964, pp. 349-367. MR**29**#2982. MR**0165702 (29:2982)****[3]**J. H. Bramble & V. C. Thomée, "Point-wise bounds for discrete Green's functions,"*SIAM I. Numer. Anal.*, v. 6, 1969, pp. 583-590. MR**0263265 (41:7870)****[4]**R. Courant & D. Hilbert,*Methoden der Mathematischen Physik.*Vol. I, Springer, Berlin, 1931; English transl., Interscience, New York, 1953. MR**16**, 426.**[5]**B. E. Hubbard, "Bounds for eigenvalues of the free and fixed membrane by finite difference methods,"*Pacific J. Math.*, v. 11, 1961, pp. 559-590. MR**25**#4633. MR**0141223 (25:4633)****[6]**J. R. Kuttler, "Finite difference approximations for eigenvalues of uniformly elliptic operators,"*SIAM J. Numer. Anal.*, v. 7, 1970. MR**0273841 (42:8717)****[7]**M. Marcus & H. Minc,*A Survey of Matrix Theory and Matrix Inequalities*, Allyn and Bacon, Boston, Mass., 1964. MR**29**#112. MR**0162808 (29:112)****[8]**H. S. Price, "Monotone and oscillation matrices applied to finite difference approximations,"*Math. Comp.*, v. 22, 1968, pp. 489-516. MR**38**#875. MR**0232550 (38:875)****[9]**H. F. Weinberger, "Lower bounds for higher eigenvalues by finite difference methods,"*Pacific J. Math.*, v. 8, 1958, pp. 339-368; erratum, 941. MR**21**#6097. MR**0107372 (21:6097)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N25

Retrieve articles in all journals with MSC: 65N25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0301955-6

Keywords:
Finite-differences,
membrane,
fixed membrane,
eigenvalues,
elliptic partial differential equations,
monotone matrices,
forced vibration problem,
discrete Green's function

Article copyright:
© Copyright 1971
American Mathematical Society