Explicit $O(h^{2})$ bounds on the eigenvalues of the half-$L$
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- by Blair K. Swartz PDF
- Math. Comp. 22 (1968), 40-59 Request permission
References
- R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1â23. MR 7838, DOI 10.1090/S0002-9904-1943-07818-4 L. Collatz, "Konvergenz des Differenzverfahrens bei Eigenwertproblemen partieller Differentialgleichungen," Deutsche Math., v. 3, 1938, pp. 200â212.
- H. F. Weinberger, Lower bounds for higher eigenvalues by finite difference methods, Pacific J. Math. 8 (1958), 339â368; erratum, 941. MR 107372, DOI 10.2140/pjm.1958.8.339 C. Moler, Finite Difference Methods for the Eigenvalues of Laplaceâs Operator, Stanford Computer Science Report CS22, 1965. (Available from Defense Document Center, Cameron Station, Alexandria, Va.)
- J. L. Synge, The hypercircle in mathematical physics: a method for the approximate solution of boundary value problems, Cambridge University Press, New York, 1957. MR 0097605
- Georges PĂłlya, Sur une interprĂ©tation de la mĂ©thode des diffĂ©rences finies qui peut fournir des bornes supĂ©rieures ou infĂ©rieures, C. R. Acad. Sci. Paris 235 (1952), 995â997 (French). MR 52674
- G. PĂłlya, Estimates for eigenvalues, Studies in mathematics and mechanics presented to Richard von Mises, Academic Press, Inc., New York, 1954, pp. 200â207. MR 0065777
- Garrett Birkhoff, C. de Boor, B. Swartz, and B. Wendroff, Rayleigh-Ritz approximation by piecewise cubic polynomials, SIAM J. Numer. Anal. 3 (1966), 188â203. MR 203926, DOI 10.1137/0703015
- George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124
- Bert Hubbard, Bounds for eigenvalues of the free and fixed membrane by finite difference methods, Pacific J. Math. 11 (1961), 559â590. MR 141223, DOI 10.2140/pjm.1961.11.559
- L. VeÄdinger, Computation of the eigenvalues of a membrane by a finite-difference method, Ćœ. VyÄisl. Mat i Mat. Fiz. 4 (1964), 1037â1044 (Russian). MR 171389
- J. K. Reid and J. E. Walsh, An elliptic eigenvalue problem for a reentrant region, J. Soc. Indust. Appl. Math. 13 (1965), 837â850. MR 182169, DOI 10.1137/0113054
- Joseph Hersch, Equations diffĂ©rentielles et fonctions de cellules, C. R. Acad. Sci. Paris 240 (1955), 1602â1604 (French). MR 68711
- Joseph Hersch, Lower bounds for all eigenvalues by cell functions: A refined form of H. F. Weinbergerâs method, Arch. Rational Mech. Anal. 12 (1963), 361â366. MR 163443, DOI 10.1007/BF00281233
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
- L. E. Payne and H. F. Weinberger, Bounds for solutions of second order elliptic equations in terms of arbitrary vector fields, Arch. Rational Mech. Anal. 20 (1965), 95â106. MR 192174, DOI 10.1007/BF00284612
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- H. F. Weinberger, Upper and lower bounds for eigenvalues by finite difference methods, Comm. Pure Appl. Math. 9 (1956), 613â623. MR 84185, DOI 10.1002/cpa.3160090329
- Paul R. Halmos, Finite-dimensional vector spaces, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton-Toronto-New York-London, 1958. 2nd ed. MR 0089819
Additional Information
- © Copyright 1968 American Mathematical Society
- Journal: Math. Comp. 22 (1968), 40-59
- MSC: Primary 65.66
- DOI: https://doi.org/10.1090/S0025-5718-1968-0223112-4
- MathSciNet review: 0223112