×

Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. I: Improved error bounds for eigenfunctions. (English) Zbl 0223.65080


MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ahlberg, J. H., Nilson, E. N., Walsh, J. L.: The theory of splines and their applications. New York: Academic Press 1967. · Zbl 0158.15901
[2] Birkhoff, G., Boor, C. de: Error bounds for spline interpolation. J. Math. Mech.13, 827-836 (1964). · Zbl 0143.28503
[3] Birkhoff, G., Boor, C. de: Piecewise polynomial interpolation and approximation. Approximation of functions (H. L. Garabedian, ed.), pp. 164-190. Amsterdam: Elsevier Pub. Co. 1965. · Zbl 0136.04703
[4] Birkhoff, G., Boor, C. de, Swartz, B., Wendroff, B.: Rayleigh-Ritz approximation by piecewise cubic polynomials. SIMA J. Numer. Anal.3, 188-203 (1966). · Zbl 0143.38002 · doi:10.1137/0703015
[5] Birkhoff, G., Schultz, M. H., Varga, R. S.: Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math.11, 232-256 (1968). · Zbl 0159.20904 · doi:10.1007/BF02161845
[6] Boor, C. de: Uniform approximation by splines. J. Approx. Theory1, 219-235 (1968). · Zbl 0193.02502 · doi:10.1016/0021-9045(68)90026-9
[7] Bramble, J. H., Hilbert, S. R.: Bounds for a class of linear functionals with applications to Hermite interpolation. Numer. Math.16, 362-369 (1971). · Zbl 0214.41405 · doi:10.1007/BF02165007
[8] Brauer, F.: Singular self-adjoint boundary value problems for the differential equationL ?=?M ?. Trans. Amer. Math. Soc.88, 331-345 (1958). · Zbl 0141.27802
[9] Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high-order accuracy for nonlinear boundary value problems. III. Eigenvalue problems. Numer. Math.12, 120-133 (1968). · Zbl 0181.18303 · doi:10.1007/BF02173406
[10] Collatz, L.: The numerical treatment of differential equations, 3rd. ed. Berlin-Göttingen-Heidelberg: Springer 1960. · Zbl 0086.32601
[11] Courant, R., Hilbert, D.: Methods of mathematical physics. I. New York: Interscience 1953. · Zbl 0051.28802
[12] Dailey, J. W., Pierce, J. G.: Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems (to appear). · Zbl 0244.65075
[13] Gould, S. H.: Variational methods for eigenvalue problems. Toronto: University of Toronto Press 1966. · Zbl 0156.12401
[14] Hald, O., Widlund, P.: On the eigenvalue problems for the finite element variational method (to appear).
[15] Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities, 2d. ed. Cambridge: Cambridge University Press 1952.
[16] Harrick, I. I.: Approximation of functions which vanish on the boundary of a region, together with their partial derivatives, by functions of special type. Akad. Nauk. SSSR Izv. Sibirsk, Otd.4, 408-425 (1963).
[17] Hedstrom, G. W., Varga, R. S.: Application of Besov spaces to spline approximation. J. Approx. Theory4, 295-327 (1971). · Zbl 0218.41001 · doi:10.1016/0021-9045(71)90018-9
[18] Hulme, B. L.: Interpolation by Ritz approximation. J. Math. Mech.18, 337-342 (1968). · Zbl 0165.38601
[19] Ince, E. L.: Ordinary differential equations. New York: Dover Publications 1948. · Zbl 0063.02971
[20] Jerome, J. W., Pierce, J. G.: On splines associated with singular self-adjoint differential operators. J. Approx. Theory (to appear). · Zbl 0228.41003
[21] Jerome, J. W., Schumaker, L. L.: OnLg-splines. J. Approx. Theory2, 29-49 (1969). · Zbl 0172.34501 · doi:10.1016/0021-9045(69)90029-X
[22] Jerome, J. W., Varga, R. S.: Generalizations of spline functions and applications to nonlinear boundary value and eigenvalue problems. Theory and applications of spline functions (T. N. E. Greville, ed.), pp. 103-155. New York: Academic Press 1969. · Zbl 0188.13004
[23] Johnson, O.: Error bounds for Sturm-Liouville eigenvalue approximations by several piecewise cubic Rayleigh-Ritz methods. SIAM J. Numer. Anal.6, 317-333 (1969). · Zbl 0183.44602 · doi:10.1137/0706030
[24] Kamke, E. A.: Über die definiten selbstadjungierten Eigenwertaufgaben bei gewöhnlichen linearen Differentialgleichungen. II, III. Math. A.46, 231-286 (1940). · JFM 66.0416.03
[25] Kamke, E. A.: Über die definiten selbstadjungierten Eigenwertaufgaben bei gewöhnlichen linearen Differentialgleichungen. IV. Math. Z.48, 67-100 (1942). · JFM 68.0197.01 · doi:10.1007/BF01180005
[26] Mikhlin, S. G.: Variational methods in mathematical physics. New York: Macmillan and Co. 1964. · Zbl 0119.19002
[27] Nitsche, J.: Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer. Math.11, 346-348 (1968). · Zbl 0175.45801 · doi:10.1007/BF02166687
[28] Nitsche, J.: Verfahren von Ritz und Spline-Interpolation bei Sturm-Liouville-Randwertproblemen. Numer. Math.13, 260-265 (1969). · Zbl 0181.18204 · doi:10.1007/BF02167557
[29] Perrin, F. M., Price, H. S., Varga, R. S.: On higher-order numerical methods for nonlinear two-point boundary value problems. Numer. Math.13, 180-198 (1969). · Zbl 0183.44501 · doi:10.1007/BF02163236
[30] Pierce, J. G.: Higher order convergence results for the Rayleigh-Ritz method applied to a special class of eigenvalue problems. Thesis, Case Western Reserve University (1969).
[31] Pierce, J. G., Varga, R. S.: Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problem. I. Estimates relating Rayleigh-Ritz and Galerkin approximations to eigenfunctions. SIAM J. Numer. Anal. (to appear, V. 9, March 1972). · Zbl 0301.65063
[32] Schultz, M. H.: Multivariate spline functions and elliptic problems, Approximation with special emphasis on spline functions (ed. I. J. Schoenberg), pp. 279-347. New York: Academic Press 1969.
[33] Schultz, M. H., Varga, R. S.:L-splines. Numer. Math.10, 345-369 (1967). · Zbl 0183.44402 · doi:10.1007/BF02162033
[34] Strang, Gilbert, Fix, George: An analysis of the finite element method. Prentice-Hall, Inc. (to appear). · Zbl 0272.65099
[35] Swartz, B. K., Varga, R. S.: Error bounds for spline andL-spline interpolation. J. Approx. Theory (to appear). · Zbl 0242.41008
[36] Wendroff, B.: Bounds for eigenvalues of some differential operators by the Rayleigh-Ritz method. Math. Comp.19, 218-224 (1965). · Zbl 0139.10703 · doi:10.1090/S0025-5718-1965-0179932-5
[37] Yosida, K.: Functional analysis. New York: Academic Press 1965. · Zbl 0126.11504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.