Geodesic
On a method of two-sided eigenvalue estimates for elliptic equations of the form $Au-\lambda Bu=0$
Applications of Mathematics, Tome 26 (1981) no. 3, pp. 211-240
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The Collatz method of twosided eigenvalue estimates was extended by K. Rektorys in his monography Variational Methods to the case of differential equations of the form $Au - \lambda Bu=0$ with elliptic operators. This method requires to solve, successively, certain boundary value problems. In the case of partial differential equations, these problems are to be solved approximately, as a rule, and this is the source of further errors. In the work, it is shown how to estimate these additional errors, or how to avoid them by a proper modification of the method. At the same time, some results of their own interest are derived.
The Collatz method of twosided eigenvalue estimates was extended by K. Rektorys in his monography Variational Methods to the case of differential equations of the form $Au - \lambda Bu=0$ with elliptic operators. This method requires to solve, successively, certain boundary value problems. In the case of partial differential equations, these problems are to be solved approximately, as a rule, and this is the source of further errors. In the work, it is shown how to estimate these additional errors, or how to avoid them by a proper modification of the method. At the same time, some results of their own interest are derived.
DOI : 10.21136/AM.1981.103913
Classification : 35P15, 49G05, 65N15, 65N25, 65N30
Keywords: Collatz method; twosided eigenvalue estimates; elliptic operators 
@article{10_21136_AM_1981_103913,
     author = {Rektorys, Karel and Vosp\v{e}l, Zden\v{e}k},
     title = {On a method of two-sided eigenvalue estimates for elliptic equations of the form $Au-\lambda Bu=0$},
     journal = {Applications of Mathematics},
     pages = {211--240},
     year = {1981},
     volume = {26},
     number = {3},
     doi = {10.21136/AM.1981.103913},
     mrnumber = {0615608},
     zbl = {0474.65080},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1981.103913/}
}
TY  - JOUR
AU  - Rektorys, Karel
AU  - Vospěl, Zdeněk
TI  - On a method of two-sided eigenvalue estimates for elliptic equations of the form $Au-\lambda Bu=0$
JO  - Applications of Mathematics
PY  - 1981
SP  - 211
EP  - 240
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1981.103913/
DO  - 10.21136/AM.1981.103913
LA  - en
ID  - 10_21136_AM_1981_103913
ER  - 
%0 Journal Article
%A Rektorys, Karel
%A Vospěl, Zdeněk
%T On a method of two-sided eigenvalue estimates for elliptic equations of the form $Au-\lambda Bu=0$
%J Applications of Mathematics
%D 1981
%P 211-240
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1981.103913/
%R 10.21136/AM.1981.103913
%G en
%F 10_21136_AM_1981_103913
Rektorys, Karel; Vospěl, Zdeněk. On a method of two-sided eigenvalue estimates for elliptic equations of the form $Au-\lambda Bu=0$. Applications of Mathematics, Tome 26 (1981) no. 3, pp. 211-240. doi: 10.21136/AM.1981.103913

[1] K. Rektorys: Variational Methods in Mathematics, Science and Engineering. Reidel Publ. Co., Dortrecht (Holland)-Boston (USA) 1977. (In Czech: Praha, SNTL, 1974.) | MR

[2] L. Collatz: Eigenwertaufgaben mit technischen Anwendungen. 2nd Ed. Leipzig, Geert and Portig 1963. | MR

[3] L. Collatz: Functional Analysis and Numerical Mathematics. New York, Academic Press 1966. | MR

[4] Z. Vospěl: Some Eigenvalue Estimates for Partial Differential Equations of the Form $Au - \lambda Bu = 0$. Dissertation, Technical University Prague, 1978. (In Czech.)

[5] P. G. Ciarlet M. H. Schulz R. S. Varga: Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problems. Part III, Eigenvalue Problems. Num. Math 12 (1968), 120-133. | DOI | MR

[6] A. K. Azis: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Part I by I. Babuška and A. K. Azis, 3-359. Academic Press, New York-London, 1972. | MR

Cité par Sources :