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Keywords:
Collatz method; twosided eigenvalue estimates; elliptic operators
Summary:
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.
References:
[1] K. Rektorys: Variational Methods in Mathematics, Science and Engineering. Reidel Publ. Co., Dortrecht (Holland)-Boston (USA) 1977. (In Czech: Praha, SNTL, 1974.) MR 0487653
[2] L. Collatz: Eigenwertaufgaben mit technischen Anwendungen. 2nd Ed. Leipzig, Geert and Portig 1963. MR 0152101
[3] L. Collatz: Functional Analysis and Numerical Mathematics. New York, Academic Press 1966. MR 0205126
[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 10.1007/BF02173406 | MR 0233517
[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 0347104

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