Error estimates for some variational methods applicable to scattering and radiation problems
Author:
M. Szałek
Journal:
Quart. Appl. Math. 27 (1970), 473-479
DOI:
https://doi.org/10.1090/qam/256637
MathSciNet review:
256637
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: We consider variational expressions helpful in calculating the approximate value of a scalar product, in Hilbert space, of an arbitrary vector $g$ with a solution $u^\circ$ of an arbitrary inhomogeneous linear equation. Error bounds for this approximate value are given. For the case where an approximate solution of an inhomogeneous equation is sought in an arbitrary subspace of a space containing $u^\circ$, conditions are specified for a best estimate of the error by the use of two trial vectors. A method is presented for an additional improvement of the error estimate by using four trial vectors.
P. Roussopoulos, Méthodes variationnelles en théorie des collisions, C. R. Acad. Sci. Paris 236, 1858–1860 (1953).
M. Becker, The principles and application of variational methods, Research monograph No. 27, MIT Press, Cambridge, Massachusetts.
- G. C. Pomraning, A variational principle for linear systems, J. Soc. Indust. Appl. Math. 13 (1965), 511–519. MR 179921
W. W. Nikol’skij, Variacionnye metody dlja vnutrennych zadač élektrodynamiki (Variational methods for electromagnetic boundary-value problems), “Nauka", Moskva 1967, p. 25.
L. A. Vajnštejn, Volny toka v tonkom cil’indričeskom provodnike, III. Variacionnyj metod (Waves of current in a thin cylindrical conductor, III. Variational method), J. Tech. Phys. 31, 29–44 (1961).
L. Cairo and T. Kahan, Techniques variationnelles en radioélectricité, Dunod, Paris 1962, p. 54.
F. E. Borgnis and C. H. Papas, Randwertprobleme der Mikrowellenphysik, Springer-Verlag, Berlin 1955.
- Robert E. Collin, Field theory of guided waves, International Series in Pure and Applied Physics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR 0116872
S. G. Michlin, Variacionnye metody v matematičeskoj fizike (Variational methods in mathematical physics), GITTL, Moskva 1957, pp. 410–417.
M. Altman, Metody przybliżone analizy funkcjonalnej (Approximate methods of functional analysis), Zakład Narodowy im. Ossolińskich, Wydawnictwo PAN, Wrocław-Warszawa-Kraków 1963, pp. 35–38.
- E. Bodewig, Matrix calculus, North-Holland Publishing Company, Amsterdam, 1956. MR 0080363
P. Roussopoulos, Méthodes variationnelles en théorie des collisions, C. R. Acad. Sci. Paris 236, 1858–1860 (1953).
M. Becker, The principles and application of variational methods, Research monograph No. 27, MIT Press, Cambridge, Massachusetts.
G. Pomraning, A variational principle for linear systems, J. Soc. Indust. Appl. Math. 13, 511 (1965).
W. W. Nikol’skij, Variacionnye metody dlja vnutrennych zadač élektrodynamiki (Variational methods for electromagnetic boundary-value problems), “Nauka", Moskva 1967, p. 25.
L. A. Vajnštejn, Volny toka v tonkom cil’indričeskom provodnike, III. Variacionnyj metod (Waves of current in a thin cylindrical conductor, III. Variational method), J. Tech. Phys. 31, 29–44 (1961).
L. Cairo and T. Kahan, Techniques variationnelles en radioélectricité, Dunod, Paris 1962, p. 54.
F. E. Borgnis and C. H. Papas, Randwertprobleme der Mikrowellenphysik, Springer-Verlag, Berlin 1955.
R. E. Collin, Field theory of guided waves, McGraw-Hill, New York, 1960, p. 314–363.
S. G. Michlin, Variacionnye metody v matematičeskoj fizike (Variational methods in mathematical physics), GITTL, Moskva 1957, pp. 410–417.
M. Altman, Metody przybliżone analizy funkcjonalnej (Approximate methods of functional analysis), Zakład Narodowy im. Ossolińskich, Wydawnictwo PAN, Wrocław-Warszawa-Kraków 1963, pp. 35–38.
E. Bodewig, Matrix calculus, North-Holland, Amsterdam, 1956, pp. 57–69.
Additional Information
Article copyright:
© Copyright 1970
American Mathematical Society
