On the stability of the Ritz-Galerkin method for Hammerstein equations

Authors:
Jörg Hertling and Alexandru I. Şchiop

Journal:
Math. Comp. **29** (1975), 484-488

MSC:
Primary 65R05

MathSciNet review:
0383797

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Abstract | References | Similar Articles | Additional Information

Abstract: For the numerical treatment of Hammerstein equations by variational methods which has been considered by Hertling, we establish the stability in the sense of Mikhlin, Stetter and Tucker.

**[1]**Jörg Hertling,*Numerical treatment of Hammerstein equations by variational methods*, Numerische Lösung nichtlinearer partieller Differential- und Integrodifferentialgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1971), Springer, Berlin, 1972, pp. 267–288. Lecture Notes in Math., Vol. 267. MR**0351133****[2]**R. H. Kasriel and M. Z. Nashed,*Stability of solutions of some classes of nonlinear operator equations*, Proc. Amer. Math. Soc.**17**(1966), 1036–1042. MR**0199753**, 10.1090/S0002-9939-1966-0199753-X**[3]**S. G. Mihlin,*On the stability of certain computational processes*, Dokl. Akad. Nauk SSSR**157**(1964), 271–273 (Russian). MR**0182138****[4]**S. G. Mikhlin,*The numerical performance of variational methods*, Translated from the Russian by R. S. Anderssen, Wolters-Noordhoff Publishing, Groningen, 1971. MR**0278506****[5]**Alexandru I. Şchiop,*Stability of Ritz procedure for nonlinear two-point boundary value problem*, Numer. Math.**20**(1972/73), 208–212. MR**0317552****[6]**Hans J. Stetter,*Analysis of discretization methods for ordinary differential equations*, Springer-Verlag, New York-Heidelberg, 1973. Springer Tracts in Natural Philosophy, Vol. 23. MR**0426438****[7]**T. S. Tucker,*Stability of nonlinear computing schemes*, SIAM J. Numer. Anal.**6**(1969), 72–81. MR**0250509****[8]**M. M. Vainberg,*Variational methods for the study of nonlinear operators*, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR**0176364****[9]**R. S. Varga,*Functional analysis and approximation theory in numerical analysis*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3. MR**0310504**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0383797-2

Keywords:
Hammerstein equation,
Ritz-Galerkin method,
stability of the numerical computation,
finite element method

Article copyright:
© Copyright 1975
American Mathematical Society