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

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

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]**J. HERTLING,*Numerical Treatment of Hammerstein-Equations by Variational Methods*, R. Ansorge & W. Törnig (Eds.), Numerische Lösung nichtlinearer partieller Differential- und Integro-differentialgleichungen, Lecture Notes in Math., no. 267, Springer-Verlag, Berlin, 1972, pp. 267-288. MR**0351133 (50:3622)****[2]**R. H. KASRIEL & M. Z. NASHED, "Stability of solutions of some classes of nonlinear operator equations,"*Proc. Amer. Math. Soc.*, v. 17, 1966, pp. 1036-1042. MR**33**#7896. MR**0199753 (33:7896)****[3]**S. G. MIKHLIN, "On the stability of certain computational processes," Dokl. Akad. Nauk SSSR, v. 157, 1964, pp. 271-273 = Soviet Math. Dokl., v. 5, 1964, pp. 931-933. MR**31**#6361. MR**0182138 (31:6361)****[4]**S. G. MIKHLIN,*The Numerical Performance of Variational Methods*, Wolters-Noordhoff, Groningen, 1971. MR**43**#4236. MR**0278506 (43:4236)****[5]**A. I. ŞCHIOP, "Stability of Ritz procedure for nonlinear two-point boundary value problem,"*Numer. Math.*, v. 20, 1973, pp. 208-212. MR**0317552 (47:6099)****[6]**H. J. STETTER,*Analysis of Discretization Methods for Ordinary Differential Equations*, Springer-Verlag, Berlin, 1973. MR**0426438 (54:14381)****[7]**T. S. TUCKER, "Stability of nonlinear computing schemes,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 72-81. MR**40**#3743. MR**0250509 (40:3743)****[8]**M. M. VAINBERG,*Variational Methods for the Study of Nonlinear Operators*, Holden-Day, San Francisco, Calif., 1964. MR**31**#638. MR**0176364 (31:638)****[9]**R. S. VARGA,*Functional Analysis and Approximation Theory in Numerical Analysis*, Regional Conference Series in Applied Mathematics, vol. 3, Soc. Indust. Appl. Math., Philadelphia, Pa., 1971. MR**0310504 (46:9602)**

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Additional Information

DOI:
https://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