Abstract
Suppose that a given complicated domain is constructed from two or more partially overlapping subdomains, for which solutions are available. The convergence in energy of the iterative method that reduces the given problem to a sequence of problems for simple subdomains is proved. Properties of Sobolev's spaces are employed, namely the theorem on equivalent norms and compactness of a bounded set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Courant and D. Hilbert, Methoden der Mathematischen Physik II, Berlin (1937).
S.L. Sobolev, On the Shwarz method in elasticity theory, Д AH CCCP 6(4) (1936) (in Russian).
K. Kalik, On convergence of the Shwarz algorithm, чИВУчЭ, Matem. (1959) (1) (in Russian).
N. Nikolsky, The Shwarz method in elasticity theory problems with the given traction vector on the boundary, ДAH CCCP 135(3) (1960) (in Russian).
S. Ja. Cogan, On alternative Shwarz method in three dimensional elasticity theory, IZV. AN SSSR (1956) (3) (in Russian).
S.G. Mikhlin, Variational Methods in Mathematical Physics, Pergamon Press, N.Y. (1964).
S.L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Amer. Math. So., Providence, R.I. (1963).
S.G. Mikhlin and K.L. Smolitsky, Approximate Methods for Solution of Differential and Integral Equations, Amer. Elsevier Publ. Co., N.Y. (1967).
R.P. Brent, Algorithms for finding zeros and extrema of functions without calculating derivatives, Stanford Univ., Tech. Report CS-71-198 (1971).
R. Fletcher and M.J. Powel, A rapidly convergent descent method for minimization, Computer J. 6(2) (1963) 163–168.
R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients, Computer J. 7(2) (1964) 149–154.
A.A. Goldstein and J.F. Price, An effective Algorithm for minimization, Numer. Math. 10 (1967) 184–189.
J. Bracken and G.P. McCormick, Selected Applications of Nonlinear Programming, J. Wiley & Sons (1968).
S.P. Timoshenko and D.H. Young, Theory of Structures, McGraw-Hill (1965).
H. Cross, Analysis of continuous frames by distributing fixed-end moments, Trans. ASCE, 96 (1932).
Additional information
Aluminum Company of America, formerly at the University of Michigan
Product Engineering Div., ALCOA Laboratories, ALCOA Center
Product Engineering Div., ALCOA Laboratories, ALCOA Center
Rights and permissions
About this article
Cite this article
Azarkhin, A. On the Shwarz alternating method in problems of elastic stability. J Elasticity 15, 233–241 (1985). https://doi.org/10.1007/BF00041422
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00041422