Abstract
This is a continuation of a previour paper [22]. Local and global convergence theorems are established for the SOR-Newton method applied to a system of mildly nonlinear equations which arises from the usual discretization of the Dirichlet problem for the equation Δu=f(u), wheref is continuously differentiable andf′≥0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W.F. Ames, Numerical Methods for Partial Differential Equations. Third Edition. Academic Press, New York, 1992.
O. Axelsson, Iterative Solution Methods. Cambridge Univ. Press, 1994.
M.E. Brewster and R. Kannan, Nonlinear successive over-relaxation. Numer., Math.,44 (1984), 309–315.
M.E. Brewster and R. Kannan, Global convergence of nonlinear successive overrelaxation via linear theory. Computing,34 (1985), 73–79.
M.E. Brewster and R. Kannan, Varying relaxation parameters in nonlinear successive overrelaxation. Computing,34 (1985), 81–85.
M.E. Brewster and R. Kannan, A computational process for choosing the relaxation parameter in nonlinear SOR. Computing,37 (1986), 19–29.
P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation. Cambridge Press, Cambridge, 1988.
D. Greenspan, Introductory Numerical Analysis of Elliptic Boundary Value Problems. Harper & Row, New York, 1965.
D. Greenspan and V. Casulli, Numerical Analysis for Applied Mathematics. Science, and Engineering. Addison-Wesley, New York, 1988.
D. Greenspan and S.V. Parter, Mildly nonlinear elliptic partial differential equations and their numerical solution. II. Numer. Math.,7 (1965), 129–146.
D. Greenspan and M. Yohe, On the approximate solution of Δu=f(u). Comm. ACM,6 (1963), 564–568.
W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations. Springer-Verlag, 1994.
K. Ishihara, Successive overrelaxation method with projection for finite element solutions of nonlinear, radiation cooling problems. Computing,38 (1987), 117–132.
K. Ishihara, Successive overrelaxation methods with projection for finite element solutions applied to the Dirichlet problem of the nonlinear elliptic equatioin Δu=bu 2. Computing,40 (1988), 51–65.
K. Ishihara, Projected successive overrelaxation method for finite element solutions to the Dirichlet problem for a system of nonlinear elliptic equations. J. Comput. Appl. Math.,38 (1991), 185–200.
K. Ishihara, Optimum SOR iterations for finite difference equations arising from nonlinear periodic boundary value problems. Math. Japon.,45 (1997) (to appear).
C.C.J. Kuo, B.C. Levy and B.R. Musicus, A local relaxation method for solving elliptic PDEs on mesh-connected arrays. SIAM J. Sci. Statist. Comput.,8 (1987), 550–573.
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
J.M. Ortega and M.L. Rockoff, Nonlinear difference equations and Gauss-Seidel type iterative methods. SIAM J. Numer. Anal.,3 (1966), 497–513.
S.V. Parter, Mildly nonlinear elliptic partial differential equations and their numerical solution. I. Numer. Math.,7 (1965), 113–128.
R.S. Varga, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962.
T. Yamamoto, On nonlinear SOR-like methods, I — Applications to simultaneous methods for polynomial zeros. Japan J. Indust. Appl. Math.,14 (1997), 87–97.
D.M. Young, Iterative Solution of Large Linear Systems. Academic Press, New York and London, 1971.
Author information
Authors and Affiliations
Additional information
This work was partially supported by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan.
About this article
Cite this article
Ishihara, K., Muroya, Y. & Yamamoto, T. On nonlinear SOR-like methods, II — Convergence of the SOR-Newton method for mildly nonlinear equations. Japan J. Indust. Appl. Math. 14, 99–110 (1997). https://doi.org/10.1007/BF03167313
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167313