Gradient method for non-injective operators in Hilbert space with application to Neumann problems

• Autorzy: János Karátson
• Języki publikacji: EN

Abstrakty

EN

The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.

Słowa kluczowe

EN

Neumann boundary value problems; non-injective non-linear operator; gradient method; Hilbert space

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1999
• Tom: 26
• Numer: 3
• Strony: 333-346

Daty

wydano

1999

otrzymano

1999-02-02

Twórcy

autor

János Karátson

• Applied Analysis Department, Loránd Eötvös University, Múzeum krt. 6-8, H-1088 Budapest, Hungary

Bibliografia

• [1] J. Céa, Lectures on Optimization. Theory and Algorithms, Springer, 1978.
• [2] J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10-26.
• [3] Yu. V. Egorov and M. A. Shubin, Partial Differential Equations I, Encyclopaedia Math. Sci., Springer, 1992.
• [4] I. Faragó and J. Karátson, The gradient-finite element method for elliptic problems, in: Conference on Numerical Mathematics and Computational Mechanics, University of Miskolc, 1998.
• [5] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
• [6] L. V. Kantorovich, On an effective method of solving extremal problems for quadratic functionals, Dokl. Akad. Nauk SSSR 48 (1945), 455-460.
• [7] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982.
• [8] J. Karátson, The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations, J. Appl. Anal. 3 (1997), 225-237.
• [9] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Wiley, 1986.
• [10] V. S. Vladimirov, A Collection of Problems on the Equations of Mathematical Physics, Mir, Moscow, 1986.