Sobolev space preconditioning for Newton's method using domain decomposition

Numerical Linear Algebra with Applications

Volumn 9, Issue 6-7, 2002, Pages 585-598

  Axelsson, O ^a   Farago I ^b   Karatson J ^b  

^a RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

^b EÖTVÖS LORÁND UNIVERSITY   (Hungary)

Author keywords

Newton's method; Preconditioning; Sobolev space background

Indexed keywords

EID: 0036391681     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.293     Document Type: Conference Paper

References (28)

1. Neuberger JW. Soboler Gradients and Differential Equations, Lecture Notes in Mathematics Vol. 1670. Springer: Berlin, 1997.
2. Richardson WB. JR. Sobolev gradient preconditioning for PDE applications. In Iterative Methods in Scientific Computation IV. Kincaid DR, Elster AC (eds). IMACS: New Brunswick. NJ, 1999;223-234.
3. Carey GF, Jiang B-N. Nonlinear preconditioned conjugate gradient and least-squares finite elements. Computer Methods in Applied Mechanics and Engineering 1987; 62:145-154.
4. Greenbaum A. Diagonal scalings of the Laplacian as preconditioners for other elliptic differential operators. SIAM Journal on Matrix Analysis and Applications 1992; 13:826-846.
5. Axelsson O, Karátson J. Double Sobolev gradient preconditioning for elliptic problems. Report 0016, Department of Mathematics, University of Nijmegen, April 2000.
6. Faragó I, Karátson J. The gradient-finite element method for elliptic problems. Computers and Mathematics in Applications 2001; 42:1043-1053.
7. Karátson J, Faragó I. Infinite-dimensional preconditioning and Sobolev gradients for nonlinear elliptic problems, submitted for publication to Comp. Math. Appl.
8. Axelsson O. On global convergence of iterative methods. In Iterative Solution of Nonlinear Systems of Equations, ed. by R. Ansorge, Lecture Notes in Math. Vol. 953. Springer: Berlin, 1982; 1-19.
9. Dembo RS, Eisenstat SC, Steihaug T. Inexact Newton methods. SIAM Journal on Numerical Analysis 1982; 19:400-408.
10. Axelsson O, Maubach J. On the updating and assembly of the Hessian matrix in finite element methods. Computer Methods in Applied Mechanics and Engineering 1988; 71:41-67.
11. Cai X, Dryja M. Domain decomposition methods for monotone nonlinear elliptic problems. In Domain Decomposition Methods in Scientific and Engineering Computing, eds. D.E. Keyes and J. Xu. Contemporary Mathematics, Vol. 180. AMS: Providence, RI. 1994;21-27.
12. Faber V, Manteuffel T, Parter SV. On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations. Advances in Applied Mathematics 1990; 11:109-163.
13. Glowinski R, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer: New York, 1984.
14. Axelsson O, Vassilevski PS. Algebraic multilevel preconditioning methods I. Numerische Mathematik 1989; 56:157-177.
15. Dryja M. An iterative substructuring method for elliptic mortar finite element problems with discontinuous coefficients. In Domain Decomposition Methods, Vol. 10 (Boulder, CO, 1997). Contemporary Mathematics, Vol. 218. AMS:Providence, RI, 1998; 94-103
16. Dryja M. An iterative substructuring method for elliptic mortar finite element problems with discontinuous coefficients. In Domain Decomposition Methods, Vol. 10 (Boulder, CO, 1997). Contemporary Mathematics, Vol. 218. AMS:Providence, RI, 1998; 94-103
17. Graham IG, Hagger MJ. Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM Journal on Scientific Computing 1999; 20:2041-2066 (electronic).
18. Axelsson O. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
19. Khoromskij BN, Wittum G. Robust interface reduction for highly anisotropic elliptic equations. In Multigrid Methods. Vol. V (Stuttgart, 1996). Lecture Notes in Computer Science and Engineering, Vol. 3, Springer: Berlin, 1998; 140-156.
20. Khoromskij BN, Wittum G. Robust interface reduction for highly anisotropic elliptic equations. In Multigrid Methods. Vol. V (Stuttgart, 1996). Lecture Notes in Computer Science and Engineering, Vol. 3, Springer: Berlin, 1998; 140-156.
21. Axelsson O, Gustafsson I. An efficient finite element method for nonlinear diffusion problems. Bulletin of the Greek Mathematical Society 1991; 32:45-61.
22. Kantorovich LV, Akilov GP. Functional Analysis. Pergamon Press: Oxford, 1982.
23. Mahavier WT. A convergence result for discrete steepest descent in weighted Sobolev spaces. Abstract and Applied Analysis 1997; 2(1-2):67-72.
24. Axelsson O, Vassilevski PS. Algebraic multilevel preconditioning methods II. SIAM Journal on Numerical Analysis 1990; 27:1569-1590.
25. Křižek M, Neittaanmäki P. Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer Academic Publishers: Dordrecht, 1996.
26. Axelsson O, Nävert U. On a graphical package for nonlinear partial differential equation problems. In Information processing. Vol. 77. Gilchrist B (ed.) North-Holland: Amsterdam, 1977; 103-108.
27. Knyazev AW, Widlund O. Uniform finite element error estimates for differential equations with jumps in the coefficients. Mathematical Computation, to appear.
28. Ainsworth M. A preconditioner based on domain decomposition for h-p finite element approximation on quasi-uniform meshes. SIAM Journal on Numerical Analysis 1996; 33:1358-1376.