Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

Abstract

Infinite-dimensional gradient method is constructed for nonlinear fourth order elliptic BVPs. Earlier results on uniformly elliptic equations are extended to strong nonlinearity when the growth conditions are only limited by the Hilbert space well-posedness. The obtained method is opposite to the usual way of first discretizing the problem. Namely, the theoretical iteration is executed for the BVP itself on the continuous level in the corresponding Sobolev space, reducing the nonlinear BVP to auxiliary linear problems. Thus we obtain a class of numerical methods, in which numerical realization is determined by the method chosen for the auxiliary problems. The biharmonic operator acts as a Sobolev space preconditioner, yielding a fixed ratio of linear convergence of the iteration (i.e. one determined by the original coefficients only, independent of the way of solution of the auxiliary problems), and at the same time reducing computational questions to those for linear problems. A numerical example is given for illustration.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages459-466
Number of pages8
Volume1988
ISBN (Print)9783540418146
Publication statusPublished - 2001
Event2nd International Conference on Numerical Analysis and Its Applications, NAA 2000 - Rousse, Bulgaria
Duration: Jun 11 2000Jun 15 2000

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1988
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other2nd International Conference on Numerical Analysis and Its Applications, NAA 2000
CountryBulgaria
CityRousse
Period6/11/006/15/00

Fingerprint

Sobolev spaces
Preconditioning
Elliptic Problems
Sobolev Spaces
Gradient methods
Hilbert spaces
Mathematical operators
Numerical methods
Biharmonic Operator
Iteration
Linear Convergence
Gradient Method
Growth Conditions
Preconditioner
Well-posedness
Elliptic Equations
Fourth Order
Hilbert space
Numerical Methods
Nonlinearity

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Karátson, J. (2001). Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1988, pp. 459-466). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1988). Springer Verlag.

Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems. / Karátson, J.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1988 Springer Verlag, 2001. p. 459-466 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1988).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Karátson, J 2001, Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 1988, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1988, Springer Verlag, pp. 459-466, 2nd International Conference on Numerical Analysis and Its Applications, NAA 2000, Rousse, Bulgaria, 6/11/00.
Karátson J. Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1988. Springer Verlag. 2001. p. 459-466. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Karátson, J. / Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1988 Springer Verlag, 2001. pp. 459-466 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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