### 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 language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 459-466 |

Number of pages | 8 |

Volume | 1988 |

ISBN (Print) | 9783540418146 |

Publication status | Published - 2001 |

Event | 2nd International Conference on Numerical Analysis and Its Applications, NAA 2000 - Rousse, Bulgaria Duration: Jun 11 2000 → Jun 15 2000 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1988 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 2nd International Conference on Numerical Analysis and Its Applications, NAA 2000 |
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Country | Bulgaria |

City | Rousse |

Period | 6/11/00 → 6/15/00 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems

AU - Karátson, J.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=23044527787&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044527787&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:23044527787

SN - 9783540418146

VL - 1988

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 459

EP - 466

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -