### Abstract

We present a Sobolev gradient type preconditioning for iterative methods used in solving second order semilinear elliptic systems; the n-tuple of independent Laplacians acts as a preconditioning operator in Sobolev spaces. The theoretical iteration is done at the continuous level, providing a linearization approach that reduces the original problem to a system of linear Poisson equations. The method obtained preserves linear convergence when a polynomial growth of the lower order reaction type terms is involved. For the proof of linear convergence for systems with mixed boundary conditions, we use suitable energy spaces. We use Sobolev embedding estimates in the construction of the exact algorithm. The numerical implementation has focus on a direct and elementary realization, for which a detailed discussion and some examples are given.

Original language | English |
---|---|

Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Electronic Journal of Differential Equations |

Volume | 2004 |

Publication status | Published - May 21 2004 |

### Fingerprint

### Keywords

- Numerical solution
- Preconditioning
- Semilinear elliptic systems
- Sobolev gradient

### ASJC Scopus subject areas

- Analysis

### Cite this

**Constructive Sobolev gradient preconditioning for semilinear elliptic systems.** / Karátson, J.

Research output: Contribution to journal › Article

*Electronic Journal of Differential Equations*, vol. 2004, pp. 1-26.

}

TY - JOUR

T1 - Constructive Sobolev gradient preconditioning for semilinear elliptic systems

AU - Karátson, J.

PY - 2004/5/21

Y1 - 2004/5/21

N2 - We present a Sobolev gradient type preconditioning for iterative methods used in solving second order semilinear elliptic systems; the n-tuple of independent Laplacians acts as a preconditioning operator in Sobolev spaces. The theoretical iteration is done at the continuous level, providing a linearization approach that reduces the original problem to a system of linear Poisson equations. The method obtained preserves linear convergence when a polynomial growth of the lower order reaction type terms is involved. For the proof of linear convergence for systems with mixed boundary conditions, we use suitable energy spaces. We use Sobolev embedding estimates in the construction of the exact algorithm. The numerical implementation has focus on a direct and elementary realization, for which a detailed discussion and some examples are given.

AB - We present a Sobolev gradient type preconditioning for iterative methods used in solving second order semilinear elliptic systems; the n-tuple of independent Laplacians acts as a preconditioning operator in Sobolev spaces. The theoretical iteration is done at the continuous level, providing a linearization approach that reduces the original problem to a system of linear Poisson equations. The method obtained preserves linear convergence when a polynomial growth of the lower order reaction type terms is involved. For the proof of linear convergence for systems with mixed boundary conditions, we use suitable energy spaces. We use Sobolev embedding estimates in the construction of the exact algorithm. The numerical implementation has focus on a direct and elementary realization, for which a detailed discussion and some examples are given.

KW - Numerical solution

KW - Preconditioning

KW - Semilinear elliptic systems

KW - Sobolev gradient

UR - http://www.scopus.com/inward/record.url?scp=3042575158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042575158&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042575158

VL - 2004

SP - 1

EP - 26

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -