Constructive Sobolev gradient preconditioning for semilinear elliptic systems

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 languageEnglish
Pages (from-to)1-26
Number of pages26
JournalElectronic Journal of Differential Equations
Volume2004
Publication statusPublished - May 21 2004

Fingerprint

Sobolev Gradient
Semilinear Elliptic Systems
Linear Convergence
Preconditioning
Sobolev Embedding
Iteration
n-tuple
Polynomial Growth
Mixed Boundary Conditions
Exact Algorithms
Poisson's equation
Sobolev Spaces
Linearization
Linear equation
Term
Operator
Energy
Estimate

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.

In: Electronic Journal of Differential Equations, Vol. 2004, 21.05.2004, p. 1-26.

Research output: Contribution to journalArticle

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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.

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