Double Sobolev gradient preconditioning for nonlinear elliptic problems

O. Axelsson, J. Karátson

Research output: Contribution to journalArticle

Abstract

A mixed variable formulation of a second-order nonlinear diffusion problem leads to a finite element matrix in a product form. This form enables the efficient updating of the nonlinearity in a Picard type iteration method, in which the preconditioner involves twice a discrete Laplacian. The article gives a conditioning analysis of this method, based on analytic investigations in the corresponding Sobolev function space that reveal the behaviour of this preconditioning. The further generalization of the preconditioner can produce arbitrarily low condition numbers by proper subdivisions of Ω, while still no differentiability of the nonlinear diffusion coefficient is required.

Original languageEnglish
Pages (from-to)1018-1036
Number of pages19
JournalNumerical Methods for Partial Differential Equations
Volume23
Issue number5
DOIs
Publication statusPublished - Sep 1 2007

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Keywords

  • Discrete laplacian
  • Nonlinear diffusion problem
  • Picard-type iteration
  • Sobolev gradient preconditioning

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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