Sobolev gradient preconditioning for elliptic reaction–diffusion problems with some nonsmooth nonlinearities

Research output: Article

Abstract

The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction–diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected.

Original languageEnglish
Pages (from-to)223-233
Number of pages11
JournalJournal of Computational and Applied Mathematics
Volume363
DOIs
Publication statusPublished - jan. 1 2020

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Sobolev Gradient
Reaction-diffusion Problems
Gradient methods
Preconditioning
Elliptic Problems
Nonlinearity
Gradient Method
Stationary States
Differentiability
Convergence Results
Elliptic Equations
Nonlinear Problem
Iteration

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Sobolev gradient preconditioning for elliptic reaction–diffusion problems with some nonsmooth nonlinearities",
abstract = "The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction–diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected.",
keywords = "Iteration, Nonlinear elliptic equation, Sobolev gradients",
author = "J. Kar{\'a}tson",
year = "2020",
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journal = "Journal of Computational and Applied Mathematics",
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AB - The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction–diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected.

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KW - Nonlinear elliptic equation

KW - Sobolev gradients

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