A numerical verification method for solutions of singularly perturbed problems with nonlinearity

Kouji Hashimoto, Ryohei Abe, Mitsuhiro T. Nakao, Yoshitaka Watanabe

Research output: Contribution to journalArticle

Abstract

In order to verify the solutions of nonlinear boundary value problems by Nakao's computer-assisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error estimates for the finite element approximation of some simple linear problems. For singularly perturbed problems, however, generally it is known that the perturbation term produces a bad effect on the a priori error estimates, i.e., leads to a large constant, if we use the usual approximation methods. In this paper, we propose some verification algorithms for solutions of singularly perturbed problems with nonlinearity by using the constant obtained in the a priori error estimates based on the exponential fitting method with Green's function. Some numerical examples which confirm us the effectiveness of our method are presented.

Original languageEnglish
Pages (from-to)111-131
Number of pages21
JournalJapan Journal of Industrial and Applied Mathematics
Volume22
Issue number1
Publication statusPublished - 2005 Feb
Externally publishedYes

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Numerical Verification
A Priori Error Estimates
Singularly Perturbed Problem
Nonlinearity
Exponential Fitting
Nonlinear Boundary Value Problems
Finite Element Approximation
Green's function
Approximation Methods
Boundary value problems
Numerical methods
Numerical Methods
Verify
Perturbation
Numerical Examples
Term

Keywords

  • A priori constant
  • Finite element method
  • Numerical verification
  • Singularly perturbed problem

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

Cite this

A numerical verification method for solutions of singularly perturbed problems with nonlinearity. / Hashimoto, Kouji; Abe, Ryohei; Nakao, Mitsuhiro T.; Watanabe, Yoshitaka.

In: Japan Journal of Industrial and Applied Mathematics, Vol. 22, No. 1, 02.2005, p. 111-131.

Research output: Contribution to journalArticle

Hashimoto, Kouji ; Abe, Ryohei ; Nakao, Mitsuhiro T. ; Watanabe, Yoshitaka. / A numerical verification method for solutions of singularly perturbed problems with nonlinearity. In: Japan Journal of Industrial and Applied Mathematics. 2005 ; Vol. 22, No. 1. pp. 111-131.
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