Approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness

K. Nagatou, N. Yamamoto, M. T. Nakao

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented.

Original languageEnglish
Pages (from-to)543-565
Number of pages23
JournalNumerical Functional Analysis and Optimization
Volume20
Issue number5-6
Publication statusPublished - 1999 Jul
Externally publishedYes

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Eigenvalue Bounds
Numerical Verification
Banach Fixed Point Theorem
Sobolev spaces
Nonlinear Elliptic Problems
Nonlinear Elliptic Equations
Uniqueness of Solutions
Finite Element Approximation
Solution Set
Absolute value
Sobolev Spaces
Error Estimates
Numerical methods
Uniqueness
Numerical Methods
Verify
Norm
Calculate
Numerical Examples
Evaluate

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

Cite this

Approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. / Nagatou, K.; Yamamoto, N.; Nakao, M. T.

In: Numerical Functional Analysis and Optimization, Vol. 20, No. 5-6, 07.1999, p. 543-565.

Research output: Contribution to journalArticle

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