An efficient approach to the numerical verification for solutions of elliptic differential equations

Mitsuhiro T. Nakao, Yoshitaka Watanabe

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

The authors and their colleagues have developed numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem using the Newton-like operator with appropriate approximation and constructive a priori error estimates for Poisson's equations. Many verification results show that the authors' methods are sufficiently useful when the equation has no first-order derivative. However, in the case that the equation includes the term of a first-order derivative, there is a possibility that the verification algorithm does not work even though we adopt a sufficiently accurate approximation subspace. The purpose of this paper is to propose an alternative method to overcome this difficulty. Numerical examples which confirm the effectiveness of the new method are presented.

Original languageEnglish
Pages (from-to)311-323
Number of pages13
JournalNumerical Algorithms
Volume37
Issue number1-4 SPEC. ISS.
DOIs
Publication statusPublished - 2004 Dec
Externally publishedYes

Fingerprint

Numerical Verification
Elliptic Differential Equations
Differential equations
Derivatives
Poisson equation
First-order
Derivative
Boundary value problems
A Priori Error Estimates
Elliptic Boundary Value Problems
Approximation
Poisson's equation
Fixed point theorem
Subspace
Numerical Examples
Alternatives
Term
Operator

Keywords

  • elliptic equations
  • fixed-point theorem
  • numerical verification

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

An efficient approach to the numerical verification for solutions of elliptic differential equations. / Nakao, Mitsuhiro T.; Watanabe, Yoshitaka.

In: Numerical Algorithms, Vol. 37, No. 1-4 SPEC. ISS., 12.2004, p. 311-323.

Research output: Contribution to journalArticle

Nakao, Mitsuhiro T. ; Watanabe, Yoshitaka. / An efficient approach to the numerical verification for solutions of elliptic differential equations. In: Numerical Algorithms. 2004 ; Vol. 37, No. 1-4 SPEC. ISS. pp. 311-323.
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