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

Mitsuhiro T. Nakao, Yoshitaka Watanabe

研究成果: ジャーナルへの寄稿記事

15 引用 (Scopus)

抄録

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.

元の言語英語
ページ(範囲)311-323
ページ数13
ジャーナルNumerical Algorithms
37
発行部数1-4 SPEC. ISS.
DOI
出版物ステータス出版済み - 12 1 2004

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

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

これを引用

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

:: Numerical Algorithms, 巻 37, 番号 1-4 SPEC. ISS., 01.12.2004, p. 311-323.

研究成果: ジャーナルへの寄稿記事

@article{62b092495f4b4a6aa96c5578e8e4ab23,
title = "An efficient approach to the numerical verification for solutions of elliptic differential equations",
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.",
author = "Nakao, {Mitsuhiro T.} and Yoshitaka Watanabe",
year = "2004",
month = "12",
day = "1",
doi = "10.1023/B:NUMA.0000049477.75366.94",
language = "English",
volume = "37",
pages = "311--323",
journal = "Numerical Algorithms",
issn = "1017-1398",
publisher = "Springer Netherlands",
number = "1-4 SPEC. ISS.",

}

TY - JOUR

T1 - An efficient approach to the numerical verification for solutions of elliptic differential equations

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

PY - 2004/12/1

Y1 - 2004/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=10044260069&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044260069&partnerID=8YFLogxK

U2 - 10.1023/B:NUMA.0000049477.75366.94

DO - 10.1023/B:NUMA.0000049477.75366.94

M3 - Article

AN - SCOPUS:10044260069

VL - 37

SP - 311

EP - 323

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

IS - 1-4 SPEC. ISS.

ER -