# Numerical verification methods for solutions of the free boundary problem

Kouji Hashimoto, Kenta Kobayashi, Mitsuhiro T. Nakao

Research output: Contribution to journalArticle

### Abstract

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.

Original language English 523-542 20 Numerical Functional Analysis and Optimization 26 4-5 https://doi.org/10.1080/01630560500248314 Published - 2005 Yes

### Fingerprint

Numerical Verification
Free Boundary Problem
Nonlinear Boundary Value Problems
Finite Element Approximation
Unknown Parameters
Boundary value problems
Error Estimates
Fixed point
Projection
Finite Element
Numerical Examples
Interval

### Keywords

• Enclosure methods
• Free boundary
• Numerical verification methods

### ASJC Scopus subject areas

• Applied Mathematics
• Control and Optimization

### Cite this

Numerical verification methods for solutions of the free boundary problem. / Hashimoto, Kouji; Kobayashi, Kenta; Nakao, Mitsuhiro T.

In: Numerical Functional Analysis and Optimization, Vol. 26, No. 4-5, 2005, p. 523-542.

Research output: Contribution to journalArticle

Hashimoto, Kouji ; Kobayashi, Kenta ; Nakao, Mitsuhiro T. / Numerical verification methods for solutions of the free boundary problem. In: Numerical Functional Analysis and Optimization. 2005 ; Vol. 26, No. 4-5. pp. 523-542.
title = "Numerical verification methods for solutions of the free boundary problem",
abstract = "We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.",
keywords = "Enclosure methods, Free boundary, Numerical verification methods",
author = "Kouji Hashimoto and Kenta Kobayashi and Nakao, {Mitsuhiro T.}",
year = "2005",
doi = "10.1080/01630560500248314",
language = "English",
volume = "26",
pages = "523--542",
journal = "Numerical Functional Analysis and Optimization",
issn = "0163-0563",
publisher = "Taylor and Francis Ltd.",
number = "4-5",

}

TY - JOUR

T1 - Numerical verification methods for solutions of the free boundary problem

AU - Hashimoto, Kouji

AU - Kobayashi, Kenta

AU - Nakao, Mitsuhiro T.

PY - 2005

Y1 - 2005

N2 - We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.

AB - We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.

KW - Enclosure methods

KW - Free boundary

KW - Numerical verification methods

UR - http://www.scopus.com/inward/record.url?scp=27744555836&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27744555836&partnerID=8YFLogxK

U2 - 10.1080/01630560500248314

DO - 10.1080/01630560500248314

M3 - Article

AN - SCOPUS:27744555836

VL - 26

SP - 523

EP - 542

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 4-5

ER -