Numerical verification methods for solutions of the free boundary problem

Kouji Hashimoto; Kenta Kobayashi; Mitsuhiro T. Nakao

doi:10.1080/01630560500248314

Numerical verification methods for solutions of the free boundary problem

Kouji Hashimoto^*, Kenta Kobayashi, Mitsuhiro T. Nakao

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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
Pages (from-to)	523-542
Number of pages	20
Journal	Numerical Functional Analysis and Optimization
Volume	26
Issue number	4-5
DOIs	https://doi.org/10.1080/01630560500248314
Publication status	Published - 2005
Externally published	Yes

Keywords

Enclosure methods
Free boundary
Numerical verification methods

ASJC Scopus subject areas

Applied Mathematics
Control and Optimization

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10.1080/01630560500248314

Cite this

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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.",

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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

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DO - 10.1080/01630560500248314

M3 - Article

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JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

IS - 4-5

ER -

Numerical verification methods for solutions of the free boundary problem

Abstract

Keywords

ASJC Scopus subject areas

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