A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems

M. T. Nakao; K. Hashimoto; Y. Watanabe

doi:10.1007/s00607-004-0111-1

A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems

M. T. Nakao, K. Hashimoto, Y. Watanabe

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

31 Citations (Scopus)

Abstract

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

Original language	English
Pages (from-to)	1-14
Number of pages	14
Journal	Computing (Vienna/New York)
Volume	75
Issue number	1 SPEC. ISS.
DOIs	https://doi.org/10.1007/s00607-004-0111-1
Publication status	Published - 2005 Jul
Externally published	Yes

Keywords

Finite element method
Numerical verification
Unique solvability of linear elliptic problem

ASJC Scopus subject areas

Theoretical Computer Science
Computational Theory and Mathematics

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Cite this

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AB - In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

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KW - Unique solvability of linear elliptic problem

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