Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation

Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, Shinichi Oishi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.

Original languageEnglish
Pages (from-to)665-679
Number of pages15
JournalJapan Journal of Industrial and Applied Mathematics
Volume31
Issue number3
DOIs
Publication statusPublished - 2014 Nov 1

Fingerprint

Elliptic Operator
Linear Operator
Mathematical operators
Eigenvalue
Norm
Evaluation
Generalized Eigenvalue Problem
Invertibility
Elliptic Boundary Value Problems
Boundary value problems
Dirichlet
Upper and Lower Bounds
Numerical methods
Numerical Methods
Boundary conditions
Numerical Examples
Target
Operator

Keywords

  • Eigenvalue problem
  • Elliptic operator
  • Finite element method
  • Inverse norm estimation
  • Numerical verification

ASJC Scopus subject areas

  • Applied Mathematics
  • Engineering(all)

Cite this

Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation. / Tanaka, Kazuaki; Takayasu, Akitoshi; Liu, Xuefeng; Oishi, Shinichi.

In: Japan Journal of Industrial and Applied Mathematics, Vol. 31, No. 3, 01.11.2014, p. 665-679.

Research output: Contribution to journalArticle

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