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

2 Citations (Scopus)


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
Issue number3
Publication statusPublished - 2014 Nov 1



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

ASJC Scopus subject areas

  • Applied Mathematics
  • Engineering(all)

Cite this