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

  • Kazuaki Tanaka
  • Akitoshi Takayasu
  • Xuefeng Liu
  • Shin’ichi Oishi
Original Paper Area 2

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.

Keywords

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

Mathematics Subject Classification

65N25 65N30 35J25 

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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2014

Authors and Affiliations

  • Kazuaki Tanaka
    • 1
  • Akitoshi Takayasu
    • 2
  • Xuefeng Liu
    • 3
    • 4
  • Shin’ichi Oishi
    • 2
    • 5
  1. 1.Graduate School of Fundamental Science and EngineeringWaseda UniversityShinjuku-kuJapan
  2. 2.Department of Applied Mathematics, Faculty of Science and EngineeringWaseda UniversityShinjuku-kuJapan
  3. 3.Research Institute for Science and EngineeringWaseda UniversityShinjuku-kuJapan
  4. 4.Graduate School of Science and Technology, Niigata UniversityNiigata CityJapan
  5. 5.JST, CRESTTokyoJapan

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