Some considerations of the invertibility verifications for linear elliptic operators

  • Mitsuhiro T. Nakao
  • Yoshitaka WatanabeEmail author
  • Takehiko Kinoshita
  • Takuma Kimura
  • Nobito Yamamoto
Original Paper Area 2


This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these procedures is an improvement of a theorem by Nakao et al. (Computing 75:1–14, 2005) that uses projection and constructive a priori error estimates and was proposed by two of the authors of this paper. Results verifying these procedures are presented for several numerical examples.


Numerical verification Solvability of linear problem Differential operators Computer-assisted proof 

Mathematics Subject Classification

65G20 47F05 35P15 



The authors heartily thank the two anonymous referees for their thorough reading and valuable comments. This work was supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 24340018, 23740074, and 24540151).


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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
  • Yoshitaka Watanabe
    • 2
    Email author
  • Takehiko Kinoshita
    • 3
    • 4
  • Takuma Kimura
    • 5
  • Nobito Yamamoto
    • 6
  1. 1.National Institute of Technology, Sasebo CollegeSaseboJapan
  2. 2.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan
  3. 3.Center for the Promotion of Interdisciplinary Education and ResearchKyoto UniversityKyotoJapan
  4. 4.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  5. 5.JST CREST/Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  6. 6.Department of Communication Engineering and Informatics, Faculty of Informatics and EngineeringUniversity of Electro-CommunicationsTokyoJapan

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