An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space

  • Yoshitaka WatanabeEmail author
  • Takehiko Kinoshita
  • Mitsuhiro T. Nakao
Original Paper


This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous sense. This approach is an improvement on a previous method (Nakao et al. in Jpn J Ind Appl Math 32:19–32, 2015) which used a projection and constructive a priori error estimates. Several examples confirming the effectiveness of the proposed procedure are reported.


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

Mathematics Subject Classification

65G20 47F05 35P15 



The authors heartily thank the anonymous referee for her/his 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. 15K05012 and 15H03637) and CREST, JST. The computations were mainly carried out using the computer facilities at the Research Institute for Information Technology, Kyushu University, Japan.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan
  2. 2.FukuokaJapan
  3. 3.Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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