Optimal estimation for the Fujino–Morley interpolation error constants

  • Shih-Kang Liao
  • Yu-Chen Shu
  • Xuefeng LiuEmail author
Original Paper


The quantitative estimation for the interpolation error constants of the Fujino–Morley interpolation operator is considered. To give concrete upper bounds for the constants, which is reduced to the problem of providing lower bounds for eigenvalues of bi-harmonic operators, a new algorithm based on the finite element method along with verified computation is proposed. In addition, the quantitative analysis for the variation of eigenvalues upon the perturbation of the shape of triangles is provided. Particularly, for triangles with longest edge length less than one, the optimal estimation for the constants is provided. An online demo with source codes of the constants calculation is available at


Fujino–Morley interpolation operator Finite element method Verified computing Eigenvalue problem 

Mathematics Subject Classification

35P15 97N50 65N30 



The authors would like to thank for the support from the Ministry of Science and Technology, Taiwan, ROC under Grant nos. MOST 106-2115-M-006-011, MOST 107-2911-M-006-506. This research is also supported by Japan Society for the Promotion of Science, Grand-in-Aid for Young Scientist (B) 26800090, Grant-in-Aid for Scientific Research (C) 18K03411 and Grant-in-Aid for Scientific Research (B) 16H03950 for the third author.


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Copyright information

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

Authors and Affiliations

  1. 1.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan
  2. 2.Department of Applied MathematicsNational Cheng Kung UniversityTainanTaiwan
  3. 3.Department of MathematicsNational Cheng Kung UniversityTainanTaiwan

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