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 http://www.xfliu.org/onlinelab/.
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Babuska, I., Osborn, J.: Eigenvalue problems. Handb. Numer. Anal. 3(1), 641–787 (1991)
Behnke, H.: The calculation of guaranteed bounds for eigenvalues using complementary variational principles. Computing 47(1), 11–27 (1991)
Carstensen, C., Gallistl, D.: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126(1), 33–51 (2014)
Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comput. 83(290), 2605–2629 (2014)
Fujino, T.: The triangular equilibrium element in the solution of plate bending problems. In: Gallagher, R.H., Yamada, Y., Tinsley Oden, J. (eds.) Recent Advaced on Matrix Methods Structural Analysis Design, pp. 725–786. The University of Alabama Press, Tuscaloosa (1971)
Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the \(P_0\) and \(P_1\) triangular finite element. Comput. Methods Appl. Mech. Eng. 196, 3750–3758 (2007)
Kobayashi, K.: On the Interpolation Constants Over Triangular Elements. Institue of Mathematics, Czech Academy of Sciences, Prague (2015)
Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341–355 (2015)
Liu, X., Kikuchi, F.: Analysis and estimation of error constants for \(P_0\) and \(P_1\) interpolations over triangular finite elements. J. Math. Sci. Univ. Tokyo 17(1), 27–78 (2010)
Liu, X., You, C.: Explicit bound for quadratic Lagrange interpolation constant on triangular finite elements. Appl. Math. Comput. 319, 693–701 (2018)
Morley, L.: The triangular equilibrium element in the solution of plate bending problems. Aeronaut. Q. 19(2), 149–169 (1968)
Morley, L.: The constant-moment plate-bending element. J. Strain Anal. 6(1), 20–24 (1971)
Rump, S.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht. http://www.ti3.tuhh.de/rump/ (1999). Accessed 4 Jan 2019
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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Liao, S., Shu, Y. & Liu, X. Optimal estimation for the Fujino–Morley interpolation error constants. Japan J. Indust. Appl. Math. 36, 521–542 (2019). https://doi.org/10.1007/s13160-019-00351-9
- Fujino–Morley interpolation operator
- Finite element method
- Verified computing
- Eigenvalue problem
Mathematics Subject Classification