# Optimal estimation for the Fujino-Morley interpolation error constants

## TL;DR

This paper develops a new finite element-based algorithm to accurately estimate the interpolation error constants for Fujino-Morley operators, providing concrete bounds and analyzing shape perturbations.

## Contribution

It introduces a verified computational method for lower bounds of eigenvalues related to interpolation error constants, including shape variation analysis.

## Key findings

- Provided concrete upper bounds for Fujino-Morley interpolation error constants.
- Developed a new algorithm based on finite element methods with verified computation.
- Analyzed how eigenvalues vary with shape perturbations of triangles.

## Abstract

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/.