Let $K$ be a triangle with the largest edge length as $h$.
The vertices of $K$ are denoted by $O$, $A$ and $B$ and
the edges by $e_1$, $e_2$, $e_3$.
Given $u\in H^2(K),$ the Fujino-Morley interpolation $\Pi^{FM}$ maps $u$ to a quadratic polynomial that satisfies
\begin{eqnarray}
&& (\Pi^{FM}u -u)(P)=0, \quad P=O,A,B;\quad \\
&& \int_{e_i} \frac{\partial }{\partial n}(\Pi^{FM}u -u) d s=0, \quad i=1,2,3.
\end{eqnarray}

In this talk, we consider the Fujino-Morley interpolation error constant $C_0$ and $C_1$, which satisfy
$$
\|\Pi^{FM}u -u\| \le C_0 |\Pi^{FM}u -u|_2,\quad
|\Pi^{FM}u -u|_1 \le C_1 |\Pi^{FM}u -u|_2\:.
$$
In [1], rough bounds for constants $C_0$ and $C_1$ are obtained.
In our research, the problem of constant estimation is transformed to
the eigenvalue problem for certain Biharmonic differential operator,
which is further solved by applying the eigenvalue estimation method developed by
Liu [2].
Particularlly, for triangle elements with longest edge length less than 1, the optimal estimation for the constants is obtained as follows,
$$
C_0 \le 0.0740 , \quad C_1 \le 0.1888 \:.
$$

[1] Carsten Carstensen and Dietmar Gallistl. Guaranteed lower eigenvalue bounds for the biharmonic equation,
Numer. Math., 126 (1):33--51, may 2014.
[2] Xuefeng Liu. A framework of verified eigenvalue bounds for self-adjoint differential operators,
Appl. Math. Comput., 267:341-355, 2015.