|TITLE||About the Optimal Estimation of Error Constants in Bounding Eigenvalues of Differential Operators|
Let K be a triangle with largest edge length as h. The vertices of K are denoted by O,A and B and the edges by e_1, e_2 and e_3. Given u∈H^2 (K), the Fujino-Morley interpolation Π^FM maps u to a quadratic polynomial that satisfies
〖(Π〗^FM u-u)(P)=0,P=O,A,B; ∫_(e_i)▒〖∂/∂n (Π^FM u-u)ds=0,〗 i=1,2,3
In this talk, we consider the Fujino-Morley interpolation error constant C_0 and C_1, which satisfy
‖Π^FM u-u‖≤C_0 |Π^FM u-u|_2,|Π^FM u-u|_1≤C_1 |Π^FM u-u|_2 .
Carsten Carstensen and Dietmar Gallistl provided rough bounds of constants C_0 and C_1. In our research, the problem of constant estimation is transformed to the eigenvalue problem for certain bi-harmonic differential operator, which is further solved by applying the eigenvalue estimation method developed by X. Liu. Particularly, for triangle elements with longest edge length less than 1, the optimal estimation for the constants is obtained as follows,
0.07349≤C_0≤0.07353, 0.18863≤C_1≤0.18868 .