Any conforming finite element scheme always gives an upper bound for elliptic eigenvalue problem. We point out that it is essentially equal to $(u_h,u_h) \le (u,u_h) \le (u,u) $, and a lower bound scheme which is called a dual scheme with the finite element projection is proposed. If $a(u-u_h,u_h) = O(\lambda \|u-u_h\|^2_{L^2})$, a series of high order schemes with the same accuracy have been obtained, such as harmonic means $\lambda_H^h=\frac{2a(u_h,u_h)}{(u,u)+(u_h,u_h)}$. For the linear and multi-linear element of rectangle, triangle and parallelpiped domains, two key conditions are investigated: $a(u-u_h,u_h)=O(\lambda \|u-u_h\|^2_{L^2})$ and the distance between the $L^2$ norm $(u,u)$ and $l^2$ norm in the discrete grid of the eigen-functions. The numerical experiments are carried out to validate the theorectical results. For Laplace problem in 2D/3D we give some high order schemes (6th, 8th and up to 10th order) to compute the first dozens of eigen-values. They are efficient to the singular eigen-functions and high-frequency eigen-values too.