  In this paper, a piecewise quadratic finite element method on rectangular
grids for the $H^1$ problems is presented. The proposed method can be viewed as
a reduced rectangular Morley element. For the source problem, the convergence
rate of this scheme is $O(h^2)$ in the energy norm on uniform grids. Besides, a
lower bound of the $L^2$-norm error is also proved, which makes the capacity
analysis of this scheme more clear. On the other hand, for the eigenvalue
problem, the numerical eigenvalues by this element are shown to be the lower
bounds of the exact ones. Some numerical results are presented, which show the
potential of the proposed finite element.
