  In this paper, we propose a novel recovery based finite element method for
the Cahn-Hilliard equation. One distinguishing feature of the method is that we
discretize the fourth-order differential operator in a standard $C^0$ linear
finite elements space. Precisely, we first transform the fourth-order
Cahn-Hilliard equation to its variational formulation in which only first-order
and second-order derivatives are involved and then we compute the and
second-order derivatives of a linear finite element function by a least-squares
fitting recovery procedure. The intrinsic link between the second-order
derivatives (Hessian matrix) recovery scheme and the finite difference method
is studied in the paper. In particular, for the first time, we discover that
the Laplace recovery scheme is exactly the well-known five-point stencil over
uniform meshes. The proposed discretization for the Cahn-Hilliard equation can
be regarded as a combination of the finite difference scheme and the finite
element scheme. In addition, special considerations are put on different
methods for imposing Neumann type boundary conditions.
  The optimal-order convergence and energy stability are numerically proved
through a series of benchmark tests.
