  In this article we present a refined convergence analysis for a second order
accurate in time, fourth order finite difference numerical scheme for the 3-D
Cahn-Hilliard equation, with an improved convergence constant. A modified
backward differentiation formula temporal discretization is applied, and a
Douglas-Dupont artificial regularization is included to ensure the energy
stability. In fact, a standard application of discrete Gronwall inequality
leads to a convergence constant dependent on the interface width parameter in
an exponential singular form. We aim to obtain an improved estimate, with such
a singular dependence only in a polynomial order. A uniform in time functional
bounds of the numerical solution, including the higher order Sobolev norms, as
well as the associated bounds for the first and second order temporal
difference stencil, have to be carefully established. Certain recursive
analysis has to be applied in the analysis for the BDF-style temporal stencil.
As a result, we are able to apply a spectrum estimate for the linearized
Cahn-Hilliard operator, and this technique leads to the refined error estimate.
A three-dimensional numerical example of accuracy check is presented as well.
