We develop multigrid algorithms for a quadratic 
interior penalty
method for a biharmonic problem with essential and natural boundary
conditions of the Cahn-Hilliard type. A multigrid solve for the Poisson
problem with homogeneous Neumann boundary condition is used as a
preconditioner in the smoothing steps for the fourth order problem, which
significantly improves the performance of the multigrid methods.
Numerical results are presented.