  We consider a non-standard finite-volume discretization of a strongly
non-linear fourth order diffusion equation on the $d$-dimensional cube, for
arbitrary $d \geq 1$. The scheme preserves two important structural properties
of the equation: the first is the interpretation as a gradient flow in a mass
transportation metric, and the second is an intimate relation to a linear
Fokker-Planck equation. Thanks to these structural properties, the scheme
possesses two discrete Lyapunov functionals. These functionals approximate the
entropy and the Fisher information, respectively, and their dissipation rates
converge to the optimal ones in the discrete-to-continuous limit. Using the
dissipation, we derive estimates on the long-time asymptotics of the discrete
solutions. Finally, we present results from numerical experiments which
indicate that our discretization is able to capture significant features of the
complex original dynamics, even with a rather coarse spatial resolution.
