  We discuss in this paper phase-field approximations of the Willmore
functional and the associated L2-flow. After recollecting known results on the
approximation of the Willmore energy and its L1-relaxation, we derive the
expression of the flows associated with various approximations, and we show
their behavior by formal arguments based on matched asymptotic expansions. We
introduce an accurate numerical scheme, whose local convergence can be proved,
to describe with more details the behavior of two flows, the classical and the
flow associated with an approximation model due to Mugnai. We propose a series
of numerical simulations in 2D and 3D to illustrate their behavior in both
smooth and singular situations.
