Phase-field approximations of the Willmore functional and flow

We discuss in this paper phase-field approximations of the Willmore functional and the associated -flow. After recollecting known results on the approximation of the Willmore energy and its 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 is 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.