  The level set method is often used to capture interface behavior in two or
three dimensions. In this paper, we present a combination of local
discontinuous Galerkin (LDG) method and level set method for simulating
Willmore flow. The LDG scheme is energy stable and mass conservative, which are
good properties comparing with other numerical methods. In addition, to enhance
the efficiency of the proposed LDG scheme and level set method, we employ a
p-adaptive local discontinuous Galerkin technique, which applies high order
polynomial approximations around the zero level set and low order ones away
from the zero level set. A major advantage of the level set method is that the
topological changes are well defined and easily performed. In particular, given
the stiffness of Willmore flow, a high order semi-implicit Runge-Kutta method
is employed for time discretization, which allows larger time step. The
equations at the implicit time level are linear, we demonstrate an efficient
and practical multi-grid solver to solve the equations. Numerical examples are
given to illustrate the combination of the LDG scheme and level set method
provides an efficient and practical approach when simulating the Willmore flow.
