A p-Adaptive Local Discontinuous Galerkin Level Set Method for Willmore Flow

The level set method is often used to capture interface behavior in two or three dimensions. In this paper, we present a combination of a local discontinuous Galerkin (LDG) method and a level set method for simulating Willmore flow. The LDG scheme is energy stable and mass conservative, which are good properties compared 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 and high nonlinearity of Willmore flow, a high order semi-implicit Runge–Kutta method is employed for time discretization, which allows larger time steps. These equations at the implicit time level are linear, we demonstrate an efficient and practical multigrid solver to solve the equations. Numerical examples are given to illustrate that the combination of the LDG scheme and level set method provides an efficient and practical approach to simulate the Willmore flow.