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.

[1]  Yan Xu,et al.  Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations , 2014, J. Sci. Comput..

[2]  Tomáš Oberhuber,et al.  Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves , 2007 .

[3]  Yan Xu,et al.  Efficient High Order Semi-implicit Time Discretization and Local Discontinuous Galerkin Methods for Highly Nonlinear PDEs , 2016, J. Sci. Comput..

[4]  Francis Filbet,et al.  High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations , 2016, Journal of Scientific Computing.

[5]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[6]  Yan Xu,et al.  Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs , 2009, J. Sci. Comput..

[7]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[8]  Gieri Simonett,et al.  The Willmore flow near spheres , 2001, Differential and Integral Equations.

[9]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[10]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations , 2009 .

[11]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[12]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[13]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[14]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[15]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[16]  D. M,et al.  A level set formulation for Willmore flow , 2004 .

[17]  T. Oberhuber,et al.  NUMERICAL SOLUTION FOR THE WILLMORE FLOW OF GRAPHS , 2006 .

[18]  S. Osher,et al.  Discontinuous Galerkin Level Set Method for Interface Capturing , 2012 .

[19]  Achi Brandt,et al.  Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition , 2011 .

[20]  Yinhua Xia,et al.  Efficient time discretization for local discontinuous Galerkin methods , 2007 .

[21]  Gerhard Dziuk,et al.  Error analysis of a finite element method for the Willmore flow of graphs , 2006 .

[22]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .