Local Discontinuous Galerkin Methods for the Functionalized Cahn-Hilliard Equation

作     者：Guo, Ruihan Xu, Yan Xu, Zhengfu 

作者机构：Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA 

出 版 物：《JOURNAL OF SCIENTIFIC COMPUTING》 (科学计算杂志)

年 卷 期：2015年第63卷第3期

页      面：913-937页

核心收录：

学科分类：08[工学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：NSFC [11371342, 11031007] Fok Ying Tung Education Foundation NSF [DMS-1316662] 

主　　题：Functionalized Cahn-Hilliard equation Local discontinuous Galerkin method Multigrid algorithm Stability 

摘      要：In this paper, we develop a local discontinuous Galerkin (LDG) method for the sixth order nonlinear functionalized Cahn-Hilliard (FCH) equation. We address the accuracy and stability issues from simulating high order stiff equations in phase-field modeling. Within the LDG framework, various boundary conditions associated with the background physics can be naturally implemented. We prove the energy stability of the LDG method for the general nonlinear case. A semi-implicit time marching method is applied to remove the severe time step restriction () for explicit methods. The adaptive capability of the LDG method allows for capturing the interfacial layers and the complicated geometric structures of the solution with high resolution. To enhance the efficiency of the proposed approach, the multigrid (MG) method is used to solve the system of linear equations resulting from the semi-implicit temporal integration at each time step. We show numerically that the MG solver has mesh-independent convergence rates. Numerical simulation results for the FCH equation in two and three dimensions are provided to illustrate that the combination of the LDG method for spatial approximation, semi-implicit temporal integration with the MG solver provides a practical and efficient approach when solving this family of problems.