In this talk, we talk about some new recovery based finite element methods for the 2D Cahn-Hilliard equation. One distinguishing feature of those methods is that we discretize the fourth-order differential operator in a standard $C^0$ linear finite elements space. Precisely, we first transform the fourth-order Cahn-Hilliard equation to its variational formulation in which only first-order and second-order derivatives are involved and then we compute the first and second-order derivatives of a linear finite element function by a least-squares fitting recovery procedure. When the underlying mesh is uniform meshes of regular pattern, our recovery scheme for the Laplacian operator coincides with the well- known five-point stencil. Another feature of the methods is some special treatments on Neumann type boundary conditions for reducing computational cost. The optimal-order convergence and energy stability are numerically proved through a series of benchmark tests. The proposed method can be regarded as a combination of the finite difference scheme and the finite element scheme.

郭海龙博士，墨尔本大学讲师，2007年和2010年分别本科和硕士毕业于湖南师范大学、北京大学，2015年8月获得韦恩州立大学数学博士学位，2015年7月-2018年6月在加州大学圣芭芭拉分校从事博士后研究，之后一直在墨尔本大学工作至今。郭海龙博士主要从事偏微分方程数值解，后验自适应方法的设计和分析，拓扑材料中的波动问题、高频波动问题等等。