Hessian recovery based finite element methods for the Cahn-Hilliard equation

JOURNAL OF COMPUTATIONAL PHYSICS

卷:386

Article

Xu, Minqiang¹  Guo, Hailong²  Zou, Qingsong³ 

[1] Sun Yat Sen Univ, Sch Data & Comp Sci, Guangzhou 510275, Guangdong, Peoples R China

[2] Univ Melbourne, Sch Math & Stat, Parkville, Vic 3010, Australia

[3] Sun Yat Sen Univ, Sch Data & Comp Sci & Guangdong Prov, Key Lab Computat Sci, Guangzhou 510275, Guangdong, Peoples R China

关键词: Hessian recovery;    Cahn-Hilliard equation;    Phase separation;    Recovery based;    Superconvergence;    Linear finite element;   

DOI  :  10.1016/j.jcp.2019.01.056

来源: Elsevier

【 摘 要 】

In this paper, we propose several novel 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free