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Mathematics of Computation

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A weak Galerkin finite element scheme for the Cahn-Hilliard equation

Authors: Junping Wang, Qilong Zhai, Ran Zhang and Shangyou Zhang
Journal: Math. Comp. 88 (2019), 211-235
MSC (2010): Primary 65N30, 65N15, 65N12, 74N20; Secondary 35B45, 35J50, 35J35
Published electronically: August 21, 2018
MathSciNet review: 3854056
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Abstract: This article presents a weak Galerkin (WG) finite element method for the Cahn-Hilliard equation. The WG method makes use of piecewise polynomials as approximating functions, with weakly defined partial derivatives (first and second order) computed locally by using the information in the interior and on the boundary of each element. A stabilizer is constructed and added to the numerical scheme for the purpose of providing certain weak continuities for the approximating function. A mathematical convergence theory is developed for the corresponding numerical solutions, and optimal order of error estimates are derived. Some numerical results are presented to illustrate the efficiency and accuracy of the method.

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Additional Information

Junping Wang
Affiliation: Division of Mathematical Sciences, National Science Foundation, Arlington, Virginia 22230
MR Author ID: 216677

Qilong Zhai
Affiliation: Department of Mathematics, Jilin University, Changchun, China
MR Author ID: 1114983

Ran Zhang
Affiliation: Department of Mathematics, Jilin University, Changchun, China
MR Author ID: 719427

Shangyou Zhang
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
MR Author ID: 261174

Keywords: Weak Galerkin finite element methods, weak Laplacian, Cahn-Hilliard equation, polytopal meshes
Received by editor(s): April 22, 2017
Received by editor(s) in revised form: October 7, 2017, and March 20, 2018
Published electronically: August 21, 2018
Additional Notes: The research of the first author was supported in part by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
The research of the third author was supported in part by China Natural National Science Foundation (U1530116,91630201,11471141), and by the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, People’s Republic of China.