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Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis

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Abstract

A postprocessing technique for mixed finite element methods for the Cahn–Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on a coarser space, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

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Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Changsha University of Science and Technology, Yuntang Campus, Changsha, 410114, China

    Wansheng Wang

  2. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Long Chen

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China

    Jie Zhou

Authors
  1. Wansheng Wang
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  2. Long Chen
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  3. Jie Zhou
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Corresponding author

Correspondence to Long Chen.

Additional information

W. S. Wang was supported by Ky and Yu-Fen Fan Fund Travel Grant from the AMS, the National Natural Science Foundation of China Grants 11001033 and 11371074, the Natural Science Foundation for Distinguished Young scholars of Hunan Province in China Grant 13JJ1020, and the Research Foundation of Education Bureau of Hunan Province in China Grant 13A108.

L. Chen was supported by NSF Grant DMS-1418934, and in part by U.S. Department of Energy (DOE) prime award # DE-SC0006903 and NIH Grant P50GM76516.

J. Zhou was supported by NSFC Project (91430213) and 2012–2013 China Scholarship Council, and partially by NSF Grant DMS-1115961.

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Wang, W., Chen, L. & Zhou, J. Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis. J Sci Comput 67, 724–746 (2016). https://doi.org/10.1007/s10915-015-0101-9

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  • DOI: https://doi.org/10.1007/s10915-015-0101-9

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