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

Wang, Wansheng; Chen, Long; Zhou, Jie

doi:10.1007/s10915-015-0101-9

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

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 the coarser mesh, 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.

Authors:

Wang, Wansheng ^[1]; Chen, Long ^[2]; Zhou, Jie ^[3]

Changsha Univ. of Science and Technology, Changsha (China)
Univ. of California, Irvine, CA (United States)
Xiangtan Univ. (China)

Publication Date:: Wed Sep 23 00:00:00 EDT 2015

Research Org.:: Pennsylvania State Univ., University Park, PA (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1467652

Grant/Contract Number:: SC0006903

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Scientific Computing

Additional Journal Information:: Journal Volume: 67; Journal Issue: 2; Journal ID: ISSN 0885-7474

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; postprocessing; mixed finite element methods; Cahn-Hilliard equation; error estimates

Citation Formats


                    Wang, Wansheng, Chen, Long, and Zhou, Jie. Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis.  United States: N. p., 2015. 
Web.  doi:10.1007/s10915-015-0101-9.

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                    Wang, Wansheng, Chen, Long, & Zhou, Jie. Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis.  United States.  https://doi.org/10.1007/s10915-015-0101-9

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                    Wang, Wansheng, Chen, Long, and Zhou, Jie. Wed .  
"Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis".  United States.  https://doi.org/10.1007/s10915-015-0101-9.  https://www.osti.gov/servlets/purl/1467652.

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@article{osti_1467652,

  title        = {Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis},

  author       = {Wang, Wansheng and Chen, Long and Zhou, Jie},

  abstractNote = {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 the coarser mesh, 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.},

  doi          = {10.1007/s10915-015-0101-9},

  journal      = {Journal of Scientific Computing},

  number       = 2,

  volume       = 67,

  place        = {United States},

  year         = {Wed Sep 23 00:00:00 EDT 2015},

  month        = {Wed Sep 23 00:00:00 EDT 2015}

}

Copy to clipboard

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

Abstract

Citation Formats

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