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POSTPROCESSING MIXED FINITE ELEMENT METHODS FOR SOLVING CAHN-HILLIARD EQUATION: METHODS AND ERROR ANALYSIS.

Wansheng Wang1, Long Chen2, Jie Zhou3

  • 1School of Mathematics and Computational Science, Changsha University of Science & Technology, Yuntang Campus, 410114 Changsha, China ( w.s.wang@163.com ).

Journal of Scientific Computing
|April 26, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a novel postprocessing technique for mixed finite element methods applied to the Cahn-Hilliard equation, enhancing computational efficiency and maintaining optimal convergence rates for approximations.

Keywords:
Cahn-Hilliard equationerror estimatesmixed finite element methodspostprocessing

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Materials science

Background:

  • The Cahn-Hilliard equation models phase separation in materials.
  • Mixed finite element methods (MFEMs) are used for solving such equations.
  • Existing MFEMs can be computationally intensive.

Purpose of the Study:

  • To develop and analyze a new postprocessing technique for MFEMs applied to the Cahn-Hilliard equation.
  • To improve the computational efficiency of these methods.
  • To maintain optimal convergence rates for key variables.

Main Methods:

  • A postprocessing step involving solving two decoupled Poisson equations in an enriched finite element space.
  • Utilizing fast Poisson solvers on finer grids or higher-order spaces.
  • Applying nonlinear iteration to a significantly smaller problem size.

Main Results:

  • The proposed technique significantly reduces computational cost.
  • Optimal rates of convergence are maintained for both concentration and chemical potential approximations.
  • Novel negative norm error estimates are derived, differing from existing literature.

Conclusions:

  • The developed postprocessing technique offers an efficient and accurate approach for solving the Cahn-Hilliard equation using MFEMs.
  • This method presents a valuable advancement in computational materials science and numerical analysis.
  • The findings provide new theoretical insights into error estimation for these types of problems.