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A partially penalty immersed Crouzeix-Raviart finite element method for interface problems.

Na An1, Xijun Yu2, Huanzhen Chen1

  • 1School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014 China.

Journal of Inequalities and Applications
|September 1, 2017
PubMed
Summary

A new partially penalty immersed finite element (PIFE) method effectively solves anisotropic flow models with discontinuous coefficients. This novel approach ensures accurate solutions for complex multi-material problems.

Keywords:
Crouzeix-Raviart elementdiscontinuous coefficientselliptic interface problemsimmersed finite element methodoptimal-order error estimatespartially penalty

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Area of Science:

  • Numerical Analysis
  • Computational Fluid Dynamics
  • Partial Differential Equations

Background:

  • Elliptic equations with discontinuous coefficients model phenomena involving multiple materials or fluids.
  • Accurate numerical methods are crucial for simulating anisotropic flow and diffusion processes.

Purpose of the Study:

  • To develop a partially penalty immersed finite element (PIFE) method for anisotropic flow models with piecewise constant diffusion coefficients.
  • To analyze the method's solvability and error estimates.
  • To validate the method's performance through numerical experiments.

Main Methods:

  • Utilizing standard linear Crouzeix-Raviart finite element space on non-interface elements.
  • Constructing a piecewise linear Crouzeix-Raviart immersed finite element (IFE) space on interface elements.
  • Employing discontinuous Galerkin formulations (symmetric, nonsymmetric, or incomplete interior penalty).

Main Results:

  • Proved the solvability of the PIFE method.
  • Obtained optimal error estimates in the energy norm.
  • Demonstrated optimal-order convergence in the L2 norm through numerical experiments.

Conclusions:

  • The developed PIFE method is effective for anisotropic elliptic interface problems.
  • The method shows validity for both isotropic and anisotropic cases.
  • Numerical results confirm the theoretical analysis and optimal convergence rates.