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Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients.

Kelin Xia1, Meng Zhan, Decheng Wan

  • 1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.

Journal of Computational Physics
|May 16, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces an adaptively deformed mesh strategy for solving elliptic interface problems with discontinuous coefficients. The new method improves accuracy and outperforms existing techniques for these challenging equations.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Mesh deformation methods are widely used for solving partial differential equations (PDEs).
  • Traditional methods fail for elliptic PDEs with discontinuous coefficients (elliptic interface problems) due to required interface jump conditions.
  • Existing numerical algorithms struggle to accurately enforce these interface conditions.

Purpose of the Study:

  • To develop a novel adaptively deformed mesh strategy for resolving elliptic interface problems.
  • To enhance accuracy and convergence for elliptic equations with discontinuous coefficients.
  • To overcome limitations of existing methods in handling interface jump conditions.

Main Methods:

  • Introduced an interface technique based on adaptively deformed meshes.
  • Leveraged the Matched Interface and Boundary (MIB) method for accuracy and robustness.
  • Employed a mesh transformation PDE with a source term and monitor function for mesh redistribution.
  • Constructed interface geometry and solution gradient based deformed meshes.

Main Results:

  • The proposed method generates deformed meshes in the physical domain while maintaining regular Cartesian meshes in the computational domain.
  • Successfully reduced L(∞) and L(2) errors in solving elliptic interface problems.
  • Demonstrated superior performance compared to the original MIB method in numerical experiments.

Conclusions:

  • The adaptively deformed mesh interface method effectively resolves elliptic interface problems with discontinuous coefficients.
  • The strategy offers improved accuracy and robustness over existing methods.
  • This approach provides a flexible and powerful tool for a range of practical applications involving elliptic PDEs.