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

  • Authors:

  • Kelin Xia;Meng Zhan;Decheng Wan;Guo-Wei Wei

  • Affiliations:

  • Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA and Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China;State Key Laboratory of Ocean Engineering, School of Naval Architecture Ocean and Civil Engineering, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China;Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA and Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2012

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Abstract

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L"~ and L"2 errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.