A coupling interface method for elliptic interface problems

  • Authors:

  • I-Liang Chern;Yu-Chen Shu

  • Affiliations:

  • Department of Mathematics, Taida Institute for Mathematical Sciences, National Center for Theoretical Science at Taipei, National Taiwan University, Taipei 106, Taiwan;Department of Mathematics, National Taiwan University, Taipei, Taiwan and Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, the CIM1 is derived from a linear approximation on both sides of the interface. The method is extended to high dimensions through a dimension-by-dimension approach. To connect information from each dimension, a coupled equation for the first-order derivatives is derived through the jump conditions in each coordinate direction. The resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions. Similarly, the CIM2 is derived from a quadratic approximation in each dimension. In high dimensions, a coupled equation for the principal second-order derivatives u"x"""k"x"""k is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by one-side interpolation. This approach reduces the number of grid points needed for one-side interpolation. The resulting stencil involves 8 grid points in two dimensions and 12-14 grid points in three dimensions. A numerical study for the condition number of the resulting linear system of the CIM2 in one dimension has been performed. It is shown that the condition number has the same behavior as that of the discrete Laplacian, independent of the relative location of the interface in a grid cell. Further, we also give a proof of the solvability of the coupling equations, provided the curvature @k of the interface satisfies @kh=