A Coupling Interface Method for Elliptic Complex Interface Problems

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary dimensions, where the coe cients, the source terms and the solutions may be discontinuous or singular across the interfaces. It is a dimension-by-dimension approach. It consisits of a rst-order version (CIM1) and a second-order version (CIM2). In one dimension, the CIM1 is derived based on 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 rst derivatives is derived through the jump conditions in each coordinate direction. The resulting stencil uses the standard 5 grid points only in two dimensions. Similarly, the CIM2 is derived based on a quadratic approximation in each dimension. In high dimensions, a coupled equation for the principal second derivatives uxkxk 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. The CIM1 requires that the interface intersects each grid segment (the segment connecting two adjacent grid points) at most once. This is a very mild restriction and can always be achievable by re ning meshes. The CIM2 requires basically that the interface does not intersect two adjacent grid segments simutaneously. In practice, we classify the underlying Cartesian grid points into interiors, normal on-fronts and exceptionals, where a standard ve-point central nite di erence method, the CIM2 and the CIM1 are adopted, respectively. This method maintains second order accuracy in most applications due to the fact that usually the number of normal on-front grid points is O(h ) and the number of the exceptional points is O(1). Here, d is the dimension and h is the mesh size. As a result, it is robust and exible to handle complex interface This work is supported by the National Science Council of the Republic of China under grant number NSC 95-2115-M-002-009. yDepartment of Mathematics, National Taiwan University, Taipei, Taiwan. National Center for Theoretical Science, Taiwan 106. Email: [email protected]. zDepartment of Mathematics, National Taiwan University, Taipei, Taiwan. Email: [email protected].