A coupling interface method for elliptic interface problems

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_{xk xk} 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 κ of the interface satisfies κh ≤ Const, where h is the mesh size. 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 is always achievable by refining meshes. The CIM2 requires basically that the interface does not intersect two adjacent grid segments simultaneously. In practice, we classify the underlying Cartesian grid points into interiors, normal on-fronts, and exceptionals, where a standard central finite difference method, the CIM2, and the CIM1 are adopted, respectively. This hybrid CIM maintains second-order accuracy in most applications due to the fact that usually in d dimensions, the number of normal on-front grid points is O(h^1-d) and the number of the exceptional points is O(1). Numerical convergence tests for the CIM1 and CIM2 are performed. A comparison study with other interface methods is also reported. Algebraic multigrid method is employed to solve the resulting linear system. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, this hybrid CIM passes many tests of complex interface problems in two and three dimensions. Therefore, we believe that it is a competitive method for complex interface problems.