High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

This paper introduces a novel high order interface scheme, the matched interface and boundary (MIB) method, for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids. By appropriate use of auxiliary line and/or fictitious points, physical jump conditions are enforced at the interface. Unlike other existing interface schemes, the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study. To construct higher order interface schemes, the proposed MIB method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions. The proposed MIB method is of arbitrarily high order, in principle. In treating straight, regular interfaces we construct MIB schemes up to 16th-order. For more general elliptic problems with curved, irregular interfaces and boundary, up to 6th-order MIB schemes have been demonstrated. By employing the standard high-order finite difference schemes to discretize the Laplacian, the present MIB method automatically reduces to the standard central difference scheme when the interface is absent. The immersed interface method (IIM) is regenerated for a comparison study of the proposed method. The robustness of the MIB method is verified against the large magnitude of the jump discontinuity across the interface. The nature of high efficiency and low memory requirement of the MIB method is extensively validated via solving various elliptic immersed interface problems in two- and three-dimensions.