On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method

This work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary (MIB) method proposed for solving elliptic interface problems. The MIB method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy. One feature of the MIB is that it disassociates the discretization of the elliptic equation from the enforcement of interface jump conditions. The other is to make iterative use of only the lowest order jump conditions to determine the fictitious values on extended domains. It is of arbitrarily high order in convergence, in principle. However, its applicability was hindered by the lack of sufficiently many grid points to determine all the fictitious values required for high order schemes at the location where the curvature of the interface is relatively large. We remove this obstacle by introducing a new concept, the disassociation between the discretization and the domain extension. We show that the improved MIB method is robust for handling general irregular interfaces by extensive numerical experiments on the Poisson equation and the Helmholtz equation. To better understand the MIB method and other potential high order interface schemes, we propose an alternative interpolation formulation of the MIB method and show that the new formulation is essentially equivalent to the improved one.