An interpolation matched interface and boundary method for elliptic interface problems

An interpolation matched interface and boundary (IMIB) method with second-order accuracy is developed for elliptic interface problems on Cartesian grids, based on original MIB method proposed by Zhou et al. [Y. Zhou, G. Wei, On the fictious-domain and interpolation formulations of the matched interface and boundary method, J. Comput. Phys. 219 (2006) 228–246]. Explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function. The difference scheme using IMIB method is shown to satisfy the discrete maximum principle for a certain class of problems. Rigorous error analyses are given for the IMIB method applied to one-dimensional (1D) problems with piecewise constant coefficients and two-dimensional (2D) problems with singular sources. Comparison functions are constructed to obtain a sharp error bound for 1D approximate solutions. Furthermore, we compare the ghost fluid method (GFM), immersed interface method (IIM), MIB and IMIB methods for 1D problems. Finally, numerical examples are provided to show the efficiency and robustness of the proposed method.