Convergence analysis of the immersed interface method

A rigorous error analysis is given for the immersed interface method (IIM) applied to elliptic problems with discontinuities and singularities. The finite difference scheme using IIM is shown to satisfy the conditions of a maximum principle for a certain class of problems. Comparison functions are constructed to obtain error bounds for some of the approximate solutions. The asymptotic error expansion provides further useful insights and details of the behaviour and convergence properties of IIM, which leads to a sharper estimate of the error bound. Second-order convergence of IIM is indicated by the analysis. Numerical examples are also given to support the analytical results.