The immersed interface method approximates the solution of the Poisson equation with interfaces on a fixed Cartesian grid by directly incorporating the necessary jump conditions into numerical schemes. It is well known that the numerical solution of this method can be second-order accurate with truncation error O(h) at points near the interface and \(O(h^2)\) elsewhere. However, the same order of accuracy can be kept by simplifying the local error assumptions. This paper contains the theory to show that we can have a \(O(h^2)\) global method, even if the local truncation errors are O(1) at points near the interface. Furthermore, this analysis indicates that this statement holds true only for specific interface locations or special grid settings. Additionally, the method reduces the required number of Cartesian jumps up to first-order derivatives. The present analysis also provides the local conditions to achieve this global second-order. We prove this fact by studying the global convergence error based on an asymptotic series technique. Finally, we present some numerical experiments to verify the theoretical results on specific example problems.
