Sharp error estimates of a fourth-order compact scheme for a Poisson interface problem

A simple and efficient fourth-order kernel-free boundary integral method was recently proposed by Xie and Ying for constant coefficients elliptic PDEs on complex domains. This method is constructed by a compact finite difference scheme and works efficiently with fourth-order accuracy in the maximum norm. But it is challenging to present the sharp error analysis of the resulting approach since the local truncation errors, at the irregular grid nodes near the interface, are only in the order of O(h(3)). The aim of this paper is to establish rigorous sharp error analysis. We prove that both the numerical solution and its gradient have fourth-order accuracy in the discrete l(2)-norm, and the scheme has fourth-order accuracy in the maximum norm based on the properties of discrete Green functions. Numerical examples are also provided to verify the error analysis.