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

Abstract

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 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.

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

“…However, it does not effect the global fourth-order accuracy of the solution. Actually, the fourth-order accuracy in the discrete maximum norms can be derived similarly as that in [17]. It is important to mention that the leading truncation errors for scheme (4.7) depend on the wave number κ, thus mesh size h should be adapted to κ so as to resolve the waves when κ becomes large.…”

“…It is worth pointing out that the local truncation errors O(h 3 ) at irregular grid nodes is sufficient to guarantee global fourth-order accuracy of the discrete solution u i,j . For the details of the proof, one can refer to [17].…”