In this article, we present and analyse an unfitted mesh method for the Poisson interface problem. By constructing a novel ansatz function in the vicinity of the interface, we are able to derive an extended Poisson problem whose interface fits a given quasi-uniform triangular mesh. Then we adopt a hybridizable discontinuous Galerkin method to solve the extended problem with an appropriate choice of flux for treating the jump conditions. In contrast with existing approaches, the ansatz function is designed through a delicate piecewise quadratic Hermite polynomial interpolation with a post-processing via a standard Lagrange polynomial interpolation. Such an explicit function offers a third-order approximation to the singular part of the underlying solution for interfaces of any shape. It is also essential for both stability and convergence of the proposed method. Moreover, we provide rigorous error analysis to show that the scheme can achieve a second-order convergence rate for the approximation of the solution and its gradient. Ample numerical examples with complex interfaces demonstrate the expected convergence order and robustness of the method.

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.

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