In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly by the MIB scheme to generate fictitious values near the interface. For the exterior part, two layers of zero-padding solutions are introduced beyond the original rectangular domain so that the FFT inversion becomes feasible. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed via the MIB scheme to generate fictitious values near boundaries. Based on fictitious values at both interfaces and boundaries, the augmented MIB method reconstructs Cartesian derivative jumps as auxiliary variables. Then, by treating such variables as unknowns, an enlarged linear system is obtained. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. Therefore, the FFT-based augmented MIB not only achieves a fourth order of accuracy in dealing with interfaces and boundaries, but also produces an overall complexity of $O(n^2\log n)$ for a $n\times n$ uniform grid. Moreover, the augmented MIB scheme can provide fourth order accurate approximations to solution gradients and fluxes.

本文基于四阶精确匹配接口和边界（MIB）方法以及快速傅立叶变换（FFT），提出了一种新的笛卡尔网格有限差分方法。所提出的增强型MIB方法由两个主要部分组成，用于解决二维椭圆界面问题。对于内部部分，在处理平滑弯曲的界面时，MIB方案反复强制执行零阶和一阶跳跃条件，以在界面附近生成虚拟值。对于外部，引入了两层零填充解决方案，超出了原始矩形域，因此FFT求反变得可行。可以通过MIB方案施加不同类型的边界条件，包括Dirichlet，Neumann，Robin及其混合组合，以在边界附近生成虚拟值。基于接口和边界处的虚拟值，增强型MIB方法将笛卡尔导数跳重构为辅助变量。然后，通过将这些变量视为未知变量，可以获得扩大的线性系统。在此类系统的Schur补解中，FFT算法将不会检测到解的不连续性，因此可以有效地离散离散的Laplacian。因此，基于FFT的增强型MIB不仅在处理接口和边界时达到了四阶精度，而且还产生了总体复杂度：FFT算法不会检测到解的不连续性，因此可以有效地离散离散的Laplacian。因此，基于FFT的增强型MIB不仅在处理接口和边界时达到了四阶精度，而且还产生了总体复杂度：FFT算法不会检测到解的不连续性，因此可以有效地离散离散的Laplacian。因此，基于FFT的增强型MIB不仅在处理接口和边界时达到了四阶精度，而且还产生了总体复杂度：$Ø（{ñ}^2\mathrm{日志}ñ）$ 为一个 $ñ\times ñ$统一网格。此外，增强型MIB方案可以为溶液梯度和通量提供四阶精确近似。