Fast Fourier Transform (FFT)-based solvers for the Poisson equation are highly efficient, exhibiting $O(N\log N)$ computational complexity and excellent parallelism. However, their application is typically restricted to simple, regular geometries due to the separability requirement of the underlying discrete operators. This paper introduces a novel domain decomposition method that extends the applicability of FFT-based solvers to complex composite domains geometries constructed from multiple sub-regions. The method transforms the global problem into a system of sub-problems coupled through Schur complements at the interfaces. A key challenge is that the Schur complement disrupts the matrix structure required for direct FFT-based inversion. To overcome this, we develop a FFT-based preconditioner to accelerate the Generalized Minimal Residual (GMRES) method for the interface system. The central innovation is a novel preconditioner based on the inverse of the block operator without the Schur complement, which can be applied efficiently using the FFT-based solver. The resulting preconditioned iteration retains an optimal complexity for each step. Numerical experiments on a cross-shaped domain demonstrate that the proposed solver achieves the expected second-order accuracy of the underlying finite difference scheme. Furthermore, it exhibits significantly improved computational performance compared to a classic sparse GMRES solver based on Eigen libeary. For a problem with $10^5$ grid points, our method achieves a speedup of over 40 times.

翻译：基于快速傅里叶变换（FFT）的泊松方程求解器具有极高的效率，展现出 $O(N\log N)$ 的计算复杂度和优异的并行性。然而，由于底层离散算子的可分离性要求，其应用通常局限于简单、规则的几何形状。本文提出了一种新颖的区域分解方法，将基于FFT的求解器的适用范围扩展到由多个子区域构成的复杂复合几何域。该方法将全局问题转化为通过界面上的Schur补耦合的子问题系统。一个关键挑战在于，Schur补破坏了直接基于FFT求逆所需的矩阵结构。为克服此困难，我们开发了一种基于FFT的预处理器，用于加速界面系统的广义最小残差（GMRES）方法。其核心创新是一种基于无Schur补的块算子逆的新型预处理器，该预处理器可利用基于FFT的求解器高效应用。由此得到的预处理迭代在每一步均保持了最优复杂度。在十字形区域上的数值实验表明，所提出的求解器达到了底层有限差分格式预期的二阶精度。此外，与基于Eigen库的经典稀疏GMRES求解器相比，其计算性能显著提升。对于一个包含 $10^5$ 个网格点的问题，我们的方法实现了超过40倍的加速比。