For Poisson problems with periodic boundary conditions, the use of the traditional SOR solution algorithm does not permit the rapid convergence of the Poisson solver. To address this limitation, a novel multigrid iterative Poisson solver (Poisson-SOR-MG) based on finite difference methods was developed, leveraging multigrid acceleration techniques to overcome the drawbacks of SOR. Additionally, a 3D Fourier spectral transform algorithm was designed by coupling 1D Fast Fourier Transform (FFT) with 3D parallel communication algorithms, eliminating cumbersome transpose techniques. Building on this, a standard 2π-periodic Poisson-Fourier solver was developed and validated, achieving spectral accuracy. However, the standard 2π-periodic Poisson-Fourier solver cannot be directly applied to general engineering problems. To resolve this, theoretical analysis based on coordinate stretching transformations was conducted, leading to the further development and validation of a generalized arbitrary-periodic Poisson-Fourier solver.

 multigrid methods, 3D Fourier spectral transformation, Poisson solvers, SOR iterative methods; coordinate mapping transformation

对于周期边界条件Poisson问题, 采用传统的SOR求解算法使Poisson求解器难以快速收敛。为解决这一问题, 针对SOR算法的缺点, 利用多网格加速技术, 开发一种新的有限差分法多网格迭代泊松求解器Poisson-SOR-MG。此外, 提出一种基于1D快速傅里叶变换(FFT)和3D并行通信算法耦合的3D傅里叶谱变换算法, 摒弃繁复的转置技术。在此基础上, 开发和验证基于标准2π周期的Poisson-Fourier求解器, 将Poisson求解器的精度提高至谱精度。然而, 基于标准2π周期的Poisson-Fourier求解器无法应用到广泛的工程问题, 因此, 基于坐标拉伸变换的理论分析, 进一步开发和验证基于一般周期的Poisson-Fourier求解器。