A FFT-based GMRES for fast solving of Poisson equation in concatenated geometry

Abstract

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.