A Fast Nonuniform Solver for the Poisson Equation over a Disk

Abstract

We study fast numerical methods for the Poisson equation on a disk within the FFTRR (Fast Fourier Transform Radial Recurrence) framework, which is built on Green's function representations. Classical FFTRR schemes apply FFTs in the azimuthal variable and evaluate mode-by-mode radial recurrences, achieving a complexity of \(O(MN\log N)\) on an \(N\times M\) uniform grid. However, they require a uniformly spaced azimuthal mesh of \(N\) points. In this work we develop a Nonuniform (NUFFTRR) solver that admits nonuniform grids in both the radial and azimuthal directions while retaining the favorable structure of the original FFTRR formulation. The azimuthal analysis-synthesis step is implemented using either a dense NUDFT least-squares solver or one of two NUFFT-based iterative schemes: a Toeplitz solver using circulant-preconditioned conjugate gradients (PCG), and a preconditioned conjugate gradient for least squares (PCGLS) solver with Pipe--Menon density compensation. These yield azimuthal complexities \(O(N^3 + MN^2)\) for the NUDFT variant and \(O(K_{\mathrm{iter}} MN\log N)\) for the NUFFT-based variants on an \(N\times M\) grid and Krylov iteration count \(K_{\mathrm{iter}}\). Numerical experiments on nonuniform meshes demonstrate that the proposed method is robust, spectrally accurate in the azimuthal variable, and competitive in runtime with existing fast Poisson solvers. We make use of vectorization and batched BLAS/GPU-accelerated linear algebra operations to eliminate explicit loops while also formulating azimuthal and radial steps entirely in terms of dense array operations, FFTs, and NUFFTs, allowing for straightforward GPU acceleration. The implementation is released as an open-source Python package at https://github.com/CharliePyle4/NUFFTRR_Poisson, and its methodology can be extended directly to related elliptic problems such as the Helmholtz equation.