Numerical Helmholtz-Hodge Decomposition for Velocity Fields via Fast Convolution

The Helmholtz-Hodge Decomposition (HHD) is a critical tool in the study and analysis of vector fields, and plays a crucial role in fluid mechanics. In this article, we present a fast and rapidly convergent numerical scheme for performing HHD of velocity fields on uniform grids. Our method is based on the direct numerical solution of Poisson equations using Green's function. To evaluate the convolution integral between the Green's function and a smooth, compact field, we employ the kernel truncation method. This approach achieves spectral order accuracy with a computational complexity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(N\log N)$</tex>. Its effectiveness is demonstrated through two-dimensional and three-dimensional examples, showcasing its ability to accurately decompose velocity fields into irrotational and solenoidal components.