Fluid simulation is a well-established research field in computer graphics that aims to generate realistic and visually appealing simulations of fluids, including water, smoke, and fire. Divergence is identified as a crucial factor in achieving realistic fluid behavior. This article presents a GPU optimization of the pointwise incompressible fluid simulation technique, which uses pointwise incompressible velocity interpolation to ensure divergence at arbitrary points, rather than grid cells. Pointwise incompressible velocity interpolation at arbitrary points is the curl of velocity potential. The velocity potential can be obtained from the divergence-free velocities by solving a Poisson equation. To enhance the computational efficiency of the simulation, an odd–even geometric multigrid method is introduced to solve Poisson’s equation on non-power-of-2 resolution grids. Additionally, to recover the potential on grid cells, a warp-level method is proposed via divergence-free velocities on grid cells. The proposed warp-level potential recovery method offers GPU optimization and enhanced efficiency, in contrast to the sweeping method that only provides CPU optimization for potential recovery. Furthermore, a locally enhanced method is proposed to recover the detailed velocity potential within small domains through the use of the odd–even multigrid. The experiments show that the odd–even multigrid outperforms Weber’s multigrid by an average speedup factor of 4.32 across various grid sizes. On the other hand, the warp-level potential recovery method demonstrates notable speedups ranging from approximately 2–6 times for 2D grids and about 1.35–2.64 times for 3D grids when compared to the parallel sweeping method on GPU. The results of these contributions demonstrate remarkable improvements in the efficiency and performance of fluid simulations in computer graphics, harnessing the computational capabilities of modern GPUs to achieve visually appealing and realistic fluid behaviors.