The Poisson equation is an often encountered problem in science and engineering, e.g., for particle-mesh solvers in CFD [1]. Specifically for flow problems, it often yields input for a transport equation. This contribution presents a quantum algorithm for solving the inhomogeneous Poisson equation for free field conditions in multiple dimensions via the Hockney method [2]. In accordance to this, the quantum algorithm is based on a multi-dimensional version of the quantum Fourier transform (QFT) [3] as a quantum computational implementation of the fast Fourier transform for which an improved number of computation steps has been found theoretically [4]. It can already be seen from minimal problems for which the quantum algorithm was simulated that the analytical solution is relatively well approximated. Concerning the required computational resources for the quantum algorithm, the conclusion is drawn that while the QFT should be an efficient procedure, the other necessary steps diminish the efficiency, which should be rooted in the used amplitude encoding so that these considerations apply more generally. References: [1] N. Hu, N. Reiche, and R. Ewert, Simulation of turbulent boundary layer wall pressure fluctuations via Poisson equation and synthetic turbulence, J. Fluid Mech. 826, 421-454 (2017). [2] R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (Taylor & Francis, 1988). [3] P. Pfeffer, Multidimensional Quantum Fourier Transformation, arXiv:2301.13835v1 [quant-ph] (2023). [4] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (10th anniversary edition, Cambridge University Press, 2010).