We present a simple and efficient fluid–solid coupling method in two and three spatial dimensions. In particular, we consider the numerical approximation of the Navier–Stokes equations on irregular domains and propose a novel approach for solving the Hodge projection step on arbitrary shaped domains. This method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity. We demonstrate the accuracy of our method in the L¹ and L^∞ norms and present its removing the errors associated with the conventional rasterization-type discretizations. We apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces. We also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate Reynolds number to reproduce the known steady axisymmetric and non-axisymmetric flow regimes. We further apply this algorithm to the coupling of flows with moving rigid bodies.