We detail and theoretically analyse the so-called fast vector (or velocity) penalty-projection methods (VPP ε) of which the main ideas and features are briefly introduced in [8,9,10]. This family of numerical schemes proves to efficiently compute the solution of unsteady Navier-Stokes/Brinkman problems governing incompressible or low Mach multi-phase viscous flows with variable mass density and/or viscosity or anisotropic permeability. In this paper, we describe in detail the connections and essential differences with usual methods to solve the Navier-Stokes equations. The key idea of the basic (VPP ε) method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This is obtained by proposing new Helmholtz-Hodge decomposition solutions of L 2-vector fields in bounded domains to get fast methods with suitable adapted right-hand sides; see [11]. This procedure only requires a few iterations of preconditioned conjugate gradients whatever the spatial mesh step. Then, the splitting (VPP ε) method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as O(ε δt), δt being the time step, with a penalty parameter ε as small as desired until the machine precision, e.g. ε = 10 −14 , whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed velocity correction step typically requires only one or two iterations of a suitable pre-conditioned Krylov solver whatever the spatial mesh step [10]. Moreover, the robustness of our method is not sensitive to large mass density ratios since the velocity penalty-projection step does not include any spatial derivative of the density. 2 In the present work, we also prove the theoretical foundations as well as global sol-vability and optimal unconditional stability results of the (VPP ε) method for Navier-Stokes problems in the case of homogeneous flows, which are the main new results. Keywords Vector penalty-projection method · divergence-free penalty-projection · penalty method · splitting prediction-correction scheme · fast Helmholtz-Hodge decompositions · Navier-Stokes/Brinkman equations · stability analysis · incompressible homogeneous flows · dilatable flows · low Mach number flows · incompressible non-homogeneous or multiphase flowsRead less <