We present an efficient parallel solver for the 3-D incompressible unsteady Navier-Stokes equations for fluid flows. These equations are discretized using second-order finite differences and the viscous terms are treated implicitly for numerical stability. The discrete formulation leads to a 3-D Helmholtz-like equation for each velocity component, which is solved by the Alternating Direction Implicit (ADI) method. This method enables us to transform the 3-D Helmholtz operator into three tridiagonal systems. An incremental prediction-projection method is used to enforce incompressibility. It results in a Poisson equation for the pressure in order to satisfy the divergence-free character of the velocity. This equation is discretized to second order accuracy and solved using a partial diagonalization method. This diagonalization transforms the Laplacean operator into a tridiagonal diagonally dominant matrix. The parallelization relies on a domain decomposition approach where the
tridiagonal systems are solved with the well-known Thomas algorithm coupled with the Schur complement procedure. Our implementation combines MPI message-passing with vectorization techniques. We provide performance results for this Navier-Stokes solver on current multicore or multicore + GPU systems.