We investigate a new technique called `pressure separation algorithm` (PSepA) which is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the CFD package FEATFLOW (www.featflow.de), we mainly focus on low order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can be also obtained for higher order discretizations, for instance with Q_2/P_1 finite elements. We analyze the application of this simple, but very efficient algorithm onto several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity, flow around obstacles), and we demonstrate its effect for spurious velocities in multiphase flow simulations (`static bubble` configuration) if combined with edge-oriented, resp., internal penalty FEM stabilization techniques.