Pressure Poisson equation (PPE) reformulations recast the incompressible Navier-Stokes equations via replacing the incompressibility constraint by an operator relation between pressure and the velocity field. As a consequence, PPE reformulations can be advanced forward in time as cheaply as classical projection methods, but without any accuracy limitations. The theme of this research is the development of efficient high-order numerical methods for incompressible flows, based on discretizations of PPE reformulations. We demonstrate that for a particularly interesting PPE reformulation, in which the velocity satisfies "electric" boundary conditions, classical nodal finite elements exhibit a Babuska paradox, i.e., they converge to a wrong solution. We then present two methodologies to correctly address this problem — mixed finite element methods and meshfree finite differences — and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time.