AbstractThe spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier–Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of both mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two‐dimensions and one element in three‐dimensions. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented h