  We consider proper orthogonal decomposition (POD) methods to approximate the
incompressible Navier-Stokes equations. We study the case in which one
discretization for the nonlinear term is used in the snapshots (that are
computed with a full order method (FOM)) and a different discretization of the
nonlinear term is applied in the POD method. We prove that an additional error
term appears in this case, compared with the case in which the same
discretization of the nonlinear term is applied for both the FOM and the POD
methods. However, the added term has the same size as the error coming from the
FOM so that the rate of convergence of the POD method is barely affected. We
analyze the case in which we add grad-div stabilization to both the FOM and the
POD methods because it allows to get error bounds with constants independent of
inverse powers of the viscosity. We also study the case in which no
stabilization is added. Some numerical experiments support the theoretical
analysis.
