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We consider proper orthogonal decomposition (POD) methods to approximate theincompressible Navier-Stokes equations. We study the case in which onediscretization for the nonlinear term is used in the snapshots (that arecomputed with a full order method (FOM)) and a different discretization of thenonlinear term is applied in the POD method. We prove that an additional errorterm appears in this case, compared with the case in which the samediscretization of the nonlinear term is applied for both the FOM and the PODmethods. However, the added term has the same size as the error coming from theFOM so that the rate of convergence of the POD method is barely affected. Weanalyze the case in which we add grad-div stabilization to both the FOM and thePOD methods because it allows to get error bounds with constants independent ofinverse powers of the viscosity. We also study the case in which nostabilization is added. Some numerical experiments support the theoreticalanalysis.

On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods