New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Model of Parabolic PDEs

The derivations of existing error bounds for reduced order models of time varying partialdi erential equations (PDEs) constructed using proper orthogonal decomposition (POD) haverelied on bounding the error between the POD data and various POD projections of that data.Furthermore, the asymptotic behavior of the model reduction error bounds depends on theasymptotic behavior of the POD data approximation error bounds. We consider time varyingdata taking values in two di erent Hilbert spacesHandV, withVH, and prove exactexpressions for the POD data approximation errors considering four di erent POD projectionsand the two di erent Hilbert space error norms. Furthermore, the exact error expressions canbe computed using only the POD eigenvalues and modes, and we prove the errors converge tozero as the number of POD modes increases. We consider the POD error estimation approachesof Kunisch and Volkwein (SIAM J. Numer. Anal., 40, pp. 492-515, 2002) and Chapelle, Gariah,and Sainte-Marie (ESAIM Math. Model. Numer. Anal., 46, pp. 731-757, 2012) and apply ourresults to derive new POD model reduction error bounds and convergence results for the twodimensional Navier-Stokes equations. We prove the new error bounds tend to zero as the numberof POD modes increases for POD spaceX=Hin both approaches; the asymptotic behaviorof existing error bounds was unknown for this case. Also, forX=H, we prove one new errorbound tends to zero without requiring time derivative data in the POD data set.