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

The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied 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 the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces H and V, with V H, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches of Kunisch and Volkwein SIAM J. Numer. Anal. , 40 (2002), pp. 492--515 and Chapelle, Gariah, and Sainte-Marie ESAIM Math. Model. Numer. Anal. , 46 (2012), pp. 731--757 and apply our results to derive new POD model reduction error bounds and convergence results for the two-dimensional Navier--Stokes equations. We prove the new error bounds tend to zero as the number of POD modes increases for POD space X = H in both approaches; the asymptotic behavior of existing error bounds was unknown for this case. Also, for X = H, we prove one new error bound tends to zero without requiring time derivative data in the POD data set.