Model Reduction of Parametrized Systems by Peter Benner Mario Ohlberger Anthony Patera Gianluigi Rozza & Karsten Urban

Author:Peter Benner, Mario Ohlberger, Anthony Patera, Gianluigi Rozza & Karsten Urban
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

For the sake of completeness, we display the results of a monolithic approach in Fig. 15.4 (center and right), where the POD basis is computed on a unique response matrix for the velocity and pressure. While velocity results are quite accurate, pressure approximation is bad, suggesting that, probably, a lack of inf-sup compatibility of the reduced basis leads to unreliable pressure approximations, independently of the dimension of the POD space.

When we turn to the segregated approach, Fig. 15.5 shows the distribution of the singular values of the response matrices R u and R p , respectively. Again the values decay is not so rapid to pinpoint a clear cut-off value (at least for significantly small dimensions of the reduced basis), as a consequence of the multiple parametrization that inhibits the redundancy of the snapshots. However, when we compare the Hi-Mod solution identified by three different choices of the POD spaces, V POD l, u and V POD l, p , with the reference approximation in Fig. 15.4 (left), we notice that the choice l = 6 is enough for a reliable reconstruction of the approximate solution (see Fig. 15.6 (center)). The horizontal velocity component—being the most predominant dynamics—is captured even with a lower size of the reduced spaces V POD l, u , while the pressure still represents the most challenging quantity to be correctly described.

Fig. 15.5Steady Navier-Stokes equations. Singular values of the response matrix R u (left) and R p (right)

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