Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier–Stokes Equations

Abstract

In this paper, proper orthogonal decomposition (POD) is used for model reduction of mixed finite element (MFE) for the nonstationary Navier–Stokes equations and error estimates between a reference solution and the POD solution of reduced MFE formulation are derived. The basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual MFE method for the nonstationary Navier–Stokes equations or from physics system trajectories by drawing samples of experiments and interpolation (or data assimilation), and then the basis functions of the usual MFE method are substituted with the POD basis functions reconstructed by the elements of the ensemble to derive the POD-reduced MFE formulation for the nonstationary Navier–Stokes equations. It is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between POD solution of reduced MFE formulation and the reference solution is consistent with theoretical results. Moreover, it is also shown that this result validates the feasibility and efficiency of the POD method.