Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Non-Stationary Navier-Stokes Equations

Zhendong Luo, Jing Chen, I.M. Navon

In this paper, proper orthogonal decomposition (POD) is used to reduce the formulation of mixed finite element (MFE) for the non-stationary Navier{Stokes equations and error estimates between a reference solution and 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 usual MFE method for the non-stationary Navier-Stokes equations or from physics system trajectories via drawing samples of experiments and interpolation (or date assimilation), and then the basis functions of usual MFE method are substituted with the POD basis functions to reconstruct the elements of the ensemble so as to derive the optimizing reduced MFE formulation based POD technique for the Navier-Stokes equations since there are few basis functions in the POD basis ensemble. It is shown by considering results obtained for numerical simulations 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 feasibility and efficiency of POD method.