A POD-Based Reduced-Order Stabilized Crank–Nicolson MFE Formulation for the Non-Stationary Parabolized Navier–Stokes Equations

Abstract

We firstly employ a proper orthogonal decomposition (POD) method, Crank–Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier–Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCNMFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier–Stokes equations.

Keyword : proper orthogonal decomposition method, reduced-order stabilized Crank– Nicolson mixed finite element formulation, non-stationary parabolized Navier–Stokes equations, error estimate