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 POD-based 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 SCN-MFE 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.

Keywords:

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

AMS Subject Classification:

• 65N12
• 65M15
• 65N30