A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the non‐stationary Navier–Stokes equations

In this paper, a proper orthogonal decomposition (POD) method is applied to reducing a classical stabilized mixed finite element (SMFE) formulation for the non‐stationary Navier–Stokes equations. Error estimates between the classical SMFE solutions and the reduced SMFE solutions based on the POD method are provided. The reduced SMFE formulation based on the POD method could not only greatly reduce its degrees of freedom but also circumvent the constraint of Brezzi–Babu ka (BB) condition so that the combination of finite element subspaces could be chosen freely and allow optimal‐order error estimates to be obtained. Numerical experiments show that the errors between the reduced SMFE solutions and the classical SMFE solutions are consistent with theoretical results. Moreover, it is shown that the reduced SMFE formulation based on the POD method is feasible and efficient for solving the non‐stationary Navier–Stokes equations.