The reduced-order method of continuous space-time finite element scheme for the non-stationary incompressible flows

Abstract

In this paper, we mainly concern with the order reduction for the unknown solution coefficient vectors about the classical continuous space-time finite element (CCSTFE) method of the two-dimensional (2D) non-stationary incompressible Navier-Stokes equations. Therefore, we first build the CCSTFE scheme for the 2D non-stationary Navier-Stokes equations with respect to stream–vorticity functions, give the existence and stability together with convergence to the CCSTFE solutions, and rewrite the CCSTFE scheme as a matrix form. And then, we use a proper orthogonal decomposition (POD) technique to build a reduced-order extrapolated continuous space-time finite element (ROECSTFE) scheme in the matrix form including very few unknowns but possessing high enough accuracy, employ matrix analysis to demonstrate the existence and stability together with convergence of the ROECSTFE solutions such that the theoretical analysis becomes very concise and convenient, and summarize the flowchart for solving the ROECSTFE scheme. Finally, we take advantage of the numerical simulations for the back-step flow and the flow around airfoil problem to verify the correctness of theoretical results.