Proper orthogonal decomposition reduced-order extrapolation continuous space-time finite element method for the two-dimensional unsteady Stokes equation

Abstract

In this study, we mainly consider the order reduction for the solution coefficient vectors of the classical continuous space-time finite element (CCSTFE) method for the two-dimensional (2D) unsteady Stokes equation. First, we construct the CCSTFE model for the 2D unsteady Stokes equation about vorticity–stream functions, and show the existence, stability, and convergence of the CCSTFE solutions. We then apply proper orthogonal decomposition (POD) to develop a reduced-order extrapolation continuous space-time finite element (ROECSTFE) model, as well as discussing the existence, stability, and convergence of the ROECSTFE solutions based on matrix analysis so the theoretical analysis is highly concise, and we use a flowchart to search for the ROECSTFE solutions. Finally, we provide two numerical examples to confirm the correctness of the theoretical results obtained. We also show that the computational results agree well with the theoretical results, thereby demonstrating the effectiveness and feasibility of the proposed ROECSTFE method. Therefore, both the theory and method are totally different from the existing POD reduced order methods.