Proper orthogonal decomposition reduced-order extrapolation continuous space-time finite element method for the two-dimensional unsteady Stokes equation (Q2633727)

The authors develop the so-called reduced-order extrapolation continuous space-time finite element (CSTFE) method for the two-dimensional unsteady Stokes equation. At first, a transformation of the equation into a form that is suitable for the derivation of the numerical method is done by using stream-vorticity functions. This transformation is followed by a characterization of an approximate solution for the time-discrete problem and some stability as well as error estimates for the solution are shown. Next, the authors introduce a proper orthogonal decomposition (POD) based reduced order method by assembling the so-called snapshot matrix. The main novelty of their method lies in constructing the snapshot matrix using the classical CSTFE solutions at the first few time nodes and use them to produce the POD basis. This approach helps to reduce the overall computational cost and eliminates the need for repetitive calculations. The existence and uniqueness of the solution are shown and stability estimates and convergence properties of the solution are investigated. Numerical results from two computations (square cavity and flow around cylinder problems) using the classical method and the proposed method demonstrate the efficiency of the proposed method.