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This manuscript is devoted to investigating the conservation laws of incompressible Navier-Stokes equations(NSEs), written in the energy-momentum-angular momentum conserving(EMAC) formulation, after being linearized by the two-level methods. With appropriate correction steps(e.g., Stoke/Newton corrections), we show that the two-level methods, discretized from EMAC NSEs, could preserve momentum, angular momentum, and asymptotically preserve energy. Error estimates and (asymptotic) conservative properties are analyzed and obtained, and numerical experiments are conducted to validate the theoretical results, mainly confirming that the two-level linearized methods indeed possess the property of (almost) retainability on conservation laws. Moreover, experimental error estimates and optimal convergence rates of two newly defined types of pressure approximation in EMAC NSEs are also obtained., Comment: 10 pages, 1 table, and 2 figures

We present a novel reduced-order pressure stabilization strategy based on continuous data assimilation(CDA) for two-dimensional incompressible Navier-Stokes equations. A feedback control term is incorporated into pressure-correction projection method to derive the Galerkin projection-based CDA proper orthogonal decomposition reduced order model(POD-ROM) that uses pressure modes as well as velocity's simultaneously to compute the reduced-order solutions. The greatest advantage over this ROM is circumventing the standard discrete inf-sup condition for the mixed POD velocity-pressure spaces with the help of CDA which also guarantees the high accuracy of reduced-order solutions; moreover, the classical projection method decouples reduced-order velocity and pressure, which further enhances computational efficiency. Unconditional stability and convergence over POD modes(up to discretization error) are presented, and a benchmark test is performed to validate the theoretical results.

In this paper, we propose an efficient proper orthogonal decomposition based reduced-order model(POD-ROM) for nonstationary Stokes equations, which combines the classical projection method with POD technique. This new scheme mainly owns two advantages: the first one is low computational costs since the classical projection method decouples the reduced-order velocity variable and reduced-order pressure variable, and POD technique further improves the computational efficiency; the second advantage consists of circumventing the verification of classical LBB/inf-sup condition for mixed POD spaces with the help of pressure stabilized Petrov-Galerkin(PSPG)-type projection method, where the pressure stabilization term is inherent which allows the use of non inf-sup stable elements without adding extra stabilization terms. We first obtain the convergence of PSPG-type finite element projection scheme, and then analyze the proposed projection POD-ROM's stability and convergence. Numerical experiments validate out theoretical results.