We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB 'deficiency' of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

Stabilization of low-order mixed finite elements for the stokes equations.

The threshold of a stochastic SIRS epidemic model with saturated incidence

This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.

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On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination Controls

This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.

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On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination

Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model

Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays

Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay

Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence

Global analysis of a delayed epidemic dynamical system with pulse vaccination and nonlinear incidence rate

SUMMARY
On the basis of two local Gauss integrations, a stabilized finite element method for transient Navier–Stokes equations is presented, which is defined by the lowest equal-order conforming finite element subspace (Xh,Mh) such as P1−P1 (or Q1−Q1) elements. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. The diffusion term in these equations is discretized by using finite element method, and the temporal differentiation and advection terms are treated by characteristic schemes. Moreover, we present some numerical simulations to demonstrate the effectiveness, good stability, and accuracy properties of our method. Especially, the rate of convergence study tells us that the stability still keeps well when the Reynolds number is increasing. Copyright © 2011 John Wiley & Sons, Ltd.

Yu Jiang

Papers

Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays

Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay

Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence

Global analysis of a delayed epidemic dynamical system with pulse vaccination and nonlinear incidence rate

A stabilized finite element method for transient Navier–Stokes equations based on two local Gauss integrations