One-order algorithm of incompressible Navier-Stokes equations in finite element method

One of the difficulties of the numerical solution of incompressible Navier-Stokes equations is the determination of the pressure field and the fulfillment of the incompressibility condition. In fact, the pressure variable is not present in continuity equation, but a constraint for the velocity field is present. In this paper, the basic variables of velocity and stress were proposed for incompressible viscous fluid, a one-order fluid dynamics equation system without pressure term was proposed and its integral form was given to handle this problem. The stress and the velocity were interpolated by equal order finite element. The Newton iterative method was used to handle the nonlinear convective term. The backward Euler method was used to discretize the time term. A steady flow of incompressible viscous fluid between two infinite parallel plates and a Benchmark problem of incompressible viscous fluid flow around a cylinder were computed on the basis of FreeFem++. The feasibility and the effectivity of the method were verified by comparing with the analytic solution and the Benchmark results respectively. The difficulty of pressure term which is not present in continuity equation is circumvented by using one-order system without pressure term.