Research On Two - Dimensional Incompressible Navier - Stokes Equations

Navier-Stokes equations are the typical partial differential equations. They are very important because they can model the weather change, ocean currents, liquid flow in the pipe and air flow around the wing. In the numerous works of studying Navier-Stokes equations, the finite element method has been the subject of very intense research activity over the past30years.Several stabilized methods for Navier-Stokes equations are proposed and analyzed by using some two-level methods, local Gauss integration and penalty mixed finite element methods.1. Two-level stabilized nonconforming finite element method for the Navier-Stokes equationsThis paper mainly concentrates on a two-level stabilized nonconforming finite element method for solving the Navier-Stokes on the three corrections, which uses the nonconforming and conforming piecewise quadratic polynomial approximations for the velocity and pressure based on local Gauss integration. Compared with the mini-element method and tradi-tional one-level method based on local Gauss integration, my method is shown to be more computationally efficient without loss of accuracy from numerical experiment.2. Two-level method based on Newton iteration for the stationary Navier-Stokes EquationsA two-level stabilized method based on Newton iteration for the stationary Navier-Stokes equations is proposed and analyzed, which uses the conforming piecewise quadratic polynomial approximations for the velocity and pressure based on local Gauss integration, the present pair is shown to be more computationally efficient without a loss of accuracy from numerical experiment.3. Two-level penalty stabilized method for Navier-Stokes equationsBased on the Mini-element, this paper’s purpose is to combine the penalty mixed finite element method with the two-level discretization for solving the Navier-Stokes equations. This penalty mixed finite element method is noted that pε can be eliminated to obtain a penalty system of με only, which is easier to solve than the original equations. The error esti-mate of two-level penalty mixed element method is obtained through using a mapping rμ and a L2-orthogonal projection operator pμ.