A Stabilized Mixed Finite Element Method For The Navier-Stokes Equations

The mixed finite element methods for the incompressible Stokes flow was the focus problem in the last twenty years of the twenties century. But it is an important convergence stability condition that the Babuska-Brezzi inequality holds for the combination of finite element subspaces. This constraint condition prevents low order velocity-pressure pairs which are a popular choice in engineering practice. To circumvent this constraint, the so-called CBB or stabilized finite element methods have been developed motivated by SUPG methods. On the other hand, when the Reynolds number grows higher, the advection term is much stronger than the diffusive one. The Navier-Stokes problem shows advection-dominated problems. The search of stabilization methods for advection-dominate problems is also a hard work for Navier-Stokes problems. Focusing on the problem mentioned above, we propose a streamline-diffusion type pressure projection stabilization method for stationary N-S equations.In chapter one, we introduce the research background.In chapter two, the pressure projection stabilization method for Navier-Stokes problem will be proposed. This technique was first developed for the Stokes problems. Now we extent it to the nonlinear stationary Navier-Stokes equations. A unified formulation for stationary Navier-Stokes equations is defined and a detail theoretical analysis is given.In chapter three, we study the advection-dominated problems. A stream-diffusion type pressure projection stabilized method to lower order element (P₁/P₁,Q₁/Q₁,P₁/P₀,Q₁/Q₀) was propose in this section. The existence and uniqueness of the discrete solution is proved and the error estimates are given.