A Two-grid Local Pressure Gradient-stabilized Finite Element Approximations Of The Navier-Stokes Equations

In 1994, R.Codina and J.Blasco developed a finite element formulation forthe stokes problem.(see [10]). In 2000, R.Codina and J.Blasco generalized thismethod to Navier-Stokes equations. (see [11]). In 2002, R.Becker and M.Braack[12] proposed and analyzed a finite element pressure gradient stabilization for thestokes equations based on local projections. However, all the methods mentionedabove are discussed when the velocity and the pressure are both in continuousH~1 space. The purpose of this thesis is to generalize this method to Navier-Stokes equations, especially when pressure in discontinuous space. Meanwhile,we introduce the two-grid algorithm to reduce the complicity in computing.This thesis is composed of two parts. In the first part, we firstly present alocal pressure gradient stabilization for the Navier-Stokes equations. Comparedto [13], which proposed and analyzed a Galerkin/least squares-type finite ele-ment method for the stationary Navier-Stokes equations, local pressure gradientstabilization is non-residual based element methods, which could avoid the cal-culation of the second order term. We prove the existence and uniqueness ofthe discrete solution by using Brouwer's fixed point theorem and give the errorestimates of velocity in H~1 and pressure in L~2 spaces.In chapter three, a two-grid local pressure gradient-stabilized finite elementmethod is presented and analyzed for the stationary Navier-Stokes equations.This method helps to solve the nonlinear equations on a coarse grid and the lin-ear equations on a fine grid. We prove this method shares the same convergence and error estimate results with the first approximation and theoretically save thetime in computing.