【作者】 张莉；

【导师】 冯民富；

【作者基本信息】 四川大学 ， 计算数学， 2007， 硕士

【摘要】 对于黏性不可压缩流动Stokes（Navier-Stokes）方程，其混合有限元方法的研究一直是个热点问题．但经典的混合方法由于LBB条件的限制，排除了低阶元的使用．为了去掉这个束缚，上世纪90年代以来，多种稳定化方法被相继提出．另一方面，当黏性系数很小或雷诺数很大时，方程的椭圆性降低，N-S方程就呈现出对流占优．针对对流占优的研究也是N-S方程的一个难点．本文针对N-S方程的这两个方面展开研究，对N-S方程提出了流线扩散型压力投影稳定化方法．这种方法既不要求有限元空间满足LBB条件，同时也克服了对流占优带来的不稳定性．第一章绪论介绍了Stokes（Navier-Stokes）方程稳定化方法的研究背景．第二章讨论定常Navier-Stokes方程的压力投影稳定化方法．将压力投影稳定化方法应用到定常N-S方程，对等阶速度压力有限元空间，定义了一个统一的有限元格式并给出了详细的理论分析．第三章讨论当黏性系数很小时，定常N-S方程的稳定化方法．在这一章针对对流占优的N-S方程采用低阶P₁／P₁，Q₁／Q₁，P₁／P₀，Q₁／Q₀元逼近，提出流线扩散型压力投影低阶稳定化有限元方法，证明了有限元格式解的存在性，唯一性，给出了最优误差估计．更多还原

【Abstract】 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.更多还原