The Exploration Of The Stabilized Method For The Fluid Dynamic Equations

With the rapid development of computer science,numerical analysis and numerical computation have been developed as very important tools in the ?elds of science researches and engineering applications,and ?nally formulated a new interdisciplinaryComputational Fluid Dynamics,It is about the computation of Navier-Stokes equations,Darcy ?uid and so on.But the incompressibility and the nonlinear term,especially the big Reynolds number, are the numerical di?culties,There are many questions as parameter disturbance when solving Darcy Fluid.So,the research of numerical methods about those equations has practical signi?cance.This thesis is to explore the stabilized projection method for the time-dependent Navier-Stokes equation.Firstly,the time is decomposed into a sequence of two steps:the?rst step has the structure of the linear elliptic problem;the second step can be seen as a generalized Stokes problem.The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity.On the other hand,a locally stabilized term is added in the second step of the schemes.It allows one to enhance the numerical stability and e?ciency by using the equal-order pairs.Convergence analysis and error estimation for the velocity and pressure of the schemes are established via the energy method.Some numerical experiments are used to demonstrate the e?ciency of this new method.In the second,a lowest equal-order stabilized ?nite element method for Darcy ?ow is studied. The stabilized method has a number of attractive computational properties:the stability analysis of the numerical solution is derived under some regularity condition;optimal error estimates of the velocity and pressure is obtained for the discrete version.Finally,numerical experiments are also used to demonstrate the theoretical result.At last,this thesis explores the stabilized projection scheme and viscosity-splitting projection scheme for the time-dependent Stokes equation,and gives comparisons between the two schemes whether have grad-div stabilization.