The Researches On Finite Element Methods For The Fluid And Coupled Fluid Problems

In this dissertation, we consider the finite element methods for some class of fluid and coupled fluid problems. The fluid and coupled fluid problems are often encoun-tered in oceanography, geophysics and fluid dynamics, such as air flow at low speed, water flow, groundwater contamination problems and the coupled atmosphere-ocean problems. When solving such problems using the standard finite element method nu-merically, the convection-dominated characteristics, high Reynolds number and the linear or nonlinear coupling conditions usually lead to the losing efficiency of the numerical methods. The aim of this dissertation is to solve fluid and coupled flu-id problems efficiently. Combining the characteristics, variational multiscale (VMS) method and stabilized finite element method, we design the efficient numerical meth-ods for the fluid and coupled fluid problems, and obtain the stability and convergence estimate for these problems.In Chapter 1, the characteristic variational multiscale (C-VMS) method is pro-posed for solving two-dimensional (2D) convection-dominated convection-diffusion-reaction problems. The scheme is combined the method of characteristics with the VMS method to create the C-VMS procedures. The stability analysis and error esti-mate of the C-VMS method are obtained. The scheme not only realizes the purpose to reduce the time-truncation error, using larger time step for solving the convection-dominated convection-diffusion-reaction problems, but also keeps the favorable sta-bility and high precision. Finally, numerical experiments in 2D and 3D cases are presented to illustrate the availability and efficiency of the scheme.In Chapter 2, based on the lowest equal-order conforming finite element sub-spaces, a novel characteristic stabilized finite element method is proposed for approx-imating solutions to the incompressible Navier-Stokes equations. We use the residuals of the momentum equation and the divergence-free equation to define the stabilization terms. The natural combination of characteristic method and stabilized finite element method retains the best features of both methods. The stability and error estimates are deduced rigorously. Finally, some numerical experiments are given to demonstrate the efficiency of our method for the nonstationary Navier-Stokes problems.In Chapter 3, we study numerical approximations for the fluid-fluid interaction problems. As a simplified model, the convection-dominated convection-diffusion-reaction equations are coupled by an interface condition. The implicit-explicit time stepping streamline diffusion method for computing numerical solutions to the above problems is presented. The stability analysis and error estimates for considered scheme are derived. Computational tests are performed to demonstrate the robust-ness of this scheme.In Chapter 4, we propose and analyse a local projection stabilized and character-istic decoupled (LPSCD) scheme for the fluid-fluid interaction problems. We use the method of characteristics type to avert the difficulties caused by the nonlinear term, and use the local projection stabilized method to control spurious oscillations in the velocities due to dominant convection, and use a geometric averaging idea to decou-ple the monolithic problems. The stability analysis is derived and numerical tests are performed to demonstrate the robustness of this new method.