Stabilized Finite Element Method For The Convection Dominant Problems And Stokes Eigenvalue Problems

A stabilized finite element method (FEM) based on the local Gauss integration technique is investigated in this thesis. This method is applied to convection dominated problems and Stokes eigenvalue problems, which is organized as follows:First, we consider a two-grid stabilized method based on two local Gauss integra-tions for the nonlinear convection dominated diffusion equation. This method combines the two-grid strategy with the stabilized FEM method. Compared with the classical variational multiscale method (CVMS), the feature of our method is using two local Gauss integrations to replace the projection operator, and introducing the stabiliza-tion at the element level without adding any new variables or storage space. Then our method keeps the effectiveness and stability as the classical variational multiscale method. Meanwhile, two grid method provide an approximate solution with the conver-gence rate of same order as the usual stabilized finite ele ment solution. So our method can save computational time. Finally, numerical examples validate the theoretical re-sults of the presented methods.Second, a stabilized finite element method based on the local Gauss integration technique for optimal control problems governed by a convection dominated diffusion equation is investigated. The control variable can be approximated by the standard discretization method or the variational discretization method. Therefore, they are ap-plied to solve optimal control problems governed by a convection dominated diffusion equation, and their advantages and disadvantages are also compared. Numerical re-sults indicate that the variational discretization method has better convergence order, although both methods can acquire accurate solutions.Third, the quadratic equal-order stabilized finite element method combined with two grid algorithm is presented. The quadratic equal-order stabilized finite element method based on two local Gaussian quadratures is applied to discretize the Stokes eigenvalue problem, and the corresponding convergence analysis is given. Furthermore, we give two space method under the frame of two grid method. The two methods are high efficiency algorithm, and is employed to reduce the computational cost. Mean-while, two stabilized finite element algorithms provide an approximate solution with the convergence rate of same order as the usual stabilized finite element solution. Numerical examples are given to confirm the theoretical results.Finally, we summarize the thesis and propose the future research work.