Analysis And Application Of The Mixed Finit Elemnt Method For Two Kinds Of Equations

In this paper, we consider the numerical simulations of two kinds of equations; one is the 2nd order parabolic problems, the other is the linear convection-dominated diffusion problems.The Chapter one is the introduction.In Chapter two, we consider the Expanded Mixed Finite Element Method for the 2nd order parabolic problemsThis method expands the standard mixed formulation in the sense that three variable are explixitly treated: the scalar unknwon, its gradient and its flux. Based on this fomulation. expanded mixed finite element approximations of the parabolic problems are considered. Error estimates for the scalar unknwon, its gradient and its flux in L2-norm are obtained for this new mixed formulation.In Chapter three, we consider the Modified Characteristics-Mixed Finite Element Method for the linear convection-dominated diffusion problemsThe method is a combination of characteristic method and a mixed finite element method. The characteristic approximation is used to handle the convection part along the direc-tion of fluid namely characteristic direction to ensure the high stability of the method in approximating the sharp fronts and reduce the numerical diffusion; The mixed finite element spatial approximation is employed to deal with diffusion part and approximate the scalar unknown and the adjoint vector function optimally and simultaneously; In order to preserve the integral conservation of the method, we introduce the modified characteristic method. Numerica analysis indicate that the modified characteristic-mixed finite element method has the desired stability, convergence and integral conservation.