Analysis And Error Estimates Of An H～1-Galerkin Mixed Finite Element Method For Two Kinds Of Equation

In this paper,we consider the numerical simulations of two kinds of equations: One is the Sobolev equation, and the other is the hyperbolic parabolic equation.This paper consists of three chapters.Chapter one is an introduction.In Chapter two, we consider an H1-Galerkin Mixed Finite Element Method for the linear Sobolev equation ,which is used largely in oil preserving problems, heat conductive problems between different media and so on. The governing equation is of the form A lot of research has been done[4,5,6,7,8] in the literature of standard Galerkin-Finite element Method. In this paper, An H1-Galerkin Mixed Finite Element Method is applied to approximat the unknown function u and its gradient p . Compared to the standard Galerkin Finite Element Method, the approximating finite element spaces Vh and Wh are allowed to be of differing polynomial degrees, and the procedure is not subject to LBB consistency condition. In addition, optimal error estimates between the approximation solution and the exact solution in L2-norm, Lp-norm, HI-norm, L∞-norm are obtained. In Chapter three, we consider the hyperbolic parabolic equation,that is ,a second time-derivative with a small coefficient is added to the heat conductive equation. Thiskind of equation have not only some characteristics of classical parabolic equations, but also some characteristics of hyperbolic equations. The initial and boundary value problem is given by the following system:An .H1-Galerkin mixed finite element method is used to approximate the unknown function and the adjoint vector value function.The semi-discrete scheme and fully-discrete scheme are presented.The optional error estimates for the unknown function and its gradient in L2-norm,H1-norm estimates are obtained .