The Study Of Nonconforming Finite Element For Evolution Type Equations

The finite element methods are one of main flows of science and engineering calculation nowadays. Compared with the conforming finite element methods, nonconforming finite element methods are appropriate, for they have the striking advantage. For example, for the nonconforming finite elements which degree of freedom is defined with the lines of elements and elements themselves, the every unknown is associated with the element face, each degree of freedom belongs to at most two elements. This results in cheap local communication. Nonconforming finite elements more easily fulfill the discrete LBB condition than conforming finite elements. Therefore, nonconforming finite element methods have drawn increasing attention. In addition, the classical finite element methods demand that the subdivisions should satisfy the regular condition or quasi-uniform, which restricts the application of the finite element methods. In fact, for problems in narrow domain, if we use conventional regular subdivisions, the increase of total degrees of freedom will make the calculated amount become very large. However, if we use anisotropic subdivisions, we will obtain the same estimates results as traditional finite element methods with less degree of freedom. At present time, the anisotropic finite element methods have been one of the hot topics in finite element domain.In this thesis, we consider some kinds of evolution equations (including Sobolev equations, parabolic integral differential equations, nonlinear Sobolev equations, nonlinear hyperbolic equations, non-stationary conduction-convection), and study the anisotropic nonconforming finite element methods, the nonconforming finite difference streamline diffusion methods, the nonconforming mixed finite element methods, ect, differential point of view, and give deep and comprehensive study form the construction of the elements, theoretical analysis and numerical computing. In chapter 3 and chapter 4, applying a kind of low order anisotropic noncon-forming elements ( including rectangular element and triangular element) are used to approximate Sobolev equations and parabolic integral differential equations. The optimal estimates of L~2-norm and H~1-norm and the supperclose and supperconver-gence results are obtained under semi-discrete scheme. And full discrete analysis of Euler-Galerkin scheme and Crank-Nicolson-Galerkin scheme are derived. Numerical results support the accuracy of our theoretical analysis. In chapter 5, The nonconforming economical finite difference- streamline diffusion methods for a class of Sobolev equations with convection-dominated term is studied. The optimal accuracy analysis of Euler-EFDSD scheme and Crank-Nicolson—EFDSD scheme are given, respectively. In chapter 6, nonconforming mixed H~1-Galerkin finite element method for a kind of hyperbolic equations is considered and the optimal estimates of H~1-noma. and H(div)-norm are obtained under semi-discrete scheme. In chapter 6, a nonconforming mixed finite element scheme is proposed for the non-stationary conduction-convection problem, the optimal error estimates of L~2(H~1)-norm for the velocity, L~2(L~2)-norm for the pressure and L~2(H~1)-norm for the temperature are derived.