Researches On Nonconforming Anisotropic Finite Element Methods

In this thesis, we consider some kinds of partial differential equations (includ-ing general neural transmission equations-. Navier-Stokes equations、second order elliptic equations、nonlinear sine-Gordon equations and nolinear parabolic integro-differential equation, etc) and study the nonconforming finite elements, moving grids and anisotropic meshes from different points of view and give the comprehensive investigations on the construction of nonconforming elements、convergence analy-sis、superclose、superconvergence and numerical computations, etc.Firstly, on anisotropic meshes, by use of moving grids ideas, we use a Crouzeix-Raviart type nonconforming triangle element to approximate the variable coeffi-cient、nonlinear general neural transmission equations and analyze a Crank-Nicolson discrete scheme. The optimal error estimates are derived by the consistent property of the Riesz projection and the interpolation operators about the element. Then, the constrained rotated Q1element on rectangular meshes is used for the nonsta-tionary Navier-Stokes equations and the convergence analysis are provided with moving grides methods, the optimal error estimates in H1-norm for the velocity and.L2-norm for the pressure are derived.Secondly, we investigate the EQrat1element on anisotropic arbitrary quadrilat-eral grids, as an attempt, we promote the results in [65] up to anisotropic nonparallel quadrilateral grids based on some new techniques and approaches and derive the optimal error estimates for second order elliptic problems. At the same time, the numerical experiments are carried out to verify the theoretical analysis.Finally, the nonconforming Quasi-Wilson finite element approximations for the general nonlinear sine-Gordon equation and the nonlinear parabolic integro-differential equation are discussed. Based on the special feature of the element, i.e., the consistency error estimate is of order O(h3) in the energy norm, which is two order higher than that of interpolation error estimate, and high accuracy analysis of the bilinear element, the optimal error estimates in L2-norm and superclose proper-ties in broken H1-norm of the corresponding unknown functions are derived for the semi-discrete and fully-discrete schemes with new approaches different from that in the previous literature, respectively. Moreover, the global superconvergence results of broken H1-norm are obtained through the postprocessing techniques and the su-perclose properties and the optimal estimates for the Backward-Euler fully-discrete scheme are derived for the nonlinear parabolic integro-differential equations.