Researches On High Accuracy Finite Element Methods For Evolution Equations

In this dissertation, we focus on several evolution equations, such as nonlinear Schr¨odinger equation, Benjamin-Bona-Mahony(BBM) equation, nonstationary incompressible Navier-Stokes equation, Cahn-Hilliard(CH) equation and the convection-dominated diffusion equation, study the nonconforming finite element methods, conforming and nonconforming mixed finite element methods from different points of view and give comprehensive investigations on the convergence analysis, superclose and superconvergence,etc.Firstly, a nonconforming quadrilateral element(named modified quasi-Wilson element) is applied to solve the nonlinear Schr ¨odinger equation. Based on a special character of this element, i.e., its consistency error is of order O(h3) for broken H1-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimates and supercloseness in broken H1-norm are derived for semidiscrete and two full-discrete schemes(Backward-Euler(B-E) and Crank-Nicolson(CN) schemes) on generalized rectangular meshes. Moreover, the global superconvergence results are deduced for rectangular meshes with the help of interpolation postprocessing technique. Finally, some numerical examples are provided to verify the theoretical analysis and validity of this method.Secondly, a low order nonconforming finite element method and a new mixed finite element scheme are developed to solve the nonlinear BBM equation. On the one hand,by use of two special properties of EQrot1element(one is that its interpolation operator is equivalent to Ritz projection operator, the other is that the consistency error can be estimated by O(h2) order, one order higher than its interpolation error), the supercloseness and global superconvergence results for semi-discrete and two full-discrete schemes(B-E and C-N schemes) are derived. On the other hand, a new nonconforming mixed finiteelement scheme is introduced, by means of high accuracy character of zero-order R-T element, the supercloseness and global superconvergence results for corresponding variables are deduced. Furthermore, the corresponding numerical experiments are carried out to verify the theoretical analysis.Thirdly, a low order mixed finite element method for the nonstationary incompressible Navier-Stokes equations is studied. The velocity and pressure are approximated by the nonconforming constrained Qrot1(CQrot1) element and the piecewise constant(Q0) element, respectively. The superclose and superconvergence results for the velocity in broken H1-norm and the pressure in L2-norm are obtained respectively. At last, a numerical experiment is executed to confirm the theoretical analysis.Then, a framework of nonconforming mixed finite element method for the CH equation is established. The superclose and global superconvergence results of the original variable u and the auxiliary variable p in broken H1-norm for the semi-discrete and B-E fully-discrete schemes are derived, respectively, where the typical character of the elements(the consistency error can be estimated as of order O(h2), which is one order higher than its interpolation error) and the interpolation postprocessing technique are employed.Finally, numerical examples are carried out to validate theoretical analysis.Finally, a new characteristic mixed finite element method is proposed for the convection-dominated diffusion problem by combining the method of characteristics with the new mixed variational formulation. The error estimates for both the original variable u and auxiliary variable p are derived. At the same time, some numerical experiments are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.