New Model Researches On The Mixed Finite Element Methods For Sobolev And Parabolic Type Equations

This paper mainly includes three parts.In the first part, H1-Galerkin mixed finite element method for Sobolev equation is studied. A new mixed finite element pattern is constructed using incomplete biquadratic element Q-2and first order BDF M element. Through Bramble-Hilbert lemma, high precision results of interpolation operators corresponding to unit are proved. Further, the superclose properties for the primitive variables u in H1-norm and the intermediate variable p in H(div)-norm are obtained respectively in semi-discrete and the backward Euler fully discrete schemes.In the second part, a new mixed finite element method is proposed for the parabolic equation of integral type boundary conditions. Compared with the traditional mixed finite element method, the method’s structure of the finite element space and theoretical analysis are much simple. We choose freedom simple bilinear element and N′ed′elec s element respectively to approximate the original variable u space and flux variable p space. The superclose and global superconverenge results of relevant variables are derived by using derivative transfer technique and boundary di?erence estimates in semi-discrete situation.Moreover, the backward Euler full-discrete scheme is given.In the third part, with the help of EQrot1 and zero order Raviart-T homas elements, a nonconforming mixed finite element approximation scheme is proposed for a class of fourthorder parabolic equations. Firstly, the existence and uniqueness of approximation solution are proved for semi-discrete scheme. Secondly, based on the high accuracy analysis of the about two elements, using derivative delivery technique with respect to the time variable and interpolated postprocessing technique, the superclose properties and superconvergence results with order O(h2) for both the primitive solution u and the intermediate variable v =-?u in H1-norm, flux p =- u in L2-norm are obtained, respectively. Finally,for backward Euler full-discrete scheme, the existence and uniqueness of approximation solution are showed. At the same time, by use of a new splitting technique, the superclose properties and superconvergence results with order O(h2+ τ) for both u and v in H1-norm,p in L2-norm are derived unconditionally with respect to h. These are the traditional analysis can’t get. Here, h and τ are parameter of the subdivision in space and time step,respectively.