Researches On Mixed Finite Element Method For Several Partial Differential Equations

In this thesis,the mixed finite element method?MFEM?for several classes of fourth order evolution equations?nonlinear Molecular Beam Epitaxy?MBE?equation,Sivashinsky and hyperbolic equations?and second order elliptic eigenvalue problem are mainly studied.From the conforming and nonconforming mixed elements,the convergence,superclose,superconvergence and extrapolation results are discussed deeply and systematically.Firstly,the conforming MFEM for two classes of fourth order nonlinear MBE equations is explored.By applying the high accuracy estimates of bilinear interpolation operator,the superclose estimates for the original variable u and intermediate variable p in H1-norm are derived in the semi-discrete scheme and two kinds of fully-discrete schemes?Backward-Euler?B-E?and Crank-Nicolson?C-N?schemes?,respectively,then the global superconvergence properties for the two variables are achieved through interpolation postprocessing technique.Secondly,a low order nonconforming MFEM and an extended new MFEM are applied to the fourth order nonlinear Sivashinsky equation.On the one hand,by using of the two special properties of the nonconforming EQ₁^rot element,i.e.,the consistency error is of order O?h²?in broken H1-norm?which is one order higher than its interpolation error?and the interpolation operator is equivalent to Ritz projection operator,the superclose and global superconvergence results for the original variable u and intermediate variable p in broken H1-norm are achieved for the semi-discrete and B-E fully-discrete schemes,respectively.On the other hand,an extended new mixed element scheme is established,according to the special nature of the lowest order Raviart-Thomas?R-T?element,techniques of integral identities and postprocessing technique,the superclose and superconvergence results are given for the related variables in the semi-discrete and B-Efully-discrete schemes,respectively.Then,the conforming bilinear MFEM is established for a fourth order wave equation.By applying the interpolation and projection simultaneously,the superclose and superconvergence results of order O?h²?for the original variable u and intermediate variable p in H1-norm are achieved for the semi-discrete and fully-discrete schemes,respectively.Compared with the existing literature for getting optimal estimates alone,the distinct advantage is that it not only can reduce the smoothness of solutions u,ut and p,but also can derive the superconvergence results.Finally,the nonconforming FEM and MFEM for Poisson eigenvalue problem are discussed.On the one hand,a nonconforming quadrilateral element?named modified quasi-Wilson element?is applied to this problem,by employing a special property of this element?when u ? H³???,the consistency error is of order O?h²?which is one orderhigher than its interpolation error O?h??and the interpolation postprocessing technique,the superclose and superconvergence results of order O?h²?for the eigenvector u in broken H1-norm are deduced on generalized rectangular meshes and rectangular meshes,respectively;Then,a new property for this element is proved that the consistency error even can reach O?h⁴?order for arbitrary quadrilateral meshes when u ? H⁵???,this is a new astonishing feature which has never been discovered before;Subsequently,based on the above characteristic and some asymptotic expansions of the conforming bilinear finite element,the extrapolation solution of order O?h⁴?for eigenvalue is derived.On the other hand,a new nonconforming MFEM is established for this equation.By use of the special properties of the nonconforming EQ₁^rot element and the lowest order of R-T element,the optimal order error estimates for both the original variable u and the auxiliary variable ?p are deduced,the lower bound of eigenvalue is estimated simultaneously.Furthermore,by applying the techniques of integral identity and interpolation postprocessing,we derive the superclose and superconvergence results of order O?h²?for u in broken H1-norm and?p in L2-norm,respectively.Finally,with the help of asymptotic expansions,the extrapolation solution of order O?h³?for eigenvalue is obtained.At the same time,the corresponding numerical results are given for each of the above parts to verify the correctness of the theoretical analysis.