Research On New Nonconforming Finite Element Schemes And Superconvergence Analysis

In this dissertation, we focus on two kinds of fourth order variational inequalities,a nonlinear reaction diffusion fourth order singular perturbation equation, second-orderelliptic equation, nonlinear sine-Gordon equation and Stokes equations, study noncon-forming Galerkin-finite element methods (FEMs), conforming and nonconforming mixedfinite element methods (MFEMs), modified penalty and anisotropic FEMs from differentpoints of view, and give comprehensive and in-depth studies on some new FE schemes’constructions, theory analysis (such as convergence, superclose properties and the globalsuperconvergence) and numerical experiments.Firstly, we consider double set parameter nonconforming FEMs for the two-sideddisplacement obstacle of clamped plate, its solution determined by a fourth order varia-tional inequality. The lack of H4regularity leads to the main difficulty in the convergenceanalysis of FEMs. As a first test, we use some nonconforming elements constructed bythe double set parameter method, which are computationally desirable and satisfy the con-vergence conditions simultaneously, to approximate the variational inequality. For conve-nience, we take a nine parameter nonconforming triangular element constructed by doubleset method (named Veubeke-Zienkiewicz element) as the approximation space, which hasthe same global dimension as Zienkiewicz element and can pass the generalized patch testfor any subdivision. In order to get optimal error estimate, we introduce an auxiliary ob-stacle problem which is a bridge between the continuous and discrete obstacle problems,then skillfully construct a enriching operator which connecting the nonconforming andthe corresponding familiar conforming element spaces. Based on the operator and somefeatures of the double set parameter element, we present the optimal estimate of the dis-cretization error in the energy norm. Furthermore, we establish a general framework ofdouble set parameter nonconforming elements for the two-sided displacement obstacle problem of clamped plate.Secondly, we focus on anisotropic noncoforming FEMs for the fourth order vari-ational inequality with curvature obstacle. Because the solution region is a convex setassociated with curvature, the convergence of FEM solutions are not always true. It isimportant to find nonconforming element which has anisotropic feature and can guaran-tee convergence simultaneously. Note that the quadratic term of Morley type rectangularnonconforming element’s interpolation meet the anisotropy and has a special mean valueproperty. Then using the function splitting method, the optimal error estimate is obtainedfor anisotropic meshes.Thirdly, we apply the nonconforming FEM to the extended Fisher-Kolmogorov (EFKfor short) equation, which is a time-dependent nonlinear reaction diffusion fourth ordersingular perturbation equation. The analysis in previous literature only concentrated onC1-conforming for regular meshes. It is well known that C1-conforming elements arecomplicated and not computationally desirable, while the regular condition can hinder thedealt with boundary layer or inner layer. We firstly employ the nonconforming FEMs tosolve the EFK equation and attempt to extend convergence results to anisotropic meshes.The main ideas are as follows: In the first step, by using Lyapunov functional and Sobolevembedding theorem, some priori bounds are established, which lead to the existence anduniqueness for finite element solutions and play an important role in the error estimateof nonlinear term. In the second step, employing the interpolation operator directly andcombining the above priori bounds, derivative transfer technique and Gronwall inequality,we obtain the convergence results uniformly with respect to the perturbation parameter forsemi-discrete and Euler fully-discrete schemes. Furthermore, we present a uniform con-vergence theorem for C0nonconforming plate element approximating to EFK equation.Moreover, the error estimates for nonstandard C0nonconforming elements constructedby double set method are discussed. At the same time, numerical experiments are carriedout, numerical results coincide with our theoretical analysis.Finally, we mainly consider the global superconvergence analysis of new MFE schemesfor four kinds of partial differential equations.(I) A new nonconforming mixed finite element scheme for the second order elliptic problem is proposed based on a new mixed variational form. The MFE space pair is con-structed by the constrained Qrot1element space and piecewise constant vectors space, it hasthe lowest degrees of freedom on rectangular meshes. The superclose property is provenby employing integral identity technique and weak BB conditions. Then global supercon-vergence result is derived through interpolation postprocessing operators. Furthermore,we give some numerical results to show the effectiveness of the new scheme.(II) Nonconforming quadrilateral finite element method of the two-dimensional non-linear sine-Gordon equation is studied. A new arbitrary quadrilateral element (namedmodified Quasi-Wilson element) is used in Crank-Nicolson fully-discrete scheme. Basedon the special feature of the element, i.e., the consistency error estimate is two ordershigher than that of interpolation error, employing Riesz projection and high accuracy re-sult of conforming part of interpolation, we obtain the optimal order error estimates forarbitrary quadrilateral meshes, the superclose and global superconvergence results forgeneralized rectangular and rectangular meshes, respectively. A numerical test is carriedout to verify the theoretical analysis.(III) Based on the integral identity technique and interpolation postprocessing op-erators, we obtain the superconvergence results of MFEMs for the EFK equation withtwo kinds of boundary conditions by employing interpolation operator directly and Rieszprojection respectively. On that basis, a new patter of high accuracy analysis of linear(bilinear) element for the EFK equations is proposed for anisotropic meshes, which isvalid to the two kinds of boundary conditions simultaneously. By use of the anisotropicfeature and integral identity result of the element, we derive the error estimates of Rieszprojection for anisotropic meshes firstly (As far as we known, up to now, there isn’t reporton anisotropic feature of Riesz projection but only on that of interpolation), and estab-lish error estimates between the interpolation and Riesz projection. Then by applying theinterpolation postprocessing operators, the superclose properties and superconvergenceresults are obtained for semi-discrete and Euler fully-discrete schemes, which can’t bededuced by the the interpolation and Riesz projection alone. Numerical results supportthe theoretical analysis.(IV) As a first attempt, we combine the modified penalty method with L2projec- tion method. Using the Crouzeix-Raviart type nonconforming linear triangular elementto approximate to the velocity and piecewise constant to the pressure, we get the super-convergence results for the velocity and pressure in the modified penalty scheme firstly.Compared with the classical penalty method, our method can achieve the high conver-gence rates with a large penalty parameter, which effectively avoids instability problemfor the penalty method resulting from the use of a small parameter. Numerical experi-ments are carried out to confirm the theoretical results.