Non-conforming Finite Element Structures And Their Applications

Several new nonconforming finite elements are constructed,the convergence analysis of these elements are discussed and their application are presented in this thesis systematically.These new nonconforming elements include:the Quasi-Carey element,the Quasi-Wilson element,the higher order Wilson element and the second order nonconforming mixed finite element.Compared with the conforming finite element methods,the finite element methods of nonconforming have many advantages.Generally speaking,nonconforming elements have fewer degrees of freedom for its simpler structure and good convergence properties,such as the Morley element and the Wilson element.In addition, the nonconforming mixed finite element methods are usually much easier to be constructed to satisfy the discrete inf-sup condition than the conforming ones. Therefore,nonconforming finite element methods have drawn increasing attention from scientists and engineers.As we know,according to the second Strang lemma, the error of every nonconforming element consists of two parts,one arises from the interpolation error and the other is the consistency error due to nonconformity of the element.In most cases,the order of the consistency error is lower than or equal to that of the interpolation error.But,in this paper,one can see that for the second order elliptic problems the consistency error of the Quasi-Carey element is of order O(h²),one order higher than that of its interpolation error O(h).We proved that the consistency error of the traditional Quasi-Wilson element is of order O(h³),two order higher than that of its interpolation error.At the same time,a new QuasiWilson element for arbitrary quadrilateral meshes possessing consistency error with order O(h³) is presented.After a careful analysis,we first show that the interpolation error of the higher order Wilson element is of order O(h³) on anisotropic meshes,one order higher than that of its consistency error.As application,in Chapter 2,we investigated the approximation of higher accuracy of the anisotropic nonconforming Quasi-Carey element for the Sobolev type equations.The superclose and global superconvergence with order O(h²) are obtained. Moveover,by virtue of the extrapolation,we improved the approximate accuracy of the related approximate solution and derive a posteriori error estimate of higher accuracy of order O(h⁴).In Chapter 3,based on the special convergence of the Quasi-Wilson element,we applied it to convection-diffusion equations and obtained the optimal convergence order O(h^3/2) as the bilinear element and the p₁^mod element.In Chapter 4,after analysing the error estimates of the higher order Wilson element on anisotropic meshes with a numerical test,the superclose properties of this element are proved.Then the interpolation postprocessing technique is used to obtain the global superconvergence and the posterior error estimate of higher accuracy.In Chapter 5,we applied the Quasi-Carey element and the modification higher order Wilson element to Maxwell's equations on the finite element scheme, the optimal convergence results are obtained.But the similar optimal convergence results can not be obtained for the nonconforming linear triangular Crouzeix-Raviart element,the Carey element and the higher order Wilson element.In Chapter 6,the new nonconforming mixed finite element schemes with second order convergence behavior are proposed for the stationary Navier-Stokes equations,the convergence analysis is presented and the error estimates of both H¹-norm of order O(h²) and L²-norm of order O(h³) with respect to velocity as well as the L²-norm of order O(h²) for the pressure are derived.At the same time,the numerical results are presented to illustrate the error analysis.