Anisotropy And Two-parameter Nonconforming Finite Element Method

Recently, a lot of studies have been focused on the anisotropic finite element method, and there appears many relative papers(see [26,27,53,54]). However, the main attention of those were paid to the study of conforming Lagrange elements and the interpolation error analysis on the second order or fourth order elliptic boundary value problems. At present, the challenge of anisotropic finite element method is as follows:(1). Since the Bramble-Hilbert Lemma [4] can not be used directly on the anisotropic meshes, so the interpolation error can not be estimated by the use of the conventional finite element methods. Furthermore, the consistency error estimate is the key of the nonconforming finite element analysis, and it will become very difficult to be dealt with because there will appear a factor |F|/|K| which tends to∞when the estimate is made on the longer sides F of the element K.(2). Some elements didn't have anisotropy [77], for example, in [53], Th.Apel proved that the rotated Q₁ element [36] can not be used on anisotropic meshes by giving a counter example.(3). The proof of the well-posed property, stability and the LBB condition are very difficult to deal with.In addition, the studies of the superconvergence analysis on anisotropic meshes is another open problem. It is also very hard to deal with since the construction of the post interpolation operator and the proof of the anisotropic property are challengeable.In this paper, we focus on the the study of anisotropic nonconforming element of the variational inequality problem with displacement obstacle, the second order elliptic problem, Stokes problem, and the planar elasticity problem. And rectangle and triangle plate elements are constructed respectively by taking advantage of the double set parameter methods. For the first element, the superconvergence result are proposed; and for the second one, we discuss its convergence and the numerical results are given to demonstrate the validity of our theoretical analysis. At the last part of the paper, the construction of two double set parameter elements and their convergence are discussed.In detail, the arrangement of the paper is as follows:Firstly, we first study the nonconforming finite element (Wilson element [20] and Carey element [9]) approximation to the second order variational obstacle problem with displacement obstacle on the anisotropic meshes. By using novel approaches, the convergence analysis is given and the optimal error estimates are obtained. The method proposed herein is also valid to the other element which can be separated into conforming part and nonconforming parts and pass the Irons patch test. Next, a class of Crouzeix-Raviart type nonconforming finite element approximations are considered for solving the above problem on anisotropic meshes. The approaches for this kind of element is totally different from that of the former, and the convergence analysis is given and the optimal error estimates are obtained under the hypothesis of the finite length of the free boundary.Secondly, the convergence behavior of a Carey triangle nonconforming element for the second order elliptic problem with lowest regularity on anisotropic meshes are discussed. There have been a lot of studies this aspect when the exact solution u∈H²(Ω) or u∈H³(Ω). [4], [52] and [50]studied the convergence of conforming linear triangle elements with minimal regularity assumptions u∈H⁴(Ω,), here u is the solutions of the second order elliptic problems. However, their is another vital deficiency of the previous studies, i.e., they relies on the regularity assumption or quasi-uniform assumption on the subdivision of meshes. By using different techniques from that of [49, 50, 52], the same results as those of [49,50,52] are obtained on the anisotropic meshes with the minimal regularity assumption.Thirdly, we discuss the the second order elliptic problems and Stokes problem. For the first one, a new element is constructed and the optimal error estimates are obtained under the lowest regularity of the exact solution. In addition, the proof of this paper is much easier in comparison with [33]. For the second one, with the use of five nodal rectangular element, the superclose results of the velocity and the pressure are obtained by novel techniques, and furthermore, by constructing a proper interpolation post operator, the superconvergence can be deduced on the anisotropic meshes. The main result of the second one is also valid to the rotated Q₁ element in [36,48] on regular meshes.Next, we study the the planar elasticity problem with pure displacement boundary condition with the use of a new anisotropic nonconforming rectangular finite element which is constructed in the paper. It is proved that this element is locking-free when the Laméconstantλ→∞. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L²-norm are obtained. Thus we get rid of the restriction of regularity assumption, quasi-uniform assumption or the inverse assumptions on the meshes required in the conventional finite element methods analysis, and extend the applicable scope of the nonconforming finite elements.Finally, we construct 12 parameters rectangular plate element(by modifying the shape function space of [47] as P₃∪span{x³,y³}) and 9 parameter triangle plate ele-ment(by modifying Morley element) by use of the double set parameter finite element methods. For the first element, we can prove that the consistency error is O(h²) order which can be obtained on anisotropic meshes. For the second one, its shape function space and the real node parameter is as same as those of Morley element and Zienkiewicz element, but the total degrees of freedoms is only three quarters of that of Morley element. The theoretical analysis and the numerical test show that the element is very excellent for the reason that the outer normal derivative is continuous, degrees of freedom is symmetrical, and it is convergent for arbitrary triangle subdivision.