| In this paper, a non C~0 nonconforming plate element with eight degrees of freedom and twelve parameters is constructed by the Double Set Parameter method, then we analyze its anisotropic convergence and superconvergence. What's more, when applied to the fourth order singular perturbation problem, it is also anisotropic convergent. Prom this paper, we can see that the double set parameter method has many new advantages.The classical finite element convergence analysis relies on the following regular condition: there exists a constant c independent of the element K and the mesh such that hK/pK≤ c, where hk and pk are diameters of K and the biggest ball contained in K, respectively. Therefore, recently, there appear some study of finite element method on anisotropic meshes, i.e., under what weak conditions, the convergence of FEM is independently of hx / pk- In this paper, we firstly summarize some exist results on this subject, which mainly include the classical results of Ciarlet and the new results of Apel [2]. We introduced a general theorem of anisotropic interpolation, by this theorem a new criterion is presented, which improves Apel's result and is easier to use.It is well-known that the interpolations of some low order plate elements are not anisotropic interpolations, e.g., the famous incomplete biquadratic plate element and its corresponding double set parameter element with the shape space span{l,x,y,xy,x2,y2,x2y,xy2}. We firstly modify the shape space of this element, then we discreticized it by the double set parameter method. We will prove the optimal anisotropic interpolation error estimate and consistency error estimate of our 8-12-2 nonconforming element.On the other hand, Professor Zhongci Shi found the consistency error of some nonconforming plate element is of O{h2) a few years ago (c.f. [44]), and reference [21] generalized this result by giving three criterions. Though there have been many plate elements enjoy this special property for the present time, all of them have at least twelve degrees of freedom and contain the complete polynomial of order three. The element presented in this paper not only does not contain the complete polynomial of order three, but also dose not satisfy the two conditions of [21]. However, by a novel approach, we prove the anisotropic consistency error of our 8-12-2 element is of O(h2) through the element cancellation technique and the special shape space of this element.Up to now, there is no study on the superconvergence of nonconforming plate elementsbecause of the absence of corresponding theory and much hardship. In this paper, we discuss the superconvergence of 8-12-2 element for the biharmonic problem. We obtain non only some important identities and superclose result, but also the natural superconvergence at cental points and the global superconvergence by a proper post-processing operator. We will note that all the results are obtained under anisotropic meshes.The finite element solving the elliptic fourth order singular perturbation problem need to be convergent plate element and to be convergent uniformly with respect to the parameter e. Recently, the references [45,46] presented the convergence criterions for C° nonconforming plate element and non C° nonconforming plate element respectively. Though many convergent plate elements are also introduced there, none of them can be applied to anisotropic meshes. In this paper, we specially analyze the convergence of ACM's element and 8-12-2 element, both of which do not satisfy the corresponding criterion conditions. Note that the meshes considered here are also anisotropic.Additionally, a lot of numerical experiments are carried out, which can be verified the theatrical results of this paper.
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