The New Progress Of The Non-conforming Element And Anisotropic Theoretical Analysis And Numerical Experiments

In this paper several new finite elements are constructed for differrent kinds of probelms by the Double Set Parameter Method and their classical and anisotropic convergences are analyzed. Anisotropic convergences of some famous finite elements are also analyzed, and a lot of numerical experiments have been carried out-Many applicable triangular and rectangular elements have been developed for the plate bending problems, some of them are conforming and some of them arc unconforming, but the researches on quadrilateral piar.e elements have been seldom reported. For the rectangular elements, the mapping from the reference element. l to a general element K is very simple. It is enough to construct, the shape function on the K. But for arbitrary quadrilateral elements, the situation is different. On the one hand, the shape function should be constructed on K because it is impossible to construct it directly on a arbitrary quadrilateral, on the other hand tin1 shape function on K, not on K, should have some continuity across the boundaries of K to ensure the convergence. But the continuities of the shape function on A dn not remain on K because the mapping from K to K is nonlinear. It. is difficult to construct the shape function on K being well posed and have the continuities on K required by convergence. An efficient 13-parameter arbitrary quadrilateral plate element is proposed by the double set parameter method and its convergence is proven. Some numerical tests are given which show the the numerical validity of this element.A finite element solving the elliptic fourth order singular perturbation problem need to be a convergent plate element and to be convergent, uniformly with respect to the parameter e. But riot everyone of the convergent plate elements is convergent uniformly with respect to the parameter e. Then, which of the convergent plate elements are convergent uniformly with respect to the parameter We will present a general convergence theorem for c?nonconforming plate elements and a general convergence theorem for non c?nonconforming plate elements for the problem re-spectively. Using the double set parameter method we have constructed several finite elements which are convergent for this problem, and the numerical tests thatwe have made have proved the efficiency of these elements.The classical finite element approximation theory relies on the regular or non-degenerate condition. This means that the length of every edges of the elments can not change greatly and this has limited applications of the finite element method in some enginering problem such as those of anisotropic structures. For such problems it is naturally wanted taht this condition is not necessary, the sizes of the elements in all directions are allowed to have great differrencs as the problems theirselves require and the convergences of the elements are still remained. This kind of finite elements are called anisotropic elements.Anisotropic finite elements have been developed recently. Among them A pel and Dobrowoiski have published a series of papers deal with anisotropic Lagvangc interpolations. However, the studies on the estimations of consistent errors for nonconforming elements have been seldom reported. We have improved the method of anisotropic interpolation error estimations and presented a method to estimate the consistent errors of nonconforming elements.Wilson's and Adini's elements are famous nonconforming rectangular elements. Using our method we have presented the anisotropic estimations of both interpola-tion errors and consistency errors for these two elements.Quasi-Wilson's element is a nonconforming element for arbitrary quadrilateral meshes. We have proved that the quasi-Wilson element is anisotropic-convergent for narrow quadrilateral meshes. Anisotropic error estimates of interpolation error and consistency error in the energy norm and the L²-norm have been given.Using the double set parameter method a noncomforming rectangular plate element with 8 degrees and 12 nodal parameters is...