Higher Order Nonconforming Element And Anisotropic Finite Element Study

In this paper, We focus on the plate problem, Stokes problem and the planar elasticity problem with pure displacement, we propose two anisotropic parallelogram nonconforming finite element methods for Stokes problem. Additionally, a lot of numerical experiments are carried out, which can be verified the theatrical results of this paper.The classical finite element convergence analysis relies on the following regular condition, that is, there exists a const C independent of the element K and the mesh, such that (?)≤C, where h_k andρ_k are diameters of K and the biggest ballcontained in K, respectively. Some works show that is unnecessary in some finite space, therefore, there appear some study of finite element method on anisotropic m(?)shes, i.e, under what weak conditions, the convergence of FEM is independent of (?). These new results mainly include the classical results of Ciarlet and the new results of Apel. We introduces a general theorem of anisotropic interpolation.By this theorem a new criterion is presented, which improves Apel's result and is easier to use.For the planar elasticity problem with pure displacement conditions, we present high order element by restricting the operator div in P₂ , show how to construct the element, and prove the optical error.Stokes problem is a typical mixed variational form, which contains pressure and velocity together. In this paper, we propose two anisotropic parallelogram nonconformingfinite elements.The anisotropic estimates of interpolation error, consistency error and LBB condition are obtained, which show that the convergence the method is independent of the regular and quasi-uniform assumptions on the meshes.Plate problem is one of main models in FEM. There are many results and practicing schemes, the regularity of subdivision is inevitable in computation.In this paper, we give a 8-12-2 nonconforming rectangular element by the double set parameter method, and prove the optimal anisotropic interpolation. error estimate and consistency error estimate.