The Superconvergence Analysis For Anisotropic Double-K Element And Structure With Double Set Parameters

Finite element method is one of the important numerical methods for solving engineering mechanics problems. The problems are solved by subdividing the differential equations solved regions, and using the field functions piecewise polynomial approximation model to transform the infinite-dimensional space into the finite-dimensional space and the continuous space into the discrete space. The basic conclusions of the traditional finite element analysis are mainly based on the regularity assumption or quasi-uniform assumption. However,the regularity assumption or quasi-uniform assumption are not always established when we use the finite element method to solve some complicated problems.In this paper, the natural super-convergence and pointwise convergence of the bicubic Hermite rectangular plate element are discussed under the anisotropic meshes—that is, without regularity assumption or quasi-uniform assumption. The methods in the article employ fully the structural characteristics of the bicubic Hermite element. By the bilinear lemma and the Bramble-Hilbert lemma, we obtain the conclusions of the bicubic Hermite rectangular plate element of the super-convergence, pointwise convergence, which are consistent with that of traditional finite element under the regularity condition. Using the same method, with the conclusions of existing literatures ,the general form of the conclusion of the Bicubic Hermite element's superconvergence in H~n(Ω).At the same time, we combine with the conclusions of the domestic related literature and summarize the general conclusions of the natural superconvergence and the general proof methods of pointwise convergence of the double-K order Lagrange rectangular plate element while solving the second-order problem under the anisotropic meshes. Finally, we discuss the plate element with double set parameters, establish two plate elements with double set parameters, proving its convergence and giving the error estimates,and calculate the disturbance quantity of the node parameters and compare with the well-known ACM element.