Superconvergence Analysis Of Obstacle Problems With Nonconforming Finite Element Method

The finite element method is a numerical technique that is widely used and has flexible applicability.The main idea of ?the finite element method is to unitize the solution domain and solve the discrete variational form on the tiny unit and then superimpose the unit from the whole area..By constructing the finite element space,the variational form of the actual problem can be discretized to obtain the finite element solution of the problem and the corresponding error analysis result.The variational inequality problem is regarded as a very important research topic because it has more flexible boundary constraints than the variational equation problem.The obstacle problem is a representative one of them.In this paper,based on the variational inequality and finite element method theory knowledge,the obstacle problem is solved.The three elements of linear element,bilinear element and nonconforming element in finite element are respectively used,and the corresponding results are obtained through theoretical derivation and numerical experiments.Superconvergence results.This paper firstly studies the background and theoretical basic knowledge,which mainly includes the research status of the finite element method and the variational inequality problem,and determines the direction of the work of this paper by enumerating the work done by the predecessors.The actual physical meaning of an obstacle problem is abstracted in addition to its corresponding mathematical expression,variational form,and discrete variational form.After determining research directions and research topics,use the definition of Sobolev space and inequality in space,embedding theorem and trace theorem,finite element approximation theory,variational method,Bramble-Hilbert lemma,finite element interpolation error estimation theorem,The theoretical knowledge of the Green formula in Sobolev space is used to derive theoretically the results of the super-approximation and super-convergence of the second-order elliptic obstacle problem using linear finite elements.After obtaining the superconvergence result of the linear element,this paper selects the Carey element in the nonconforming element for theoretical derivation,which uses the correspondence between the coordination part and the linear element in the Carey element and the integral character of the noncoordinated part.After obtaining the results of the superconvergence theory,this paper validates the theoretical results by selecting two numerical examples with/without exact solutions.In this process,appropriate examples are selected,a numerical experiment procedure is established,and the unit coordinate transformation is consulted.As well as the corresponding theories of basis function and shape function,each step in the numerical experiment process is programmed and implemented,which mainly includes the area division,the solution of the discrete unit stiffness matrix and the element load synthesis,and the combination of the right end item and the obstacle.Solve the finite element solution to the obstacle problem.In the case of using linear elements and bilinear elements,the superconvergence results that are more consistent with the theoretical results are obtained and the values ?of finite element solutions in the case of using Carey elements are obtained.Through the super-approximation and super-convergence theorem of both elements,the superconvergence result of 3/2 order is obtained,and by using the numerical experiments for the three units,the linear unit and the bilinear unit are used.The superconvergence results that are more consistent with the theoretical results are obtained and the values ?of the finite element solutions in the case of using Carey elements are obtained.