Study On P-type Superconvergent Recovery Method For Fe Analysis Of One-dimensional Nonlinear Boundary Value Problems

The finite element method(FEM)is efficient for solving differential equations.It has the characteristics of good numerical stability,strong universality and wide applicability.The accuracy of the FEM solutions depends on the mesh size and the degree of shape functions,generally,the smaller mesh size and the higher order element shape funtions,the better accuracy.In order to obtain high precision solutions,the tranditional method requires densify mesh or improve element order,which accordingly leading to the rapid increase of computational cost,resulting in very big calculating cost.In order to solve the contradiction between the accuracy of solutions and computational cost,the finite element superconvergence recovery has become the focus and hotspot in the field of finite element research.Theoretical and numerical results show that the solutions on element end node has higher convergence order compared with the element inner solutions.This means the nodal solutions has the property of superconvergence.As the one-dimensional nonlinear problem,that superconvergence property also exists.Based on the superconvergence property of the nodal solutions,the p-type superconvergence algorithm for the one-dimensional nonlinear finite element is studied in this thesis.The main work of this thesis is as follows:Firstly,the p-type superconvergence algorithm for one-dimensional nonlinear finite element is proposed.The Newton method is adopted to solve the nonlinear problems iteratively,and the Newton iterative scheme is derived.The the idea of p-type superconvergence calculation in linear problems is extended to solve nonlinear boundary value problems(BVP).The governing BVP on each element for the p-type superconvergence recovery algorithm is set up by setting the nodal solutions as element boundary conditions.Then it is linearized by making the Taylor expansion to the original nonlinear differential equation.This linearized BVP is solved by using a single high order element,from which the superconvergence solution is obtained.The superconvergence solution on the whole domain can be obtained by implementing the process on each element.Secondly,this thesis applies the basic strategy successfully solving boundary value problems of second order nonlinear ordinary differential equations,fourth order nonlinear ordinary differential equations,first order nonlinear ordinary differential equations and nonlinear mixed order ordinary differential equations.Thirdly,a large number of numerical experiments are carried out for each type of nonlinear model problem,and a large number of classical nonlinear problems are solved.The finite element solutions and the superconvergence solutions of each nonlinear problem are analyzed(error and the order of convergence),and the general rules of superconvergence solutions are obtained through theoretical analysis and numerical experiments.The numerical results show that the error estimation of the superconvergence solutions is correct and reliable,and the superconvergence algorithm is simple and efficient.Finally,this thesis studies the nonlinear boundary conditions tentatively,and puts forward a set of simple and efficient methods to deal with the nonlinear boundary conditions.In addition,this thesis summarizes the work and prospects.