RBF-PS Method With Wendland Compactly Supported Radial Basis Function And Its Convergence Analysis

In this paper,the RBF-PS(radial basis function pseudospectral)method based on Wendland compactly supported radial basis function is introduced in detail.Some differential equations,two kinds of elliptic variational inequalities and the quasistatic viscoelastic frictionless contact problems with long memory involving normal compliance are solved by this method.The convergence analysis is given and numerical examples are realized.The main work of this paper is as follows:In chapter 2,the common global radial basis functions and Wendland compactly supported radial basis functions are introduced,the RBF-PS method and its implementation process are described in detail,and the first convergence theorem of the RBF-PS method for linear differential equation problems is given;Secondly,for the RBF-PS method of Poisson problem with Dirichlet boundary condition,the second convergence theorem with higher convergence order is given;Finally,two numerical examples are solved by this method including discussing the influence of different shape parameters ε.Compared with the linear finite element method,the Wendland RBF-PS method has the advantages of higher accuracy and faster convergence.In chapter 3,the first and second elliptic variational inequality problems described by static elastoplastic torsion problems and simplified friction problems are studied.Through the coupling method,an iterative calculation scheme for the coupling of Uzawa algorithm and RBF-PS for the corresponding problem is constructed.In the framework of variation,the error estimation theorem of single-step RBF-PS approximation is given,and then combined with Uzawa iteration,the error estimation of the whole RBF-PS coupling iteration method is given.Finally,some numerical examples are completed,and the numerical results show that the convergence order is consistent with the results of the error estimation theorem.In chapter 4.a quasistatic viscoelastic frictionless contact problems with long-term memory involving normal compliance is studied,and the existence and uniqueness theorem of its weak solution is proved under some reasonable assumptions.We construct corresponding RBF-PS full-discrete computational schemes based on the classical form of the problem and provide error estimates for the semi-discrete in time and RBF-PS full-discrete schemes under the variational framework respectively.Finally,two numerical examples are completed,and the numerical results show that the computational scheme has good convergence and accuracy.