Error Estimates Of The Complex Variable Moving Least Squares Approximation

The meshless method is a new numerical method which is developed after the finite element method.The core of the meshless method is the construction of shape function.The moving least squares(MLS)approximation is one of the most widely used approximation scheme.However,the calculation of meshless methods based on moving least square approximation is expensive in general.By introducing the complex variable theory into the MLS approximation,a complex variable moving least squares(CVMLS)approximation has been developed to approximate vector functions.The trial function of a two-dimensional problem is formed with a one-dimensional basis function in the CVMLS approximation.The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the MLS approximation.Thus,we can select fewer nodes in the meshless method that is formed from the CVMLS approximation with no loss of accuracy.The meshless method based on the CVMLS approximation has been widely used in the field of engineering,but the mathematical theory is not perfect.In order to promote the application of meshless method based on the CVMLS approximation,it is necessary to analyze the error of the CVMLS approximation.This paper discusses the error of the CVMLS approximation in detail.The main contents are as follows.This dissertation first introduces several important numerical methods for partial differential equation,and the developed history and research status of meshless method.The second chapter introduces the MLS approximation and the CVMLS approximation.In the third chapter,we analyze the error of the approximation function and its partial derivatives for smooth function.The results show that the error has a great relation with the node spacing.Some numerical examples are given to verify the theoretical results.In the fourth chapter,we analyze the error of the CVMLS approximation in Sobolev space.