Error Estimates Of Interpolating Moving Least-Squares Method And It’s Meshless Methods

Meshless method is a new numerical method which is developed after the finiteelement method. Meshless method has some advantages such as simplepre-processing and high computing accuracy. Now Meshless method is a hotdirection of computing science.There were a few investigations on corresponding mathematical theory ofmeshless method. Without the corresponding mathematical theory, meshless methodcan not been researched and applied well. Then it is very necessary to study themathematical theory of the meshless method. In this dissertation, the interpolatingmoving least-square (IMLS) method is discussed in details, an improved expressionof the shape function of the IMLS method is obtained, and the convergence and errorestimates of the IMLS method and some meshless methods based on it areinvestigated. The main contents of this paper are as follows.To decrease the singularity of the coefficient matrix in the IMLS method andimprove the accuracy, the expression of the singular weight function of the IMLSmethod is improved. Then some inner products are proved to be omitted, thus thecorresponding simpler expression of the shape function of the IMLS method isobtained. The shape function of the IMLS method satisfies the property ofKronecker function, and then the meshless method based on the IMLS methodcan apply the essential boundary condition directly. And the number of the unknowncoefficients in the trial function of the IMLS method is less than that of the MLSapproximation, then the computational efficiency is higher.The error estimate of the IMLS method is investigated. The error estimate ofthe approximation function and its first and second derivatives of the IMLS methodin one-dimensional and n-dimensional space are presented respectively. Thetheoretical results show that the accuracy of the approximation function and itsderivatives are closely related to the smoothness of the original function, the order ofthe polynomial basis functions, and the radii of the domains of influence of nodes.The interpolating element-free Galerkin (IEFG) method based on the IMLSmethod is presented to solve the two-point boundary value problems. Comparingwith the contraditional element-free Galerkin (EFG) method, in the IEFG method theessential boundary conditions can be applied directly. By proving the inverse property of the function in the shape function space, the error estimates of thenumerical solution and its derivatives of the IEFG method for two-point boundaryvalue problems are also presented. The superconvergence of the IMLS method isstudied. The superconvergence point for the approximation function and its firstderivative of the IMLS method are presented in one-dimensional space. Then fromnumerical examples, it is shown that the IEFG method for the two boundary pointproblems also has superconvergence at the superconvergent points of the IMLSmethod. And by using these superconvergent points, the numerical solution withhigher precision can be reproduced.Based on the IMLS method, the IEFG method for two-dimensional potentialproblems can apply the boundary conditions directly. Since the numerical solutionspace of the IEFG method for two-dimensional potential problems is not thesubspace of the solution space of the corresponding variational problem, then theCe lemma dose not hold. Then the error estimate of the IEFG method fortwo-dimensional potential problems is more complicated than it in one-dimensionalcase. Based on the abstract error estimates, the error estimates of the numericalsolution and its first derivative of the IEFG method for the two-dimensional potentialproblems are presented in this paper. And some numerical examples are given toshow the validity of the theories in this paper.Based on the error analysis of the IMLS method, the error estimates of theinterpolating boundary element-free method (IBEFM) are studied fortwo-dimensional potential problems. We show that the relationship of the error of thenumerical solution and the radii of the domains of influence of nodes, and the one ofthe error of the numerical solution and the condition number of the coefficient matrix.And some numerical examples are presented to demonstrate the error estimate theoryin this paper.The researches in this thesis provide some mathematic theories, and canpromote the further developments and applications of meshless method andcorresponding mathematical theory.