Error estimates for the interpolating moving least-squares method

In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in details. The advantage of the IMLS method is that the meshless method which is constructed based on the IMLS method can apply the essential boundary conditions directly and easily. A simpler expression of the approximation function of the IMLS method is obtained. Then the error estimate of the approximation function and its first and second order derivatives of the IMLS method are presented in one-dimensional case in this paper. The theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth, then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, some selected numerical examples are given to prove the theories in this paper.