Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces

Abstract

The moving least square (MLS) approximation is one of the most important methods to construct approximation functions in meshless methods. For the error analysis of the MLS-based meshless methods it is fundamental to have error estimates of the MLS approximation in the generic n-dimensional Sobolev spaces. In this paper, error estimates for the MLS approximation are obtained in the Wk,p norm in arbitrary n dimensions when weight functions satisfy certain conditions. The element-free Galerkin (EFG) method is a typical Galerkin method combined with the use of the MLS approximation. The error results of the MLS approximation are then used to yield error estimates of the EFG method for solving both Neumann and Dirichlet boundary value problems. Finally, some numerical examples are given to confirm the theoretical analysis.