Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method

The improved moving least-square (IMLS) approximation is a method to form shape functions in meshless methods. For the application of IMLS-based meshless methods to the numerical solution of boundary value problems, it is fundamental to analyze error of the IMLS approximation in Sobolev spaces. This paper begins by discussing properties of the IMLS shape function. Under appropriate assumption on weight functions, error estimates for the IMLS approximation are then established in Sobolev spaces in multiple dimensions. The improved element-free Galerkin (IEFG) method is a typical meshless Galerkin method based on coupling the IMLS approximation and Galerkin weak form. Error analysis of the IEFG method is also provided. Numerical examples are finally presented to prove the theoretical error results.