An Approximation of the Error in Element Free Galerkin Method

The diffuse element method developed by Nayroles et al. is a new way for solving partial differential equations. In this method, only a mesh of nodes and a boundary description is needed to develop the Galerkin equations. The approximating functions are polynomials fitted to the nodal values of each local domain by a weighted least squares approximation. Belytschko et al. developed an alternative implementation using moving least squares approximation as were defined by Lancaster and Salkauskas. They called their approach the Element Free Galerkin (EFG) method. In their work, Belytschko and his coworkers have introduced a background cell structure in order to carry out integration by numerical quadrature and Lagrange multipliers to enforce essential boundary conditions. Liu et al. has recently proposed a different kind of "griddles" multiple scale methods based on reproducing kernel and wavelet analysis (RPKM method). Oñate et al. focused on the application to fluid flow problems with a standard point collocation technique. All these methods can be considered as Finite Point or Meshless Methods. In this paper we present a new approximation of the error for Element Free Galerkin (EFG) method, whose evaluation is computationally so simple that it can be readily implemented in existing EFG codes. The estimator allows the global energy norm error to be well apprroximated and also gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for grid redesign which allows the user to obtain a desired accuracy. L. Gavete, S. Falcón, and J.C.Bellido 2