Numerical integration of the Galerkin weak form in meshfree methods

The numerical integration of Galerkin weak forms for meshfree methods is investigated and some improvements are presented. The character of the shape functions in meshfree methods is reviewed and compared to those used in the Finite Element Method (FEM). Emphasis is placed on the relationship between the supports of the shape functions and the subdomains used to integrate the discrete equations. The construction of quadrature cells without regard to the local supports of the shape functions is shown to result in the possibility of considerable integration error. Numerical studies using the meshfree Element Free Galerkin (EFG) method illustrate the effect of these errors on solutions to elliptic problems. A construct for integration cells which reduces quadrature error is presented. The observations and conclusions apply to all Galerkin methods which use meshfree approximations.