Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients

In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k ≥ 1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green's formula, which is not problem dependent, i. e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green's formula.