Quadrature in meshless methods for general elliptic problems with natural boundary conditions

In this paper, we explore the effect of numerical integration on the meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered 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 from the meshless method under the presence of numerical integration. This result was established under the assumption that the numerical integration rule satisfied 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 in two dimensions satisfying the discrete Green’s formula.