Numerical integration with constraints for meshless local petrov-galerkin methods

We propose numerical integration rules for meshless local Petrov- Galerkin methods (MLPG) employed to solve elliptic partial different equations (PDE) with Neumann boundary conditions. The integration rules are required to satisfy an integration constraint condition of Green's formula type (GIC). GIC was first developed in [Babuska, Banerjee, Osborn, and Zhang (2009)] for Galerkin meshless method, and we will show in this paper that it has better features for MLPG due to flexibility of MLPG in choosing different trial and test function spaces. A general constructive algorithm is presented to design the integration rules satisfying GIC. We also present a useful situation, where GIC holds automatically for Gaussian rules. According to this, we conclude that the conical weight is suggested to adopt in MLPG from viewpoint of reducing integration complexity. Approach to extending GIC in [Babuska, Banerjee, Osborn, and Zhang (2009)] to more general elliptic PDE, such as elasticity equation, is discussed. The 1D and 2D numerical results illuminate that GIC reduces the errors in the approximate solutions of MLPG significantly.