Keywords: error analysis, domain discretization, meshless methods, meshless local Petrov-Galerkin method, diffuse approximate method, weak form, strong form, finite element method, diffusion equation.

The errors of the meshless local Petrov-Galerkin (MLPG) method, the diffuse approximate method (DAM) and the standard finite element method (FEM) are compared for application to the diffusion equation. The MLPG method is based on the local weak form. The shape functions are obtained by using a moving least squares (MLS) approximation on the local circular support domain with the thirteen closest nodes and a polynomial weight function of the fourth order. The test functions are similar to the MLS weight function but defined over the square quadrature domain, which enables local integration. The DAM is formulated in the local strong form. The support domain and nodal shape functions have been determined using the same MLS weight functions and with the same number of support nodes as in the MLPG. The methods are tested on uniform and non-uniform node arrangements on a Dirichlet jump problem with a square domain. The classical FEM method with second order shape functions is given for a comparative reference. The time integration is performed for both meshless methods in a fully implicit mode. The MLPG method involves numerical integration of the weak form that is performed using Gaussian integration with nine integration points while the DAM does not involve numerical integration because of its strong formulation. In the DAM the solution is known at the nodes only, whereas in MLPG, the smooth solution is known over the whole computational domain. It has been found that the MLPG method was in our test case slightly less accurate than the DAM with similar convergence, but much more complex to calculate as a result of the numerical integration. The FEM is more accurate than the DAM. We show that the MLPG errors arise as a result of the inaccurate numerical integration over the local domain in cases with inappropriate nodal distribution. As highly randomized nodes the MLPG method and the DAM solutions diverge, while the FEM works in all cases. Because of the DAM simplicity the error could be diminished by a four-fold increase in the number of discretization nodes.

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