Comparison of local weak and strong form meshless methods for 2-D diffusion equation

Abstract

A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.