Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation

The Meshless Local Petrov–Galerkin (MLPG) method is one of the popular meshless methods that has been used very successfully to solve several types of boundary value problems since the late nineties. In this paper, using a generalized moving least squares (GMLS) approximation, a new direct MLPG technique, called DMLPG, is presented. Following the principle of meshless methods to express everything “entirely in terms of nodes”, the generalized MLS recovers test functionals directly from values at nodes, without any detour via shape functions. This leads to a cheaper and even more accurate scheme. In particular, the complete absence of shape functions allows numerical integrations in the weak forms of the problem to be done over low-degree polynomials instead of complicated shape functions. Hence, the standard MLS shape function subroutines are not called at all. Numerical examples illustrate the superiority of the new technique over the classical MLPG. On the theoretical side, this paper discusses stability and convergence for the new discretizations that replace those of the standard MLPG. However, it does not treat stability, convergence, or error estimation for the MLPG as a whole. This should be taken from the literature on MLPG.