MLPG Formulation of the Multipoint Meshless FDM

Discussed is here the multipoint meshless finite difference method (MMFDM) following the original Collatz [2] multipoint concept, and the essential ideas of the meshless FDM [3]. The Collatz approach was based on interpolation, regular meshes and the local formulation of b.v. problems. On the other hand in the MFDM we deal with the moving weighted least squares (MWLS) approximation, arbitrarily irregular cloud of nodes, and arbitrary global or local formulations of b.v. problems analysed. The new multipoint method [4,5,6] improves the FD solution without increasing the number of nodes in the domain. Two versions of the multipoint MFDM, namely general and specific are considered. Further extension of the multipoint MFDM is developed here. It allows for analysis of b.v. problems posed in the meshless local Petrov-Galerkin (MLPG) formulations. Several versions of the multipoint method in MLPG formulation are proposed and examined, especially the MLPG5 one, reducing amount of calculations involved. Results of tests done for the MLPG formulations, and their comparison with those obtained for the local or global ones are encouraging. Further research is planned.