Global-local Petrov-Galerkin formulations in the Meshless Finite Difference Method

Abstract

The paper presents the recent developments in both the Local Petrov- Galerkin (LPG) formulations of the boundary value problems of mechanics, and the Meshless Finite Difference Method MFDM of numerical analysis. The MLPG formulations use the well-known concept of the Petrov-Galerkin weak approach, where the test function may be different from the trial function. The support of such test function is limited to chosen subdomains, usually of regular shape, rather than to the whole domain. This significantly simplifies the numerical integration. MLPG discretization is performed here for the first time ever, in combination with the MFDM, the oldest and possibly the most developed meshless method. It is based on arbitrarily irregular clouds of nodes and moving weighted least squares approximation (MWLS), using here additional Higher Order correction terms. These Higher Order terms, originated from the Taylor series expansion, are considered in order to raise the local approximation rank in the most efficient manner, as well as to estimate both the a-posteriori solution and residual errors. Some new concepts of development of the original MLPG formulations are proposed as well. Several benchmark problems are analysed. Results of preliminary tests are very encouraging.