Numerical investigation on direct MLPG for 2D and 3D potential problems

Pure meshless techniques are promising methods for solving Partial Differential Equations (PDE). They alleviate difficulties both in designing discretization meshes, and in refining/coarsening, a task which is demanded e.g. in adaptive strategies. Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques that receive increasing attention. Very recently, new methods, called Direct MLPG (DMLPG), have been proposed. They rely upon approximating PDE via the Generalized Moving Least Square method. DMLPG methods alleviate some difficulties of MLPG, e.g. numerical integration of tricky, non-polynomial factors, in weak forms. DMLPG techniques require lower computational costs respect to their MLPG counterparts. In this paper we numerically analyze the solution of test 2D problems via DMLPG. We report about our expansion of meshless techniques to 3D problems. Finally, we perform comparisons between DMLPG and two MLPG techniques, when solving 3D problems.