Discretization coarsening for the accurate meshless solution of Poisson problems

Meshless methods are nowadays emerging, alternative or subsidiary techniques to classical Finite Element (FE) and Finite Difference (FD) methods, for the numerical solution of Partial Differential Equations (PDE). Among the huge number of proposed meshless methods, the Meshless Petrov–Galerkin (MLPG) class of methods is one of the most promising. Recently, the Direct MLPG (DMLPG) methods were added to the MLPG class. MLPG is a Generalized FE method, while DMLPG is a Generalized FD method. Notwithstanding elegant theoretical anal- ysis of meshless methods have been performed, effective, practical ap- plications rely upon numerical experiments. That is why our paper is focused on performing neat numerical experiments on simple test prob- lems. Adaptive methods are the most efficient for solving many prob- lems, and MLPG techniques are well apt for adaptivity. The adaptive methods for MLPG techniques that one can find in the literature are based upon intricate norm estimations. This paper aims at proposing a simple yet effective technique for coarsening a discretization cloud, by deleting only “useless” nodes, hence allowing for reducing the computa- tional cost without loosing accuracy. We analyze the effectiveness and efficiency of MLPG and DMLPG methods when coupled with our coarsening procedure. We point out some differences in the performances of these two methods.