A DMLPG Refinement Technique for 2D and 3D Potential Problems

Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques for solving Partial Differential Equations (PDE). MLPG techniques are nowadays used for solving a huge number of complex, real-life problems. While MLPG aims to approximate the solution of a given differential problem, its "dual" Direct MLPG (DMLPG) technique relies upon approximating linear functionals. Assume adaptive methods are to be implemented. When using a mesh-based method, inserting and/or deleting a node implies complex adjustment of connections. Meshless methods are more apt to implement adaptivity, since they does not require such adjustments. Nevertheless, ad-hoc insertion and/or deletion algorithms must be devised, in order to attain a good accuracy. In this paper we introduce a fresh refinement technique for DMLPG methods. Nodes are inserted in a discretization cloud where the local variation in the solution is supposed to be "large". The variation is estimated using the (local) Total Variation (TV). DMLPG allows to directly estimate the partial derivatives, in order to compute the TV. MLPG must rely upon approximating the derivatives of the shape functions, hence MLPG refinement results to be more involved than its DMLPG counterpart. We show that our DMLPG refinement procedure allows one to efficiently solve a given diffusion problem whose solution undergoes large variations on a small portion of the domain. The accuracy afforded by a fine uniform cloud can be attained by using far less non-uniformly arranged nodes.