MLPG Refinement Techniques for 2D and 3D Diffusion Problems

Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques for solving Partial Differential Equations. One of pure meshless methods main applications is for implementing Adaptive Discretization Techniques. In this paper, we describe our fresh node-wise refinement technique, based upon estimations of the "local" Total Variation of the approximating function. We numerically analyze the accuracy and efficiency of our MLPG-based refinement. Solutions to test Poisson problems are approximated, which undergo large variations inside small portions of the domain. We show that 2D problems can be accurately solved. The gain in accuracy with respect to uniform discretizations is shown to be appreciable. By extending our procedure to 3D problems, we prove by experiments that good improvements in efficiency can be obtained.