Monomial augmentation guidelines for RBF-FD from accuracy vs. computational time perspective

Abstract: Local meshless methods using RBFs augmented with monomials have become increasingly popular, due to the fact that they can be used to solve PDEs on scattered node sets in a dimension-independent way, with the ability to easily control the order of the method, but at a greater cost to execution time. We analyze this ability on a Poisson problem with mixed boundary conditions in 1D, 2D and 3D, and reproduce theoretical convergence orders practically, also in a dimension-independent manner, as demonstrated with a solution of Poisson's equation in an irregular 4D domain. The results are further combined with theoretical complexity analyses and with conforming execution time measurements, into a study of accuracy vs. execution time trade-off for each dimension. Optimal regimes of order for given target accuracy ranges are extracted and presented, along with guidelines for generalization.