An Improved Adaptive Meshless Refinement for the RBF-FD Method for 2D Elliptic Equations

In this paper, we improve the adaptive meshless discretization method to solve the 2D elliptic equations using the radial basis function (RBF) interpolation. To avoid the initial center being dependent on the original discretization of the finite element method (FEM), we introduce an initial center set based on a finite difference grid. To enhance efficiency, we introduce a strategy to determine the boundary candidate points for domains with complex geometries. We also show how to determine the error indicator threshold for the initial center set, and adjust this value at subsequent refinements. To ensure the candidate points corresponding to the edges with a higher error indicator value have a better chance of being accepted, we introduce a formula that determines each edge’s adaptive separation on tolerance value based on its error indicator value. Furthermore, we prioritize the refinements in descending order of local distances to make the distribution of the local centers more uniformly distributed. Our experiments show that the algorithm presented in this paper significantly improves our earlier algorithms in that the accuracy and stability of its approximation solutions are better for problems with sharp peaks and rapid oscillations in the neighborhood of an isolated point or with complex geometries.