Enhanced Adaptive RBF-FD Methods for 2D Elliptic Equations with New Thresholding Strategies

Abstract

In this paper, we present strategies for determining the error indicator threshold value at each refinement step. The error indicator threshold is not directly computed from the maximum error indicator, improving refinement efficiency. Moreover, to enhance refinement efficiency, we propose three new candidate center structures, which include a greater number of centers and may double the distribution density. Additionally, we improve the variable support size algorithm for making it more flexible. Moreover, we introduce a measure to evaluate the average recursive preprocessing cost per new center generated by the adaptive RBF-FD (Radial Basis Function-Finite Difference) meshless method. This metric is used to assess the preprocessing cost efficiency of the adaptive RBF-FD meshless method and to compare it with our previously published adaptive RBF-FD methods for solving 2D elliptic equations. The results demonstrate that the numerical solutions obtained using the methods proposed in this paper are significantly more accurate, more stable, and more effective in terms of refinement than those from our earlier studies and the adaptive FEM (Finite Element Method), while achieving the lowest computational cost.