  Radial basis function generated finite-difference (RBF-FD) methods have
recently gained popularity due to their flexibility with irregular node
distributions. However, the convergence theories in the literature, when
applied to nonuniform node distributions, require shrinking fill distance and
do not take advantage of areas with high data density. Non-adaptive approach
using same stencil size and degree of appended polynomial will have higher
local accuracy at high density region, but has no effect on the overall order
of convergence and could be a waste of computational power. This work proposes
an adaptive RBF-FD method that utilizes the local data density to achieve a
desirable order accuracy. By performing polynomial refinement and using
adaptive stencil size based on data density, the adaptive RBF-FD method yields
differentiation matrices with higher sparsity while achieving the same
user-specified convergence order for nonuniform point distributions. This
allows the method to better leverage regions with higher node density,
maintaining both accuracy and efficiency compared to standard non-adaptive
RBF-FD methods.
