Adaptive radial basis function generated finite-difference on non-uniform nodes using $p$-refinement

Radial basis function generated finite-difference (RBF-FD) methods have been gained popularity recently. For the approximation order of RBF-FDs' weights on scattered nodes, one can find mathematical theories in the literature. Many practical problems in numerical analysis, however, do not have a uniform node-distribution. Instead, it would be better suited if a relatively higher node-density is imposed on specific areas of domain where complicated physics neeeded to be resolved. In this paper, we propose a practical adaptive RBF-FD with a user defined convergence order with respect to the total number data points $N$. Our algorithm can output a sparse differentiation matrix system with the desired approximation order. Numerical examples about elliptic and parabolic equations are provided to show that the proposed adaptive RBF-FD method yields the expected convergence order. The proposed method reduces the number of non-zero elements in the linear system without sacrificing the accuracy. Furthermore, we apply our adaptive RBFFD method to the elastic wave models and obtain the desired convergence order.