Adaptive radial basis function generated finite-difference (RBF-FD) on non-uniform nodes using p—refinement
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Radial basis functions-generated finite difference methods (RBF-FDs) have been gaining popularity recently. In particular, the RBF-FD based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly) has been found significantly effective. For the approximation order of RBF-FDs' weights on scattered nodes, one can already find mathematical theories in the literature. Many practical problems in numerical analysis, however, do not require a uniform node-distribution. Instead, they would be better suited if specific areas of the domain, where complicated physics needed to be resolved, had a relatively higher node-density compared to the rest of the domain. In this work, we proposed a practical adaptive RBF-FD with a user-defined order of convergence with respect to the total number of (possibly scattered and non-uniform) data points N. Our algorithm outputs a sparse differentiation matrix with the desired approximation order. Numerical examples are provided to show that the proposed adaptive RBF-FD method yields the expected N-convergence even for highly non-uniform node-distributions. The proposed method also reduces the number of non-zero elements in the linear system without sacrificing accuracy.
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