The D-RBF-PU method for solving surface PDEs - Mir, Reyhaneh; Mirzaei, Davoud (Journal of computational physics, 2023)
In this paper a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces which are discretized with sets of scattered nodes and with approximations to normal vectors at each of the nodes. The accuracy, stability and efficiency of the new method are studied through some theoretical and experimental results. This method is a localized RBF based technique, results in a perfectly sparse final linear system, uses only scattered nodes on the surface rather than a connected mesh, and is applicable for a large class of PDEs on manifolds. Applications to some biological and chemical reaction-diffusion models are also given. Results show that the new method outperforms other comparable techniques for surface PDEs. •Development of a novel localized RBF-based method for solving surface PDEs.•Being based on scattered nodes instead of a connected mesh.•Overcoming a major disadvantage of previous methods for handling the surface derivatives of partition of unity weights.•Using a larger class of weights leading to more accurate and faster algorithms.•Observing average speedups of 10x compared to well-known available techniques.