Journal
NUMERICAL COMPUTATIONS: THEORY AND ALGORITHMS (NUMTA-2016)
Volume 1776, Issue -, Pages -Publisher
AMER INST PHYSICS
DOI: 10.1063/1.4965351
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The Radial Basis Functions Orthogonal Gradients method (RBF-OGr) was introduced in [1] to discretize differential operators defined on arbitrary manifolds defined only by a point cloud. We take advantage of the meshfree character of RBFs, which give us a high accuracy and the flexibility to represent complex geometries in any spatial dimension. A large limitation of the RBF-OGr method was its large computational complexity, which greatly restricted the size of the point cloud. In this paper, we apply the RBF-Finite Difference (RBF-FD) technique to the RBF-OGr method for building sparse differentiation matrices discretizing continuous differential operators such as the Laplace-Beltrami operator. This method can be applied to solving PDEs on arbitrary surfaces embedded in R-3. We illustrate the accuracy of our new method by solving the heat equation on the unit sphere.
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