Fast RBF OGr for solving PDEs on arbitrary surfaces
The Radial Basis Functions Orthogonal Gradients method (RBF-OGr) was introduced in  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 ℛ3. We illustrate the accuracy of our new method by solving the heat equation on the unit sphere.
AIP Conference Proceedings
Piret, Cecile M. and Dunn, Jarrett, "Fast RBF OGr for solving PDEs on arbitrary surfaces" (2016). Department of Mathematical Sciences Publications. 8.