Title

Fast RBF OGr for solving PDEs on arbitrary surfaces

Authors

Cecile M. Piret, Michigan Technological UniversityFollow
Jarrett Dunn, Michigan Technological University

Document Type

Article

Publication Date

10-2016

Abstract

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 ℛ³. We illustrate the accuracy of our new method by solving the heat equation on the unit sphere.

Publication Title

AIP Conference Proceedings

Recommended Citation

Piret, C. M., & Dunn, J. (2016). Fast RBF OGr for solving PDEs on arbitrary surfaces. AIP Conference Proceedings, 1776(070005). http://doi.org/10.1063/1.4965351
Retrieved from: https://digitalcommons.mtu.edu/math-fp/8