A Meshfree Collocation Scheme for Surface Differential Operators on Point Clouds

We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on discretization-corrected particle strength exchange (DC-PSE), whi...

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Bibliographic Details
Published in:Journal of scientific computing 2023-09, Vol.96 (3), p.89, Article 89
Main Authors: Singh, Abhinav, Foggia, Alejandra, Incardona, Pietro, Sbalzarini, Ivo F.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Collocation
Computational Mathematics and Numerical Analysis
Differential equations
Divergence
Fields (mathematics)
Finite difference method
Finite element method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Meshless methods
Methods
Neighborhoods
Numerical analysis
Operators (mathematics)
Partial differential equations
Scalars
Theoretical
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Summary:We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on discretization-corrected particle strength exchange (DC-PSE), which generalizes finite difference methods to meshfree point clouds. The proposed Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along surface normals to obtain a purely intrinsic computational scheme over surface point clouds. We benchmark Surface DC-PSE by discretizing the Laplace–Beltrami operator on a circle and a sphere, and we present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing Gauss and mean curvature of an ellipsoid and of the Stanford Bunny by approximating the intrinsic divergence of the normal vector field. Finally, we compare Surface DC-PSE with surface finite elements (SFEM) and diffuse-interface finite elements (DI FEM) in a validation case.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-023-02313-3