Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives

• Published in: Computers & mathematics with applications (1987) Vol. 147; pp. 1 - 13
• Main Authors: Jones, Andrew M., Bosler, Peter A., Kuberry, Paul A., Wright, Grady B.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Ltd 01-10-2023; Elsevier
• Subjects: GMLS; MATHEMATICS AND COMPUTING; Meshfree; Meshless; PDEs on surfaces; Polyharmonic spline; RBF-FD; RBF-FD; Polyharmonic spline; GMLS; PDEs on surfaces; Meshfree; Meshless
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2023.07.015

Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational cost, and they can be applied to surfaces defined only by point clouds. However, there have yet to be any studies that perform a direct comparison of these methods for approximating surface differential operators (SDOs). The first purpose of this work is to fill that gap. For this comparison, we focus on an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly) since they are most closely related to the GMLS method. Additionally, we use a relatively new technique for approximating SDOs with RBF-FD called the tangent plane method since it is simpler than previous techniques and natural to use with PHS+Poly RBF-FD. The second purpose of this work is to relate the tangent plane formulation of SDOs to the local coordinate formulation used in GMLS and to show that they are equivalent when the tangent space to the surface is known exactly. The final purpose is to use ideas from the GMLS SDO formulation to derive a new RBF-FD method for approximating the tangent space for a point cloud surface when it is unknown. For the numerical comparisons of the methods, we examine their convergence rates for approximating the surface gradient, divergence, and Laplacian as the point clouds are refined for various parameter choices. We also compare their efficiency in terms of accuracy per computational cost, both when including and excluding setup costs.
•Show the equivalence of the surface differential operators formulation used in MLS methods and a new one based only on the tangent plane.•Develop a new RBF-FD method for approximating the tangent space of surfaces defined only by point clouds.•Perform the first comparison of GMLS and RBF-FD for approximating derivatives on surfaces defined by point clouds.