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.

翻译：定义在二维曲面上的微分算子逼近是科学与工程多个领域中的重要问题。近十年来，基于广义移动最小二乘法（GMLS）和径向基函数有限差分法（RBF-FD）的局部无网格方法已被证明能高效解决该问题，它们能以较低计算成本实现高阶精度，且适用于仅由点云定义的曲面。然而，目前尚未有研究直接比较这两种方法在逼近曲面微分算子（SDOs）时的表现。本文的首要目的即是填补这一空白。在比较中，我们重点关注基于多调和样条核与多项式（PHS+Poly）的RBF-FD方法，因其与GMLS方法关联最为紧密。同时采用一种名为切平面法的RBF-FD新技术逼近SDOs，该法较以往技术更简洁，且天然适用于PHS+Poly RBF-FD。本文的第二个目的是建立RBF-FD中SDOs的切平面表述与GMLS中使用的局部坐标表述之间的关联，并证明当曲面切空间精确已知时二者等价。最终目的则是借鉴GMLS-SDO表述的思想，推导一种新的RBF-FD方法，用于在曲面切空间未知时逼近点云曲面的切空间。在数值方法比较中，我们考察了不同参数选择下点云加密时曲面梯度、散度和拉普拉斯算子逼近的收敛速率，并进一步在计入与不计入初始化成本两种情形下，比较了两种方法在单位计算成本下的精度效率。