Meshfree, kernel-based spatial discretisations are recent tools to discretise partial differential equations on surfaces. The goals of this paper are to analyse and compare three different meshfree kernel-based methods for the spatial discretisation of semi-linear parabolic partial differential equations (PDEs) on surfaces, i.e. on smooth, compact, connected, orientable, and closed (d − 1)-dimensional submanifolds of \(\mathbb {R}^{d}\). The three different methods are collocation, the Galerkin, and the RBF-FD method, respectively. Their advantages and drawbacks are discussed, and previously known theoretical results are extended and numerically verified. Finally, a significant part of this paper is devoted to solving PDEs on evolving surfaces with RBF-FD, which has not been done previously.

中文翻译：

---

用径向基函数求解（演化的）表面上的偏微分方程

基于网格的无网格空间离散化是用于离散化曲面上偏微分方程的最新工具。本文的目标是分析和比较用于半线性抛物线偏微分方程（PDE的）上表面上，即在平滑，结构紧凑，连接，定向，和封闭（在空间离散化三个不同的无网格的基于内核的方法d - 1）\（\ mathbb {R} ^ {d} \）的维子流形。三种不同的方法分别是并置，Galerkin和RBF-FD方法。讨论了它们的优缺点，并扩展了先前已知的理论结果并进行了数值验证。最后，本文的重要部分致力于用RBF-FD求解演化表面上的PDE，这是以前没有做过的。