An Overview of Meshfree Collocation Methods

TL;DR

This survey presents a unifying mathematical framework for meshfree collocation methods used to approximate differential operators on irregular point clouds. It classifies approaches into four derivation families—approximation of moments, $\ell_2$ minimization of the discretization error, $\ell_2$ minimization of weights, and generalized $\ell_2$ minimization—and demonstrates how many well-known methods (e.g., MLS, GFDM, DC-PSE, RKCM, LABFM) fit within a common operator form $L_i^{\bm{\alpha}} u = \sum_{j\in\mathcal{N}_i} u_j w_{ji}^{\bm{\alpha}}$ with consistency up to order $m$. The paper emphasizes a unifying matrix notation, clarifies historical connections, and highlights practical implications such as stencil compactness, conditioning, and boundary treatment. By drawing formal links among seemingly disparate techniques, it enables transfer of theoretical insights and numerical results across method families and outlines a generalized path forward for meshfree collocation theory. The compilation also surveys software implementations and prior reviews, providing a resource to practitioners for selecting and implementing appropriate methods. Overall, the work bridges communities and sets a foundation for future generalized derivations and cross-method innovations.

Abstract

We provide a comprehensive overview of meshfree collocation methods for numerically approximating differential operators on continuously labeled unstructured point clouds. Meshfree collocation methods do not require a computational grid or mesh. Instead, they approximate smooth functions and their derivatives at potentially irregularly distributed collocation points, often called particles, to a desired order of consistency. We review several meshfree collocation methods from the literature, trace the historical development of key concepts, and propose a classification of methods according to their principle of derivation. Although some of the methods reviewed are similar or identical, there are subtle yet important differences between many, which we highlight and discuss. We present a unifying formulation of meshfree collocation methods that renders these differences apparent and show how each method can be derived from this formulation. Finally, we propose a generalized derivation for meshfree collocation methods going forward.