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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.20056 |
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| _version_ | 1866912622125776896 |
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| author | Halada, Tomas Yaskovets, Serhii Singh, Abhinav Benes, Ludek Suchde, Pratik Sbalzarini, Ivo F. |
| author_facet | Halada, Tomas Yaskovets, Serhii Singh, Abhinav Benes, Ludek Suchde, Pratik Sbalzarini, Ivo F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Overview of Meshfree Collocation Methods Halada, Tomas Yaskovets, Serhii Singh, Abhinav Benes, Ludek Suchde, Pratik Sbalzarini, Ivo F. Numerical Analysis Computational Engineering, Finance, and Science 65-02, 65N06, 65N35 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. |
| title | An Overview of Meshfree Collocation Methods |
| topic | Numerical Analysis Computational Engineering, Finance, and Science 65-02, 65N06, 65N35 |
| url | https://arxiv.org/abs/2509.20056 |