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Appendix G - Node Sets for FD and RBF-FD-based PDE Discretizations

Published online by Cambridge University Press:  16 May 2025

Bengt Fornberg
Affiliation:
University of Colorado Boulder
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Summary

The two main types of node sets when solving PDEs are grid based and mesh free. Grid-based discretizations were traditionally the only option in FD contexts. In contrast, mesh-free methods do not require any direct node-to-node connections. The nodes can be placed irregularly, thereby offering simple opportunities for gradual local refinement in critical spatial areas. Finite element methods are in this sense not mesh free but rather based on irregular meshes (typically triangular or tetrahedral), as they require specific node-to-node connections in forming their elements. In the context of solving PDEs with RBF-FD methods, random and Halton node sets are excessively irregular, while quasi-uniform node sets typically give near-optimal accuracies. Several methods are available for creating such node sets, allowing for problem-specific node density variations over different regions. Effective methods are also available for subsampling such node sets, as needed for scattered-node counterparts to multigrid methods for solving elliptic-type PDEs.

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Publisher: Cambridge University Press
Print publication year: 2025

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