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Why Do We Need Voronoi Cells and Delaunay Meshes? Essential Properties of the Voronoi Finite Volume Method

Abstract

Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the geometry of a problem and are able to preserve the essential qualitative properties of the solution for any given resolution in space and time as well as changes in time scales of multiple orders of magnitude. This work provides a brief description of the essential and useful properties of the Voronoi FVM, application examples, and a motivation why Voronoi FVM deserve to be used more often in practice than they are currently.

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Notes

  1. In the case of a degenerate tetraheder (d + 1 vertices on a common empty d – 1 hypersphere, the so-called “sliver”), Delaunay’s perturbation argument (Proposition 1 in [6]) can be used to effectively forbid its appearance in a triangulation.

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Gärtner, K., Kamenski, L. Why Do We Need Voronoi Cells and Delaunay Meshes? Essential Properties of the Voronoi Finite Volume Method. Comput. Math. and Math. Phys. 59, 1930–1944 (2019). https://doi.org/10.1134/S096554251912008X

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  • DOI: https://doi.org/10.1134/S096554251912008X

Keywords:

  • finite volume method
  • boundary conforming Delaunay triangulation
  • Voronoi cells