Applied Numerical Mathematics 55 (2005) 458–472 www.elsevier.com/locate/apnum Non-degeneracy study of the 8-tetrahedra longest-edge partition Angel Plaza a,∗ , Miguel A. Padrón b , José P. Suárez c a Department of Mathematics, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain b Department of Civil Engineering, ULPGC, 35017 Las Palmas de Gran Canaria, Spain c Department of Graphic Engineering, ULPGC, 35017 Las Palmas de Gran Canaria, Spain Available online 26 January 2005 Abstract In this paper we show empirical evidence on the non-degeneracy property of the tetrahedral meshes obtained by iterative application of the 8-tetrahedra longest-edge (8T-LE) partition. The 8T-LE partition of an initial tetra- hedron t yields an infinite sequence of tetrahedral meshes τ 1 ={t },τ 2 ={t 2 i },τ 3 ={t 3 i },... . We give numerical experiments showing that for a standard shape measure introduced by Liu and Joe (η), the non-degeneracy conver- gence to a fixed positive value is guaranteed, that is, for any tetrahedron t n i in τ n , n  1, η(t n i )  cη(t) where c is a positive constant independent of i and n. Based on our experiments, estimates of c are provided.  2005 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Mesh quality; Degeneracy; 8-tetrahedra longest-edge partition 1. Introduction Unstructured mesh generation and adaptive mesh refinement methods for two- and three-dimensional complex domains are very successful tools for the efficient solution of numerical application problems. A major drawback of these methods is that they may produce poorly shaped elements causing the nu- merical solution to be less accurate and more difficult to compute [1,19]. In [14] Rivara and Levin considered a pure three-dimensional longest-edge refinement method. Em- pirical experimentation was provided showing that the solid angle decreases slowly with the refinement iteration, and that a quality-element improvement behavior, holds in practice, as in the two-dimensional * Corresponding author. E-mail address: aplaza@dmat.ulpgc.es (A. Plaza). 0168-9274/$30.00  2005 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2004.12.003