Properties of the longest-edge n-section refinement scheme for triangular meshes

The stability condition or non-degeneracy property means that the interior angles of all elements have to be bounded uniformly away from zero. Non-degeneracy is essential, for example, for the approximation properties of finite element spaces and the convergence behavior of multigrid and multilevel algorithms. Rosenberg and Stenger [1] showed the non-degeneracy property for LE-bisection: if α0 is the minimum angle of initial given triangle, and αk is the minimum interior angle in new triangles appeared at iteration k, then αk > α0/2. A similar bound has been obtained recently for the LE-trisection: αk > α0/c where c = π/3 arctan( √ 3 11 ) [2]. Theorem 1. The iterative application of longest-edge n-section when n > 4 to a given arbitrary triangle △ABC generates a sequence of new triangles in which limk→∞ αk = 0, αk being the minimum triangle angle in iteration k. Proof. It is enough to prove that there exists a sequence {τk}∞k=0 such that: (1) τk is the value of the interior angle obtained after kth iteration of the LE n-section of the given triangle △ABC . (2) limk→∞ τk = 0. In fact, for all k > 1 we have αk 6 τk, then: 0 6 limk→∞ αk 6 limk→∞ τk = 0, where, clearly, limk→∞ αk = 0. We now prove that there exists such a sequence {τk}∞k=0. Let n > 4 and △ABC be an arbitrary triangle with sides |AB| 6 |AC | 6 |BC |. We consider a triangle sequence {∆k}∞k=0 such that ∆0 = △A0B0C0, A0 = A, B0 = B, C0 = C . For all k > 0 let ∆k+1 = △Ak+1Bk+1Ck+1 where Ak+1 ∈ BkCk such that |Ak+1Ck| = n |BkCk|, Bk+1 = Ck and Ck+1 = Ak. It should be noted that for all k > 1, |AkBk| 6 |AkCk| < |BkCk| and that ∆k is one of the triangles generated by applying the LE n-section to triangle ∆k−1. ∗ Correspondence to: University of Las Palmas de Gran Canaria, Escuela de Ingenierías Industriales y Civiles, 35017-Las Palmas de Gran Canaria, Spain. Tel.: +34 928 45 72 68; fax: +34 928 45 18 72. E-mail address: [email protected] (J.P. Suárez). 0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.04.014 2038 J.P. Suárez et al. / Applied Mathematics Letters 25 (2012) 2037–2039 Fig. 1. Scheme for the constructed triangle sequence in the LE n-section.