Corrigendum to A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes [Applied Mathematics and Computation 219 (4) (2012) 2342-2344]

Abstract

© 2015 Elsevier Inc. In the article [Applied Mathematics and Computation 219 (4) (2012) 2342-2344] there exists a minor error in the case n = 4. We correct the error and give a proof for the case n = 4. The argument in reference [1] used that “Since [Formula prsented] then [Formula prsented]”. It should be noted that the later inequality does not hold. Notice that [Formula prsented]. So, if [Formula prsented] then [Formula prsented]. Even though, the subsequent argument in [1] is correct for the case n > 4. But, the case n = 4 needs a closer look. Let us consider the semi-circle passing through point z and tangent to the real axis at point z = 1/2, see Fig. 1. Let r be the radius of the circle, so its center is at point [Formula prsented]. Then the equation of the circle is [Formula prsented] that is [Formula prsented]. Now, using that [Formula prsented] it follows[Formula prsented] This circle is invariant under the action of the Moebius transform [Formula prsented]: If we change [Formula prsented] in Eq. (1), it is obtained: [Formula prsented]where after clearing denominators we have [Formula prsented] The Moebius transform [Formula prsented] has a unique fixed point, [Formula prsented]. The sequence {z, w(z), w 2 (z), …} is on the semicircle and {Im z, Im w(z), Im w 2 (z), …} is decreasing. Hence it has an accumulation point which corresponds to the fixed point [Formula prsented]. This completes the argument of the paper [1] for the case n = 4.