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On faster convergence of the bisection method for all triangles

Author: Martin Stynes
Journal: Math. Comp. 35 (1980), 1195-1201
MSC: Primary 51N99; Secondary 65B99, 65N30
MathSciNet review: 583497
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Abstract: Let $ \Delta ABC$ be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of $ \Delta ABC$, let D be the midpoint of AB, then replace $ \Delta ABC$ by two triangles $ \Delta ADC$ and $ \Delta DBC$.

Let $ {\Delta _{01}}$ be a given triangle. Bisect $ {\Delta _{01}}$ into two triangles $ {\Delta _{11}}$ and $ {\Delta _{12}}$. Next bisect each $ {\Delta _{1i}},\;i = 1,2$, forming four new triangles $ {\Delta _{2i}},\;i = 1,2,3,4$. Continue thus, forming an infinite sequence $ {T_j},\;j = 0,1,2, \ldots $, of sets of triangles, where $ {T_j} = \left\{ {{\Delta _{ji}}:1 \leqslant i \leqslant {2^j}} \right\}$. Let $ {m_j}$ denote the mesh of $ {T_j}$. It is shown that there exists $ N = N({\Delta _{01}})$ such that, for $ j \geqslant N$, $ {m_{2j}} \leqslant {(\sqrt 3 /2)^N}{(1/2)^{j - N}}{m_0}$, thus greatly improving the previous best known bound of $ {m_{2j}} \leqslant {(\sqrt 3 /2)^j}{m_0}$.

It is also shown that only a finite number of distinct shapes occur among the triangles produced, and that, as the method proceeds, $ {\Delta _{01}}$ tends to become covered by triangles which are approximately equilateral in a certain sense.

References [Enhancements On Off] (What's this?)

  • [1] R. B. KEARFOTT, "A proof of convergence and an error bound for the method of bisection in $ {R^n}$," Math. Comp., v. 32, 1978, pp. 1147-1153. MR 0494897 (58:13677)
  • [2] I. G. ROSENBERG & F. STENGER, "A lower bound on the angles of triangles constructed by bisecting the longest side," Math. Comp., v. 29, 1975, pp. 390-395. MR 0375068 (51:11264)
  • [3] M. STYNES, "On faster convergence of the bisection method for certain triangles," Math. Comp., v. 33, 1979, pp. 717-721. MR 521285 (80c:51020)
  • [4] M. STYNES, "Why Stenger's topological degree algorithm usually works in $ {R^3}$." (In preparation.)
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Article copyright: © Copyright 1980 American Mathematical Society

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