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

DOI:
https://doi.org/10.1090/S0025-5718-1980-0583497-1

MathSciNet review:
583497

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Abstract | References | Similar Articles | Additional Information

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.

- Baker Kearfott,
*A proof of convergence and an error bound for the method of bisection in ${\bf R}^{n}$*, Math. Comp.**32**(1978), no. 144, 1147–1153. MR**494897**, DOI https://doi.org/10.1090/S0025-5718-1978-0494897-3 - Ivo G. Rosenberg and Frank Stenger,
*A lower bound on the angles of triangles constructed by bisecting the longest side*, Math. Comp.**29**(1975), 390–395. MR**375068**, DOI https://doi.org/10.1090/S0025-5718-1975-0375068-5 - Martin Stynes,
*On faster convergence of the bisection method for certain triangles*, Math. Comp.**33**(1979), no. 146, 717–721. MR**521285**, DOI https://doi.org/10.1090/S0025-5718-1979-0521285-4
M. STYNES, "Why Stenger’s topological degree algorithm usually works in ${R^3}$." (In preparation.)
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*On $C^1$-complexes*, Ann. of Math. (2)**41**(1940), 809–824. MR**2545**, DOI https://doi.org/10.2307/1968861

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Article copyright:
© Copyright 1980
American Mathematical Society