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 | References | Similar Articles | Additional Information

Abstract: Let be a triangle with vertices *A, B*, and *C*. It is "bisected" as follows: choose a/the longest side (say *AB*) of , let *D* be the midpoint of *AB*, then replace by two triangles and .

Let be a given triangle. Bisect into two triangles and . Next bisect each , forming four new triangles . Continue thus, forming an infinite sequence , of sets of triangles, where . Let denote the mesh of . It is shown that there exists such that, for , , thus greatly improving the previous best known bound of .

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

**[1]**Baker Kearfott,*A proof of convergence and an error bound for the method of bisection in 𝑅ⁿ*, Math. Comp.**32**(1978), no. 144, 1147–1153. MR**0494897**, 10.1090/S0025-5718-1978-0494897-3**[2]**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**0375068**, 10.1090/S0025-5718-1975-0375068-5**[3]**Martin Stynes,*On faster convergence of the bisection method for certain triangles*, Math. Comp.**33**(1979), no. 146, 717–721. MR**521285**, 10.1090/S0025-5718-1979-0521285-4**[4]**M. STYNES, "Why Stenger's topological degree algorithm usually works in ." (In preparation.)**[5]**J. H. C. Whitehead,*On 𝐶¹-complexes*, Ann. of Math. (2)**41**(1940), 809–824. MR**0002545**

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

Article copyright:
© Copyright 1980
American Mathematical Society