On faster convergence of the bisection method for all triangles
Author:
Martin Stynes
Journal:
Math. Comp. 35 (1980), 11951201
MSC:
Primary 51N99; Secondary 65B99, 65N30
MathSciNet review:
583497
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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.
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Baker
Kearfott, A proof of convergence and an error
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Stynes, On faster convergence of the bisection
method for certain triangles, Math. Comp.
33 (1979), no. 146, 717–721. MR 521285
(80c:51020), http://dx.doi.org/10.1090/S00255718197905212854
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M. STYNES, "Why Stenger's topological degree algorithm usually works in ." (In preparation.)
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J.
H. C. Whitehead, On 𝐶¹complexes, Ann. of Math.
(2) 41 (1940), 809–824. MR 0002545
(2,73d)
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 R. B. KEARFOTT, "A proof of convergence and an error bound for the method of bisection in ," Math. Comp., v. 32, 1978, pp. 11471153. 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. 390395. MR 0375068 (51:11264)
 [3]
 M. STYNES, "On faster convergence of the bisection method for certain triangles," Math. Comp., v. 33, 1979, pp. 717721. MR 521285 (80c:51020)
 [4]
 M. STYNES, "Why Stenger's topological degree algorithm usually works in ." (In preparation.)
 [5]
 J. H. C. WHITEHEAD, "On complexes," Ann. of Math., v. 41, 1940, pp. 809824. MR 0002545 (2:73d)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005834971
PII:
S 00255718(1980)05834971
Article copyright:
© Copyright 1980
American Mathematical Society
