On faster convergence of the bisection method for all triangles
Math. Comp. 35 (1980), 1195-1201
Primary 51N99; Secondary 65B99, 65N30
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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.
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
(58 #13677), http://dx.doi.org/10.1090/S0025-5718-1978-0494897-3
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
(51 #11264), http://dx.doi.org/10.1090/S0025-5718-1975-0375068-5
Stynes, On faster convergence of the bisection
method for certain triangles, Math. Comp.
33 (1979), no. 146, 717–721. MR 521285
M. STYNES, "Why Stenger's topological degree algorithm usually works in ." (In preparation.)
H. C. Whitehead, On 𝐶¹-complexes, Ann. of Math.
(2) 41 (1940), 809–824. MR 0002545
- R. B. KEARFOTT, "A proof of convergence and an error bound for the method of bisection in ," Math. Comp., v. 32, 1978, pp. 1147-1153. MR 0494897 (58:13677)
- 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)
- M. STYNES, "On faster convergence of the bisection method for certain triangles," Math. Comp., v. 33, 1979, pp. 717-721. MR 521285 (80c:51020)
- M. STYNES, "Why Stenger's topological degree algorithm usually works in ." (In preparation.)
- J. H. C. WHITEHEAD, "On -complexes," Ann. of Math., v. 41, 1940, pp. 809-824. MR 0002545 (2:73d)
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