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.
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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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