On faster convergence of the bisection method for certain triangles

Author:
Martin Stynes

Journal:
Math. Comp. **33** (1979), 717-721

MSC:
Primary 51N99; Secondary 41A63

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521285-4

MathSciNet review:
521285

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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 . Next, bisect each , forming four new triangles . Continue thus, forming an infinite sequence , of sets of triangles, where . It is known that the mesh of tends to zero as . It is shown here that if satisfies any of four certain properties, the rate of convergence of the mesh to zero is much faster than that predicted by the general case.

**[1]**G. DAHLQUIST & A. BJORCK,*Numerical Methods*, translated by N. Anderson, Prentice-Hall, Englewood Cliffs, N. J., 1974. MR**0368379 (51:4620)****[2]**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)****[3]**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)****[4]**M. STYNES, "An algorithm for numerical calculation of topological degree,"*Applicable Anal.*(To appear.) MR**536692 (82g:55001)****[5]**M. STYNES, "On the construction of sufficient refinements for computation of topological degree,"*Numer. Math.*(Submitted.) MR**627117 (82i:55001)**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0521285-4

Article copyright:
© Copyright 1979
American Mathematical Society