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Mathematics of Computation

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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 $ \Delta ABC$ be a triangle with vertices A, B and C. It is "bisected" as follows: choose a/the longest side (say AB) of $ \Delta ABC$, let D be the midpoint of AB, then replace $ \Delta ABC$ by two triangles, $ \Delta ADC$ and $ \Delta DBC$. Let $ {\Delta _{01}}$, be a given triangle. Bisect $ {\Delta _{01}}$, into two triangles $ {\Delta _{11}},{\Delta _{12}}$. Next, bisect each $ {\Delta _{1i}},i = 1,2$, forming four new triangles $ {\Delta _{2i}},i = 1,2,3,4$. Continue thus, forming an infinite sequence $ {T_j},j = 0,1,2, \ldots $, of sets of triangles, where $ {T_j} = \{ {\Delta _{ji}}:1 \leqslant i \leqslant {2^j}\} $. It is known that the mesh of $ {T_j}$ tends to zero as $ j \to \infty $. It is shown here that if $ {\Delta _{01}}$ 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.


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DOI: https://doi.org/10.1090/S0025-5718-1979-0521285-4
Article copyright: © Copyright 1979 American Mathematical Society