On faster convergence of the bisection method for certain triangles
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- by Martin Stynes PDF
- Math. Comp. 33 (1979), 717-721 Request permission
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.References
- Ȧke Björck and Germund Dahlquist, Numerical methods, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson. MR 0368379
- Baker Kearfott, A proof of convergence and an error bound for the method of bisection in $\textbf {R}^{n}$, Math. Comp. 32 (1978), no. 144, 1147–1153. MR 494897, DOI 10.1090/S0025-5718-1978-0494897-3
- Ivo 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 375068, DOI 10.1090/S0025-5718-1975-0375068-5
- Martin Stynes, An algorithm for numerical calculation of topological degree, Applicable Anal. 9 (1979), no. 1, 63–77. MR 536692, DOI 10.1080/00036817908839252
- Martin Stynes, On the construction of sufficient refinements for computation of topological degree, Numer. Math. 37 (1981), no. 3, 453–462. MR 627117, DOI 10.1007/BF01400322
Additional Information
- © Copyright 1979 American Mathematical Society
- 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