# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## On the shape of tetrahedra from bisectionHTML articles powered by AMS MathViewer

by Anwei Liu and Barry Joe
Math. Comp. 63 (1994), 141-154 Request permission

## Abstract:

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes ${\mathcal {T}^0},{\mathcal {T}^1},{\mathcal {T}^2}, \ldots$, which has the following properties: (1) Each mesh ${\mathcal {T}^n}$ is conforming. (2) There are a finite number of classes of similar tetrahedra in all the ${\mathcal {T}^n},n \geq 0$. (3) For any tetrahedron ${\mathbf {T}}_i^n$ in ${\mathcal {T}^n},\eta ({\mathbf {T}}_i^n) \geq {c_1}\eta ({\mathbf {T}})$, where $\eta$ is a tetrahedron shape measure and ${c_1}$ is a constant. (4) $\delta ({\mathbf {T}}_i^n) \leq {c_2}{(1/2)^{n/3}}\delta ({\mathbf {T}})$, where $\delta ({\mathbf {T’}})$ denotes the diameter of tetrahedron ${\mathbf {T’}}$ and ${c_2}$ is a constant. Estimates of ${c_1}$ and ${c_2}$ are provided. Properties (2) and (3) extend similar results of Stynes and Adler, and of Rosenberg and Stenger, respectively, for the 2-D case. The diameter bound in property (4) is better than one given by Kearfott.
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