Triangulation of open sets in ℝ ⁿ

As triangulation open sets in ${\displaystyle \mathbb {R} ^{n}}$ certain simplicial decompositions are called of areas. One speaks therefore of a decomposition of open sets into ${\displaystyle \mathbb {R} ^{n}}$ . With the is -dimensional coordinate space with the real numbers meant as coordinates. Such triangulations are further classified and are particularly important in numerical calculation (such as the finite element method ). ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$

Allowable triangulation

Let be a domain , i.e. an open , connected subset. Next is a triangulation of , so a decomposition into simplices . ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {T}}=\left\{{\mathit {T}}_{1},{\mathit {T}}_{2},\dotsc ,{\mathit {T}}_{M}\right\}}$ ${\displaystyle \Omega }$

${\displaystyle {\mathcal {T}}}$is now called admissible if:

1. If the intersection consists of exactly one point, this point is a corner point from and from . ${\displaystyle {\mathit {T}}_{i}\cap {\mathit {T}}_{j}}$ ${\displaystyle {\mathit {T}}_{i}}$ ${\displaystyle {\mathit {T}}_{j}}$
2. If the intersection for consists of more than one point, then is an edge of and . ${\displaystyle {\mathit {T}}_{i}\cap {\mathit {T}}_{j}}$ ${\displaystyle i\neq j}$ ${\displaystyle {\mathit {T}}_{i}\cap {\mathit {T}}_{j}}$ ${\displaystyle {\mathit {T}}_{i}}$ ${\displaystyle {\mathit {T}}_{j}}$

Quasi-uniform triangulation

The family of triangulations is called quasi-uniform if there is a number such that holds for each . Here are half the diameter of and the inner diameter of the element . may have at most one diameter (where the grid width is). ${\displaystyle {\mathcal {T}}_{h}}$ ${\displaystyle \kappa >0}$ ${\displaystyle {\mathit {T}}\in {\mathcal {T}}_{h}}$ ${\displaystyle \kappa \geq {\tfrac {h_{T}}{\rho _{T}}}}$ ${\displaystyle h_{T}}$ ${\displaystyle {\mathit {T}}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle {\mathit {T}}}$ ${\displaystyle {\mathcal {T}}_{h}}$ ${\displaystyle 2h}$ ${\displaystyle h}$

Uniform triangulation

The family of triangulations is called uniform if there is a number such that holds for each . may have at most one diameter . ${\displaystyle {\mathcal {T}}_{h}}$ ${\displaystyle \kappa >0}$ ${\displaystyle {\mathit {T}}\in {\mathcal {T}}_{h}}$ ${\displaystyle \kappa \geq {\tfrac {h}{\rho _{T}}}}$ ${\displaystyle {\mathcal {T}}_{h}}$ ${\displaystyle 2h}$

Individual evidence

1. ↑ Wolfgang Arendt, Karsten Urban: Partial differential equations - An introduction to analytical and numerical methods . Spektrum Akademischer Verlag, 2010, ISBN 978-3-8274-2237-8 , pp. 298 , doi : 10.1007 / 978-3-8274-2237-8 ( springer.com ).
2. ↑ ^{a b c} Dietrich Braess: Finite element theory, fast solvers and applications in elasticity theory . 4th edition. Springer-Verlag Berlin Heidelberg, Berlin, Heiderberg 2007, ISBN 978-3-540-72450-6 , pp. 58 , doi : 10.1007 / 978-3-540-72450-6 .