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Zeitschrift für Analysis und ihre Anwendungen


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Volume 3, Issue 4, 1984, pp. 337–355
DOI: 10.4171/ZAA/112

Published online: 1984-08-31

Zur numerischen Bestimmung des Abbildungsgrades im $\mathbb R^n$ I

Wolfgang Kliesch

Two formulas computing the topological degree of a continuous function $\Phi$ relative to an $n$-dimensional polyhedron $P^n$ are presented. Both formulas are based on the same idea of construction. Let T be a triangulation of the boundary of $P^n$ and let f$(c)$ = sgn $\Phi (e), e \in$ E(T), be a simplicial mapping from T into a boundary triangulation of the $n$-dimensional unit cube, then the topological degree of the function $\Phi$ relative to $P^n$ is given by $$\mathrm {deg} (\Phi, \mathrm {int} \; P^n) = k^{-1} \sum_{\sigma \in \mathrm T_{n-1}} \mathrm {sgn \; d}(\sigma, \Phi),$$ $$\mathrm d(\sigma, \Phi := \mathrm {det} (\mathrm f (a^1) \dots \mathrm f (a^n)), \quad \sigma = [a^1 \dots a^n],$$ if T is oriented in a suitable manner.

The second computation formula is based on a simplicial mapping from T into the natural boundary triangulation of the $n$-dimensional unit octahedron.

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Kliesch Wolfgang: Zur numerischen Bestimmung des Abbildungsgrades im $\mathbb R^n$ I. Z. Anal. Anwend. 3 (1984), 337-355. doi: 10.4171/ZAA/112