# cite as¶

## .bib¶

```
@InProceedings{Zhang2021spline,
author="Zhang, Yi
and Jain, Varun
and Palha, Artur
and Gerritsma, Marc",
editor="van Brummelen, Harald
and Vuik, Cornelis
and M{\"o}ller, Matthias
and Verhoosel, Clemens
and Simeon, Bernd
and J{\"u}ttler, Bert",
title="The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms",
booktitle="Isogeometric Analysis and Applications 2018",
year="2021",
publisher="Springer International Publishing",
address="Cham",
pages="227--242",
abstract="In ℝn{\$}{\$} {\backslash}mathbb {\{}R{\}}^n {\$}{\$}, let $\Lambda$k({\thinspace}$\Omega$) represent the space of smooth differential k-forms in $\Omega$. The de Rham complex consists of a sequence of spaces, $\Lambda$k({\thinspace}$\Omega$), k{\thinspace}={\thinspace}0, 1{\ldots}, n, connected by the exterior derivative, d: $\Lambda$k({\thinspace}$\Omega$){\thinspace}{\textrightarrow} $\Lambda$k+1({\thinspace}$\Omega$). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.",
isbn="978-3-030-49836-8"
}
```

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