Notation

Due to the wide variety of material in this work drawn from different fields in scientific computing, some conflicts or ambiguitites in the notation cannot be avoided. The following table summarized the used notations:

$\mathbb{D}^n$	$n$ -dimensional closed unit ball $\{x \in \mathbb{R}^{n} \vert \Vert x\Vert \leq 1\}$
$\mathbb{S}^n$	$n$ -dimensional unit sphere $\{x \in \mathbb{R}^{n+1} \vert \Vert x\Vert=1\}$
$\langle a , b \rangle$	scalar product ${\ensuremath{\varsigma}}$ of $a,b$
${\ensuremath{\mathbb{F}}}$	field, e.g., $\mathbb{R}$
${\ensuremath{\mathcal{V}}}$	vector space
${\ensuremath{\mathcal{V}^*}}$	dual vector space
$({\ensuremath{X}}, \mathcal{T})$	topological space
${\ensuremath{\mathfrak{M}}}$	manifold
${\ensuremath{\Omega}}$	domain, subset of $\mathbb{R}^n$
${\ensuremath{\Gamma}}$	boundary of a domain $\partial {\ensuremath{\Omega}}$
${\ensuremath{\mathfrak{K}}}$	cell complex
$\mathcal{L}$	linear operator
${\ensuremath{\mathcal{T_P}}}$	tangent space
${\ensuremath{\mathcal{T_P}^*}}$	cotangent space
$O$	(globally) ordered set
$P$	partially ordered set (poset)
$(B\times F,B, pr_1)$	trivial fiber bundle
$(F,B, pr_1)$	fiber bundle
$\mathcal{T}({\ensuremath{\mathfrak{M}}})$	tangent bundle
$C_j^i$	connection matrix
$\ensuremath{\tau_{p}^{i}}$	$i$ -th $p$ -cell
$c_p$	$p$ -chain
$c^p$	$p$ -cochain
$C_*$	chain complex
$C^*$	cochain complex
$\partial$	boundary operator, partial derivative
$\delta$	coboundary operator
$\ensuremath{\mathbf{E}}$	electric field
$\ensuremath{\mathbf{H}}$	magnetic field
$\ensuremath{\mathbf{D}}$	electric flux
$\ensuremath{\mathbf{B}}$	magnetic flux
$\ensuremath{\mathbf{J}}$	current flux density vector
$\ensuremath{\mathbf{A}}$	magnetic vector potential
$\varepsilon$	dielectric permittivity
$\mu$	magnetic permeability
$\varrho$	electrical charge density
${\ensuremath{\phi}}$	basis function for finite elements
${\ensuremath{\mathrm{u}}}$	generic solution quantity