List of Abbreviations

${\mathbb{R}}$The real numbers
${\mathbb{R}}^n$An $n$-dimensional vector space over the real numbers
$a$A scalar
$\bm{a}$A vector, point, or vertex of ${\mathbb{R}}^n$
$\bm{0}$The zero vector or the origin

$f$A function
$\operatorname{img}(f)$The image of a function
$\operatorname{dom}(f)$The domain of a function

$A$A subset of ${\mathbb{R}}^n$
${\mathcal{P}}(A)$The power set of a set
${\mathcal{P}}^\star(A)$The power set of a set without the empty set: ${\mathcal{P}}^\star(A) = {\mathcal{P}}(A) \setminus \emptyset$
$\operatorname{cl}(A)$The closure of a set (cf. Definition A.1)
$\operatorname{int}(A)$The interior of a set (cf. Definition A.1)
${\operatorname{bnd}}(A)$The boundary of a set (cf. Definition A.1)
$\operatorname{int}^\star (A)$The relative interior of a set (cf. Definition A.8)
${\operatorname{bnd}}^\star (A)$The relative boundary of a set (cf. Definition A.8)
${\operatorname{DIM}}(X)$The dimension of a set (cf. Definition A.6)

$\mathbb{H}_{\bm{n},d}$A hyperplane with normal vector $\bm{n}$ and distance to origin $d$ (cf. Definition A.5)
$\overline{\mathcal{B}}_r^n(\bm{x})$An $n$-ball with radius $r$ and center $\bm{(}x)$ (cf. Definition A.3)
${\mathcal{B}}_r^n(\bm{x})$An open $n$-ball with radius $r$ and center $\bm{(}x)$ (cf. Definition A.3)
${\mathcal{H}}_r^n(\bm{x})$An $n$-half-ball with radius $r$ and center $\bm{(}x)$ (cf. Definition A.3)

${\mathcal{G}}$A geometry (cf. Definition 2.1)
${({\mathcal{G}}, {\widetilde{\xi}})}$A multi-region geometry (cf. Definition 2.2)
${\mathcal{E}}$An element space (cf. Definition A.6)
${\mathcal{M}}$A mesh (cf. Definition 2.4)
${({\mathcal{M}}, {\xi})}$A multi-region mesh (cf. Definition 2.6)
${\operatorname{elem}}_k({\mathcal{E}})$The set of all $k$-dimensional elements of an element space (cf. Definition A.7)
${\operatorname{facets}}({\mathcal{E}})$The set of all facets of an element space (cf. Definition A.7)
${\operatorname{facets}}_{\mathcal{E}}(E)$The set of all facets of an element (based on an element space) (cf. Definition A.9)
${\operatorname{faces}}_{\mathcal{E}}(E)$The set of all faces of an element (based on an element space) (cf. Definition A.9)

${\operatorname{aff}}(X)$The affine hull of a set of points (cf. Definition A.15)
${\operatorname{conv}}(X)$The convex hull of a set of points (cf. Definition A.16)
${\operatorname{simplex}}(X)$A simplex based on a set of points (cf. Definition A.17)

$\mathfrak{M}^n$The $n$-dimensional manifold space (cf. Definition A.6)
${\mathfrak{E}}^n$The $n$-dimensional mesh element space (cf. Definition A.25)
$\mathfrak{L}^n$The $n$-dimensional geometry space (cf. Definition A.25)
$\operatorname{ip}(A,B)$The intersection of two partitions (cf. Definition A.29)
$\operatorname{refine}(S, P)$The partition refinement of two partitions (cf. Definition A.30)

${\operatorname{us}}({\mathcal{M}})$The underlying space of a mesh (cf. Definition A.6)
${\operatorname{geo}}({\mathcal{M}})$The geometry of a mesh (cf. Definition 2.11)

${\mathbb{X}}$A templated structure (cf. Definition 4.3)
$\operatorname{templ}({\mathbb{X}}, i)$The $i$-th template of a templated structure (cf. Section 4.1)
$\operatorname{inst}({\mathbb{X}},i,j), \operatorname{inst}({\mathbb{X}},T_{i,j})$The $j$-th instance of the $i$-th template (cf. Section 4.1)
$\operatorname{tf}_{i,j}$The transformation function of the $j$-th instance of the $i$-th template (cf. Definition 4.3)
$\operatorname{rid}_{i,j}$The region indicator of the $j$-th instance of the $i$-th template (cf. Definition 4.3)

${\Gamma}$A templated mesh (cf. Definition 4.4) ${\Lambda}$A templated geometry (cf. Definition 4.4)

${\operatorname{AT}}({\mathbb{X}})$The apply-template operator (cf. Section 4.1)
${\operatorname{interf}}({\mathbb{X}}, I_1, \dots, I_k)$The instance interface of a templated structure based on a set of instances (cf. Definition 4.6)
${\operatorname{interf}}_{\operatorname{geo}}({\mathbb{X}}, I_1, \dots, I_k)$The geometry instance interface of a templated structure based on a set of instances (cf. Definition 4.6)

${\mathbb{T}}_{T,U}$A template boundary mapping (cf. Definition 4.7)
${\mathfrak{T}}_{\mathbb{X}}$All valid template boundary mappings of a templated structure (cf. Section 4.3)
${\operatorname{bndpart}}({\mathbb{X}})$The boundary patch partition of a templated structure (cf. Section 4.3)
${\sim}_{\mathbb{X}}$The boundary patch relation (cf. Definition 4.8)