# Discrete Weber inequalities and related Maxwell compactness for hybrid
spaces over polyhedral partitions of domains with general topology

Simon<EMAIL_ADDRESS>(corresponding author) Inria, Univ.
Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France
Silvano<EMAIL_ADDRESS>Inria, Univ. Lille, CNRS, UMR 8524
– Laboratoire Paul Painlevé, 59000 Lille, France

###### Abstract

We prove discrete versions of the first and second Weber inequalities on
$\boldsymbol{H}(\operatorname{\bf
curl})\cap\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid spaces
spanned by polynomials attached to the faces and to the cells of a polyhedral
mesh. The proven hybrid Weber inequalities are optimal in the sense that (i)
they are formulated in terms of $\boldsymbol{H}(\operatorname{\bf curl})$\-
and $\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid semi-norms
designed so as to embed optimally (polynomially) consistent face penalty
terms, and (ii) they are valid for face polynomials in the smallest possible
stability-compatible spaces. Our results are valid on domains with general,
possibly non-trivial topology. In a second part we also prove, within a
general topological setting, related discrete Maxwell compactness properties.
Keywords: Weber inequalities; Maxwell compactness; Hybrid polynomial spaces;
Polyhedral meshes; de Rham cohomology; Topology; Maxwell’s equations.
AMS Subject Classification 2020: 65N12, 14F40, 35Q60.

##  1 Introduction

Let $\Omega$ be a domain in $\mathbb{R}^{3}$, i.e. a bounded and connected
Lipschitz open set of $\mathbb{R}^{3}$. Our main motivation in this work comes
from the study of time-harmonic Maxwell’s equations. We are interested in PDE
models, posed in $\Omega$, for which the main (vector) unknown
$\boldsymbol{u}$ lies in a subspace $\boldsymbol{X}_{\star}(\Omega)$ of the
space

$\boldsymbol{X}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{div}_{\eta};\Omega),$

where $\eta$ is a physical parameter (possibly varying inside $\Omega$) and
$\operatorname{div}_{\eta}\mathrel{\mathop{:}}=\operatorname{div}(\eta\,\cdot)$.
In practice, the unknown $\boldsymbol{u}$ is the (vector) complex amplitude of
a time-harmonic electromagnetic field, which may either be the electric field
$\boldsymbol{e}$ (then $\eta$ is the electric permittivity $\varepsilon$), or
the magnetic field $\boldsymbol{h}$ (then $\eta$ is the magnetic permeability
$\mu$), or some vector potential $\boldsymbol{a}$ (then $\eta\equiv 1$ if one
adopts the Coulomb gauge). Vector fields in $\boldsymbol{X}_{\star}(\Omega)$
bear, almost everywhere on the boundary of $\Omega$, either zero tangential or
zero $\eta$-weighted normal trace. When zero tangential trace is imposed on
the whole boundary of $\Omega$, we let
$\boldsymbol{X}_{\star}(\Omega)=\mathrel{\mathop{:}}\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$,
and we refer to this case as fully tangential. In the same vein, when zero
$\eta$-weighted normal trace is imposed on the whole boundary of $\Omega$, we
let
$\boldsymbol{X}_{\star}(\Omega)=\mathrel{\mathop{:}}\boldsymbol{X}_{n}(\Omega)$,
and we refer to this case as fully normal. All other cases are referred to as
mixed.

It is nowadays well-understood that the well-posedness of variational problems
set in $\boldsymbol{X}_{\star}(\Omega)$ is topology-depending, and directly
relates to the (co)homology of the underlying de Rham complex. For standard
(i.e. either fully tangential, or fully normal) boundary conditions, the
homology of the de Rham complex has been extensively studied in the
literature, and the homology spaces, spanned by harmonic vector fields, have
been fully characterized [29, 21, 1, 30]. Based on the latter
characterizations, topologically general Poincaré-type inequalities, the so-
called Weber inequalities, can be established. These inequalities are named
after Christian Weber, in relation to his seminal contributions to the topic
[42]. The first Weber inequality corresponds to the fully tangential case of
vector fields in $\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$, whereas the
second corresponds to the fully normal case (of vector fields in
$\boldsymbol{X}_{n}(\Omega)$). Whenever $\Omega$ has trivial topology (think
of a contractible domain), letting $\|{\cdot}\|_{0}$ indifferently denote the
$\\{L^{2}(\Omega),\boldsymbol{L}^{2}(\Omega)\\}$-norm, the first and second
Weber inequalities read:

$\|\boldsymbol{v}\|_{0}\leq
C_{\eta,\star}\big{(}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}+\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}\big{)}\qquad\forall\boldsymbol{v}\in\boldsymbol{X}_{\star}(\Omega),$
(1)

for a real number $C_{\eta,\star}>0$ only depending on $\Omega$, $\eta$, and
on the boundary conditions type. Weber inequalities are closely related to the
compact embedding of $\boldsymbol{X}_{\star}(\Omega)$ into
$\boldsymbol{L}^{2}(\Omega)$, also known in the literature under the name of
Maxwell compactness property (see e.g. [6]). Weber inequalities must not be
confused with Gaffney inequalities, which are related to the continuous
embedding of $\boldsymbol{X}_{\star}(\Omega)$ into
$\boldsymbol{H}^{1}(\Omega)$, only valid for smooth $\eta$ and smooth or
convex domains (see [1, Sect. 2.3] for further details). On general Lipschitz
domains $\Omega$, even under the additional assumption that $\Omega$ is a
polyhedron, one can solely prove that $\boldsymbol{X}_{\star}(\Omega)$ is
continuously embedded (i) into $\boldsymbol{H}^{\frac{1}{2}+s}(\Omega)$ for
some $s>0$ if $\eta$ is globally smooth (cf. [19] and [1, Prop. 3.7]), or (ii)
into $\boldsymbol{H}^{s}(\Omega)$ (still for some $s>0$) if $\eta$ is
piecewise smooth (see [20, 9]). When departing from standard boundary
conditions, the (co)homology of the de Rham complex can be more complicated.
For mixed boundary conditions, the characterization of the homology spaces has
been investigated in [28], $\boldsymbol{H}^{s}(\Omega)$-regularity for
piecewise smooth $\eta$ proven in [32], and Maxwell compactness in [31].

Let us briefly comment at this point on the terminology. We endow
$\boldsymbol{X}(\Omega)$ with its standard norm
$\|\boldsymbol{v}\|_{\boldsymbol{X}}^{2}\mathrel{\mathop{:}}=\left(\|\boldsymbol{v}\|_{0}^{2}+\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}^{2}\right)+\left(\|\boldsymbol{v}\|_{0}^{2}+\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}^{2}\right)$.
From an historical perspective, what Weber actually proved in [42] (based on a
preliminary work by Weck [43]) are not exactly inequalities of the form (1),
but rather Maxwell compactness properties in
$\left(\boldsymbol{X}_{\star}(\Omega),\|{\cdot}\|_{\boldsymbol{X}}\right)$.
Whereas the $\|{\cdot}\|_{\boldsymbol{X}}$-norm embeds an
$\boldsymbol{L}^{2}(\Omega)$-norm control on the function, at the opposite,
remark that such a control is absent from the right-hand side of (1). To prove
Poincaré-type inequalities of the form (1) from Maxwell compactness, we know
(by Peetre–Tartar’s lemma) that it is sufficient to identify, for some Banach
space $\boldsymbol{\mathcal{Y}}$, an injective bounded linear mapping
$\boldsymbol{\mathcal{A}}:\left(\boldsymbol{X}_{\star}(\Omega),\|{\cdot}\|_{\boldsymbol{X}}\right)\to\left(\boldsymbol{\mathcal{Y}},\|{\cdot}\|_{\boldsymbol{\mathcal{Y}}}\right)$
such that

$\|\boldsymbol{v}\|_{\boldsymbol{X}}\leq
C_{1}\big{(}\|\boldsymbol{\mathcal{A}}(\boldsymbol{v})\|_{\boldsymbol{\mathcal{Y}}}+\|\boldsymbol{v}\|_{0}\big{)}\qquad\forall\boldsymbol{v}\in\boldsymbol{X}_{\star}(\Omega).$

It then follows that

$\|\boldsymbol{v}\|_{0}\leq\|\boldsymbol{v}\|_{\boldsymbol{X}}\leq
C_{2}\|\boldsymbol{\mathcal{A}}(\boldsymbol{v})\|_{\boldsymbol{\mathcal{Y}}}\qquad\forall\boldsymbol{v}\in\boldsymbol{X}_{\star}(\Omega).$

For instance, when $\Omega$ has trivial topology, the mapping

$\boldsymbol{\mathcal{A}}:\boldsymbol{X}_{\star}(\Omega)\to\boldsymbol{\mathcal{Y}}\mathrel{\mathop{:}}=L^{2}(\Omega)\times\boldsymbol{L}^{2}(\Omega);\boldsymbol{v}\mapsto\big{(}\operatorname{div}(\eta\boldsymbol{v}),\operatorname{\bf
curl}\boldsymbol{v}\big{)}$

satisfies all the required properties, whence the inequality (1). Within a
general topological setting, identifying $\boldsymbol{\mathcal{A}}$ crucially
requires a precise characterization of the (co)homology spaces of the de Rham
complex. Such a characterization has not been provided by Weber in his seminal
work, but by different authors later on (cf. e.g. [29, 21, 1, 30]). In this
respect, naming Poincaré-type inequalities of the form (1) “Weber
inequalities” is imperfect. However, considering that Weber was the first key
player in their discovery, and following [4], we will adopt this terminology
throughout this article.

The focus in this work is on (arbitrary-order) skeletal methods over
polyhedral partitions. Contrary to plain vanilla Discontinuous Galerkin (DG)
methods, skeletal methods attach unknowns to the mesh skeleton, thus allowing
for the static condensation of (potential) cell unknowns (cf. [34] for further
insight). In what follows, for $\mathcal{X}(\Omega)$ a given functional space
on $\Omega$, we will term $\mathcal{X}(\Omega)$-conforming any skeletal method
attaching unknowns to the boundary of the cells so as to mimic the
“continuity” properties of the underlying continuous space
$\mathcal{X}(\Omega)$. For instance, in the lowest-order case,
$H^{1}(\Omega)$-conforming methods attach unknowns to the mesh vertices so as
to emulate (full) continuity of traces, whereas
$\boldsymbol{H}(\operatorname{\bf curl};\Omega)$-conforming methods attach
unknowns to the mesh edges so as to reproduce tangential continuity of
(vector) traces. As opposed, we will term non-conforming any other kind of
skeletal method. In the polyhedral context, this dichotomy is particularly
convenient to discriminate between the approaches, for which, anyhow at the
end, only fully discontinuous polynomial proxies are computable. We
exclusively focus in this work on non-conforming (skeletal) methods, often
referred to in the literature as hybrid methods. Hybrid methods only attach
unknowns to the faces and to the cells of the mesh. Examples of such
approaches include the Hybridizable Discontinuous Galerkin (HDG) [18], the
Weak Galerkin (WG) [41], the Hybrid High-Order (HHO) [24, 25], or the non-
conforming Virtual Element (nc-VE) [5] methods. All these technologies share
tight links, as first observed in [17]. At the opposite side of the skeletal
spectrum lie conforming polyhedral methods, whose salient examples include the
conforming Virtual Element (c-VE) [7] or the Discrete De Rham (DDR) [23]
approaches, between which strong connections also exist.

In this work, we aim at deriving, within a general topological setting,
discrete functional analysis tools for hybrid approximations of div-curl
systems over polyhedral mesh families. To our knowledge, hybrid methods have
only been studied yet within a trivial topological setting. In the HDG and WG
contexts, these contributions respectively include [40, 14, 13] (cf. also
[26]) and [37]. In the HHO context, a discrete (first) Weber inequality was
proven in [12] for $\boldsymbol{H}(\operatorname{\bf curl})$-like hybrid
spaces satisfying a discrete divergence-free constraint in trivial domains,
and leveraged to perform the analysis of HHO approximations of first- and
second-order magnetostatics models. In this paper, building on [12], we prove
discrete versions of the first and second Weber inequalities on generic
$\boldsymbol{H}(\operatorname{\bf
curl})\cap\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid spaces, which
are valid for domains with general, possibly non-trivial topology. We also
establish the related discrete Maxwell compactness properties. The results are
valid for complex-valued fields and piecewise constant (over the polyhedral
partition), real-valued, isotropic parameters $\eta$. The results seamlessly
extend to the case of complex-valued, Hermitian anisotropic parameters. The
hybrid Weber inequalities we prove are optimal in the following sense:

  * (i)

they are formulated in terms of $\boldsymbol{H}(\operatorname{\bf curl})$\-
and $\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid semi-norms
designed so as to embed optimally (polynomially) consistent face penalty
terms;

  * (ii)

they are valid for face polynomials in the smallest possible stability-
compatible spaces.

The first property above guarantees that the corresponding stabilization
Hermitian forms, of the so-called Lehrenfeld–Schöberl type [33] (cf. also [16]
for further insight), shall provide superconvergence when used within hybrid
numerical schemes (cf. [14, 12]). The second property, in turn, allows for
substantial savings when considering the approximation of first-order systems,
for which discrete differential operators need not be reconstructed (cf. [12,
Sect. 3.1]). More generally, our work paves the way to the analysis of hybrid
polyhedral approximations of general div-curl systems on domains with
arbitrary topology, including non-linear (Hilbertian) problems as they may
arise e.g. in the modeling of ferromagnetic materials.

Let us finally relate our work to the so-called compatible approaches. In the
Finite Element (FE) context over matching tetrahedral partitions, Nédélec
elements [38, 39] can be used to define discrete (polynomial) de Rham
complexes [10, 36]. Through the prism of Whitney forms [44] and, more broadly,
of the Finite Element Exterior Calculus (FEEC) [3, 2], the notion of discrete
de Rham complex can be extended from the framework of vector calculus to the
one of exterior calculus over differential forms. The corresponding numerical
methods then lie in an analytical setting which is referred to as compatible.
In a compatible FE setting, Poincaré-type inequalities are straightforward
consequences of the (co)homological properties of the underlying polynomial de
Rham complex. In the polyhedral context, following the DDR approach [23, 8],
discrete de Rham complexes can also be written and analyzed, but at the price
of some complications. These complications include the use of discrete
(reconstructed) differential operators over algebraic spaces of discrete
unknowns, and the tedious tracking in the analysis of mesh-dependent
quantities. Notice that all the above-mentioned compatible approaches are
related to conforming (FE or polyhedral) methods. For non-conforming methods,
the only contributions we are aware of towards the developement of a
compatible analytical setting are [35, 15] (based on seminal ideas from [11]),
in which the convenient notion of discrete distributional differential form is
introduced. The framework considered in [35, 15] is the one of DG spaces over
matching simplicial meshes. The subject thus still seems largely open for
extending such ideas to hybrid spaces over polyhedral partitions. We believe
our work may constitute, in the language of vector calculus, a first step in
this direction.

The rest of the article is organized as follows. In Section 2, after
introducing both the topological and functional frameworks, we state and
prove, based on ad hoc Helmholtz–Hodge decompositions, the continuous first
and second Weber inequalities in $\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)$, as well as
the related Maxwell compactness properties. In Section 3 we introduce the
discrete setting, and define the notions of polyhedral mesh family, face and
cell polynomial decompositions, and $\boldsymbol{H}(\operatorname{\bf
curl})\cap\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid spaces.
Section 4 is dedicated to the statements and proofs of the first and second
hybrid Weber inequalities. The related discrete Maxwell compactness properties
are, in turn, established in Section 5. Finally, Appendix A collects a
connected technical result.

##  2 Preliminaries

###  2.1 Topological framework

Let $\Omega$ be a domain in $\mathbb{R}^{3}$, i.e. a bounded and connected
(strongly) Lipschitz open set of $\mathbb{R}^{3}$. We let
$\Gamma\mathrel{\mathop{:}}=\partial\Omega$ denote the boundary of $\Omega$.
By the Rademacher theorem, one can define almost everywhere on $\Gamma$ a unit
vector field $\boldsymbol{n}$ normal to $\Gamma$, which we further assume to
point outward from $\Omega$. We recall that the Betti numbers $\beta_{0}$,
$\beta_{1}$ and $\beta_{2}$ of $\Omega$ respectively denote the number of
(maximally) connected components of $\Omega$ (here, $\beta_{0}=1$), the number
of tunnels crossing through $\Omega$ ($\beta_{1}\in\mathbb{N}$), and the
number of voids enclosed in $\Omega$ ($\beta_{2}\in\mathbb{N}$). For instance,
for $\Omega$ simply-connected, $\beta_{1}=0$. In the same manner, when the
boundary $\Gamma$ of $\Omega$ is connected, $\beta_{2}=0$. In the following,
when both $\beta_{1}$ and $\beta_{2}$ are equal to zero, we say that $\Omega$
has trivial topology.

When $\beta_{1}>0$, we make the following classical assumption: there exist
$\beta_{1}$ non-intersecting, orientable, two-dimensional manifolds
$\Sigma_{1},\ldots,\Sigma_{\beta_{1}}$ with boundary, called cutting surfaces,
satisfying $\partial\Sigma_{i}\subset\Gamma$ for all
$i\in\\{1,\ldots,\beta_{1}\\}$, such that the open set
$\hat{\Omega}\mathrel{\mathop{:}}=\Omega\setminus\cup_{i\in\\{1,\ldots,\beta_{1}\\}}\Sigma_{i}$
is not crossed by any tunnel (its first Betti number is then zero). Remark
that
$\partial\hat{\Omega}=\Gamma\cup\cup_{i\in\\{1,\ldots,\beta_{1}\\}}\Sigma_{i}$.
We will assume in what follows that $\hat{\Omega}$ is connected (which is
generally the case starting from a connected domain $\Omega$), and that the
cutting surfaces are sufficiently regular so that the set $\hat{\Omega}$ is
pseudo-Lipschitz (cf. e.g. [4, Def. 3.2.2]). Since the cutting surfaces
$\Sigma_{i}$ are orientable, one can then define almost everywhere on
$\Sigma_{i}$, for any $i\in\\{1,\ldots,\beta_{1}\\}$, a unit vector field
$\boldsymbol{n}_{\Sigma_{i}}$ normal to $\Sigma_{i}$, whose orientation is
arbitrary but prescribed once and for all.

When $\beta_{2}>0$, letting $\Gamma_{0}$ be the (connected) boundary of the
only unbounded component of the exterior open set
$\mathbb{R}^{3}\setminus\overline{\Omega}$, there exist $\beta_{2}$
(maximally) connected components $\Gamma_{1},\ldots,\Gamma_{\beta_{2}}$ of
$\Gamma$ such that $\Gamma=\cup_{j\in\\{0,\ldots,\beta_{2}\\}}\Gamma_{j}$.
When $\beta_{2}=0$, there holds $\Gamma=\Gamma_{0}$.

###  2.2 Functional setting

Let $\eta:\Omega\to\mathbb{R}$ be a given function satisfying, for real
numbers $0<\eta_{\flat}\leq\eta_{\sharp}<\infty$,

$\eta_{\flat}\leq\eta(\boldsymbol{x})\leq\eta_{\sharp}\qquad\text{for
a.e.}\leavevmode\nobreak\ \boldsymbol{x}\in\Omega.$ (2)

We also introduce
$\kappa_{\eta}\mathrel{\mathop{:}}=\eta_{\sharp}/\eta_{\flat}\geq 1$ the
(global) heterogeneity ratio of the parameter $\eta$.

For $m\in\\{2,3\\}$, and for $X$ an $m$-dimensional, (relatively) open
Lipschitz subset of $\overline{\Omega}$, we let $L^{2}(X)$ (respectively,
$\boldsymbol{L}^{2}(X)$) denote the Lebesgue space of $\mathbb{C}$-valued
functions (respectively, $\mathbb{C}^{m}$-valued vector fields) with square-
integrable modulus on $X$. The standard Hermitian inner products (and norms)
in $L^{2}(X)$ and $\boldsymbol{L}^{2}(X)$ are irrespectively denoted by
$(\mathfrak{f},\mathfrak{g})_{X}\mathrel{\mathop{:}}=\int_{X}\mathfrak{f}{\cdot}\overline{\mathfrak{g}}$
(and $\|{\cdot}\|_{0,X}\mathrel{\mathop{:}}=\sqrt{({\cdot},{\cdot})_{X}}$,
with the convention that
$\|{\cdot}\|_{0}\mathrel{\mathop{:}}=\|{\cdot}\|_{0,\Omega}$). We also define

$L^{2}_{0}(\Omega)\mathrel{\mathop{:}}=\left\\{v\in
L^{2}(\Omega)\mid\int_{\Omega}v=0\right\\}\qquad\text{and}\qquad\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)\mathrel{\mathop{:}}=\left\\{\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega)\mid\int_{\Omega}\boldsymbol{v}=\boldsymbol{0}\right\\},$

as well as the space
$\boldsymbol{L}^{2}_{\eta}(\Omega)\mathrel{\mathop{:}}=\left(\boldsymbol{L}^{2}(\Omega),(\eta\,\cdot,{\cdot})_{\Omega}\right)$
(remark that the norm $\|\eta^{\frac{1}{2}}{\cdot}\|_{0}$ is equivalent, by
(2), to the norm $\|{\cdot}\|_{0}$ in $\boldsymbol{L}^{2}(\Omega)$). For
$s\in\mathbb{N}^{\star}$, we let $H^{s}(X)$ (respectively,
$\boldsymbol{H}^{s}(X)$) denote the Sobolev space of $\mathbb{C}$-valued
functions in $L^{2}(X)$ (respectively, $\mathbb{C}^{m}$-valued vector fields
in $\boldsymbol{L}^{2}(X)$) whose partial weak derivatives of order up to $s$
have square-integrable modulus on $X$. The standard norms (and semi-norms) in
$H^{s}(X)$ and $\boldsymbol{H}^{s}(X)$ are irrespectively denoted by
$\|{\cdot}\|_{s,X}$ (and $|{\cdot}|_{s,X}$), with the convention that
$\|{\cdot}\|_{s}\mathrel{\mathop{:}}=\|{\cdot}\|_{s,\Omega}$ (and
$|{\cdot}|_{s}\mathrel{\mathop{:}}=|{\cdot}|_{s,\Omega}$). We also define
$H^{1}_{0}(\Omega)\mathrel{\mathop{:}}=\\{v\in H^{1}(\Omega)\mid
v_{\mid\Gamma}=0\\}$.

Let $Y$ be a three-dimensional, open Lipschitz subset of $\Omega$.
Classically, we let

$\displaystyle\boldsymbol{H}(\operatorname{\bf curl};Y)$
$\displaystyle\mathrel{\mathop{:}}=\big{\\{}\boldsymbol{v}\in\boldsymbol{L}^{2}(Y)\mid\operatorname{\bf
curl}\boldsymbol{v}\in\boldsymbol{L}^{2}(Y)\big{\\}},$
$\displaystyle\boldsymbol{H}(\operatorname{div}_{\eta};Y)$
$\displaystyle\mathrel{\mathop{:}}=\big{\\{}\boldsymbol{v}\in\boldsymbol{L}^{2}(Y)\mid\operatorname{div}(\eta\boldsymbol{v})\in
L^{2}(Y)\big{\\}},$

as well as their subspaces

$\displaystyle\boldsymbol{H}(\operatorname{\bf curl}^{\boldsymbol{0}};Y)$
$\displaystyle\mathrel{\mathop{:}}=\big{\\{}\boldsymbol{v}\in\boldsymbol{H}(\operatorname{\bf
curl};Y)\mid\operatorname{\bf
curl}\boldsymbol{v}\equiv\boldsymbol{0}\big{\\}},$
$\displaystyle\boldsymbol{H}(\operatorname{div}^{0}_{\eta};Y)$
$\displaystyle\mathrel{\mathop{:}}=\big{\\{}\boldsymbol{v}\in\boldsymbol{H}(\operatorname{div}_{\eta};Y)\mid\operatorname{div}(\eta\boldsymbol{v})\equiv
0\big{\\}}.$

The spaces $\boldsymbol{H}(\operatorname{\bf curl};Y)$ and
$\boldsymbol{H}(\operatorname{div}_{\eta};Y)$ are endowed with the following
weighted norms:

$\|{\cdot}\|_{\operatorname{\bf
curl},Y}^{2}\mathrel{\mathop{:}}=\|\eta^{\frac{1}{2}}{\cdot}\|_{0,Y}^{2}+\eta_{\sharp}\|{\operatorname{\bf
curl}\cdot}\|_{0,Y}^{2},\qquad\|{\cdot}\|_{\operatorname{div},Y}^{2}\mathrel{\mathop{:}}=\|\eta^{\frac{1}{2}}{\cdot}\|_{0,Y}^{2}+\eta_{\flat}^{-1}\|{\operatorname{div}(\eta\,\cdot)}\|_{0,Y}^{2},$

with the convention that $\|{\cdot}\|_{\operatorname{\bf
curl}}\mathrel{\mathop{:}}=\|{\cdot}\|_{\operatorname{\bf curl},\Omega}$ and
$\|{\cdot}\|_{\operatorname{div}}\mathrel{\mathop{:}}=\|{\cdot}\|_{\operatorname{div},\Omega}$.
Let $\boldsymbol{n}_{\partial Y}$ denote the unit outward normal vector field
to $\partial Y$, defined almost everywhere on $\partial Y$. For
$\boldsymbol{v}\in\boldsymbol{H}(\operatorname{div}_{\eta};Y)$, one can give a
sense to the normal trace of $\eta\boldsymbol{v}$ on $\partial Y$, denoted
$(\eta\boldsymbol{v})_{\mid\partial Y}{\cdot}\boldsymbol{n}_{\partial Y}$, as
an element of $H^{-\frac{1}{2}}(\partial Y)$ (space of bounded antilinear
forms on $H^{\frac{1}{2}}(\partial Y)$). In addition, the mapping
$\boldsymbol{v}\mapsto(\eta\boldsymbol{v})_{\mid\partial
Y}{\cdot}\boldsymbol{n}_{\partial Y}$ is continuous from
$\boldsymbol{H}(\operatorname{div}_{\eta};Y)$ to $H^{-\frac{1}{2}}(\partial
Y)$. Likewise, for $\boldsymbol{v}\in\boldsymbol{H}(\operatorname{\bf
curl};Y)$, one can give a sense to the rotated tangential trace of
$\boldsymbol{v}$ on $\partial Y$, denoted $\boldsymbol{v}_{\mid\partial
Y}{\times}\boldsymbol{n}_{\partial Y}$, as an element of
$H^{-\frac{1}{2}}(\partial Y)^{3}$ (space of bounded antilinear forms on
$H^{\frac{1}{2}}(\partial Y)^{3}$), and the mapping
$\boldsymbol{v}\mapsto\boldsymbol{v}_{\mid\partial
Y}{\times}\boldsymbol{n}_{\partial Y}$ is continuous from
$\boldsymbol{H}(\operatorname{\bf curl};Y)$ to $H^{-\frac{1}{2}}(\partial
Y)^{3}$.

######  Remark 1 (Tangential vector fields).

Remark that, with our above choice of notation, $\boldsymbol{L}^{2}(\partial
Y)=L^{2}(\partial Y)^{2}$. Define $\boldsymbol{H}^{\frac{1}{2}}(\partial
Y)\mathrel{\mathop{:}}=H^{\frac{1}{2}}(\partial Y)^{2}$ and
$\boldsymbol{H}^{-\frac{1}{2}}(\partial
Y)\mathrel{\mathop{:}}=H^{-\frac{1}{2}}(\partial Y)^{2}$. Letting
$H_{\boldsymbol{x}}$ denote the tangent hyperplane to $\partial Y$ at point
$\boldsymbol{x}$, and considering
$\boldsymbol{\tau}_{1}(\boldsymbol{x}),\boldsymbol{\tau}_{2}(\boldsymbol{x})\subset
H_{\boldsymbol{x}}$ two (orthonormal) vectors such that
$\big{(}\boldsymbol{\tau}_{1}(\boldsymbol{x}),\boldsymbol{\tau}_{2}(\boldsymbol{x}),\boldsymbol{n}_{\partial
Y}(\boldsymbol{x})\big{)}$ forms a right-handed orthonormal basis of
$\mathbb{R}^{3}$, one can identify $(\boldsymbol{v}_{\mid\partial
Y}{\times}\boldsymbol{n}_{\partial Y})(\boldsymbol{x})$ with the vector of
$\mathbb{C}^{2}$ of its local coordinates in the basis
$\big{(}\boldsymbol{\tau}_{1}(\boldsymbol{x}),\boldsymbol{\tau}_{2}(\boldsymbol{x})\big{)}$
of $H_{\boldsymbol{x}}$. With a slight abuse of notation, one can then write
that $\boldsymbol{v}_{\mid\partial Y}{\times}\boldsymbol{n}_{\partial
Y}\in\boldsymbol{H}^{-\frac{1}{2}}(\partial Y)$. We will extensively make use
of this observation in the sequel.

In what follows, the (sesquilinear) duality pairings between
$H^{-\frac{1}{2}}(\partial Y)$ and $H^{\frac{1}{2}}(\partial Y)$ on the one
side, and between $\boldsymbol{H}^{-\frac{1}{2}}(\partial Y)$ and
$\boldsymbol{H}^{\frac{1}{2}}(\partial Y)$ on the other side, are
irrespectively denoted by $\langle{\cdot},{\cdot}\rangle_{\partial Y}$.
Whenever
$\boldsymbol{v}\in\boldsymbol{H}^{1}(Y)\subset\boldsymbol{H}(\operatorname{\bf
curl};Y)\cap\boldsymbol{H}(\operatorname{div};Y)$ with
$\boldsymbol{H}(\operatorname{div};Y)\mathrel{\mathop{:}}=\boldsymbol{H}(\operatorname{div}_{1};Y)$,
for almost every $\boldsymbol{x}\in\partial Y$,

$\boldsymbol{v}_{\mid\partial Y}(\boldsymbol{x})=(\boldsymbol{v}_{\mid\partial
Y}{\cdot}\boldsymbol{n}_{\partial Y})(\boldsymbol{x})\boldsymbol{n}_{\partial
Y}(\boldsymbol{x})+\boldsymbol{n}_{\partial
Y}(\boldsymbol{x}){\times}(\boldsymbol{v}_{\mid\partial
Y}{\times}\boldsymbol{n}_{\partial Y})(\boldsymbol{x}).$

In this case, $\boldsymbol{v}_{\mid\partial Y}{\cdot}\boldsymbol{n}_{\partial
Y}\in L^{2}(\partial Y)$ and $\boldsymbol{v}_{\mid\partial
Y}{\times}\boldsymbol{n}_{\partial Y}\in\boldsymbol{L}^{2}(\partial Y)$.

We finally introduce the following subspaces of
$\boldsymbol{H}(\operatorname{\bf curl};\Omega)$ and
$\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)$:

$\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)\mathrel{\mathop{:}}=\left\\{\boldsymbol{v}\in\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\mid\boldsymbol{v}_{\mid\Gamma}{\times}\boldsymbol{n}\equiv\boldsymbol{0}\right\\}$

and $\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl}^{\boldsymbol{0}};\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{\bf
curl}^{\boldsymbol{0}};\Omega)$, as well as

$\boldsymbol{H}_{0}(\operatorname{div}_{\eta};\Omega)\mathrel{\mathop{:}}=\left\\{\boldsymbol{v}\in\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)\mid(\eta\boldsymbol{v})_{\mid\Gamma}{\cdot}\boldsymbol{n}\equiv
0\right\\}$

and
$\boldsymbol{H}_{0}(\operatorname{div}^{0}_{\eta};\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}_{0}(\operatorname{div}_{\eta};\Omega)\cap\boldsymbol{H}(\operatorname{div}^{0}_{\eta};\Omega)$.

Assume that the first Betti number $\beta_{1}$ of $\Omega$ is positive. Then,
for $v\in L^{2}(\hat{\Omega})$ (resp.
$\boldsymbol{v}\in\boldsymbol{L}^{2}(\hat{\Omega})$), we denote by $\check{v}$
(resp. $\check{\boldsymbol{v}}$) its continuation to $L^{2}(\Omega)$ (resp.
$\boldsymbol{L}^{2}(\Omega)$). Also, for each cutting surface $\Sigma_{i}$ of
$\Omega$, $i\in\\{1,\ldots,\beta_{1}\\}$, we associate the superscript $+$ to
the side of $\hat{\Omega}$ (with respect to $\Sigma_{i}$) for which
$\boldsymbol{n}_{\Sigma_{i}}$ is outward, and the superscript $-$ to the side
of $\hat{\Omega}$ for which $\boldsymbol{n}_{\Sigma_{i}}$ is inward. Let $v$
be some function defined on $\hat{\Omega}$ and, with obvious notation, let
$v^{+}_{\mid\Sigma_{i}}$ and $v^{-}_{\mid\Sigma_{i}}$ denote its two traces on
$\Sigma_{i}$ defined (if need be, in a weak sense) from both sides of
$\hat{\Omega}$. Then, the jump of $v$ on $\Sigma_{i}$ is defined by

$\llbracket
v\rrbracket_{\Sigma_{i}}\mathrel{\mathop{:}}=v^{+}_{\mid\Sigma_{i}}-v^{-}_{\mid\Sigma_{i}}.$
(3)

###  2.3 Helmholtz–Hodge decompositions

We introduce in this section two Helmholtz–Hodge decompositions for vector
fields in $\boldsymbol{L}^{2}(\Omega)$.

####  2.3.1 First Helmholtz–Hodge decomposition

Let us introduce the following space of harmonic vector fields:

$\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl}^{\boldsymbol{0}};\Omega)\cap\boldsymbol{H}(\operatorname{div}^{0}_{\eta};\Omega).$
(4)

By [4, Prop. 3.3.10], the harmonic space
$\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ has dimension $\beta_{2}$, and a
basis for $\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ is given by
$\big{(}\operatorname{\bf
grad}\omega_{j}\big{)}_{j\in\\{1,\ldots,\beta_{2}\\}}$, where, for
$j\in\\{1,\ldots,\beta_{2}\\}$, $\omega_{j}\in H^{1}(\Omega)$ is the unique
solution to

$\displaystyle-\operatorname{div}(\eta\operatorname{\bf grad}\omega_{j})$
$\displaystyle=0$ $\displaystyle\text{in $\Omega$},$ (5a)
$\displaystyle\omega_{j}$ $\displaystyle=0$ $\displaystyle\text{on
$\Gamma\setminus\Gamma_{j}$},$ (5b) $\displaystyle\omega_{j}$
$\displaystyle=1$ $\displaystyle\text{on $\Gamma_{j}$}.$ (5c)

Remark that the $\omega_{j}$’s are real-valued (since $\overline{\omega_{j}}$
also solves the uniquely solvable Problem (5)), and

$\mathfrak{C}_{j^{\prime}j}\mathrel{\mathop{:}}=\big{(}\eta\operatorname{\bf
grad}\omega_{j},\operatorname{\bf
grad}\omega_{j^{\prime}}\big{)}_{\Omega}=\big{\langle}(\eta\operatorname{\bf
grad}\omega_{j})_{\mid\Gamma_{j^{\prime}}}{\cdot}\boldsymbol{n},1\big{\rangle}_{\Gamma_{j^{\prime}}},$

which implies that $\operatorname{\bf grad}\omega_{j}$ and $\operatorname{\bf
grad}\omega_{j^{\prime}}$ are not
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal in general. The real-valued
matrix $\mathfrak{C}\in\mathbb{R}^{\beta_{2}\times\beta_{2}}$ is called
capacitance matrix, and is symmetric positive-definite (cf. [4, Cor. 3.3.8]).
For $\boldsymbol{w}\in\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ writing
$\boldsymbol{w}=\operatorname{\bf grad}\omega$, with $\omega\in H^{1}(\Omega)$
such that
$\omega\mathrel{\mathop{:}}=\sum_{j=1}^{\beta_{2}}\alpha_{j}\omega_{j}$,
defining the vectors
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{j}\in\mathbb{C})_{j\in\\{1,\ldots,\beta_{2}\\}}$
and
$\boldsymbol{\beta}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\langle(\eta\boldsymbol{w})_{\mid\Gamma_{j^{\prime}}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j^{\prime}}}\in\mathbb{C})_{j^{\prime}\in\\{1,\ldots,\beta_{2}\\}}$,
there holds

$\boldsymbol{\alpha}_{\boldsymbol{w}}=\mathfrak{C}^{-1}\boldsymbol{\beta}_{\boldsymbol{w}}.$
(6)

As a consequence, $\boldsymbol{w}\equiv\boldsymbol{0}$ if and only if
$\langle(\eta\boldsymbol{w})_{\mid\Gamma_{j^{\prime}}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j^{\prime}}}=0$
for all $j^{\prime}\in\\{1,\ldots,\beta_{2}\\}$ (in fact, for all
$j^{\prime}\in\\{0,\ldots,\beta_{2}\\}$ since
$\boldsymbol{w}\in\boldsymbol{H}(\operatorname{div}^{0}_{\eta};\Omega)$).

Building on [4, Prop. 3.7.1], and using [4, Thm. 3.4.1], we infer the
following first $\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal Helmholtz–Hodge
decomposition:

$\boldsymbol{L}^{2}(\Omega)=\operatorname{\bf
grad}\big{(}H^{1}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\big{(}\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega).$
(7)

Furthermore, by [4, Thm. 3.4.1], we know that there is $C_{\Omega,1}>0$ such
that
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)$
satisfying $\operatorname{\bf curl}\boldsymbol{\psi}=\boldsymbol{z}$ can
always be chosen so that

$\|\boldsymbol{\psi}\|_{1}\leq C_{\Omega,1}\|\boldsymbol{z}\|_{0}.$ (8)

####  2.3.2 Second Helmholtz–Hodge decomposition

Let us now introduce the following space of harmonic vector fields:

$\boldsymbol{H}_{n}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}(\operatorname{\bf
curl}^{\boldsymbol{0}};\Omega)\cap\boldsymbol{H}_{0}(\operatorname{div}^{0}_{\eta};\Omega).$
(9)

By [4, Prop. 3.3.13], the harmonic space $\boldsymbol{H}_{n}(\Omega)$ has
dimension $\beta_{1}$, and a basis for $\boldsymbol{H}_{n}(\Omega)$ is given
by $\big{(}\check{\operatorname{\bf
grad}\pi_{i}}\big{)}_{i\in\\{1,\ldots,\beta_{1}\\}}$, where, for
$i\in\\{1,\ldots,\beta_{1}\\}$, $\pi_{i}\in H^{1}(\hat{\Omega})\cap
L^{2}_{0}(\hat{\Omega})$ is the unique solution (recall that $\hat{\Omega}$ is
assumed to be connected) to

$\displaystyle-\operatorname{div}(\eta\operatorname{\bf grad}\pi_{i})$
$\displaystyle=0$ $\displaystyle\text{in $\hat{\Omega}$},$ (10a)
$\displaystyle(\eta\operatorname{\bf grad}\pi_{i}){\cdot}\boldsymbol{n}$
$\displaystyle=0$ $\displaystyle\text{on $\Gamma$},$ (10b)
$\displaystyle\llbracket(\eta\operatorname{\bf
grad}\pi_{i}){\cdot}\boldsymbol{n}_{\Sigma_{i^{\prime}}}\rrbracket_{\Sigma_{i^{\prime}}}$
$\displaystyle=0$ $\displaystyle\text{for all
$i^{\prime}\in\\{1,\ldots,\beta_{1}\\}$},$ (10c)
$\displaystyle\llbracket\pi_{i}\rrbracket_{\Sigma_{i^{\prime}}}$
$\displaystyle=\delta_{ii^{\prime}}$ $\displaystyle\text{for all
$i^{\prime}\in\\{1,\ldots,\beta_{1}\\}$}.$ (10d)

We remind the reader that $\check{\operatorname{\bf grad}\pi_{i}}$ is the
continuation to $\boldsymbol{L}^{2}(\Omega)$ of $\operatorname{\bf
grad}\pi_{i}\in\boldsymbol{L}^{2}(\hat{\Omega})$, and that the jump operator
$\llbracket{\cdot}\rrbracket_{\Sigma_{i^{\prime}}}$ is defined in (3). Notice
that the $\pi_{i}$’s are real-valued (since $\overline{\pi_{i}}$ also solves
the uniquely solvable Problem (10)) and, following the convention (3),

$\mathfrak{L}_{i^{\prime}i}\mathrel{\mathop{:}}=\big{(}\eta\,\check{\operatorname{\bf
grad}\pi_{i}},\check{\operatorname{\bf
grad}\pi_{i^{\prime}}}\big{)}_{\Omega}=\big{(}\eta\operatorname{\bf
grad}\pi_{i},\operatorname{\bf
grad}\pi_{i^{\prime}}\big{)}_{\hat{\Omega}}=\big{\langle}(\eta\operatorname{\bf
grad}\pi_{i})_{\mid\Sigma_{i^{\prime}}}{\cdot}\boldsymbol{n}_{\Sigma_{i^{\prime}}},1\big{\rangle}_{\Sigma_{i^{\prime}}},$

which implies that $\check{\operatorname{\bf grad}\pi_{i}}$ and
$\check{\operatorname{\bf grad}\pi_{i^{\prime}}}$ are not
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal in general. The real-valued
matrix $\mathfrak{L}\in\mathbb{R}^{\beta_{1}\times\beta_{1}}$ is called
inductance matrix, and is symmetric positive-definite (see [4, Cor. 3.3.14]).
For $\boldsymbol{w}\in\boldsymbol{H}_{n}(\Omega)$ writing
$\boldsymbol{w}=\check{\operatorname{\bf grad}\pi}$, with $\pi\in
H^{1}(\hat{\Omega})\cap L^{2}_{0}(\hat{\Omega})$ such that
$\pi\mathrel{\mathop{:}}=\sum_{i=1}^{\beta_{1}}\alpha_{i}\pi_{i}$, letting
this time
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{i}\in\mathbb{C})_{i\in\\{1,\ldots,\beta_{1}\\}}$
and
$\boldsymbol{\beta}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\langle(\eta\boldsymbol{w})_{\mid\Sigma_{i^{\prime}}}{\cdot}\boldsymbol{n}_{\Sigma_{i^{\prime}}},1\rangle_{\Sigma_{i^{\prime}}}\in\mathbb{C})_{i^{\prime}\in\\{1,\ldots,\beta_{1}\\}}$,
there holds

$\boldsymbol{\alpha}_{\boldsymbol{w}}=\mathfrak{L}^{-1}\boldsymbol{\beta}_{\boldsymbol{w}}.$
(11)

As a consequence, $\boldsymbol{w}\equiv\boldsymbol{0}$ if and only if
$\langle(\eta\boldsymbol{w})_{\mid\Sigma_{i^{\prime}}}{\cdot}\boldsymbol{n}_{\Sigma_{i^{\prime}}},1\rangle_{\Sigma_{i^{\prime}}}=0$
for all $i^{\prime}\in\\{1,\ldots,\beta_{1}\\}$.

Building on [4, Prop. 3.7.3], and using [4, Thm. 3.5.1], we infer the
following second $\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal
Helmholtz–Hodge decomposition:

$\boldsymbol{L}^{2}(\Omega)=\operatorname{\bf grad}\big{(}H^{1}(\Omega)\cap
L^{2}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\big{(}\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{n}(\Omega).$
(12)

Furthermore, combining the results of [4, Thm. 3.5.1] and [4, Thm. 3.6.7], we
know that there is $C_{\Omega,2}>0$ such that
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$ satisfying $\operatorname{\bf
curl}\boldsymbol{\psi}=\boldsymbol{z}$ can always be chosen so that

$\|\boldsymbol{\psi}\|_{1}\leq C_{\Omega,2}\|\boldsymbol{z}\|_{0}.$ (13)

###  2.4 Weber inequalities and Maxwell compactness

Recall the definition
$\boldsymbol{X}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)$. Endowed
with the (weighted) norm

$\|\boldsymbol{v}\|^{2}_{\boldsymbol{X}}\mathrel{\mathop{:}}=\|\boldsymbol{v}\|_{\operatorname{div}}^{2}+\|\boldsymbol{v}\|_{\operatorname{\bf
curl}}^{2},$

$\boldsymbol{X}(\Omega)$ is a Hilbert space. The first and second Weber
inequalities, as well as the related Maxwell compactness properties, are
direct consequences of the Helmholtz–Hodge decompositions (7) and (12).

####  2.4.1 First Weber inequality

Let

$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{div}_{\eta};\Omega).$ (14)

As a closed subspace of $\boldsymbol{X}(\Omega)$ (recall that the rotated
tangential trace map is continuous),
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$, endowed with the
$\|{\cdot}\|_{\boldsymbol{X}}$-norm, is a Hilbert space. We state and prove
the first Weber inequality.

######  Proposition 2 (First Weber inequality).

There is $C_{W,1}>0$ only depending on $\Omega$ such that, for all
$\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$,

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}\leq
C_{W,1}\Bigg{(}\eta_{\flat}^{-\frac{1}{2}}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}+\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\left(\sum_{j=1}^{\beta_{2}}|\langle(\eta\boldsymbol{v})_{\mid\Gamma_{j}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j}}|^{2}\right)^{\nicefrac{{1}}{{2}}}\Bigg{)}.$
(15)

###### Proof.

The proof of (15) builds on the first Helmholtz–Hodge decomposition (7). Since
$\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)\subset\boldsymbol{L}^{2}(\Omega)$,
there exist $\varphi\in H^{1}_{0}(\Omega)$,
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)$,
and $\boldsymbol{w}\in\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ (writing
$\boldsymbol{w}=\operatorname{\bf grad}\omega$, with $\omega\in H^{1}(\Omega)$
such that
$\omega\mathrel{\mathop{:}}=\sum_{j=1}^{\beta_{2}}\alpha_{j}\omega_{j}$), such
that

$\boldsymbol{v}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}.$ (16)

From the previous decomposition, there holds

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}=(\eta\boldsymbol{v},\operatorname{\bf
grad}\varphi)_{\Omega}+\eta_{\sharp}(\boldsymbol{v},\operatorname{\bf
curl}\boldsymbol{\psi})_{\Omega}+(\eta\boldsymbol{v},\operatorname{\bf
grad}\omega)_{\Omega}.$

By integration by parts, since $\varphi\in H^{1}_{0}(\Omega)$,
$\boldsymbol{v}\in\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$, and each $\omega_{j}$ satisfies (5), we infer that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}=-(\operatorname{div}(\eta\boldsymbol{v}),\varphi)_{\Omega}+\eta_{\sharp}(\operatorname{\bf
curl}\boldsymbol{v},\boldsymbol{\psi})_{\Omega}-(\operatorname{div}(\eta\boldsymbol{v}),\omega)_{\Omega}+\sum_{j=1}^{\beta_{2}}\langle(\eta\boldsymbol{v})_{\mid\Gamma_{j}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j}}\overline{\alpha_{j}}.$

The triangle and Cauchy–Schwarz inequalities, followed by the Poincaré
inequality applied to both $\varphi\in H^{1}_{0}(\Omega)$ and $\omega\in
H^{1}(\Omega)$ (remark that $\omega_{\mid\Gamma_{0}}=0$), and the estimate
(8), then yield

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}\leq
C\Bigg{(}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\|\operatorname{\bf
grad}\varphi\|_{0}+\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}\,\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\\\
+\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\|\operatorname{\bf
grad}\omega\|_{0}+\left(\sum_{j=1}^{\beta_{2}}|\langle(\eta\boldsymbol{v})_{\mid\Gamma_{j}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j}}|^{2}\right)^{\nicefrac{{1}}{{2}}}\\!\\!|\boldsymbol{\alpha}_{\boldsymbol{w}}|\Bigg{)},$

where $|\boldsymbol{\alpha}_{\boldsymbol{w}}|$ denotes the Euclidean norm of
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{j}\in\mathbb{C})_{j\in\\{1,\dots,\beta_{2}\\}}$.
Starting from the expression (6) of $\boldsymbol{\alpha}_{\boldsymbol{w}}$,
and since
$\langle(\eta\boldsymbol{w})_{\mid\Gamma_{j^{\prime}}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j^{\prime}}}=(\eta\operatorname{\bf
grad}\omega,\operatorname{\bf grad}\omega_{j^{\prime}})_{\Omega}$, we infer
that

$|\boldsymbol{\alpha}_{\boldsymbol{w}}|=|\mathfrak{C}^{-1}\boldsymbol{\beta}_{\boldsymbol{w}}|\leq\rho(\mathfrak{C}^{-1})|\boldsymbol{\beta}_{\boldsymbol{w}}|\leq\rho(\mathfrak{C}^{-1})\left(\sum_{j^{\prime}=1}^{\beta_{2}}\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\omega_{j^{\prime}}\|_{0}^{2}\right)^{\nicefrac{{1}}{{2}}}\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\omega\|_{0},$ (17)

where $\rho(\mathfrak{C}^{-1})>0$ denotes the spectral radius of the symmetric
positive-definite (real-valued) matrix $\mathfrak{C}^{-1}$, which is
proportional to $\eta_{\flat}^{-1}$. The conclusion then follows from the
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonality of the decomposition (16),
so that $\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\varphi\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$,
$\eta_{\sharp}\|\eta^{-\frac{1}{2}}\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$, and
$\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\omega\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$, and from the
fact that
$\|{\cdot}\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}{\cdot}\|_{0}$
and
$\eta_{\sharp}^{\frac{1}{2}}\|{\cdot}\|_{0}\leq\eta_{\sharp}\|\eta^{-\frac{1}{2}}{\cdot}\|_{0}$.
∎

The above proof of (15) is constructive. An alternative proof (in the case
$\eta\equiv 1$), based on a contradiction argument, can be found in [4, Thm.
3.4.3]. Another proof (in the case of a symmetric tensor field $\eta$),
leveraging the inverse mapping theorem, is available in [4, Thm. 6.1.6
($s=0$)].

######  Remark 3 (Useful variant).

Assume that
$\boldsymbol{v}\in\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$ satisfies

$(\eta\boldsymbol{v},\boldsymbol{z})_{\Omega}=0\qquad\forall\boldsymbol{z}\in\operatorname{\bf
grad}\big{(}H^{1}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega).$

Then, following the above proof, one can show that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}\leq
C_{\Omega,1}\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0},$

where $C_{\Omega,1}>0$ is the multiplicative constant from (8).

The first Weber inequality enables us to define the following norm on
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$:

$\|\boldsymbol{v}\|^{2}_{\boldsymbol{X}_{\boldsymbol{\tau}}}\mathrel{\mathop{:}}=\eta_{\flat}^{-1}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}^{2}+\eta_{\sharp}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}^{2}+\eta_{\flat}^{-1}\sum_{j=1}^{\beta_{2}}|\langle(\eta\boldsymbol{v})_{\mid\Gamma_{j}}{\cdot}\boldsymbol{n},1\rangle_{\Gamma_{j}}|^{2},$

which is equivalent to the $\|{\cdot}\|_{\boldsymbol{X}}$-norm: there is
$C_{\boldsymbol{\tau}}>0$, only depending on $\Omega$, such that

$(C_{\boldsymbol{\tau}}\kappa_{\eta})^{-1}\|\boldsymbol{v}\|_{\boldsymbol{X}}^{2}\leq\|\boldsymbol{v}\|_{\boldsymbol{X}_{\boldsymbol{\tau}}}^{2}\leq
C_{\boldsymbol{\tau}}\kappa_{\eta}\|\boldsymbol{v}\|_{\boldsymbol{X}}^{2}\qquad\forall\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega).$

Endowed with the $\|{\cdot}\|_{\boldsymbol{X}_{\boldsymbol{\tau}}}$-norm (more
precisely, with the corresponding Hermitian inner product),
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ is a Hilbert space. We are now in
position to state and prove Maxwell compactness in
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$. Our proof partly takes
inspiration from the one of [4, Thm. 3.4.4] (see also [4, Thm. 7.5.1 ($s=0$)]
in the case of a symmetric tensor field $\eta$).

######  Proposition 4 (Maxwell compactness in
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$).

Let $(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ be a sequence of elements of
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ for which there exists a real
number $C_{M}>0$ such that
$\eta_{\sharp}^{-\frac{1}{2}}\|\boldsymbol{v}_{m}\|_{\boldsymbol{X}_{\boldsymbol{\tau}}}\leq
C_{M}$ for all $m\in\mathbb{N}$. Then, there exists an element
$\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ such that, along
a subsequence as $m\to\infty$, $\boldsymbol{v}_{m}\to\boldsymbol{v}$ strongly
in $\boldsymbol{L}^{2}(\Omega)$.

###### Proof.

The proof proceeds in three steps.

Step 1 (Weak convergence): By assumption,
$\|\boldsymbol{v}_{m}\|_{\boldsymbol{X}_{\boldsymbol{\tau}}}\leq
C_{M}\eta_{\sharp}^{\frac{1}{2}}$ for all $m\in\mathbb{N}$. By definition of
the $\|{\cdot}\|_{\boldsymbol{X}_{\boldsymbol{\tau}}}$-norm, and by the first
Weber inequality (15), this implies that (i)
$(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ and $(\operatorname{\bf
curl}\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ are uniformly (in $m$) bounded in
$\boldsymbol{L}^{2}(\Omega)$, and (ii)
$(\operatorname{div}(\eta\boldsymbol{v}_{m}))_{m\in\mathbb{N}}$ is uniformly
(in $m$) bounded in $L^{2}(\Omega)$. Using standard (weak compactness and
limit regularity) arguments, we then infer the existence of
$\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ such that, along
a subsequence (not relabelled), (i)
$\boldsymbol{v}_{m}\rightharpoonup\boldsymbol{v}$ and $\operatorname{\bf
curl}\boldsymbol{v}_{m}\rightharpoonup\operatorname{\bf curl}\boldsymbol{v}$
weakly in $\boldsymbol{L}^{2}(\Omega)$, and (ii)
$\operatorname{div}(\eta\boldsymbol{v}_{m})\rightharpoonup\operatorname{div}(\eta\boldsymbol{v})$
weakly in $L^{2}(\Omega)$.

Step 2 (Characterization of the limit): For any $m\in\mathbb{N}$, since
$\boldsymbol{v}_{m}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)\subset\boldsymbol{L}^{2}(\Omega)$,
invoking the first Helmholtz–Hodge decomposition (7), there exist
$\varphi_{m}\in H^{1}_{0}(\Omega)$,
$\boldsymbol{\psi}_{m}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)$,
and $\boldsymbol{w}_{m}\in\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$, such
that

$\boldsymbol{v}_{m}=\operatorname{\bf
grad}\varphi_{m}+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}_{m}+\boldsymbol{w}_{m},$ (18)

with $\|\boldsymbol{\psi}_{m}\|_{1}\leq C_{\Omega,1}\|\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\|_{0}$ by (8). Since the sequence
$(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ is uniformly (in $m$) bounded in
$\boldsymbol{L}^{2}(\Omega)$, and the decomposition (18) is
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal with $\eta$ essentially bounded
by above and by below away from zero in $\Omega$, the three sequences
$(\operatorname{\bf grad}\varphi_{m})_{m\in\mathbb{N}}$, $(\operatorname{\bf
curl}\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$, and
$(\boldsymbol{w}_{m})_{m\in\mathbb{N}}$ are also uniformly (in $m$) bounded in
$\boldsymbol{L}^{2}(\Omega)$. Given that
$(\boldsymbol{w}_{m})_{m\in\mathbb{N}}\subset\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$,
and that the harmonic space $\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ has
finite dimension, we directly infer from the Bolzano–Weierstraß theorem the
existence of $\boldsymbol{w}\in\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$
such that, up to extraction (not relabelled),
$\boldsymbol{w}_{m}\to\boldsymbol{w}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. Let us now deal with the two remaining terms of
the decomposition. By the Poincaré inequality applied to $\varphi_{m}\in
H^{1}_{0}(\Omega)$, and the estimate $\|\boldsymbol{\psi}_{m}\|_{1}\leq
C_{\Omega,1}\|\operatorname{\bf curl}\boldsymbol{\psi}_{m}\|_{0}$, there holds
that (i) $(\varphi_{m})_{m\in\mathbb{N}}$ is uniformly bounded in
$H^{1}(\Omega)$, and (ii) $(\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$ is
uniformly bounded in $\boldsymbol{H}^{1}(\Omega)$. Invoking standard (weak
compactness and limit regularity) arguments, one can infer the existence of
$\varphi\in H^{1}_{0}(\Omega)$ and
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)$
such that, up to extractions (not relabelled), $\operatorname{\bf
grad}\varphi_{m}\rightharpoonup\operatorname{\bf grad}\varphi$ and
$\operatorname{\bf curl}\boldsymbol{\psi}_{m}\rightharpoonup\operatorname{\bf
curl}\boldsymbol{\psi}$ weakly in $\boldsymbol{L}^{2}(\Omega)$. By linearity,
we have thus proven that, along a subsequence (not relabelled),
$\boldsymbol{v}_{m}\rightharpoonup\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}$ weakly in $\boldsymbol{L}^{2}(\Omega)$.
The uniqueness of the weak limit then yields that
$\boldsymbol{v}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}$.

Step 3 (Strong convergence): By Rellich’s compactness theorem, it actually
holds that, along the same subsequence (not relabelled) as in Step 2,
$\varphi_{m}\to\varphi$ strongly in $L^{2}(\Omega)$ and
$\boldsymbol{\psi}_{m}\to\boldsymbol{\psi}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. We now want to prove the strong convergences of
$(\operatorname{\bf grad}\varphi_{m})_{m\in\mathbb{N}}$ and
$(\operatorname{\bf curl}\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$ in
$\boldsymbol{L}^{2}(\Omega)$ (we remind that we already proved strong
convergence for $(\boldsymbol{w}_{m})_{m\in\mathbb{N}}$). Recalling the
expression of $\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ we
derived in Step 2, since the decomposition (18) is
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal, we have

$\|\eta^{\frac{1}{2}}(\operatorname{\bf grad}\varphi_{m}-\operatorname{\bf
grad}\varphi)\|_{0}^{2}=\big{(}\eta(\boldsymbol{v}_{m}-\boldsymbol{v}),\operatorname{\bf
grad}(\varphi_{m}-\varphi)\big{)}_{\Omega}=-\big{(}\operatorname{div}(\eta(\boldsymbol{v}_{m}-\boldsymbol{v})),\varphi_{m}-\varphi\big{)}_{\Omega},$

where we have used that $\varphi_{m},\varphi\in H^{1}_{0}(\Omega)$. By
Cauchy–Schwarz inequality, combined with the fact that
$\|\operatorname{div}(\eta\boldsymbol{v}_{m})\|_{0}\leq C_{M}\eta_{\sharp}$
for all $m\in\mathbb{N}$ and
$\operatorname{div}(\eta\boldsymbol{v}_{m})\rightharpoonup\operatorname{div}(\eta\boldsymbol{v})$
weakly in $L^{2}(\Omega)$ (so that
$\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\leq C_{M}\eta_{\sharp}$), we
infer that

$\|\eta^{\frac{1}{2}}(\operatorname{\bf grad}\varphi_{m}-\operatorname{\bf
grad}\varphi)\|_{0}^{2}\leq 2C_{M}\eta_{\sharp}\|\varphi_{m}-\varphi\|_{0}.$

Since $\varphi_{m}\to\varphi$ strongly in $L^{2}(\Omega)$, passing to the
limit $m\to\infty$ we conclude that $\operatorname{\bf
grad}\varphi_{m}\to\operatorname{\bf grad}\varphi$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. By the very same arguments, there holds

$\|\eta^{-\frac{1}{2}}(\operatorname{\bf
curl}\boldsymbol{\psi}_{m}-\operatorname{\bf
curl}\boldsymbol{\psi})\|_{0}^{2}=\eta_{\sharp}^{-1}\big{(}\boldsymbol{v}_{m}-\boldsymbol{v},\operatorname{\bf
curl}(\boldsymbol{\psi}_{m}-\boldsymbol{\psi})\big{)}_{\Omega}=\eta_{\sharp}^{-1}\big{(}\operatorname{\bf
curl}(\boldsymbol{v}_{m}-\boldsymbol{v}),\boldsymbol{\psi}_{m}-\boldsymbol{\psi}\big{)}_{\Omega},$

where we have used that
$\boldsymbol{v}_{m},\boldsymbol{v}\in\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$. This implies that

$\|\eta^{-\frac{1}{2}}(\operatorname{\bf
curl}\boldsymbol{\psi}_{m}-\operatorname{\bf
curl}\boldsymbol{\psi})\|_{0}^{2}\leq
2C_{M}\eta_{\sharp}^{-1}\|\boldsymbol{\psi}_{m}-\boldsymbol{\psi}\|_{0},$

which eventually yields that $\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\to\operatorname{\bf curl}\boldsymbol{\psi}$
strongly in $\boldsymbol{L}^{2}(\Omega)$. Thus, along a subsequence (not
relabelled), $\boldsymbol{v}_{m}\to\boldsymbol{v}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$, which concludes the proof. ∎

####  2.4.2 Second Weber inequality

Let

$\boldsymbol{X}_{n}(\Omega)\mathrel{\mathop{:}}=\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}_{0}(\operatorname{div}_{\eta};\Omega).$ (19)

As a closed subspace of $\boldsymbol{X}(\Omega)$ (recall that the
$\eta$-weighted normal trace map is continuous), $\boldsymbol{X}_{n}(\Omega)$,
endowed with the $\|{\cdot}\|_{\boldsymbol{X}}$-norm, is a Hilbert space. We
now state and prove the second Weber inequality.

######  Proposition 5 (Second Weber inequality).

There is $C_{W,2}>0$ only depending on $\Omega$ such that, for all
$\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega)$,

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}\leq
C_{W,2}\Bigg{(}\eta_{\flat}^{-\frac{1}{2}}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}+\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\left(\sum_{i=1}^{\beta_{1}}|\langle(\eta\boldsymbol{v})_{\mid\Sigma_{i}}{\cdot}\boldsymbol{n}_{\Sigma_{i}},1\rangle_{\Sigma_{i}}|^{2}\right)^{\nicefrac{{1}}{{2}}}\Bigg{)}.$
(20)

###### Proof.

The proof of (20) builds on the second Helmholtz–Hodge decomposition (12).
Since
$\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega)\subset\boldsymbol{L}^{2}(\Omega)$,
there exist $\varphi\in H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$,
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$, and $\boldsymbol{w}\in\boldsymbol{H}_{n}(\Omega)$ (writing
$\boldsymbol{w}=\check{\operatorname{\bf grad}\pi}$, with $\pi\in
H^{1}(\hat{\Omega})\cap L^{2}_{0}(\hat{\Omega})$ such that
$\pi\mathrel{\mathop{:}}=\sum_{i=1}^{\beta_{1}}\alpha_{i}\pi_{i}$), such that

$\boldsymbol{v}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}.$ (21)

From the previous decomposition, there holds

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}=(\eta\boldsymbol{v},\operatorname{\bf
grad}\varphi)_{\Omega}+\eta_{\sharp}(\boldsymbol{v},\operatorname{\bf
curl}\boldsymbol{\psi})_{\Omega}+(\eta\boldsymbol{v},\operatorname{\bf
grad}\pi)_{\hat{\Omega}}.$

By integration by parts, since
$\boldsymbol{v}\in\boldsymbol{H}_{0}(\operatorname{div}_{\eta};\Omega)$,
$\boldsymbol{\psi}\in\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$, and each $\pi_{i}$ satisfies (10), we infer

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}=-(\operatorname{div}(\eta\boldsymbol{v}),\varphi)_{\Omega}+\eta_{\sharp}(\operatorname{\bf
curl}\boldsymbol{v},\boldsymbol{\psi})_{\Omega}-(\operatorname{div}(\eta\boldsymbol{v}),\pi)_{\hat{\Omega}}+\sum_{i=1}^{\beta_{1}}\langle(\eta\boldsymbol{v})_{\mid\Sigma_{i}}{\cdot}\boldsymbol{n}_{\Sigma_{i}},1\rangle_{\Sigma_{i}}\overline{\alpha_{i}}.$

The triangle and Cauchy–Schwarz inequalities, followed by the Poincaré–Steklov
inequality applied to both $\varphi\in H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$
and $\pi\in H^{1}(\hat{\Omega})\cap L^{2}_{0}(\hat{\Omega})$ (recall that
$\hat{\Omega}$ is assumed to be connected), and the estimate (13), then yield

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}^{2}\leq
C\Bigg{(}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\|\operatorname{\bf
grad}\varphi\|_{0}+\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}\,\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\\\
+\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\|\check{\operatorname{\bf
grad}\pi}\|_{0}+\left(\sum_{i=1}^{\beta_{1}}|\langle(\eta\boldsymbol{v})_{\mid\Sigma_{i}}{\cdot}\boldsymbol{n}_{\Sigma_{i}},1\rangle_{\Sigma_{i}}|^{2}\right)^{\nicefrac{{1}}{{2}}}\\!\\!|\boldsymbol{\alpha}_{\boldsymbol{w}}|\Bigg{)},$

where $|\boldsymbol{\alpha}_{\boldsymbol{w}}|$ denotes the Euclidean norm of
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{i}\in\mathbb{C})_{i\in\\{1,\dots,\beta_{1}\\}}$,
and where we also used that
$\|\operatorname{div}(\eta\boldsymbol{v})\|_{0,\hat{\Omega}}=\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}$
since $\boldsymbol{v}\in\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)$, and
that $\|\operatorname{\bf
grad}\pi\|_{0,\hat{\Omega}}=\|\check{\operatorname{\bf grad}\pi}\|_{0}$.
Starting from the expression (11) of $\boldsymbol{\alpha}_{\boldsymbol{w}}$,
and since
$\langle(\eta\boldsymbol{w})_{\mid\Sigma_{i^{\prime}}}{\cdot}\boldsymbol{n}_{\Sigma_{i^{\prime}}},1\rangle_{\Sigma_{i^{\prime}}}=(\eta\,\check{\operatorname{\bf
grad}\pi},\check{\operatorname{\bf grad}\pi_{i^{\prime}}})_{\Omega}$, we infer
that

$|\boldsymbol{\alpha}_{\boldsymbol{w}}|=|\mathfrak{L}^{-1}\boldsymbol{\beta}_{\boldsymbol{w}}|\leq\rho(\mathfrak{L}^{-1})|\boldsymbol{\beta}_{\boldsymbol{w}}|\leq\rho(\mathfrak{L}^{-1})\left(\sum_{i^{\prime}=1}^{\beta_{1}}\|\eta^{\frac{1}{2}}\check{\operatorname{\bf
grad}\pi_{i^{\prime}}}\|_{0}^{2}\right)^{\nicefrac{{1}}{{2}}}\|\eta^{\frac{1}{2}}\check{\operatorname{\bf
grad}\pi}\|_{0},$ (22)

where $\rho(\mathfrak{L}^{-1})>0$ denotes the spectral radius of the symmetric
positive-definite (real-valued) matrix $\mathfrak{L}^{-1}$, which is
proportional to $\eta_{\flat}^{-1}$. The conclusion then follows from the
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonality of the decomposition (21),
so that $\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\varphi\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$,
$\eta_{\sharp}\|\eta^{-\frac{1}{2}}\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$, and
$\|\eta^{\frac{1}{2}}\check{\operatorname{\bf
grad}\pi}\|_{0}\leq\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}$, and from the
fact that
$\|{\cdot}\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}{\cdot}\|_{0}$
and
$\eta_{\sharp}^{\frac{1}{2}}\|{\cdot}\|_{0}\leq\eta_{\sharp}\|\eta^{-\frac{1}{2}}{\cdot}\|_{0}$.
∎

As for (15), the above proof of (20) is constructive. An alternative proof (in
the case $\eta\equiv 1$), based on a contradiction argument, can be found in
[4, Thm. 3.5.3]. Another proof (in the case of a symmetric tensor field
$\eta$), leveraging the inverse mapping theorem, is available in [4, Thm.
6.2.5].

######  Remark 6 (Useful variant).

Assume that $\boldsymbol{v}\in\boldsymbol{H}(\operatorname{\bf curl};\Omega)$
satisfies

$(\eta\boldsymbol{v},\boldsymbol{z})_{\Omega}=0\qquad\forall\boldsymbol{z}\in\operatorname{\bf
grad}\big{(}H^{1}(\Omega)\cap
L^{2}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{n}(\Omega).$

Then, revisiting the above proof, one can show that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}\|_{0}\leq
C_{\Omega,2}\eta_{\sharp}^{\frac{1}{2}}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0},$

where $C_{\Omega,2}>0$ is the multiplicative constant from (13).

The second Weber inequality enables us to define the following norm on
$\boldsymbol{X}_{n}(\Omega)$:

$\|\boldsymbol{v}\|^{2}_{\boldsymbol{X}_{n}}\mathrel{\mathop{:}}=\eta_{\flat}^{-1}\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}^{2}+\eta_{\sharp}\|\operatorname{\bf
curl}\boldsymbol{v}\|_{0}^{2}+\eta_{\flat}^{-1}\sum_{i=1}^{\beta_{1}}|\langle(\eta\boldsymbol{v})_{\mid\Sigma_{i}}{\cdot}\boldsymbol{n}_{\Sigma_{i}},1\rangle_{\Sigma_{i}}|^{2},$

which is equivalent to the $\|{\cdot}\|_{\boldsymbol{X}}$-norm: there is
$C_{n}>0$, only depending on $\Omega$, such that

$(C_{n}\kappa_{\eta})^{-1}\|\boldsymbol{v}\|_{\boldsymbol{X}}^{2}\leq\|\boldsymbol{v}\|_{\boldsymbol{X}_{n}}^{2}\leq
C_{n}\kappa_{\eta}\|\boldsymbol{v}\|_{\boldsymbol{X}}^{2}\qquad\forall\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega).$

Endowed with the $\|{\cdot}\|_{\boldsymbol{X}_{n}}$-norm (more precisely, with
the corresponding Hermitian inner product), $\boldsymbol{X}_{n}(\Omega)$ is a
Hilbert space. We are now in position to state and prove Maxwell compactness
in $\boldsymbol{X}_{n}(\Omega)$. Our proof partly takes inspiration from the
one of [4, Thm. 3.5.4] (see also [4, Thm. 7.5.3] in the case of a symmetric
tensor field $\eta$).

######  Proposition 7 (Maxwell compactness in $\boldsymbol{X}_{n}(\Omega)$).

Let $(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ be a sequence of elements of
$\boldsymbol{X}_{n}(\Omega)$ for which there exists a real number $C_{M}>0$
such that
$\eta_{\sharp}^{-\frac{1}{2}}\|\boldsymbol{v}_{m}\|_{\boldsymbol{X}_{n}}\leq
C_{M}$ for all $m\in\mathbb{N}$. Then, there exists an element
$\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega)$ such that, along a subsequence
as $m\to\infty$, $\boldsymbol{v}_{m}\to\boldsymbol{v}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$.

###### Proof.

The proof proceeds in three steps.

Step 1 (Weak convergence): By assumption,
$\|\boldsymbol{v}_{m}\|_{\boldsymbol{X}_{n}}\leq
C_{M}\eta_{\sharp}^{\frac{1}{2}}$ for all $m\in\mathbb{N}$. By definition of
the $\|{\cdot}\|_{\boldsymbol{X}_{n}}$-norm, and by the second Weber
inequality (20), this implies that (i) $(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$
and $(\operatorname{\bf curl}\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ are
uniformly (in $m$) bounded in $\boldsymbol{L}^{2}(\Omega)$, and (ii)
$(\operatorname{div}(\eta\boldsymbol{v}_{m}))_{m\in\mathbb{N}}$ is uniformly
(in $m$) bounded in $L^{2}(\Omega)$. Using standard (weak compactness and
limit regularity) arguments, we then infer the existence of
$\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega)$ such that, along a subsequence
(not relabelled), (i) $\boldsymbol{v}_{m}\rightharpoonup\boldsymbol{v}$ and
$\operatorname{\bf curl}\boldsymbol{v}_{m}\rightharpoonup\operatorname{\bf
curl}\boldsymbol{v}$ weakly in $\boldsymbol{L}^{2}(\Omega)$, and (ii)
$\operatorname{div}(\eta\boldsymbol{v}_{m})\rightharpoonup\operatorname{div}(\eta\boldsymbol{v})$
weakly in $L^{2}(\Omega)$.

Step 2 (Characterization of the limit): For any $m\in\mathbb{N}$, since
$\boldsymbol{v}_{m}\in\boldsymbol{X}_{n}(\Omega)\subset\boldsymbol{L}^{2}(\Omega)$,
invoking the second Helmholtz–Hodge decomposition (12), there exist
$\varphi_{m}\in H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$,
$\boldsymbol{\psi}_{m}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$, and $\boldsymbol{w}_{m}\in\boldsymbol{H}_{n}(\Omega)$, such
that

$\boldsymbol{v}_{m}=\operatorname{\bf
grad}\varphi_{m}+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}_{m}+\boldsymbol{w}_{m},$ (23)

with $\|\boldsymbol{\psi}_{m}\|_{1}\leq C_{\Omega,2}\|\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\|_{0}$ by (13). Since the sequence
$(\boldsymbol{v}_{m})_{m\in\mathbb{N}}$ is uniformly (in $m$) bounded in
$\boldsymbol{L}^{2}(\Omega)$, and the decomposition (23) is
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal with $\eta$ essentially bounded
by above and by below away from zero in $\Omega$, the three sequences
$(\operatorname{\bf grad}\varphi_{m})_{m\in\mathbb{N}}$, $(\operatorname{\bf
curl}\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$, and
$(\boldsymbol{w}_{m})_{m\in\mathbb{N}}$ are also uniformly (in $m$) bounded in
$\boldsymbol{L}^{2}(\Omega)$. Given that
$(\boldsymbol{w}_{m})_{m\in\mathbb{N}}\subset\boldsymbol{H}_{n}(\Omega)$, and
that the harmonic space $\boldsymbol{H}_{n}(\Omega)$ has finite dimension, we
straightforwardly infer from the Bolzano–Weierstraß theorem the existence of
$\boldsymbol{w}\in\boldsymbol{H}_{n}(\Omega)$ such that, up to extraction (not
relabelled), $\boldsymbol{w}_{m}\to\boldsymbol{w}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. Let us now deal with the two remaining terms of
the decomposition. By the Poincaré–Steklov inequality applied to
$\varphi_{m}\in H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$, and the estimate
$\|\boldsymbol{\psi}_{m}\|_{1}\leq C_{\Omega,2}\|\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\|_{0}$, there holds that (i)
$(\varphi_{m})_{m\in\mathbb{N}}$ is uniformly bounded in $H^{1}(\Omega)$, and
(ii) $(\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$ is uniformly bounded in
$\boldsymbol{H}^{1}(\Omega)$. Invoking standard (weak compactness and limit
regularity) arguments, one can infer the existence of $\varphi\in
H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$ and
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$ such that, up to extractions (not relabelled),
$\operatorname{\bf grad}\varphi_{m}\rightharpoonup\operatorname{\bf
grad}\varphi$ and $\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\rightharpoonup\operatorname{\bf
curl}\boldsymbol{\psi}$ weakly in $\boldsymbol{L}^{2}(\Omega)$. By linearity,
we have thus proven that, along a subsequence (not relabelled),
$\boldsymbol{v}_{m}\rightharpoonup\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}$ weakly in $\boldsymbol{L}^{2}(\Omega)$.
The uniqueness of the weak limit then yields that
$\boldsymbol{v}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w}$.

Step 3 (Strong convergence): By Rellich’s compactness theorem, it actually
holds that, along the same subsequence (not relabelled) as in Step 2,
$\varphi_{m}\to\varphi$ strongly in $L^{2}(\Omega)$ and
$\boldsymbol{\psi}_{m}\to\boldsymbol{\psi}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. We now want to prove the strong convergences of
$(\operatorname{\bf grad}\varphi_{m})_{m\in\mathbb{N}}$ and
$(\operatorname{\bf curl}\boldsymbol{\psi}_{m})_{m\in\mathbb{N}}$ in
$\boldsymbol{L}^{2}(\Omega)$ (we remind that we already proved strong
convergence for $(\boldsymbol{w}_{m})_{m\in\mathbb{N}}$). Recalling the
expression of $\boldsymbol{v}\in\boldsymbol{X}_{n}(\Omega)$ we derived in Step
2, since the decomposition (23) is
$\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonal, we have

$\|\eta^{\frac{1}{2}}(\operatorname{\bf grad}\varphi_{m}-\operatorname{\bf
grad}\varphi)\|_{0}^{2}=\big{(}\eta(\boldsymbol{v}_{m}-\boldsymbol{v}),\operatorname{\bf
grad}(\varphi_{m}-\varphi)\big{)}_{\Omega}=-\big{(}\operatorname{div}(\eta(\boldsymbol{v}_{m}-\boldsymbol{v})),\varphi_{m}-\varphi\big{)}_{\Omega},$

where we have used that
$\boldsymbol{v}_{m},\boldsymbol{v}\in\boldsymbol{H}_{0}(\operatorname{div}_{\eta};\Omega)$.
By Cauchy–Schwarz inequality, combined with the fact that
$\|\operatorname{div}(\eta\boldsymbol{v}_{m})\|_{0}\leq C_{M}\eta_{\sharp}$
for all $m\in\mathbb{N}$ and
$\operatorname{div}(\eta\boldsymbol{v}_{m})\rightharpoonup\operatorname{div}(\eta\boldsymbol{v})$
weakly in $L^{2}(\Omega)$ (so that
$\|\operatorname{div}(\eta\boldsymbol{v})\|_{0}\leq C_{M}\eta_{\sharp}$), we
infer that

$\|\eta^{\frac{1}{2}}(\operatorname{\bf grad}\varphi_{m}-\operatorname{\bf
grad}\varphi)\|_{0}^{2}\leq 2C_{M}\eta_{\sharp}\|\varphi_{m}-\varphi\|_{0}.$

Since $\varphi_{m}\to\varphi$ strongly in $L^{2}(\Omega)$, passing to the
limit $m\to\infty$ we conclude that $\operatorname{\bf
grad}\varphi_{m}\to\operatorname{\bf grad}\varphi$ strongly in
$\boldsymbol{L}^{2}(\Omega)$. By the very same arguments, there holds

$\|\eta^{-\frac{1}{2}}(\operatorname{\bf
curl}\boldsymbol{\psi}_{m}-\operatorname{\bf
curl}\boldsymbol{\psi})\|_{0}^{2}=\eta_{\sharp}^{-1}\big{(}\boldsymbol{v}_{m}-\boldsymbol{v},\operatorname{\bf
curl}(\boldsymbol{\psi}_{m}-\boldsymbol{\psi})\big{)}_{\Omega}=\eta_{\sharp}^{-1}\big{(}\operatorname{\bf
curl}(\boldsymbol{v}_{m}-\boldsymbol{v}),\boldsymbol{\psi}_{m}-\boldsymbol{\psi}\big{)}_{\Omega},$

where we have used that
$\boldsymbol{\psi}_{m},\boldsymbol{\psi}\in\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$. This implies that

$\|\eta^{-\frac{1}{2}}(\operatorname{\bf
curl}\boldsymbol{\psi}_{m}-\operatorname{\bf
curl}\boldsymbol{\psi})\|_{0}^{2}\leq
2C_{M}\eta_{\sharp}^{-1}\|\boldsymbol{\psi}_{m}-\boldsymbol{\psi}\|_{0},$

which eventually yields that $\operatorname{\bf
curl}\boldsymbol{\psi}_{m}\to\operatorname{\bf curl}\boldsymbol{\psi}$
strongly in $\boldsymbol{L}^{2}(\Omega)$. Thus, along a subsequence (not
relabelled), $\boldsymbol{v}_{m}\to\boldsymbol{v}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$, which concludes the proof. ∎

##  3 Discrete setting

From now on, we assume that the domain $\Omega\subset\mathbb{R}^{3}$ is a
(Lipschitz) polyhedron.

###  3.1 Polyhedral meshes

We consider meshes
$\mathcal{M}_{h}\mathrel{\mathop{:}}=(\mathcal{T}_{h},\mathcal{F}_{h})$ of
$\Omega\subset\mathbb{R}^{3}$ in the sense of [22, Def. 1.4].

The set $\mathcal{T}_{h}$ is a finite collection of disjoint, open Lipschitz
polyhedra $T$ (called mesh cells) such that
$\overline{\Omega}=\bigcup_{T\in\mathcal{T}_{h}}\overline{T}$. The subscript
$h$ here refers to the mesh size, defined by
$h\mathrel{\mathop{:}}=\max_{T\in\mathcal{T}_{h}}h_{T}$, where
$h_{T}\mathrel{\mathop{:}}=\max_{\boldsymbol{x},\boldsymbol{y}\in\overline{T}}|\boldsymbol{x}-\boldsymbol{y}|$
denotes the diameter of the mesh cell $T$. The set $\mathcal{F}_{h}$ is a
finite collection of disjoint, two-dimensional connected subsets of
$\overline{\Omega}$ (called mesh faces) such that, for any
$F\in\mathcal{F}_{h}$, (i) $F$ is a relatively open Lipschitz polygonal subset
of an affine hyperplane, and (ii) either there exist two distinct mesh cells
$T^{+},T^{-}\in\mathcal{T}_{h}$ such that $\overline{F}\subseteq\partial
T^{+}\cap\partial T^{-}$ ($F$ is then an interface), or there exists one mesh
cell $T\in\mathcal{T}_{h}$ such that $\overline{F}\subseteq\partial
T\cap\Gamma$ ($F$ is then a boundary face). The set of mesh faces is further
assumed to satisfy $\bigcup_{T\in\mathcal{T}_{h}}\partial
T=\bigcup_{F\in\mathcal{F}_{h}}\overline{F}$. For all $F\in\mathcal{F}_{h}$,
we let
$h_{F}\mathrel{\mathop{:}}=\max_{\boldsymbol{x},\boldsymbol{y}\in\overline{F}}|\boldsymbol{x}-\boldsymbol{y}|$
denote the diameter of the face $F$. Interfaces are collected in the set
$\mathcal{F}_{h}^{\circ}$, whereas boundary faces are collected in the set
$\mathcal{F}_{h}^{\partial}$. For all $T\in\mathcal{T}_{h}$, we denote by
$\mathcal{F}_{T}$ the subset of $\mathcal{F}_{h}$ collecting those mesh faces
lying on the boundary of $T$, so that $\partial
T=\bigcup_{F\in\mathcal{F}_{T}}\overline{F}$. For all $T\in\mathcal{T}_{h}$,
consistently with our notation so far, we let $\boldsymbol{n}_{\partial T}$
denote the unit vector field, defined almost everywhere on $\partial T$,
normal to $\partial T$ and pointing outward from $T$. For all
$F\in\mathcal{F}_{T}$, we also let
$\boldsymbol{n}_{T,F}\mathrel{\mathop{:}}=\boldsymbol{n}_{\partial T\mid F}$
denote the (constant) unit vector normal to the hyperplane containing $F$ and
pointing outward from $T$. Finally, for all $F\in\mathcal{F}_{h}$, we define
$\boldsymbol{n}_{F}$ as the unit (constant) vector normal to $F$ such that
either $\boldsymbol{n}_{F}\mathrel{\mathop{:}}=\boldsymbol{n}_{T^{+},F}$ if
$F\subset\partial T^{+}\cap\partial T^{-}\in\mathcal{F}_{h}^{\circ}$, or
$\boldsymbol{n}_{F}\mathrel{\mathop{:}}=\boldsymbol{n}_{T,F}(=\boldsymbol{n}_{\mid
F})$ if $F\subset\partial T\cap\Gamma\in\mathcal{F}_{h}^{\partial}$. For
further use, we also let, for all $T\in\mathcal{T}_{h}$ and all
$F\in\mathcal{F}_{T}$, $\varepsilon_{T,F}\in\\{-1,1\\}$ be such that
$\varepsilon_{T,F}\mathrel{\mathop{:}}=\boldsymbol{n}_{T,F}{\cdot}\boldsymbol{n}_{F}$.

When $\beta_{1}>0$, for all $i\in\\{1,\ldots,\beta_{1}\\}$, we assume that
there exists a subset $\mathcal{F}^{\circ}_{h,\Sigma_{i}}$ of
$\mathcal{F}_{h}^{\circ}$ such that
$\overline{\Sigma_{i}}=\bigcup_{F\in\mathcal{F}^{\circ}_{h,\Sigma_{i}}}\overline{F}$,
and for which $\boldsymbol{n}_{F}=\boldsymbol{n}_{\Sigma_{i}\mid F}$ for all
$F\in\mathcal{F}^{\circ}_{h,\Sigma_{i}}$. This ensures that we do associate
the superscript $+$ to the side of $\hat{\Omega}$ (with respect to
$\Sigma_{i}$) for which $\boldsymbol{n}_{\Sigma_{i}}$ is outward, consistently
with our assumption from Section 2.2. Remark that, since the cutting surfaces
are piecewise planar, the set $\hat{\Omega}$ is indeed pseudo-Lipschitz. We
also set

$\hat{\mathcal{F}}_{h}^{\circ}\mathrel{\mathop{:}}=\mathcal{F}_{h}^{\circ}\setminus\bigcup_{i=1}^{\beta_{1}}\mathcal{F}_{h,\Sigma_{i}}^{\circ}.$

When $\beta_{2}>0$, for all $j\in\\{0,\ldots,\beta_{2}\\}$, we let
$\mathcal{F}^{\partial}_{h,\Gamma_{j}}$ denote the subset of
$\mathcal{F}_{h}^{\partial}$ such that
$\Gamma_{j}=\bigcup_{F\in\mathcal{F}^{\partial}_{h,\Gamma_{j}}}\overline{F}$.

At the discrete level, the (real-valued) parameter
$\eta:\Omega\to[\eta_{\flat},\eta_{\sharp}]$ introduced in (2) is assumed to
be piecewise constant on the partition $\mathcal{T}_{h}$ of the domain
$\Omega$, and we let

$\eta_{\flat}\leq\eta_{T}\mathrel{\mathop{:}}=\eta_{\mid
T}\leq\eta_{\sharp}\qquad\forall T\in\mathcal{T}_{h}.$ (24)

Let $\mathcal{H}\subset(0,\infty)$ be a countable set of mesh sizes having $0$
as its unique accumulation point. When studying asymptotic properties with
respect to the mesh size, one has to adopt a measure of regularity for refined
mesh families $(\mathcal{M}_{h})_{h\in\mathcal{H}}$. We classically follow
[22, Def. 1.9], in which regularity for refined mesh families is quantified
through a uniform (with respect to $h$) parameter $\varrho\in(0,1)$, called
mesh regularity parameter. In a nutshell, it is assumed that, for all
$h\in\mathcal{H}$, there exists a matching tetrahedral submesh of
$\mathcal{M}_{h}$, (i) which is uniformly shape-regular, and (ii) whose
elements have a diameter that is uniformly comparable to the diameter of the
cell in $\mathcal{M}_{h}$ they belong to. In what follows, we write $a\lesssim
b$ (resp. $a\gtrsim b$) in place of $a\leq Cb$ (resp. $a\geq Cb$), if $C>0$
only depends on $\Omega$, on the mesh regularity parameter $\varrho$, and (if
need be) on the underlying polynomial degree, but is independent of both $h$
and $\eta$. In particular, for regular mesh families
$(\mathcal{M}_{h})_{h\in\mathcal{H}}$, for all $h\in\mathcal{H}$, and for all
$T\in\mathcal{T}_{h}$, there holds ${\rm card}(\mathcal{F}_{T})\lesssim 1$, as
well as $h_{T}\lesssim h_{F}\leq h_{T}$ for all $F\in\mathcal{F}_{T}$ (cf.
[22, Lem. 1.12]). Finally, for $X\in\mathcal{T}_{h}\cup\mathcal{F}_{h}$, we
let $\boldsymbol{x}_{X}\in\mathbb{R}^{3}$ be some point inside $X$ such that
$X$ contains a ball/disk centered at $\boldsymbol{x}_{X}$ of radius
$h_{X}\lesssim r_{X}\leq h_{X}$. The existence of such a point is guaranteed
for regular mesh families.

###  3.2 Polynomial spaces

For $\ell\in\mathbb{N}$ and $m\in\\{2,3\\}$, we let $\mathcal{P}^{\ell}_{m}$
denote the linear space of $m$-variate $\mathbb{C}$-valued polynomials of
total degree at most $\ell$, with the convention that, for any $m$,
$\mathcal{P}^{0}_{m}$ is identified to $\mathbb{C}$, and
$\mathcal{P}^{-1}_{m}\mathrel{\mathop{:}}=\\{0\\}$. For any
$X\in\mathcal{T}_{h}\cup\mathcal{F}_{h}$, we denote by $\mathcal{P}^{\ell}(X)$
the linear space spanned by the restrictions to $X$ of the polynomials in
$\mathcal{P}^{\ell}_{3}$. Letting $m\in\\{2,3\\}$ be the dimension of $X$,
$\mathcal{P}^{\ell}(X)$ is isomorphic to $\mathcal{P}^{\ell}_{m}$ (cf. [22,
Prop. 1.23]). We let $\pi^{\ell}_{X}$ denote the $L^{2}(X)$-orthogonal
projector onto $\mathcal{P}^{\ell}(X)$. For convenience, we also let
$\boldsymbol{\mathcal{P}}^{\ell}(X)\mathrel{\mathop{:}}=\mathcal{P}^{\ell}(X)^{m}$
(i.e. for all $T\in\mathcal{T}_{h}$,
$\boldsymbol{\mathcal{P}}^{\ell}(T)=\mathcal{P}^{\ell}(T)^{3}$, and for all
$F\in\mathcal{F}_{h}$,
$\boldsymbol{\mathcal{P}}^{\ell}(F)=\mathcal{P}^{\ell}(F)^{2}$), and we define
$\boldsymbol{\pi}^{\ell}_{X}$ as the $\boldsymbol{L}^{2}(X)$-orthogonal
projector onto $\boldsymbol{\mathcal{P}}^{\ell}(X)$.

For any $F\in\mathcal{F}_{h}$, we let $H_{F}$ denote the hyperplane containing
$F$, that we orient according to the normal $\boldsymbol{n}_{F}$. For
$w:F\to\mathbb{C}$, we let $\operatorname{\bf grad}_{F}w:F\to\mathbb{C}^{2}$
denote the (tangential) gradient of $w$. Likewise, for
$\boldsymbol{w}:F\to\mathbb{C}^{2}$,
$\operatorname{div}_{F}\boldsymbol{w}:F\to\mathbb{C}$ denotes the (tangential)
divergence of $\boldsymbol{w}$. We also let
$\mathbf{rot}_{F}w:F\to\mathbb{C}^{2}$ be such that

$\mathbf{rot}_{F}w\mathrel{\mathop{:}}=\big{(}\operatorname{\bf
grad}_{F}w\big{)}^{\perp},$

where $\boldsymbol{z}^{\perp}$ is the rotation of angle $-\pi/2$ of
$\boldsymbol{z}$ in the oriented hyperplane $H_{F}$. Let us introduce the
following subspaces of $\boldsymbol{\mathcal{P}}^{\ell}(F)$,
$\ell\in\mathbb{N}$:

$\boldsymbol{\mathcal{R}}^{\ell}(F)\mathrel{\mathop{:}}=\mathbf{rot}_{F}\big{(}\mathcal{P}^{\ell+1}(F)\big{)},\qquad\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(F)\mathrel{\mathop{:}}=\mathcal{P}^{\ell-1}(F)(\boldsymbol{x}-\boldsymbol{x}_{F}),$

where, for $\boldsymbol{x}\in F$, $(\boldsymbol{x}-\boldsymbol{x}_{F})\subset
H_{F}$ is identified to its two-dimensional tangential counterpart (cf. Remark
1). The polynomial space $\boldsymbol{\mathcal{R}}^{{\rm c},\ell}(F)$ is the
so-called Koszul complement of $\boldsymbol{\mathcal{R}}^{\ell}(F)$. As a
consequence of the homotopy formula (cf. [2, Thm. 7.1]), the following (non
$\boldsymbol{L}^{2}(F)$-orthogonal) polynomial decomposition holds true:

$\boldsymbol{\mathcal{P}}^{\ell}(F)=\boldsymbol{\mathcal{R}}^{\ell}(F)\oplus\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(F).$ (25)

Furthermore, by exactness of the (tangential) polynomial de Rham complex, the
differential mapping $\operatorname{div}_{F}:\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(F)\to\mathcal{P}^{\ell-1}(F)$ is an isomorphism.

For any $T\in\mathcal{T}_{h}$ now, we introduce the following subspaces of
$\boldsymbol{\mathcal{P}}^{\ell}(T)$, $\ell\in\mathbb{N}$:

$\displaystyle\boldsymbol{\mathcal{G}}^{\ell}(T)$
$\displaystyle\mathrel{\mathop{:}}=\operatorname{\bf
grad}\big{(}\mathcal{P}^{\ell+1}(T)\big{)},\qquad$
$\displaystyle\boldsymbol{\mathcal{G}}^{{\rm c},\ell}(T)$
$\displaystyle\mathrel{\mathop{:}}=\boldsymbol{\mathcal{P}}^{\ell-1}(T){\times}(\boldsymbol{x}-\boldsymbol{x}_{T}),$
$\displaystyle\boldsymbol{\mathcal{R}}^{\ell}(T)$
$\displaystyle\mathrel{\mathop{:}}=\operatorname{\bf
curl}\big{(}\boldsymbol{\mathcal{P}}^{\ell+1}(T)\big{)},\qquad$
$\displaystyle\boldsymbol{\mathcal{R}}^{{\rm c},\ell}(T)$
$\displaystyle\mathrel{\mathop{:}}=\mathcal{P}^{\ell-1}(T)(\boldsymbol{x}-\boldsymbol{x}_{T}).$

The two polynomial spaces $\boldsymbol{\mathcal{G}}^{{\rm c},\ell}(T)$ and
$\boldsymbol{\mathcal{R}}^{{\rm c},\ell}(T)$ are the Koszul complements of
$\boldsymbol{\mathcal{G}}^{\ell}(T)$ and $\boldsymbol{\mathcal{R}}^{\ell}(T)$,
respectively. By the homotopy formula (cf. [2, Thm. 7.1]), the two following
(non $\boldsymbol{L}^{2}(T)$-orthogonal) polynomial decompositions hold true:

$\boldsymbol{\mathcal{P}}^{\ell}(T)=\boldsymbol{\mathcal{G}}^{\ell}(T)\oplus\boldsymbol{\mathcal{G}}^{{\rm
c},\ell}(T)=\boldsymbol{\mathcal{R}}^{\ell}(T)\oplus\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(T).$ (26)

By exactness of the polynomial de Rham complex, the differential mappings
$\operatorname{\bf curl}:\boldsymbol{\mathcal{G}}^{{\rm
c},\ell}(T)\to\boldsymbol{\mathcal{R}}^{\ell-1}(T)$ and
$\operatorname{div}:\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(T)\to\mathcal{P}^{\ell-1}(T)$ are isomorphisms.

######  Remark 8 (Inverses of cell-wise differential mappings).

By the inverse mapping theorem, we know that the inverse mappings
$\operatorname{\bf curl}^{-1}\mathrel{\mathop{:}}=(\operatorname{\bf
curl})^{-1}:\boldsymbol{\mathcal{R}}^{\ell-1}(T)\to\boldsymbol{\mathcal{G}}^{{\rm
c},\ell}(T)$ and
$\boldsymbol{\operatorname{div}}^{-1}\mathrel{\mathop{:}}=(\operatorname{div})^{-1}:\mathcal{P}^{\ell-1}(T)\to\boldsymbol{\mathcal{R}}^{{\rm
c},\ell}(T)$ are bounded. From a quantitative viewpoint, one can show that

$\|\operatorname{\bf curl}^{-1}\boldsymbol{c}\|_{0,T}\lesssim
h_{T}\|\boldsymbol{c}\|_{0,T}\quad\forall\boldsymbol{c}\in\boldsymbol{\mathcal{R}}^{\ell-1}(T),\qquad\|\boldsymbol{\operatorname{div}}^{-1}d\|_{0,T}\lesssim
h_{T}\|d\|_{0,T}\quad\forall d\in\mathcal{P}^{\ell-1}(T).$ (27)

This has been proven for general polyhedral cells by a transport/scaling
argument in [23, Lem. 9]. For star-shaped polyhedral cells, it is possible to
obtain explicit multiplicative constants (cf. [12, Lem. 2.2] for
$\operatorname{\bf curl}^{-1}$, and Appendix A for
$\boldsymbol{\operatorname{div}}^{-1}$).

For all $T\in\mathcal{T}_{h}$ and all $F\in\mathcal{F}_{T}$, since for
$v:T\to\mathbb{C}$ we have $\boldsymbol{n}_{F}{\times}((\operatorname{\bf
grad}v)_{\mid F}{\times}\boldsymbol{n}_{F})=\operatorname{\bf
grad}_{F}(v_{\mid F})$ (where $\boldsymbol{n}_{F}{\times}((\operatorname{\bf
grad}v)_{\mid F}{\times}\boldsymbol{n}_{F})$ is identified to its two-
dimensional tangential counterpart), there holds

$\boldsymbol{\mathcal{G}}^{\ell}(T)_{\mid
F}{\times}\boldsymbol{n}_{F}=\boldsymbol{\mathcal{R}}^{\ell}(F),$ (28)

where vectors in $\boldsymbol{\mathcal{G}}^{\ell}(T)_{\mid
F}{\times}\boldsymbol{n}_{F}$ are identified to their two-dimensional
counterparts. In the same vein, since for $\boldsymbol{v}:T\to\mathbb{C}^{3}$
we have $(\operatorname{\bf curl}\boldsymbol{v})_{\mid
F}{\cdot}\boldsymbol{n}_{F}=\operatorname{div}_{F}(\boldsymbol{v}_{\mid
F}{\times}\boldsymbol{n}_{F})$ (where $\boldsymbol{v}_{\mid
F}{\times}\boldsymbol{n}_{F}$ is identified to its two-dimensional tangential
counterpart), and
$\operatorname{div}_{F}\big{(}\boldsymbol{\mathcal{P}}^{\ell+1}(F)\big{)}=\mathcal{P}^{\ell}(F)$
(by (25) with $\ell\leftarrow\ell+1$ and recalling that
$\operatorname{div}_{F}:\boldsymbol{\mathcal{R}}^{{\rm
c},\ell+1}(F)\to\mathcal{P}^{\ell}(F)$ is an isomorphism), there holds

$\boldsymbol{\mathcal{R}}^{\ell}(T)_{\mid
F}{\cdot}\boldsymbol{n}_{F}=\mathcal{P}^{\ell}(F).$ (29)

For the $3$-variate $\mathbb{C}$-valued polynomial space $\mathcal{P}^{\ell}$,
and $\mathbb{C}^{3}$-valued polynomial space
$\boldsymbol{\mathcal{P}}^{\ell}$, we introduce the following broken versions:

$\displaystyle\mathcal{P}^{\ell}(\mathcal{T}_{h})\mathrel{\mathop{:}}=$
$\displaystyle\\{v\in L^{2}(\Omega)\mid v_{\mid
T}\in\mathcal{P}^{\ell}(T)\,\forall T\in\mathcal{T}_{h}\\},$
$\displaystyle\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})\mathrel{\mathop{:}}=$
$\displaystyle\\{\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega)\mid\boldsymbol{v}_{\mid
T}\in\boldsymbol{\mathcal{P}}^{\ell}(T)\,\forall T\in\mathcal{T}_{h}\\}.$

We classically define on $\mathcal{P}^{\ell}(\mathcal{T}_{h})$ the broken
gradient operator $\operatorname{\bf grad}_{h}$, and on
$\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})$ the broken rotational and
divergence operators $\operatorname{\bf curl}_{h}$ and
$\operatorname{div}_{h}$. We let $\pi^{\ell}_{h}$ (resp.
$\boldsymbol{\pi}^{\ell}_{h}$) be the $L^{2}(\Omega)$-orthogonal (resp.
$\boldsymbol{L}^{2}(\Omega)$-orthogonal) projector onto
$\mathcal{P}^{\ell}(\mathcal{T}_{h})$ (resp.
$\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})$).

###  3.3 $\boldsymbol{H}(\operatorname{\bf
curl})\cap\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid spaces

Let $\ell\in\mathbb{N}$ be a given polynomial degree. We define the following
discrete counterpart of the space
$\boldsymbol{X}(\Omega)=\boldsymbol{H}(\operatorname{\bf
curl};\Omega)\cap\boldsymbol{H}(\operatorname{div}_{\eta};\Omega)$:

$\underline{\mathbbmsl{X}}_{h}^{\ell}\mathrel{\mathop{:}}=\left\\{\underline{\mathbbmsl{v}}_{h}\mathrel{\mathop{:}}=\Big{(}(\boldsymbol{v}_{T})_{T\in\mathcal{T}_{h}},(\boldsymbol{v}_{F,\boldsymbol{\tau}})_{F\in\mathcal{F}_{h}},([\eta
v]_{F,n})_{F\in\mathcal{F}_{h}}\Big{)}\leavevmode\nobreak\
:\leavevmode\nobreak\ \begin{aligned}
\boldsymbol{v}_{T}&\in\boldsymbol{\mathcal{P}}^{\ell}(T)&\quad&\forall
T\in\mathcal{T}_{h}\\\
\boldsymbol{v}_{F,\boldsymbol{\tau}}&\in\boldsymbol{\mathcal{Q}}^{\ell}(F)&\quad&\forall
F\in\mathcal{F}_{h}\\\ [\eta v]_{F,n}&\in\mathcal{P}^{\ell}(F)&\quad&\forall
F\in\mathcal{F}_{h}\end{aligned}\right\\},$ (30)

where the (possibly trimmed) polynomial space
$\boldsymbol{\mathcal{Q}}^{\ell}(F)$ satisfies

$\boldsymbol{\mathcal{R}}^{\ell}(F)\subseteq\boldsymbol{\mathcal{Q}}^{\ell}(F)\subseteq\boldsymbol{\mathcal{P}}^{\ell}(F).$
(31)

We let $\boldsymbol{\pi}_{\boldsymbol{\mathcal{Q}},F}^{\ell}$ denote the
$\boldsymbol{L}^{2}(F)$-orthogonal projector onto
$\boldsymbol{\mathcal{Q}}^{\ell}(F)$. In (30),
$\boldsymbol{v}_{F,\boldsymbol{\tau}}$ stands for the rotated tangential trace
unknown, whereas $[\eta v]_{F,n}$ stands for the $\eta$-weighted normal trace
unknown.

Given a mesh cell $T\in\mathcal{T}_{h}$, we denote by
$\underline{\mathbbmsl{X}}_{T}^{\ell}$ the restriction of
$\underline{\mathbbmsl{X}}_{h}^{\ell}$ to $T$, and by

$\underline{\mathbbmsl{v}}_{T}\mathrel{\mathop{:}}=\big{(}\boldsymbol{v}_{T},(\boldsymbol{v}_{F,\boldsymbol{\tau}})_{F\in\mathcal{F}_{T}},([\eta
v]_{F,n})_{F\in\mathcal{F}_{T}}\big{)}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$

the restriction of the generic element
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}$ to the
cell $T$. For
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}$, we let
$\boldsymbol{v}_{h}$ (not underlined) be the broken polynomial vector field in
$\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})$ such that

$(\boldsymbol{v}_{h})_{\mid
T}\mathrel{\mathop{:}}=\boldsymbol{v}_{T}\qquad\forall T\in\mathcal{T}_{h}.$

Also, we let
$\boldsymbol{\gamma}_{\boldsymbol{\tau},h}:\underline{\mathbbmsl{X}}_{h}^{\ell}\to\boldsymbol{L}^{2}(\Gamma)$
denote the discrete rotated tangential trace operator, and
$\gamma_{n,h}:\underline{\mathbbmsl{X}}_{h}^{\ell}\to L^{2}(\Gamma)$ be the
discrete $\eta$-weighted normal trace operator such that, for all
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}$,

$\boldsymbol{\gamma}_{\boldsymbol{\tau},h}(\underline{\mathbbmsl{v}}_{h})_{\mid
F}\mathrel{\mathop{:}}=\boldsymbol{v}_{F,\boldsymbol{\tau}}\qquad\text{and}\qquad\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h})_{\mid
F}\mathrel{\mathop{:}}=[\eta v]_{F,n}\qquad\forall
F\in\mathcal{F}_{h}^{\partial}.$ (32)

Note that, for almost every $\boldsymbol{x}\in\Gamma$,
$\boldsymbol{\gamma}_{\boldsymbol{\tau},h}(\underline{\mathbbmsl{v}}_{h})(\boldsymbol{x})$
is a vector in $\mathbb{C}^{2}$. The discrete trace operators enable us to
define the following subspaces of $\underline{\mathbbmsl{X}}_{h}^{\ell}$,
which are the discrete counterparts of
$\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ and $\boldsymbol{X}_{n}(\Omega)$,
respectively defined in (14) and (19):

$\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}\mathrel{\mathop{:}}=\left\\{\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}\mid\boldsymbol{\gamma}_{\boldsymbol{\tau},h}(\underline{\mathbbmsl{v}}_{h})\equiv\boldsymbol{0}\right\\},\qquad\underline{\mathbbmsl{X}}_{h,n}^{\ell}\mathrel{\mathop{:}}=\left\\{\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}\mid\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h})\equiv
0\right\\}.$ (33)

For any $T\in\mathcal{T}_{h}$, we introduce the following local Hermitian,
positive semi-definite, sesquilinear forms: for all
$\underline{\mathbbmsl{w}}_{T},\underline{\mathbbmsl{v}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$,

$\displaystyle s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{w}}_{T},\underline{\mathbbmsl{v}}_{T})$
$\displaystyle\mathrel{\mathop{:}}=\sum_{F\in\mathcal{F}_{T}}h_{F}^{-1}\left(\big{(}\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{w}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}-\boldsymbol{w}_{F,\boldsymbol{\tau}}\big{)},\big{(}\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\big{)}\right)_{F},$
(34) $\displaystyle
s_{\operatorname{div},T}(\underline{\mathbbmsl{w}}_{T},\underline{\mathbbmsl{v}}_{T})$
$\displaystyle\mathrel{\mathop{:}}=\sum_{F\in\mathcal{F}_{T}}h_{F}^{-1}\Big{(}\big{(}\eta_{T}\boldsymbol{w}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}-[\eta
w]_{F,n}\big{)},\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}-[\eta v]_{F,n}\big{)}\Big{)}_{F},$

where $\boldsymbol{w}_{T\mid F}{\times}\boldsymbol{n}_{F}$ and
$\boldsymbol{v}_{T\mid F}{\times}\boldsymbol{n}_{F}$ are identified to their
two-dimensional (tangential) counterparts. Based on (34), we next define
$\boldsymbol{H}(\operatorname{\bf curl})$\- and
$\boldsymbol{H}(\operatorname{div}_{\eta})$-like hybrid semi-norms: for all
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}$,

$|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}^{2}\mathrel{\mathop{:}}=\sum_{T\in\mathcal{T}_{h}}\left(\|\operatorname{\bf
curl}\boldsymbol{v}_{T}\|_{0,T}^{2}+s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right),$
(35)

and

$|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}^{2}\mathrel{\mathop{:}}=\sum_{T\in\mathcal{T}_{h}}\left(\|\operatorname{div}(\eta_{T}\boldsymbol{v}_{T})\|_{0,T}^{2}+s_{\operatorname{div},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right).$
(36)

Finally, we equip the discrete space $\underline{\mathbbmsl{X}}_{h}^{\ell}$
with the following weighted norm:

$\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X},h}^{2}\mathrel{\mathop{:}}=\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}^{2}+\eta_{\flat}^{-1}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}^{2}+\eta_{\sharp}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}^{2}.$ (37)

Notice in (35) the deliberate presence, within the least-squares face penalty
terms of $s_{\operatorname{\bf curl},T}$, of the tangential face projection
$\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}$. As will be made precise in Lemma 12,
this projection aims at guaranteeing the optimal (polynomial) consistency of
the local Hermitian form $s_{\operatorname{\bf curl},T}$, making of the latter
a ready-to-use (optimal) stabilization for hybrid schemes. It is quite crucial
to note that the introduction of this face projection does not jeopardize
stability, as is made clear in Lemma 9 below. This latter fact essentially
relies on the assumption (31) on the tangential face space. Similar
considerations actually apply to (36) and $s_{\operatorname{div},T}$ but with
the difference that, since the normal face space is chosen to be
$\mathcal{P}^{\ell}(F)$ and $\eta$ is piecewise constant on $\mathcal{T}_{h}$,
then $\pi^{\ell}_{F}(\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F})=\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}$ for
$\boldsymbol{v}_{T}\in\boldsymbol{\mathcal{P}}^{\ell}(T)$. The heuristics
behind the choice of face spaces is further motivated in Remark 10.

######  Lemma 9 (Full control of tangential jumps).

For all
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}$, and
all $T\in\mathcal{T}_{h}$, the following holds true:

$\left(\sum_{F\in\mathcal{F}_{T}}h_{F}^{-1}\|\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\|_{0,F}^{2}\right)^{\nicefrac{{1}}{{2}}}\lesssim\|\operatorname{\bf
curl}\boldsymbol{v}_{T}\|_{0,T}+s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})^{\nicefrac{{1}}{{2}}}.$
(38)

###### Proof.

By Remark 8, notice that $\big{(}\boldsymbol{v}_{T}-\operatorname{\bf
curl}^{-1}(\operatorname{\bf
curl}\boldsymbol{v}_{T})\big{)}\in\boldsymbol{\mathcal{G}}^{\ell}(T)$. Hence,
by (28), for any $F\in\mathcal{F}_{T}$,

$\big{(}\boldsymbol{v}_{T\mid F}{\times}\boldsymbol{n}_{F}-\operatorname{\bf
curl}^{-1}(\operatorname{\bf curl}\boldsymbol{v}_{T})_{\mid
F}{\times}\boldsymbol{n}_{F}\big{)}\in\boldsymbol{\mathcal{R}}^{\ell}(F).$

Since
$\boldsymbol{\mathcal{R}}^{\ell}(F)\subseteq\boldsymbol{\mathcal{Q}}^{\ell}(F)$
owing to (31), we then infer that

$\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}}=\big{(}\boldsymbol{I}-\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{)}\big{(}\operatorname{\bf
curl}^{-1}(\operatorname{\bf curl}\boldsymbol{v}_{T})_{\mid
F}{\times}\boldsymbol{n}_{F}\big{)}+\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}-\boldsymbol{v}_{F,\boldsymbol{\tau}}.$

By the triangle inequality, and the $\boldsymbol{L}^{2}(F)$-optimality of the
orthogonal projector $\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}$,
we get

$h_{F}^{-\frac{1}{2}}\|\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\|_{0,F}\leq
h_{F}^{-\frac{1}{2}}\|\operatorname{\bf curl}^{-1}(\operatorname{\bf
curl}\boldsymbol{v}_{T})_{\mid F}{\times}\boldsymbol{n}_{F}\|_{0,F}\\\
+h_{F}^{-\frac{1}{2}}\|\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\|_{0,F},$

which, in turn, by a discrete trace inequality (cf. e.g. [22, Lem. 1.32])
combined to mesh regularity, the estimate (27) on the local norm of
$\operatorname{\bf curl}^{-1}$, and the expression (34) of
$s_{\operatorname{\bf curl},T}$, yields (38). ∎

######  Remark 10 (Face spaces).

For a cell unknown $\boldsymbol{v}_{T}$ in
$\boldsymbol{\mathcal{P}}^{\ell}(T)$, the choice of the tangential face space
$\boldsymbol{\mathcal{Q}}^{\ell}(F)$ satisfying (31) is driven by the
following stability argument. Locally to a mesh cell $T\in\mathcal{T}_{h}$,
the face projection
$\boldsymbol{\pi}^{\ell}_{\boldsymbol{\mathcal{Q}},F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}\big{)}$ for $F\in\mathcal{F}_{T}$ must enable to
control, via least-squares face penalty, (at least) the rotated tangential
trace of the $\boldsymbol{\mathcal{G}}^{\ell}(T)$-part of $\boldsymbol{v}_{T}$
(i.e. $\boldsymbol{v}_{T}-\operatorname{\bf curl}^{-1}(\operatorname{\bf
curl}\boldsymbol{v}_{T})$). As a matter of fact,
$\boldsymbol{\mathcal{G}}^{\ell}(T)$ is precisely the kernel of the rotational
operator in $\boldsymbol{\mathcal{P}}^{\ell}(T)$. Since, by (28),
$\boldsymbol{\mathcal{G}}^{\ell}(T)_{\mid
F}{\times}\boldsymbol{n}_{F}=\boldsymbol{\mathcal{R}}^{\ell}(F)$, this
motivates the choice of $\boldsymbol{\mathcal{Q}}^{\ell}(F)$ such that
$\boldsymbol{\mathcal{R}}^{\ell}(F)\subseteq\boldsymbol{\mathcal{Q}}^{\ell}(F)\subseteq\boldsymbol{\mathcal{P}}^{\ell}(F)$,
the space $\boldsymbol{\mathcal{R}}^{\ell}(F)$ being the smallest admissible
set ensuring stability. If one wants to adapt the argument to the normal face
space, one must now control with a face projection of
$\eta_{T}\boldsymbol{v}_{T\mid F}{\cdot}\boldsymbol{n}_{F}$ (at least) the
normal trace of the $\boldsymbol{\mathcal{R}}^{\ell}(T)$-part of
$\eta_{T}\boldsymbol{v}_{T}$ (i.e.
$\eta_{T}\boldsymbol{v}_{T}-\boldsymbol{\operatorname{div}}^{-1}(\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}))$).
As a matter of fact, $\eta_{T}$ is a constant, so that
$\eta_{T}\boldsymbol{v}_{T}\in\boldsymbol{\mathcal{P}}^{\ell}(T)$, and
$\boldsymbol{\mathcal{R}}^{\ell}(T)$ is precisely the kernel of the divergence
operator in $\boldsymbol{\mathcal{P}}^{\ell}(T)$. Since, by (29),
$\boldsymbol{\mathcal{R}}^{\ell}(T)_{\mid
F}{\cdot}\boldsymbol{n}_{F}=\mathcal{P}^{\ell}(F)$, this means that choosing
$\mathcal{P}^{\ell}(F)$ as the normal face space is actually optimal.

For $q>2$, we define the following global reduction operator:
$\underline{\mathbbmsl{I}}^{\ell}_{h}:\boldsymbol{X}(\Omega)\cap\boldsymbol{L}^{q}(\Omega)\to\underline{\mathbbmsl{X}}_{h}^{\ell}$
is such that, for all
$\boldsymbol{v}\in\boldsymbol{X}(\Omega)\cap\boldsymbol{L}^{q}(\Omega)$,

$\underline{\mathbbmsl{I}}^{\ell}_{h}(\boldsymbol{v})\mathrel{\mathop{:}}=\left(\big{(}\boldsymbol{\pi}_{T}^{\ell}(\boldsymbol{v}_{\mid
T})\big{)}_{T\in\mathcal{T}_{h}},\big{(}\boldsymbol{\pi}_{\boldsymbol{\mathcal{Q}},F}^{\ell}(\boldsymbol{v}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F\in\mathcal{F}_{h}},\big{(}\pi_{F}^{\ell}((\eta\boldsymbol{v})_{\mid
F}{\cdot}\boldsymbol{n}_{F})\big{)}_{F\in\mathcal{F}_{h}}\right),$ (39)

where $\boldsymbol{v}_{\mid F}{\times}\boldsymbol{n}_{F}$ is identified to its
two-dimensional (tangential) counterpart.

######  Remark 11 (Regularity).

Following [27], for $\eta$ satisfying (2), the regularity
$\boldsymbol{v}\in\boldsymbol{X}(\Omega)\cap\boldsymbol{L}^{q}(\Omega)$ is
sufficient to give a (weak) meaning to the face polynomial projections in
(39). Besides, this regularity is typically satisfied by the solutions to the
Maxwell system with piecewise smooth parameter $\eta$ as in (24). Indeed,
functions in $\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ or
$\boldsymbol{X}_{n}(\Omega)$ with piecewise smooth $\eta$ belong to
$\boldsymbol{H}^{s}(\Omega)$ for some $s>0$ (cf. [20, 32, 9]), i.e. by Sobolev
embedding, to $\boldsymbol{L}^{q}(\Omega)$ for some $q>2$.

For all $T\in\mathcal{T}_{h}$, we also introduce the local reduction operator:
$\underline{\mathbbmsl{I}}^{\ell}_{T}:\boldsymbol{H}(\operatorname{\bf
curl};T)\cap\boldsymbol{H}(\operatorname{div}_{\eta};T)\cap\boldsymbol{L}^{q}(T)\to\underline{\mathbbmsl{X}}_{T}^{\ell}$
is such that, for all $\boldsymbol{v}\in\boldsymbol{H}(\operatorname{\bf
curl};T)\cap\boldsymbol{H}(\operatorname{div}_{\eta};T)\cap\boldsymbol{L}^{q}(T)$,

$\underline{\mathbbmsl{I}}^{\ell}_{T}(\boldsymbol{v})\mathrel{\mathop{:}}=\left(\boldsymbol{\pi}_{T}^{\ell}(\boldsymbol{v}),\big{(}\boldsymbol{\pi}_{\boldsymbol{\mathcal{Q}},F}^{\ell}(\boldsymbol{v}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F\in\mathcal{F}_{T}},\big{(}\pi_{F}^{\ell}(\eta_{T}\boldsymbol{v}_{\mid
F}{\cdot}\boldsymbol{n}_{F})\big{)}_{F\in\mathcal{F}_{T}}\right).$ (40)

The proof of the next lemma is straightforward based on the definitions (34)
and (40).

######  Lemma 12 (Optimal polynomial consistency).

Let $T\in\mathcal{T}_{h}$. For all
$\boldsymbol{p}\in\boldsymbol{\mathcal{P}}^{\ell}(T)$, letting
$\underline{\mathbbmsl{p}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$ be such
that
$\underline{\mathbbmsl{p}}_{T}\mathrel{\mathop{:}}=\underline{\mathbbmsl{I}}^{\ell}_{T}(\boldsymbol{p})$,
the following holds true:

  * $\bullet$

$s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{p}}_{T},\underline{\mathbbmsl{v}}_{T})=0$ for
all $\underline{\mathbbmsl{v}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$;

  * $\bullet$

$s_{\operatorname{div},T}(\underline{\mathbbmsl{p}}_{T},\underline{\mathbbmsl{v}}_{T})=0$
for all
$\underline{\mathbbmsl{v}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$.

The important Lemma 12 above guarantees that the stabilization Hermitian forms
(34) shall provide superconvergence when used within hybrid numerical schemes
(cf. [14, 12]). In the literature, this kind of stabilization is often
referred to as of Lehrenfeld–Schöberl type [33] (see also [16]).

##  4 Hybrid Weber inequalities

Let $\ell\in\mathbb{N}$ be a given polynomial degree.

###  4.1 First Weber inequality

Recall the definitions (33) of the discrete space
$\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$, (35)–(36) of the
hybrid semi-norms $|{\cdot}|_{\operatorname{\bf curl},h}$ and
$|{\cdot}|_{\operatorname{div},h}$, and (32) of the discrete $\eta$-weighted
normal trace operator $\gamma_{n,h}$.

######  Theorem 13 (First hybrid Weber inequality).

There exists $c_{W,1}>0$ independent of $h$ and $\eta$ such that, for all
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$,
one has

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}\leq
c_{W,1}\left(\eta_{\flat}^{-\frac{1}{2}}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}+\eta_{\sharp}^{\frac{1}{2}}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\Bigg{(}\sum_{j=1}^{\beta_{2}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\right).$
(41)

###### Proof.

Since
$\boldsymbol{v}_{h}\in\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})\subset\boldsymbol{L}^{2}(\Omega)$,
by the first Helmholtz–Hodge decomposition (7), there exist $\varphi\in
H^{1}_{0}(\Omega)$,
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{L}^{2}_{\boldsymbol{0}}(\Omega)$,
and $\boldsymbol{w}\in\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega)$ (writing
$\boldsymbol{w}=\operatorname{\bf grad}\omega$, with $\omega\in H^{1}(\Omega)$
such that
$\omega\mathrel{\mathop{:}}=\sum_{j=1}^{\beta_{2}}\alpha_{j}\omega_{j}$), such
that

$\boldsymbol{v}_{h}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w},$ (42)

with $\|\boldsymbol{\psi}\|_{1}\leq C_{\Omega,1}\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}$ by (8). From the decomposition (42), we directly
infer that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}^{2}=(\eta\boldsymbol{v}_{h},\operatorname{\bf
grad}\varphi)_{\Omega}+\eta_{\sharp}\big{(}\boldsymbol{v}_{h},\operatorname{\bf
curl}\boldsymbol{\psi}\big{)}_{\Omega}+\big{(}\eta\boldsymbol{v}_{h},\operatorname{\bf
grad}\omega\big{)}_{\Omega}=\mathrel{\mathop{:}}\mathcal{I}_{1}+\eta_{\sharp}\mathcal{I}_{2}+\mathcal{I}_{3}.$
(43)

Let us now estimate the three addends in the right-hand side of (43).

For $\mathcal{I}_{1}$, first remark that

$\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}[\eta
v]_{F,n},\varphi_{\mid F}\big{)}_{F}=0,$

as a consequence of the fact that $\varphi\in H^{1}_{0}(\Omega)$ (so that
$\varphi$ is single-valued at interfaces and vanishes on boundary faces) and
that, for any $F\subset\partial T^{+}\cap\partial
T^{-}\in\mathcal{F}_{h}^{\circ}$,
$\varepsilon_{T^{+},F}+\varepsilon_{T^{-},F}=0$. Hence, performing cell-by-
cell integration by parts, we get

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\big{(}\eta_{T}\boldsymbol{v}_{T},\operatorname{\bf
grad}\varphi)_{T}=\sum_{T\in\mathcal{T}_{h}}\bigg{(}-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\varphi\big{)}_{T}+\sum_{F\in\mathcal{F}_{T}}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{T,F},\varphi_{\mid F}\big{)}_{F}\bigg{)}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\varphi\big{)}_{T}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}-[\eta v]_{F,n},\varphi_{\mid F}\big{)}_{F}.$

Repeated applications of the Cauchy–Schwarz inequality then yield

$\mathcal{I}_{1}\leq|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\left(\|\varphi\|^{2}_{0}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}\|\varphi_{\mid
F}\|^{2}_{0,F}\right)^{\nicefrac{{1}}{{2}}}.$

Applying a continuous trace inequality (cf. e.g. [22, Lem. 1.31]) combined
with mesh regularity, using the fact that $h_{T}\leq{\rm diam}(\Omega)$ for
all $T\in\mathcal{T}_{h}$, and concluding with the Poincaré inequality applied
to $\varphi\in H^{1}_{0}(\Omega)$, we finally infer that

$\mathcal{I}_{1}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\|\operatorname{\bf
grad}\varphi\|_{0}.$ (44)

Let us now turn to the estimation of $\mathcal{I}_{2}$. In what follows,
following Remark 1, all tangential vector fields are pointwise identified to
their counterparts in $\mathbb{C}^{2}$. First, notice that

$\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}\boldsymbol{v}_{F,\boldsymbol{\tau}},\boldsymbol{n}_{F}{\times}(\boldsymbol{\psi}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F}=0,$

as a consequence of the fact that
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)$ (so that all components of
$\boldsymbol{\psi}$ are square-integrable and single-valued at interfaces),
that
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$
(so that $\boldsymbol{v}_{F,\boldsymbol{\tau}}\equiv\boldsymbol{0}$ for all
$F\in\mathcal{F}_{h}^{\partial}$), and that, for any $F\subset\partial
T^{+}\cap\partial T^{-}\in\mathcal{F}_{h}^{\circ}$,
$\varepsilon_{T^{+},F}+\varepsilon_{T^{-},F}=0$. Hence, performing cell-by-
cell integration by parts yields

$\displaystyle\mathcal{I}_{2}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\big{(}\boldsymbol{v}_{T},\operatorname{\bf
curl}\boldsymbol{\psi}\big{)}_{T}=\sum_{T\in\mathcal{T}_{h}}\bigg{(}\big{(}\operatorname{\bf
curl}\boldsymbol{v}_{T},\boldsymbol{\psi}\big{)}_{T}+\sum_{F\in\mathcal{F}_{T}}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{T,F},\boldsymbol{n}_{F}{\times}(\boldsymbol{\psi}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F}\bigg{)}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\big{(}\operatorname{\bf
curl}\boldsymbol{v}_{T},\boldsymbol{\psi}\big{)}_{T}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}},\boldsymbol{n}_{F}{\times}(\boldsymbol{\psi}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F}.$

By repeated applications of the Cauchy–Schwarz inequality, we then get

$\mathcal{I}_{2}\leq\Bigg{(}\|\operatorname{\bf
curl}_{h}\\!\boldsymbol{v}_{h}\|_{0}^{2}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}^{-1}\|\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\|_{0,F}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\\\
\times\Bigg{(}\|\boldsymbol{\psi}\|_{0}^{2}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}\|\boldsymbol{n}_{F}{\times}(\boldsymbol{\psi}_{\mid
F}{\times}\boldsymbol{n}_{F})\|_{0,F}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}.$ (45)

Let us focus on the first factor in the right-hand side of (45). By the
estimate (38), and the definition (35) of $|{\cdot}|_{\operatorname{\bf
curl},h}$, we infer that

$\Bigg{(}\|\operatorname{\bf
curl}_{h}\\!\boldsymbol{v}_{h}\|_{0}^{2}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}^{-1}\|\boldsymbol{v}_{T\mid
F}{\times}\boldsymbol{n}_{F}-\boldsymbol{v}_{F,\boldsymbol{\tau}}\|_{0,F}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}.$

For the second factor in the right-hand side of (45), applying a continuous
trace inequality combined with mesh regularity, using the fact that
$h_{T}\leq{\rm diam}(\Omega)$ for all $T\in\mathcal{T}_{h}$, and concluding
with the estimate $\|\boldsymbol{\psi}\|_{1}\leq
C_{\Omega,1}\|\operatorname{\bf curl}\boldsymbol{\psi}\|_{0}$, there holds

$\left(\|\boldsymbol{\psi}\|_{0}^{2}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}\|\boldsymbol{n}_{F}{\times}(\boldsymbol{\psi}_{\mid
F}{\times}\boldsymbol{n}_{F})\|_{0,F}^{2}\right)^{\nicefrac{{1}}{{2}}}\lesssim\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}.$

Collecting the different results, we finally infer that

$\mathcal{I}_{2}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}\|\operatorname{\bf curl}\boldsymbol{\psi}\|_{0}.$ (46)

We are now left with estimating $\mathcal{I}_{3}$. First, remark that

$\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}[\eta
v]_{F,n},\omega_{\mid
F}\big{)}_{F}=\sum_{j=1}^{\beta_{2}}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\overline{\alpha_{j}},$

as a consequence of the fact that $\omega\in H^{1}(\Omega)$ (so that $\omega$
is single-valued at interfaces), that
$\omega=\sum_{j=1}^{\beta_{2}}\alpha_{j}\omega_{j}$ with
$\omega_{j\mid\Gamma_{j^{\prime}}}=\delta_{jj^{\prime}}$ for all
$j^{\prime}\in\\{0,\ldots,\beta_{2}\\}$ (see (5)), and that, for any
$F\subset\partial T^{+}\cap\partial T^{-}\in\mathcal{F}_{h}^{\circ}$,
$\varepsilon_{T^{+},F}+\varepsilon_{T^{-},F}=0$, whereas for any
$F\subset\partial T\cap\Gamma\in\mathcal{F}_{h}^{\partial}$,
$\varepsilon_{T,F}=1$. Hence, performing cell-by-cell integration by parts, we
get

$\displaystyle\mathcal{I}_{3}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\big{(}\eta_{T}\boldsymbol{v}_{T},\operatorname{\bf
grad}\omega\big{)}_{T}=\sum_{T\in\mathcal{T}_{h}}\left(-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\omega\big{)}_{T}+\sum_{F\in\mathcal{F}_{T}}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{T,F},\omega_{\mid F}\big{)}_{F}\right)$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\omega\big{)}_{T}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}-[\eta v]_{F,n},\omega_{\mid F}\big{)}_{F}$
$\displaystyle+\sum_{j=1}^{\beta_{2}}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\overline{\alpha_{j}}.$

Repeated applications of the Cauchy–Schwarz inequality then yield

$\mathcal{I}_{3}\leq|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\left(\|\omega\|^{2}_{0}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}\|\omega_{\mid
F}\|^{2}_{0,F}\right)^{\nicefrac{{1}}{{2}}}+\Bigg{(}\sum_{j=1}^{\beta_{2}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}|\boldsymbol{\alpha}_{\boldsymbol{w}}|,$

where $|\boldsymbol{\alpha}_{\boldsymbol{w}}|$ denotes the Euclidean norm of
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{j}\in\mathbb{C})_{j\in\\{1,\dots,\beta_{2}\\}}$.
Proceeding as in the continuous setting to estimate
$|\boldsymbol{\alpha}_{\boldsymbol{w}}|$ (cf. (17)), and as for the term
$\mathcal{I}_{1}$ to estimate the $\omega$-factor (recall that
$\omega_{\mid\Gamma_{0}}=0$), we finally infer that

$\mathcal{I}_{3}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\|\operatorname{\bf
grad}\omega\|_{0}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\Bigg{(}\sum_{j=1}^{\beta_{2}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\omega\|_{0}.$ (47)

By the $\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonality of the decomposition
(42), there holds $\|\operatorname{\bf
grad}\varphi\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$,
$\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\leq\eta_{\sharp}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$,
and $\|\operatorname{\bf
grad}\omega\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\operatorname{\bf
grad}\omega\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$.
Plugging the three estimates (44), (46), and (47) into (43), we eventually
obtain (41). ∎

######  Remark 14 (Useful variant).

Let us introduce the following hybrid counterpart of the space
$\boldsymbol{H}(\operatorname{\bf curl};\Omega)$:

$\underline{\boldsymbol{X}}_{h}^{\ell}\mathrel{\mathop{:}}=\left\\{\underline{\boldsymbol{v}}_{h}\mathrel{\mathop{:}}=\Big{(}(\boldsymbol{v}_{T})_{T\in\mathcal{T}_{h}},(\boldsymbol{v}_{F,\boldsymbol{\tau}})_{F\in\mathcal{F}_{h}}\Big{)}\leavevmode\nobreak\
:\leavevmode\nobreak\ \begin{aligned}
\boldsymbol{v}_{T}&\in\boldsymbol{\mathcal{P}}^{\ell}(T)&\quad&\forall
T\in\mathcal{T}_{h}\\\
\boldsymbol{v}_{F,\boldsymbol{\tau}}&\in\boldsymbol{\mathcal{Q}}^{\ell}(F)&\quad&\forall
F\in\mathcal{F}_{h}\end{aligned}\right\\},$ (48)

along with its subspace
$\underline{\boldsymbol{X}}_{h,\boldsymbol{\tau}}^{\ell}\mathrel{\mathop{:}}=\left\\{\underline{\boldsymbol{v}}_{h}\in\underline{\boldsymbol{X}}_{h}^{\ell}\mid\boldsymbol{v}_{F,\boldsymbol{\tau}}\equiv\boldsymbol{0}\,\forall
F\in\mathcal{F}_{h}^{\partial}\right\\}$ (hybrid counterpart of
$\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf curl};\Omega)$). Following
Remark 3, assume that
$\underline{\boldsymbol{v}}_{h}\in\underline{\boldsymbol{X}}_{h,\boldsymbol{\tau}}^{\ell}$
is such that

$(\eta\boldsymbol{v}_{h},\boldsymbol{z})_{\Omega}=0\qquad\forall\boldsymbol{z}\in\operatorname{\bf
grad}\big{(}H^{1}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{\boldsymbol{\tau}}(\Omega).$
(49)

Then, following the proof of Theorem 13, and (seamlessly) restricting the
definition (35) of $|{\cdot}|_{\operatorname{\bf curl},h}$ to vectors in
$\underline{\boldsymbol{X}}_{h}^{\ell}$, one can show that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}\lesssim\eta_{\sharp}^{\frac{1}{2}}|\underline{\boldsymbol{v}}_{h}|_{\operatorname{\bf
curl},h},$ (50)

which is a complex-valued, variable-$\eta$ extension to possibly non-trivial
topologies of [12, Thm. 2.1]. We refer the reader to [12] for an example, in
the magnetostatics framework on trivial domains, of how the orthogonality
condition (49) may be (possibly approximately) imposed at the discrete level
on general polyhedral mesh families.

###  4.2 Second Weber inequality

Recall the definitions (33) of the discrete space
$\underline{\mathbbmsl{X}}_{h,n}^{\ell}$, (35)–(36) of the hybrid semi-norms
$|{\cdot}|_{\operatorname{\bf curl},h}$ and
$|{\cdot}|_{\operatorname{div},h}$, and (32) of the discrete $\eta$-weighted
normal trace operator $\gamma_{n,h}$.

######  Theorem 15 (Second hybrid Weber inequality).

There exists $c_{W,2}>0$ independent of $h$ and $\eta$ such that, for all
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h,n}^{\ell}$, one
has

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}\leq
c_{W,2}\left(\eta_{\flat}^{-\frac{1}{2}}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}+\eta_{\sharp}^{\frac{1}{2}}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\Bigg{(}\sum_{i=1}^{\beta_{1}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Sigma_{i}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\right).$
(51)

###### Proof.

Since
$\boldsymbol{v}_{h}\in\boldsymbol{\mathcal{P}}^{\ell}(\mathcal{T}_{h})\subset\boldsymbol{L}^{2}(\Omega)$,
by the second Helmholtz–Hodge decomposition (12), there exist $\varphi\in
H^{1}(\Omega)\cap L^{2}_{0}(\Omega)$,
$\boldsymbol{\psi}\in\boldsymbol{H}^{1}(\Omega)\cap\boldsymbol{H}_{\boldsymbol{0}}(\operatorname{\bf
curl};\Omega)$, and $\boldsymbol{w}\in\boldsymbol{H}_{n}(\Omega)$ (writing
$\boldsymbol{w}=\check{\operatorname{\bf grad}\pi}$, with $\pi\in
H^{1}(\hat{\Omega})\cap L^{2}_{0}(\hat{\Omega})$ such that
$\pi\mathrel{\mathop{:}}=\sum_{i=1}^{\beta_{1}}\alpha_{i}\pi_{i}$), such that

$\boldsymbol{v}_{h}=\operatorname{\bf
grad}\varphi+\frac{\eta_{\sharp}}{\eta}\operatorname{\bf
curl}\boldsymbol{\psi}+\boldsymbol{w},$ (52)

with $\|\boldsymbol{\psi}\|_{1}\leq C_{\Omega,2}\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}$ by (13). From the decomposition (52), we
directly infer that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}^{2}=(\eta\boldsymbol{v}_{h},\operatorname{\bf
grad}\varphi)_{\Omega}+\eta_{\sharp}\big{(}\boldsymbol{v}_{h},\operatorname{\bf
curl}\boldsymbol{\psi}\big{)}_{\Omega}+\big{(}\eta\boldsymbol{v}_{h},\operatorname{\bf
grad}\pi\big{)}_{\hat{\Omega}}=\mathrel{\mathop{:}}\mathcal{I}_{1}+\eta_{\sharp}\mathcal{I}_{2}+\mathcal{I}_{3}.$
(53)

Let us now estimate the three addends in the right-hand side of (53).

For $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$, it is straightforward to prove,
adapting the arguments advocated in the proof of Theorem 13, that

$\mathcal{I}_{1}+\eta_{\sharp}\mathcal{I}_{2}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\|\operatorname{\bf
grad}\varphi\|_{0}+\eta_{\sharp}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}\|\operatorname{\bf curl}\boldsymbol{\psi}\|_{0}.$ (54)

Let us now focus on $\mathcal{I}_{3}$. First, remark that

$\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}[\eta
v]_{F,n},(\pi_{\mid T})_{\mid
F}\big{)}_{F}=\sum_{i=1}^{\beta_{1}}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Sigma_{i}}\overline{\alpha_{i}},$

as a consequence of the fact that $\pi\in H^{1}(\hat{\Omega})$ (so that $\pi$
is single-valued at interfaces $F\in\hat{\mathcal{F}}_{h}^{\circ}$), that
$\pi=\sum_{i=1}^{\beta_{1}}\alpha_{i}\pi_{i}$ with
$\llbracket\pi_{i}\rrbracket_{\Sigma_{i^{\prime}}}=\delta_{ii^{\prime}}$ for
all $i^{\prime}\in\\{1,\ldots,\beta_{1}\\}$ (see (10)), that
$\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h,n}^{\ell}$ (so
that $[\eta v]_{F,n}=0$ for all $F\in\mathcal{F}_{h}^{\partial}$), and that,
for any $F\subset\partial T^{+}\cap\partial T^{-}\in\mathcal{F}_{h}^{\circ}$,
$\varepsilon_{T^{+},F}+\varepsilon_{T^{-},F}=0$ with, for any
$i\in\\{1,\ldots,\beta_{1}\\}$, and any $F\subset\partial T^{+}\cap\partial
T^{-}\in\mathcal{F}_{h,\Sigma_{i}}^{\circ}$, $\varepsilon_{T^{+},F}=1$. Hence,
performing cell-by-cell integration by parts, we get

$\displaystyle\mathcal{I}_{3}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\big{(}\eta_{T}\boldsymbol{v}_{T},\operatorname{\bf
grad}\pi)_{T}=\sum_{T\in\mathcal{T}_{h}}\left(-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\pi\big{)}_{T}+\sum_{F\in\mathcal{F}_{T}}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{T,F},(\pi_{\mid T})_{\mid F}\big{)}_{F}\right)$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}-\big{(}\operatorname{div}(\eta_{T}\boldsymbol{v}_{T}),\pi\big{)}_{T}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}\varepsilon_{T,F}\big{(}\eta_{T}\boldsymbol{v}_{T\mid
F}{\cdot}\boldsymbol{n}_{F}-[\eta v]_{F,n},(\pi_{\mid T})_{\mid F}\big{)}_{F}$
$\displaystyle+\sum_{i=1}^{\beta_{1}}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Sigma_{i}}\overline{\alpha_{i}}.$

Repeated applications of the Cauchy–Schwarz inequality then yield

$\mathcal{I}_{3}\leq|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\left(\|\pi\|^{2}_{0,\hat{\Omega}}+\sum_{T\in\mathcal{T}_{h}}\sum_{F\in\mathcal{F}_{T}}h_{F}\|(\pi_{\mid
T})_{\mid
F}\|^{2}_{0,F}\right)^{\nicefrac{{1}}{{2}}}+\Bigg{(}\sum_{i=1}^{\beta_{1}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Sigma_{i}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}|\boldsymbol{\alpha}_{\boldsymbol{w}}|,$

where $|\boldsymbol{\alpha}_{\boldsymbol{w}}|$ denotes the Euclidean norm of
$\boldsymbol{\alpha}_{\boldsymbol{w}}\mathrel{\mathop{:}}=(\alpha_{i}\in\mathbb{C})_{i\in\\{1,\dots,\beta_{1}\\}}$.
Proceeding as in the continuous setting (cf. (22)) to estimate
$|\boldsymbol{\alpha}_{\boldsymbol{w}}|$, and applying to the $\pi$-factor a
continuous trace inequality combined with mesh regularity (and the fact that
$h_{T}\leq{\rm diam}(\Omega)$ for all $T\in\mathcal{T}_{h}$), along with a
Poincaré–Steklov inequality (recall that $\pi\in H^{1}(\hat{\Omega})\cap
L^{2}_{0}(\hat{\Omega})$ and that $\hat{\Omega}$ is assumed to be connected),
we finally infer

$\mathcal{I}_{3}\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}\|\check{\operatorname{\bf
grad}\pi}\|_{0}+\eta_{\flat}^{-\frac{1}{2}}\kappa_{\eta}^{\frac{1}{2}}\Bigg{(}\sum_{i=1}^{\beta_{1}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Sigma_{i}}\big{|}^{2}\Bigg{)}^{\nicefrac{{1}}{{2}}}\|\eta^{\nicefrac{{1}}{{2}}}\check{\operatorname{\bf
grad}\pi}\|_{0},$ (55)

where we also used that $\|\operatorname{\bf
grad}\pi\|_{0,\hat{\Omega}}=\|\check{\operatorname{\bf grad}\pi}\|_{0}$.

By the $\boldsymbol{L}^{2}_{\eta}(\Omega)$-orthogonality of the decomposition
(52), there holds $\|\operatorname{\bf
grad}\varphi\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$,
$\|\operatorname{\bf
curl}\boldsymbol{\psi}\|_{0}\leq\eta_{\sharp}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$,
and $\|\check{\operatorname{\bf
grad}\pi}\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\check{\operatorname{\bf
grad}\pi}\|_{0}\leq\eta_{\flat}^{-\frac{1}{2}}\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}$.
Plugging the two estimates (54) and (55) into (53), we eventually obtain (51).
∎

######  Remark 16 (Useful variant).

Recall the definition (48) of the hybrid counterpart
$\underline{\boldsymbol{X}}_{h}^{\ell}$ of the space
$\boldsymbol{H}(\operatorname{\bf curl};\Omega)$. Following Remark 6, assume
that $\underline{\boldsymbol{v}}_{h}\in\underline{\boldsymbol{X}}_{h}^{\ell}$
is such that

$(\eta\boldsymbol{v}_{h},\boldsymbol{z})_{\Omega}=0\qquad\forall\boldsymbol{z}\in\operatorname{\bf
grad}\big{(}H^{1}(\Omega)\cap
L^{2}_{0}(\Omega)\big{)}\overset{\perp_{\eta}}{\oplus}\boldsymbol{H}_{n}(\Omega).$
(56)

Then, revisiting the proof of Theorem 15, one can show that

$\|\eta^{\frac{1}{2}}\boldsymbol{v}_{h}\|_{0}\lesssim\eta_{\sharp}^{\frac{1}{2}}|\underline{\boldsymbol{v}}_{h}|_{\operatorname{\bf
curl},h}.$ (57)

##  5 Discrete Maxwell compactness

Let $\ell,m,p\in\mathbb{N}$ be three given polynomial degrees.

###  5.1 Discrete differential operators

Recall the definitions (30) of the discrete space
$\underline{\mathbbmsl{X}}_{h}^{\ell}$, (34) of the local Hermitian forms
$s_{\operatorname{\bf curl},T}$ and $s_{\operatorname{div},T}$, and (35)–(36)
of the hybrid semi-norms $|{\cdot}|_{\operatorname{\bf curl},h}$ and
$|{\cdot}|_{\operatorname{div},h}$.

We define the (global) rotational reconstruction operator
$\boldsymbol{C}_{h}^{m}:\underline{\mathbbmsl{X}}_{h}^{\ell}\to\boldsymbol{\mathcal{P}}^{m}(\mathcal{T}_{h})$
through its local restrictions
$\boldsymbol{C}_{T}^{m}:\underline{\mathbbmsl{X}}_{T}^{\ell}\to\boldsymbol{\mathcal{P}}^{m}(T)$
to any mesh cell $T\in\mathcal{T}_{h}$: for all
$\underline{\mathbbmsl{v}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$,
$\boldsymbol{C}_{T}^{m}(\underline{\mathbbmsl{v}}_{T})\in\boldsymbol{\mathcal{P}}^{m}(T)$
is the unique solution to

$\big{(}\boldsymbol{C}_{T}^{m}(\underline{\mathbbmsl{v}}_{T}),\boldsymbol{q}\big{)}_{T}=(\boldsymbol{v}_{T},\operatorname{\bf
curl}\boldsymbol{q})_{T}-\\!\\!\\!\sum_{F\in\mathcal{F}_{T}}\\!\varepsilon_{T,F}\big{(}\boldsymbol{v}_{F,\boldsymbol{\tau}},\boldsymbol{n}_{F}{\times}(\boldsymbol{q}_{\mid
F}{\times}\boldsymbol{n}_{F})\big{)}_{F}\qquad\forall\boldsymbol{q}\in\boldsymbol{\mathcal{P}}^{m}(T),$
(58)

where, as now standard, $\boldsymbol{n}_{F}{\times}(\boldsymbol{q}_{\mid
F}{\times}\boldsymbol{n}_{F})$ is identified to its two-dimensional
(tangential) counterpart. Performing an integration by parts of the term
$(\boldsymbol{v}_{T},\operatorname{\bf curl}\boldsymbol{q})_{T}$, and using a
discrete trace inequality along with Lemma 9, it is an easy matter to prove
that

$\sum_{T\in\mathcal{T}_{h}}\left(\|\boldsymbol{C}_{T}^{m}(\underline{\mathbbmsl{v}}_{T})\|_{0,T}^{2}+s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right)\lesssim|\underline{\mathbbmsl{v}}_{h}|^{2}_{\operatorname{\bf
curl},h}\qquad\forall\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}.$
(59)

If $m\geq\ell-1$, and $\boldsymbol{\mathcal{Q}}^{\ell}(F)$ (already satisfying
(31)) also satisfies
$\boldsymbol{\mathcal{P}}^{\ell-1}(F)\subset\boldsymbol{\mathcal{Q}}^{\ell}(F)$
for all $F\in\mathcal{F}_{h}$, one can also prove that
$|\underline{\mathbbmsl{v}}_{h}|^{2}_{\operatorname{\bf
curl},h}\lesssim\sum_{T\in\mathcal{T}_{h}}\left(\|\boldsymbol{C}_{T}^{m}(\underline{\mathbbmsl{v}}_{T})\|_{0,T}^{2}+s_{\operatorname{\bf
curl},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right)$.

In the same vein, we define the (global) divergence reconstruction operator
$D_{h}^{p}:\underline{\mathbbmsl{X}}_{h}^{\ell}\to\mathcal{P}^{p}(\mathcal{T}_{h})$
through its local restrictions
$D_{T}^{p}:\underline{\mathbbmsl{X}}_{T}^{\ell}\to\mathcal{P}^{p}(T)$ to any
mesh cell $T\in\mathcal{T}_{h}$: for all
$\underline{\mathbbmsl{v}}_{T}\in\underline{\mathbbmsl{X}}_{T}^{\ell}$,
$D_{T}^{p}(\underline{\mathbbmsl{v}}_{T})\in\mathcal{P}^{p}(T)$ is the unique
solution to

$\big{(}D_{T}^{p}(\underline{\mathbbmsl{v}}_{T}),q\big{)}_{T}=-(\eta_{T}\boldsymbol{v}_{T},\operatorname{\bf
grad}q)_{T}+\\!\\!\sum_{F\in\mathcal{F}_{T}}\\!\varepsilon_{T,F}\big{(}[\eta
v]_{F,n},q_{\mid F}\big{)}_{F}\qquad\forall q\in\mathcal{P}^{p}(T).$ (60)

Performing an integration by parts of the term
$(\eta_{T}\boldsymbol{v}_{T},\operatorname{\bf grad}q)_{T}$, and using a
discrete trace inequality, it is an easy matter to prove that

$\sum_{T\in\mathcal{T}_{h}}\left(\|D_{T}^{p}(\underline{\mathbbmsl{v}}_{T})\|_{0,T}^{2}+s_{\operatorname{div},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right)\lesssim|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}^{2}\qquad\forall\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h}^{\ell}.$
(61)

If $p\geq\ell-1$, one can also prove that
$|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}^{2}\lesssim\sum_{T\in\mathcal{T}_{h}}\left(\|D_{T}^{p}(\underline{\mathbbmsl{v}}_{T})\|_{0,T}^{2}+s_{\operatorname{div},T}(\underline{\mathbbmsl{v}}_{T},\underline{\mathbbmsl{v}}_{T})\right)$.

###  5.2 Compactness in
$\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$

Recall the definitions (33) of the discrete space
$\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$, and (35)–(36) of the
hybrid semi-norms $|{\cdot}|_{\operatorname{\bf curl},h}$ and
$|{\cdot}|_{\operatorname{div},h}$. Leveraging the first hybrid Weber
inequality from Theorem 13, we infer that

$\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X}_{\boldsymbol{\tau}},h}^{2}\mathrel{\mathop{:}}=\eta_{\flat}^{-1}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{div},h}^{2}+\eta_{\sharp}|\underline{\mathbbmsl{v}}_{h}|_{\operatorname{\bf
curl},h}^{2}+\eta_{\flat}^{-1}\sum_{j=1}^{\beta_{2}}\big{|}\big{(}\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h}),1\big{)}_{\Gamma_{j}}\big{|}^{2}$
(62)

defines a norm on $\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$,
which can be proven uniformly (in $h$) equivalent to the
$\|{\cdot}\|_{\boldsymbol{X},h}$-norm (37): there is
$c_{\boldsymbol{\tau}}>0$, independent of $h$ and $\eta$, such that

$(c_{\boldsymbol{\tau}}\kappa_{\eta})^{-1}\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X},h}^{2}\leq\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X}_{\boldsymbol{\tau}},h}^{2}\leq
c_{\boldsymbol{\tau}}\kappa_{\eta}\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X},h}^{2}\qquad\forall\underline{\mathbbmsl{v}}_{h}\in\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}.$

######  Theorem 17 (Maxwell compactness in
$\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$).

Let $\big{(}\mathcal{M}_{h}\big{)}_{h\in\mathcal{H}}$ be a regular mesh family
as defined in Section 3.1. Let
$(\underline{\mathbbmsl{v}}_{h})_{h\in\mathcal{H}}$ be a sequence of elements
of $\underline{\mathbbmsl{X}}_{h,\boldsymbol{\tau}}^{\ell}$ for which there
exists a real number $c_{M}>0$ (independent of $h$) such that
$\eta_{\sharp}^{-\frac{1}{2}}\|\underline{\mathbbmsl{v}}_{h}\|_{\boldsymbol{X}_{\boldsymbol{\tau}},h}\leq
c_{M}$ for all $h\in\mathcal{H}$. Then, there exists an element
$\boldsymbol{v}\in\boldsymbol{X}_{\boldsymbol{\tau}}(\Omega)$ such that, along
a subsequence, as $h\to 0$,

  1. (a)

$\boldsymbol{v}_{h}\rightarrow\boldsymbol{v}$ strongly in
$\boldsymbol{L}^{2}(\Omega)$;

  2. (b)

$\boldsymbol{C}_{h}^{m}(\underline{\mathbbmsl{v}}_{h})\rightharpoonup\operatorname{\bf
curl}\boldsymbol{v}$ weakly in $\boldsymbol{L}^{2}(\Omega)$;

  3. (c)

$D_{h}^{p}(\underline{\mathbbmsl{v}}_{h})\rightharpoonup\operatorname{div}(\eta\boldsymbol{v})$
weakly in $L^{2}(\Omega)$;

  4. (d)

$\gamma_{n,h}(\underline{\mathbbmsl{v}}_{h})\rightharpoonup(\eta\boldsymbol{v})_{\mid\Gamma}{\cdot}\boldsymbol{n}$
weakly in $H^{-\frac{1}{2}}(\Gamma)$.

###### Proof.