# A weak Galerkin finite element method for solving the asymptotic lower bound
of Maxwell eigenvalue problem

Shusheng Li School of Mathematics, Jilin University, Changchun, 130012, China
(ssli22@mails.jlu.edu.cn). Qilong Zhai School of Mathematics, Jilin
University, Changchun, 130012, China (zhaiql@jlu.edu.cn).

###### Abstract

In this paper, we propose a weak Galerkin (WG) finite element method for the
Maxwell eigenvalue problem. By restricting subspaces, we transform the mixed
form of Maxwell eigenvalue problem into simple elliptic equation. Then we give
the WG numerical scheme for the Maxwell eigenvalue problem. Furthermore, we
obtain the optimal error estimates of arbitrarily high convergence order and
prove the lower bound property of numerical solutions for eigenvalues.
Numerical experiments show the accuracy of theoretical analysis and the
property of lower bound.

###### keywords:

weak Galerkin finite element method, Maxwell eigenvalue problem, asymptotic
lower bound, error estimate.

###### AMS:

65N15, 65N25, 65N30

##  1 Introduction

The computation of eigenvalues in Maxwell equations is crucial in
computational electromagnetics. Since Silvester introduced the nodal-based
finite element method to electrical engineering [23], it has demonstrated
several advantages for solving homogeneous waveguide problems. However, when
applied to inhomogeneous waveguide problems, this method can yield spurious
modes with nonzero eigenvalues. Considerable effort has been dedicated to
reducing or eliminating these unwanted numerical artifacts caused by spurious
modes [8]. Rahman and Winkler noted that spurious modes fail to satisfy the
zero divergence condition on the electric or magnetic field, and they proposed
a penalty function to enforce this condition [10, 22, 26]. Unfortunately, this
approach does not completely eliminate the spurious modes and requires users
to select an appropriate penalty function parameter.

Another approach to eliminating spurious modes in finite element waveguide
problems involves selecting appropriate finite element approximation functions
[8, 20, 21]. It is well-established that using
$H(\rm{curl;\Omega})$-conforming basis functions (also known as edge elements)
for the electric field [20, 21] ensure the continuity of tangential field
components across interfaces between different media, while allowing normal
field components to jump across such interfaces. With the edge element method,
spurious modes with nonzero eigenvalues were eliminated. However, the number
of spurious modes with zero eigenvalues equal the number of nodal points
inside the computational domain, due to the violation of Gauss’s law [2].
Therefore, to remove these zero eigenvalues, it is necessary to enforce the
divergence-free property (Gauss’s law) of the eigenfunction in addition to
using the proper finite element space. One effective approach for computing
eigenvalues in waveguide problems is to employ
$H(\rm{curl;\Omega})$-conforming basis functions to approximate the electric
field, while imposing the divergence-free condition using a Lagrange
multiplier, as suggested by Kikuchi [9].

Meanwhile, higher-order methods like the spectral element method [11, 15] have
been proposed to solve electromagnetic eigenvalue problems. Despite their high
convergence rates, these methods also suffer from the presence of spurious
zero eigenvalues. In [14], Liu et al. introduced the mixed spectral element
method, which incorporates the divergence-free equation from Kikuchi’s scheme
into the spectral element method.

There are many methods to study the Maxwell eigenvalue problems. Based on a
Helmholtz decomposition of the error, Boffi et al. [5] presented an a
posteriori estimator of the error in the $L^{2}$-norm for the numerical
approximation of the Maxwell eigenvalue problems using
N${\rm\acute{e}}$d${\rm\acute{e}}$lec finite elements. Furthermore, based on
edge finite elements, Boffi and Gastaldi [4] proved the optimal convergence of
a adaptive scheme. In [6], Boffi et al. introduced a residual error indicator
for the N${\rm\acute{e}}$d${\rm\acute{e}}$lec finite element approximation for
Maxwell eigenvalue problems. In addition, based on a domain decomposition
method, Liang and Xu [12] proposed a two-level preconditioned Helmholtz
subspace iterative method for solving algebraic eigenvalue problems arising
from edge element approximations of Maxwell eigenvalue problems. In certain
settings, Boffi et al. [7] prove the convergence of the discrete eigenvalues
using Lagrange finite elements.

The outline of this paper is as follows. In Section 2, we introduce the
Maxwell eigenvalue problem and three equivalent variational forms are given.
In Section 3, we introduce the weak Galerkin (WG) finite element method for
the Maxwell eigenvalue problem, then propose the WG numerical scheme. The
error estimates of eigenfunctions and eigenvalues are analyzed in Section 4.
Section 5 is devoted to prove the lower bounda of eigenvalues. Numerical
examples are presented to our theoretical analysis in Section 6.

##  2 The Maxwell eigenvalue problem

In this paper, for simplicity, we consider the following Maxwell eigenvalue
problem

(2.1) $\left\\{\begin{aligned}
\nabla\times\mu_{r}^{-1}\nabla\times\mathbf{u}&=k^{2}\varepsilon_{r}\mathbf{u},&\quad&\text{in
}\Omega,&\\\ \nabla\cdot\varepsilon_{r}\mathbf{u}&=0,&\quad&\text{in
}\Omega,&\\\ \mathbf{u}\times\mathbf{n}&=0,&\quad&\text{on
}\partial\Omega,&\end{aligned}\right.$

where $\Omega$ is a polygon region in $\mathbb{R}^{d}$ $(d=2,3)$, and
$\varepsilon_{r}$ and $\mu_{r}$ denote the relative dielectric permittivity
and magnetic permeability tensors, respectively, and $\mathbf{n}$ is the
outward unit normal. For convenience, we consider the case where
$\varepsilon_{r}$ and $\mu_{r}$ are constants in this paper.

The standard Sobolev space notation are also used in this paper. Let $D$ be
any open bounded domain with Lipschitz continuous boundary in $\mathbb{R}^{d}$
$(d=2,3)$. We use the standard definition for the Sobolev space $H^{s}(D)$ and
their associated inner products $(\cdot,\cdot)_{s,D}$, norms
$\|\cdot\|_{s,D}$, and seminorms $|\cdot|_{s,D}$ for any $s\geq 0$. When
$D=\Omega$, we shall drop the subscript $D$ in the norm and in the inner
product notation. Next, we define the curl operators in 2D:

$\nabla\times\mathbf{v}=\partial_{x}v_{2}-\partial_{y}v_{1},\quad\nabla\times
w=(\partial_{y}w,-\partial_{x}w)^{T}.$

In addition, we define some Sobolev space used in this paper.

$\displaystyle H({\rm
curl};\Omega)=\left\\{\mathbf{v}\in[L^{2}(\Omega)]^{d}:\nabla\times\mathbf{v}\in[L^{2}(\Omega)]^{2d-3}\right\\},$
$\displaystyle H_{0}({\rm curl};\Omega)=\\{\mathbf{v}\in H({\rm
curl};\Omega):\mathbf{v}\times\mathbf{n}=0\text{ on }\partial\Omega\\},$
$\displaystyle H_{0}({\rm curl^{0}};\Omega)=\\{\mathbf{v}\in H_{0}({\rm
curl};\Omega):\nabla\times\mathbf{v}=0\\},$ $\displaystyle H({\rm
div};\Omega)=\left\\{\mathbf{v}\in[L^{2}(\Omega)]^{d}:\nabla\cdot\mathbf{v}\in
L^{2}(\Omega)\right\\},$ $\displaystyle H({\rm
div^{0}};\Omega)=\\{\mathbf{v}\in H({\rm
div};\Omega):\nabla\cdot\mathbf{v}=0\\}.$

Then we introduce the following Helmholtz decomposition:

$H_{0}({\rm curl};\Omega)=U_{0}\oplus\nabla H_{0}^{1}(\Omega),$

where $U_{0}=\\{\mathbf{v}\in H_{0}({\rm curl};\Omega):(\mathbf{v},\nabla
p)=0,\forall p\in H_{0}^{1}(\Omega)\\}=\\{\mathbf{v}\in H_{0}({\rm
curl};\Omega):\nabla\cdot\mathbf{v}=0\\}$ and $\nabla
H_{0}^{1}(\Omega)=H_{0}({\rm curl^{0}};\Omega)$.

The corresponding variational form of (2.1) is: Find $\lambda\in\mathbb{R}$
and $\mathbf{u}\in H_{0}({\rm curl};\Omega)\setminus\\{0\\}$ such that

(2.2) $\left\\{\begin{aligned}
(\mu_{r}^{-1}\nabla\times\mathbf{u},\nabla\times\mathbf{v})&=\lambda(\varepsilon_{r}\mathbf{u},\mathbf{v}),&\quad&\forall\mathbf{v}\in
H_{0}({\rm curl};\Omega),&\\\ (\varepsilon_{r}\mathbf{u},\nabla
p)&=0,&\quad&\forall p\in H_{0}^{1}(\Omega).&\end{aligned}\right.$

Here we introduce a mixed form [9] that is equivalent to (2.2): Find
$\lambda\in\mathbb{R}$ and $\mathbf{u}\in H_{0}({\rm
curl};\Omega)\setminus\\{0\\}$, $p\in H_{0}^{1}(\Omega)$ such that

(2.3) $\left\\{\begin{aligned}
(\mu_{r}^{-1}\nabla\times\mathbf{u},\nabla\times\mathbf{v})+(\nabla
p,\varepsilon_{r}\mathbf{v})&=\lambda(\varepsilon_{r}\mathbf{u},\mathbf{v})&\quad&\forall\mathbf{v}\in
H_{0}({\rm curl};\Omega),&\\\ (\varepsilon_{r}\mathbf{u},\nabla
q)&=0&\quad&\forall q\in H_{0}^{1}(\Omega).&\end{aligned}\right.$

Using the Helmholtz decomposition, we have the following equivalent form: Find
$\lambda\in\mathbb{R}$ and $\mathbf{u}\in U_{0}\setminus\\{0\\}$ such that

(2.4)
$(\mu_{r}^{-1}\nabla\times\mathbf{u},\nabla\times\mathbf{v})=\lambda(\varepsilon_{r}\mathbf{u},\mathbf{v}),\quad\forall\mathbf{v}\in
U_{0}.$

In this paper, we only consider form (2.4). Therefore, we give the variational
form of Maxwell eigenvalue problem (2.1): Find $\lambda\in\mathbb{R}$ and
$\mathbf{u}\in U_{0}\setminus\\{0\\}$ such that

(2.5) $a(\mathbf{u},\mathbf{v})=\lambda
b(\mathbf{u},\mathbf{v}),\quad\forall\mathbf{v}\in U_{0},$

where

$\displaystyle a(\mathbf{u},\mathbf{v})=$
$\displaystyle(\mu_{r}^{-1}\nabla\times\mathbf{u},\nabla\times\mathbf{v}),$
$\displaystyle b(\mathbf{u},\mathbf{v})=$
$\displaystyle(\varepsilon_{r}\mathbf{u},\mathbf{v}).$

##  3 The weak Galerkin finite element method

In this section, the weak Galerkin finite element numerical scheme of Maxwell
eigenvalue problem is given.

Let ${\mathcal{T}}_{h}$ be a partition of the domain $\Omega$, and the
elements in ${\mathcal{T}}_{h}$ are polygons satisfying the regular
assumptions specified in [24]. Denote by ${\mathcal{E}}_{h}$ the edges in
${\mathcal{T}}_{h}$, and by ${\mathcal{E}}_{h}^{0}$ the interior edges
${\mathcal{E}}_{h}\backslash\partial\Omega$. For each element
$T\in{\mathcal{T}}_{h}$, $h_{T}$ represents the diameter of $T$, and
$h=\max_{T\in{\mathcal{T}}_{h}}h_{T}$ denotes the mesh size. Let $C$ be a
positive constant which is independent of the mesh size and $a\lesssim b$
denote $a\leq Cb$.

Now we introduce the WG space for the eigenvalue problem (2.1). For a given
integer $k\geq 1$, define the WG finite element space

$\displaystyle
V_{h}=\\{\mathbf{v}_{h}=\\{\mathbf{v}_{0},\mathbf{v}_{b}\\}:\mathbf{v}_{0}|_{T}\in[P_{k}(T)]^{d},\mathbf{v}_{b}|_{e}\in[P_{k}(e)]^{d},e\in\mathcal{E}_{h},\mathbf{v}_{b}\times\mathbf{n}=0\text{
on }\partial\Omega\\},$ $\displaystyle
W_{h}=\\{w=\\{w_{0},w_{b}\\}:w_{0}|_{T}\in P_{k-1}(T),w_{b}|_{e}\in
P_{k}(e),e\in\mathcal{E}_{h},w_{b}=0\text{ on }\partial\Omega\\}.$

Next, we define the following weak curl operator.

######  Definition 1.

For any $\mathbf{v}_{h}\in V_{h}$, its weak curl is defined as the polynomial
$\nabla_{w}\times\mathbf{v}_{h}|_{T}$ is the unique polynomial in
$[P_{k-1}(T)]^{2d-3}$ satisfying

(3.1)
$\displaystyle(\nabla_{w}\times\mathbf{v}_{h},\mathbf{q})_{T}=(\mathbf{v}_{0},\nabla\times\mathbf{q})_{T}-\langle\mathbf{v}_{b}\times\mathbf{n},\mathbf{q}\rangle_{\partial
T},\quad\forall\mathbf{q}\in[P_{k-1}(T)]^{2d-3}.$

In addition, we define the following weak gradient operator.

######  Definition 2.

For any $p_{h}\in W_{h}$, its weak gradient is defined as the polynomial
$\nabla_{w}p_{h}|_{T}$ is the unique polynomial in $[P_{k}(T)]^{d}$ satisfying

(3.2)
$\displaystyle(\nabla_{w}p_{h},\phi)_{T}=-(p_{0},\nabla\cdot\phi)_{T}+\langle
p_{b},\phi\cdot\mathbf{n}\rangle_{\partial
T},\quad\forall\phi\in[P_{k}(T)]^{d}.$

Then we define the subspace $V_{h}^{0}$ of $V_{h}$:

$V_{h}^{0}=\\{\mathbf{v}_{h}\in V_{h}:(\mathbf{v}_{0},\nabla_{w}q)=0,\forall
q\in W_{h}\\}.$

For any $\mathbf{u}_{h},\mathbf{v}_{h}\in V_{h}^{0}$, define the following
bilinear forms:

$\displaystyle
s(\mathbf{u}_{h},\mathbf{v}_{h})=\gamma(h)\sum\limits_{T\in\mathcal{T}_{h}}h_{T}^{-1}\l\langle(\mathbf{u}_{0}-\mathbf{u}_{b})\times\mathbf{n},(\mathbf{v}_{0}-\mathbf{v}_{b})\times\mathbf{n}\rangle_{\partial
T},$ $\displaystyle
a_{w}(\mathbf{u}_{h},\mathbf{v}_{h})=\sum\limits_{T\in\mathcal{T}_{h}}(\mu_{r}^{-1}\nabla_{w}\times\mathbf{u}_{h},\nabla_{w}\times\mathbf{v}_{h})_{T}+s(\mathbf{u}_{h},\mathbf{v}_{h})$
$\displaystyle
b_{w}(\mathbf{u}_{h},\mathbf{v}_{h})=(\varepsilon_{r}\mathbf{u}_{0},\mathbf{v}_{0}),$

where $\gamma(h)=h^{\varepsilon},0<\varepsilon<1,\text{ or
}\gamma(h)=-\frac{1}{\log(h)}$.

For further error analysis, we introduce some projection operators used in
this paper. Let $\mathbf{Q}_{0}$ be the $L^{2}$ projection from
$[L^{2}(T)]^{d}$ onto $[P_{k}(T)]^{d}$, $Q_{0}$ be the $L^{2}$ projection from
$L^{2}(T)$ onto $P_{k-1}(T)$. Let $\mathbf{Q}_{b}$ denote the $L^{2}$
projection from $[L^{2}(e)]^{d}$ onto $[P_{k}(e)]^{d}$, and $Q_{b}$ denote the
$L^{2}$ projection from $L^{2}(T)$ onto $P_{k}(T)$. In addition, let
$\mathbb{Q}_{h}$ be the $L^{2}$ projection from $[L^{2}(T)]^{2d-3}$ onto
$[P_{k}(T)]^{2d-3}$. Combining $\mathbf{Q}_{0}$ and $\mathbf{Q}_{b}$ together,
we can define $\mathbf{Q}_{h}=\\{\mathbf{Q}_{0},\mathbf{Q}_{b}\\}$.
Analogously, denote $Q_{h}=\\{Q_{0},Q_{b}\\}$.

The following commutativity properties are crucial in error analysis.

######  Lemma 3.

[19, Lemma 5.1] For any $\mathbf{v}\in H({\rm curl};\Omega)$, we have

$\nabla_{w}\mathbf{Q}_{h}\mathbf{v}=\mathbb{Q}_{h}(\nabla\times\mathbf{v}),$

and for any $q\in H^{1}(\Omega)$, there holds

$\nabla_{w}Q_{h}q=\mathbf{Q}_{0}(\nabla q).$

Now we give the numerical scheme of (2.1) corresponding to (2.4).

######  Weak Galerkin Algorithm 1.

Find $(\lambda_{h},\mathbf{u}_{h})\in\mathbb{R}\times V_{h}^{0}$ such that
$\|\mathbf{u}_{h}\|=1$ and

(3.3) $\displaystyle
a_{w}(\mathbf{u}_{h},\mathbf{v}_{h})=\lambda_{h}b_{w}(\mathbf{u}_{0},\mathbf{u}_{0}),\quad\forall\mathbf{v}_{h}\in
V_{h}^{0}.$

Define the sum space $V=U_{0}+V_{h}^{0}$. Now we define the following semi-
norm on $V$. For any $\mathbf{v}\in V$,

$\displaystyle\|\mathbf{v}\|_{V}^{2}=\sum_{T\in\mathcal{T}_{h}}\Big{(}\mu_{r}^{-1}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}+h_{T}^{-1}\|(\mathbf{v}_{0}-\mathbf{v}_{b})\times\mathbf{n}\|^{2}_{\partial
T}\Big{)},$

where $\mathbf{v}_{0}$ is the internal value and $\mathbf{v}_{b}$ is the
boundary value.

The following Friedrichs inequality [3] holds on $U_{0}$.

######  Lemma 4.

Let $\Omega$ be a bounded Lipschitz domain, there exists a positive constant
$C$ such that

$\|\mathbf{u}\|\leq C\|\nabla\times\mathbf{u}\|,\quad\forall\mathbf{u}\in
U_{0}.$

Therefore, from Lemma 4, $\|\cdot\|_{V}$ is a norm on $U_{0}$.

######  Lemma 5.

$\|\cdot\|_{V}$ defines a norm on $V_{h}^{0}$.

###### Proof.

We prove that $\|\mathbf{v}_{h}\|_{V}=0$ if and only if $\mathbf{v}_{h}=0$. If
$\mathbf{v}_{h}=0$, obviously, $\|\mathbf{v}_{h}\|_{V}=0$. Next, assume that
$\|\mathbf{v}_{h}\|_{V}=0$. Then we have $\nabla\times\mathbf{v}_{0}=0$ for
any $T\in\mathcal{T}_{h}$, and
$(\mathbf{v}_{0}-\mathbf{v}_{b})\times\mathbf{n}=0$ for any
$e\in\mathcal{E}_{h}$. Notice that $\mathbf{v}_{h}\in V_{h}^{0}$, then for any
$q_{h}\in W_{h}$, there holds

$\displaystyle b_{w}(\mathbf{v}_{0},q_{h})$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}(\varepsilon_{r}\mathbf{v}_{0},\nabla_{w}q_{h})_{T}$
$\displaystyle=-\sum_{T\in\mathcal{T}_{h}}(\nabla\cdot(\varepsilon_{r}\mathbf{v}_{0}),q_{0})_{T}+\sum_{T\in\mathcal{T}_{h}}\langle\varepsilon_{r}\mathbf{v}_{0}\cdot\mathbf{n},q_{b}\rangle_{\partial
T}$ $\displaystyle=0.$

Specially, let $q_{0}=\nabla\cdot\mathbf{v}_{0}$ and $q_{b}=0$, then we have
$\nabla\cdot\mathbf{v}_{0}=0$ for any $T\in\mathcal{T}_{h}$. Let $q_{0}=0$ and
$q_{b}$ is the jump of $\mathbf{v}_{0}\cdot\mathbf{n}$ on $e$, then we can get
that $\mathbf{v}_{0}\cdot\mathbf{n}$ is continuous on $e$. Notice that
$\nabla\times\mathbf{v}_{0}=0$, then there exists a potential function $\phi$
such that $\mathbf{v}_{0}=\nabla\phi$ in $\Omega$. In addition,
$\nabla\cdot\mathbf{v}_{0}=0$ and $\mathbf{v}_{0}\cdot\mathbf{n}$ is
continuous, there holds $\phi=0$ in $\Omega$. Notice that
$\mathbf{v}_{b}\times\mathbf{n}=0$ on $\partial\Omega$, and
$(\mathbf{v}_{0}-\mathbf{v}_{b})\times\mathbf{n}=0$ for any
$e\in\mathcal{E}_{h}$, Thus, $\mathbf{v}_{0}\times\mathbf{n}=\nabla\phi=0$ on
$\partial\Omega$. Therefore, $\phi$ must be a constant on $\partial\Omega$.
From the uniqueness of the solution of the Laplace equation, if $\phi$ is
simply connected, $\phi$ must be a constant in $\Omega$. Then
$\mathbf{v}_{0}=\nabla\phi=0$ and $\mathbf{v}_{b}=0$. ∎

##  4 Error analysis

By limiting to subspaces, we transform the Maxwell eigenvalue problem into an
elliptic eigenvalue problem. Therefore, we can use the framework presented in
[27] and just prove Assumption 1 (A1)-Assumption 7 (A7) in Section 2 of [27].

First we verify (A1): $a(\cdot,\cdot)$ and $a_{w}(\cdot,\cdot)$ are symmetric,
and

$\displaystyle
a(\mathbf{w},\mathbf{w})\geq\gamma\|\mathbf{w}\|_{V}^{2},\quad\forall\mathbf{w}\in
U_{0},$ $\displaystyle
a_{w}(\mathbf{v}_{h},\mathbf{v}_{h})\geq\gamma(h)\|\mathbf{v}_{h}\|_{V}^{2},\quad\forall\mathbf{v}_{h}\in
V_{h},$

where $\gamma$ and $\gamma(h)$ are two positive constants for given $h$.

Obviously, $a(\cdot,\cdot)$ is coercive and continuous on $U_{0}$. And
$a_{w}(\cdot,\cdot)$ is continuous on $V_{h}^{0}$, we only prove that
$a_{w}(\cdot,\cdot)$ is coercive on $V_{h}^{0}$.

######  Lemma 6.

For any $\mathbf{v}_{h}\in V^{0}_{h}$, we have

$a_{w}(\mathbf{v}_{h},\mathbf{v}_{h})\geq
C\gamma(h)\|\mathbf{v}_{h}\|_{V}^{2}.$

###### Proof.

It is easy to prove

$\sum_{T\in\mathcal{T}_{h}}h_{T}^{-1}\|(\mathbf{v}_{0}-\mathbf{v}_{b})\times\mathbf{n}\|^{2}_{\partial
T}\leq C\gamma(h)^{-1}a_{w}(\mathbf{v}_{h},\mathbf{v}_{h}).$

From integration by parts, (3.1), Young inequality, trace inequality [24] and
inverse inequality [24], we have

$\displaystyle\sum_{T\in\mathcal{T}_{h}}(\nabla\times\mathbf{v}_{0},\nabla\times\mathbf{v}_{0})_{T}$
$\displaystyle=$
$\displaystyle\sum_{T\in\mathcal{T}_{h}}(\nabla\times\nabla\times\mathbf{v}_{0},\mathbf{v}_{0})_{T}-\sum_{T\in\mathcal{T}_{h}}\langle\nabla\times\mathbf{v}_{0},\mathbf{v}_{0}\times\mathbf{n}\rangle_{\partial
T}$ $\displaystyle=$
$\displaystyle\sum_{T\in\mathcal{T}_{h}}(\nabla_{w}\times\mathbf{v}_{h},\nabla\times\mathbf{v}_{0})_{T}+\sum_{T\in\mathcal{T}_{h}}\langle\nabla\times\mathbf{v}_{0},(\mathbf{v}_{b}-\mathbf{v}_{0})\times\mathbf{n}\rangle_{\partial
T}$ $\displaystyle\leq$ $\displaystyle
C\left(\sum_{T\in\mathcal{T}_{h}}\|\nabla_{w}\times\mathbf{v}_{h}\|_{T}^{2}\right)^{1/2}\left(\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}\right)^{1/2}$
$\displaystyle+C\left(\sum_{T\in\mathcal{T}_{h}}h_{T}\|\nabla\times\mathbf{v}_{0}\|_{\partial
T}^{2}\right)^{1/2}\left(h_{T}^{-1}\|(\mathbf{v}_{b}-\mathbf{v}_{0})\times\mathbf{n}\|_{\partial
T}^{2}\right)^{1/2}$ $\displaystyle\leq$ $\displaystyle
C\sum_{T\in\mathcal{T}_{h}}\|\nabla_{w}\times\mathbf{v}_{h}\|_{T}^{2}+\frac{1}{4}\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}$
$\displaystyle+C\sum_{T\in\mathcal{T}_{h}}h_{T}^{-1}\|(\mathbf{v}_{b}-\mathbf{v}_{0})\times\mathbf{n}\|_{\partial
T}^{2}+\frac{1}{4}\sum_{T\in\mathcal{T}_{h}}h_{T}\|\nabla\times\mathbf{v}_{0}\|_{\partial
T}^{2}$ $\displaystyle\leq$ $\displaystyle
C\sum_{T\in\mathcal{T}_{h}}\|\nabla_{w}\times\mathbf{v}_{h}\|_{T}^{2}+\frac{1}{4}\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}$
$\displaystyle+C\sum_{T\in\mathcal{T}_{h}}h_{T}^{-1}\|(\mathbf{v}_{b}-\mathbf{v}_{0})\times\mathbf{n}\|_{\partial
T}^{2}+\frac{1}{4}\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}$
$\displaystyle\leq\frac{1}{2}\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}+C\gamma(h)^{-1}a_{w}(\mathbf{v}_{h},\mathbf{v}_{h}),$

which implies that

$\sum_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{v}_{0}\|_{T}^{2}\leq
C\gamma(h)^{-1}a_{w}(\mathbf{v}_{h},\mathbf{v}_{h}).$

Combining the two equations above, we have

$\|\mathbf{v}_{h}\|_{V}^{2}\leq
C\gamma(h)^{-1}a_{w}(\mathbf{v}_{h},\mathbf{v}_{h}),$

which completes the proof. ∎

Before verifying (A2), we define the following two operators $K$ and $K_{h}$.

$\displaystyle K:L^{2}(\Omega)\rightarrow U_{0},\quad
a(K\mathbf{f},\mathbf{v})=b(\mathbf{f},\mathbf{v}),\quad\forall\mathbf{v}\in
U_{0},$ $\displaystyle K_{h}:V^{0}_{h}+L^{2}(\Omega)\rightarrow
V_{h}^{0},\quad
a_{w}(K_{h}\mathbf{f}_{h},\mathbf{v}_{h})=b_{w}(\mathbf{f}_{h},\mathbf{v}_{h}),\quad\forall\mathbf{v}_{h}\in
V^{0}_{h}.$

It follows from [17, Lemma 5.3] and [18, Lemma 7.1] that

$\|\mathbf{v}_{0}\|\leq C\|\mathbf{v}_{h}\|_{V}.$

Therefore, it is easy to prove that $K$ and $K_{h}$ are well-defined by the
Lax-Milgram theorem.

Next we verify (A2): $K$ and $K_{h}$ are compact.

######  Lemma 7.

$K$ and $K_{h}$ are compact operators from $U_{0}$ onto $U_{0}$ and
$V_{h}^{0}$ onto $V_{h}^{0}$, respectively.

###### Proof.

As we all know, it follows from [16, Theorem 4.7] that $U_{0}$ is compactly
embedded in $L^{2}(\Omega)$. By the definition of $K$, we know that $K$ is a
bounded operator from $L^{2}(\Omega)$ onto $U_{0}$. Because bounded operator
compound compact operator is still compact operator, $K$ is compact operator
from $U_{0}$ onto $U_{0}$. For $V_{h}^{0}$, it is bounded by Lax-Milgram
Theorem, and it is also finite rank, so $K_{h}$ is compact operator from
$V_{h}^{0}$ onto $V_{h}^{0}$. Then we complete the proof. ∎

Now we verify (A3): there exists a bounded linear operator
$\mathbf{Q}_{h}:V\rightarrow V_{h}^{0}$ satisfying

$\displaystyle\mathbf{Q}_{h}\mathbf{w}_{h}=\mathbf{w}_{h},\quad\forall\mathbf{w}_{h}\in
V_{h}^{0},$ $\displaystyle
b(\mathbf{Q}_{h}\mathbf{f},\mathbf{w}_{h})=b(\mathbf{f},\mathbf{w}_{h}),\quad\forall\mathbf{f}\in
V,\mathbf{w}_{h}\in V_{h}^{0}.$

Obviously, for any $\mathbf{w}_{h}\in V_{h}^{0}$,
$\mathbf{Q}_{h}\mathbf{w}_{h}=\mathbf{w}_{h}$, and

$b(\mathbf{f},\mathbf{w}_{h})=(\varepsilon_{r}\mathbf{f}_{0},\mathbf{w}_{0})=(\varepsilon_{r}\mathbf{Q}_{0}\mathbf{f}_{0},\mathbf{w}_{0})=b(\mathbf{Q}_{h}\mathbf{f},\mathbf{w}_{h}).$

Further we need to prove $\mathbf{Q}_{h}\mathbf{f}\in V_{h}^{0}$. For any
$q_{h}\in W_{h}$, it follows from (3.2) that

$\displaystyle(\varepsilon_{r}\mathbf{Q}_{h}\mathbf{f},\nabla_{w}q_{h})$
$\displaystyle=(\varepsilon_{r}\mathbf{Q}_{0}\mathbf{f},\nabla_{w}q_{h})$
$\displaystyle=(\varepsilon_{r}\mathbf{f},\nabla_{w}q_{h})$
$\displaystyle=-(\nabla\cdot(\varepsilon_{r}\mathbf{f}),q_{0})+\sum_{T\in\mathcal{T}_{h}}\langle\varepsilon_{r}\mathbf{f}\cdot\mathbf{n},q_{b}\rangle_{\partial
T}$ $\displaystyle=0,$

which uses the fact that
$\sum\limits_{T\in\mathcal{T}_{h}}\langle\mathbf{f}\cdot\mathbf{n},q_{b}\rangle_{\partial
T}=0$. Then (A3) is verified.

Denote

$\displaystyle\delta_{h,\mu}=\sup\limits_{\mathbf{v}\in
R(E_{\mu}(K)),\atop\|\mathbf{v}\|_{V}=1}\|\mathbf{v}-Q_{h}\mathbf{v}\|_{V},$
$\displaystyle\delta_{h,\mu}^{\prime}=\sup\limits_{\mathbf{v}\in
R(E_{\mu}(K)),\atop\|\mathbf{v}\|_{V}=1}\|\mathbf{v}-\mathbf{Q}_{h}\mathbf{v}\|,$

where $R(E_{\mu}(K))$ is the space of eigenfunctions corresponding to the
eigenvalue $\mu$, and can be found in Section 2 of [27].

Let $\Pi_{0}$ be the projection operator from $V$ onto $L^{2}(\Omega)$ under
$b(\cdot,\cdot)$, denote

$\displaystyle e_{h,\mu}=\|(K-K_{h}Q_{h})|_{R(E_{\mu}(K)}\|_{V},$
$\displaystyle
e_{h,\mu}^{\prime}=\|(\Pi_{0}K-\Pi_{0}K_{h}Q_{h})|_{R(E_{\mu}(K))}\|_{X}.$

Next we verify (A4) and (A5):

$\displaystyle(A4):e_{h,\mu}\rightarrow 0~{}{\rm as}~{}h\rightarrow 0,{\rm
and}~{}\delta_{h,\mu}\gamma(h)^{-1}\rightarrow 0~{}{\rm as}~{}h\rightarrow 0,$
$\displaystyle(A5):e_{h,\mu}^{\prime}\rightarrow 0~{}{\rm as}~{}h\rightarrow
0,{\rm and}~{}\delta_{h,\mu}^{\prime}\gamma(h)^{-1}\rightarrow 0~{}{\rm
as}~{}h\rightarrow 0.$

######  Lemma 8.

Assume $R(E_{\mu}(K))\subset H^{k+1}(\Omega)$, there hold the following
estimates

$\displaystyle\delta_{h,\mu}\lesssim h^{k},$
$\displaystyle\delta_{h,\mu}^{\prime}\lesssim h^{k+1}.$

###### Proof.

It follows from the trace inequality and the projection inequality [25] that

$\displaystyle\|\mathbf{u}-\mathbf{Q}_{h}\mathbf{u}\|_{V}^{2}$
$\displaystyle=$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\mu_{r}^{-1}\nabla\times(\mathbf{u}-\mathbf{Q}_{0}\mathbf{u})\|_{T}^{2}+\sum\limits_{T\in\mathcal{T}_{h}}h_{T}^{-1}\|(\mathbf{Q}_{0}\mathbf{u}-\mathbf{Q}_{b}\mathbf{u})\times\mathbf{n}\|_{\partial
T}^{2}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\mu_{r}^{-1}\nabla\times(\mathbf{u}-\mathbf{Q}_{0}\mathbf{u})\|_{T}^{2}+\sum\limits_{T\in\mathcal{T}_{h}}h_{T}^{-1}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{u}\|_{\partial
T}^{2}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\mu_{r}^{-1}\nabla\times(\mathbf{u}-\mathbf{Q}_{0}\mathbf{u})\|_{T}^{2}+\sum\limits_{T\in\mathcal{T}_{h}}C(h_{T}^{-2}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{u}\|_{T}^{2}+\|\nabla(\mathbf{Q}_{0}\mathbf{u}-\mathbf{u})\|_{T}^{2})$
$\displaystyle\lesssim$ $\displaystyle h^{2k},$

and

$\|\mathbf{u}-\mathbf{Q}_{h}\mathbf{u}\|^{2}\lesssim h^{2(k+1)}.$

Then we complete the proof. ∎

For further analysis, we consider the following source problem corresponding
the eigenvalue problem (2.1):

(4.1) $\left\\{\begin{aligned}
\nabla\times\mu_{r}^{-1}\nabla\times\mathbf{u}&=\varepsilon_{r}\mathbf{f},&\quad&\text{in
}\Omega,&\\\ \nabla\cdot\varepsilon_{r}\mathbf{u}&=0,&\quad&\text{in
}\Omega,&\\\ \mathbf{u}\times\mathbf{n}&=0,&\quad&\text{on
}\partial\Omega,&\end{aligned}\right.$

where $\mathbf{f}\in H({\rm div^{0}};\Omega)$.

The corresponding variational form is: Find $\mathbf{u}\in H_{0}({\rm
curl};\Omega)$ and $p\in H_{0}^{1}(\Omega)$ such that

(4.2) $\left\\{\begin{aligned}
(\mu_{r}^{-1}\nabla\times\mathbf{u},\nabla\times\mathbf{v})+(\nabla
p,\varepsilon_{r}\mathbf{v})&=(\varepsilon_{r}\mathbf{f},\mathbf{v})&\quad&\forall\mathbf{v}\in
H_{0}({\rm curl};\Omega),&\\\ (\varepsilon_{r}\mathbf{u},\nabla
q)&=0&\quad&\forall q\in H_{0}^{1}(\Omega).&\end{aligned}\right.$

Then we give the WG numerical scheme for (4.2): Find $\mathbf{u}_{h}\in V_{h}$
and $p_{h}\in W_{h}$ such that

(4.3) $\left\\{\begin{aligned}
&a_{w}(\mathbf{u}_{h},\mathbf{v}_{h})+(\varepsilon_{r}\mathbf{v}_{0},\nabla_{w}p_{h})=(\varepsilon_{r}\mathbf{f},\mathbf{v}_{0})&\quad&\forall\mathbf{v}_{h}\in
V_{h},&\\\
&(\varepsilon_{r}\mathbf{u}_{0},\nabla_{w}q_{h})+\sum_{T\in\mathcal{T}_{h}}h_{T}\langle
p_{0}-p_{b},q_{0}-q_{b}\rangle_{\partial T}=0&\quad&\forall q_{h}\in
W_{h}.&\end{aligned}\right.$

######  Lemma 9.

[19, Theorem 7.2] Let
$(\mathbf{u};p)\in[H^{k+1}(\Omega)]^{d}\times[H^{1}_{0}(\Omega)\cap
H^{k}(\Omega)]$ be the solution of (4.2) and $(\mathbf{u}_{h};p_{h})\in
V_{h}\times W_{h}$ be the numerical solution of (4.3), then

$\|\mathbf{Q}_{h}\mathbf{u}-\mathbf{u}_{h}\|_{V}\leq
C\gamma(h)^{-1}h^{k}\|\mathbf{u}\|_{k+1}.$

######  Lemma 10.

[19, Theorem 8.1] Let
$(\mathbf{u};p)\in[H^{k+1}(\Omega)]^{d}\times[H^{1}_{0}(\Omega)\cap
H^{k}(\Omega)]$ be the solution of (4.2) and $(\mathbf{u}_{h};p_{h})\in
V_{h}\times W_{h}$ be the numerical solution of (4.3), then

$\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{u}_{0}\|_{V}\leq
C\gamma(h)^{-1}h^{k+1}\|\mathbf{u}\|_{k+1}.$

By triangle inequality, we have

$\displaystyle\|\mathbf{u}-\mathbf{u}_{h}\|_{V}\lesssim\gamma(h)^{-1}h^{k},$
$\displaystyle\|\mathbf{u}-\mathbf{u}_{h}\|\lesssim\gamma(h)^{-1}h^{k+1}.$

Notice that $e_{h,\mu}$ and $e_{h,\mu}^{\prime}$ are the
$\|\mathbf{u}-\mathbf{u}_{h}\|_{V}$ and $\|\mathbf{u}-\mathbf{u}_{h}\|$ error
estimates of the source problem (4.1), respectively. Then the following lemma
holds.

######  Lemma 11.

Assume $R(E_{\mu}(K))\subset H^{k+1}(\Omega)$, then

$\displaystyle e_{h,\mu}\lesssim\gamma(h)^{-1}h^{k},$ $\displaystyle
e_{h,\mu}^{\prime}\lesssim\gamma(h)^{-1}h^{k+1}.$

According to [27, Theorem 2.1, Theorem 2.2], the following theorem holds. The
details can be seen in [1, Theorem 7.1].

######  Theorem 12.

Let $\lambda$ be an eigenvalue of $(\ref{weak form})$ with multiplicity $m$,
and $\\{\mathbf{u}_{j}\\}_{j=1}^{m}$ denote a basis of the corresponding
m-dimensional eigenspace $R(E_{\mu}(K))\subset H^{k+1}(\Omega)$. Similarly,
let $\\{\lambda_{j,h}\\}_{j=1}^{m}$ be the eigenvalues of $(\ref{WG-scheme})$,
and $\\{\mathbf{u}_{j,h}\\}_{j=1}^{m}$ denote a basis of the corresponding
eigenspace $R(E_{\mu,h}(K_{h}))$. For any $j=1,\cdots,m$, there exists an
eigenfunction $\mathbf{u}_{j}\in R(E_{\mu}(K))$ such that

$\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|_{V}\lesssim\gamma(h)^{-1}h^{k},$
$\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|\lesssim\gamma(h)^{-1}h^{k+1}.$

Notice that the parameter $\gamma(h)$ is chosen as either
$\gamma(h)=h^{\varepsilon}(0<\varepsilon<1)$, or $\gamma(h)=-1/\log(h)$, then
the conclusions in Theorem 12 can be written as follows

$\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|_{V}\lesssim
h^{k-\varepsilon},$ $\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|\lesssim
h^{k+1-\varepsilon},$

or

$\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|_{V}\lesssim-\log(h)h^{k},$
$\displaystyle\|\mathbf{u}_{j}-\mathbf{u}_{j,h}\|\lesssim-\log(h)h^{k+1}.$

Let
$\varepsilon_{h,\mathbf{u}}=a(\mathbf{u},\mathbf{u})-a_{w}(\mathbf{Q}_{h}\mathbf{u},\mathbf{Q}_{h}\mathbf{u})$.
Then we verify (A6): $\forall\mu\in\sigma_{K}$, $\forall\mathbf{u}\in
R(E_{\mu}(K))$, $\varepsilon_{h,\mu}\rightarrow 0$ as $h\rightarrow 0$, where
$\sigma_{K}$ is the spectrum of $K$.

######  Lemma 13.

Let $\mathbf{u}\in H^{k+1}(\Omega)$, there holds

$|\varepsilon_{h,\mathbf{u}}|\lesssim h^{2k}.$

###### Proof.

It follows from Lemma 3 and projection inequalities [19, Lemma 6.1] that

$\displaystyle|\varepsilon_{h,\mathbf{u}}|$ $\displaystyle=$
$\displaystyle\left|\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}\|_{T}^{2}-\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla_{w}\times\mathbf{Q}_{h}\mathbf{u}\|_{T}^{2}-s(\mathbf{Q}_{h}\mathbf{u},\mathbf{Q}_{h}\mathbf{u})\right|$
$\displaystyle=$
$\displaystyle\left|\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}\|_{T}^{2}-\sum\limits_{T\in\mathcal{T}_{h}}\|\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}-s(\mathbf{Q}_{h}\mathbf{u},\mathbf{Q}_{h}\mathbf{u})\right|$
$\displaystyle\leq$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}-\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}+\sum\limits_{T\in\mathcal{T}_{h}}\gamma(h)h_{T}^{-1}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{u}\|_{\partial
T}^{2}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}-\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}+\sum\limits_{T\in\mathcal{T}_{h}}C\gamma(h)(h_{T}^{-2}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{u}\|_{T}^{2}+\|\nabla(\mathbf{Q}_{0}\mathbf{u}-\mathbf{u})\|_{T}^{2})$
$\displaystyle\lesssim$ $\displaystyle h^{2k},$

which completes the proof of this lemma. ∎

Then the following error estimate of eigenvalues holds.

######  Theorem 14.

Let $\lambda$ be an eigenvalue of $(\ref{weak form})$ with multiplicity $m$,
and $\\{\mathbf{u}_{j}\\}_{j=1}^{m}$ be a basis of the corresponding
m-dimensional eigenspace $R(E_{\mu}(K))\subset H^{k+1}(\Omega)$. Let
$\\{\lambda_{j,h}\\}_{j=1}^{m}$ denote the eigenvalues of $(\ref{WG-scheme})$,
and $\\{\mathbf{u}_{j,h}\\}_{j=1}^{m}$ denote a basis of the corresponding
eigenspace $R(E_{\mu,h}(K_{h}))$. When $h$ is sufficiently small, for any
$j=1,\cdots,m$, the following estimate holds

$|\lambda-\lambda_{j,h}|\lesssim\gamma(h)^{-2}h^{2k}.$

###### Proof.

It follows from [27, Theorem 2.3] that

$|\lambda-\lambda_{j,h}|\leq
C(\varepsilon_{h,\mathbf{u}_{j}}+e_{h,\lambda^{-1}}^{2}+e_{h,\lambda^{-1}}^{\prime
2}+\delta_{h,\lambda^{-1}}^{2}\gamma(h)^{-2}+\delta_{h,\lambda^{-1}}^{\prime
2}\gamma(h)^{-2}).$

From Lemmas 8, 11 and 13, we have

$\displaystyle\delta_{h,\lambda^{-1}}\lesssim
h^{k},~{}\delta_{h,\lambda^{-1}}^{\prime}\lesssim h^{k+1},$ $\displaystyle
e_{h,\lambda^{-1}}\lesssim\gamma(h)^{-1}h^{k},~{}e_{h,\lambda^{-1}}^{\prime}\lesssim\gamma(h)^{-1}h^{k+1},$
$\displaystyle|\varepsilon_{h,\mathbf{u}}|\lesssim h^{2k},$

which implies that

$|\lambda-\lambda_{j,h}|\lesssim\gamma(h)^{-2}h^{2k}.$

Then we complete the proof. ∎

Notice that the parameter $\gamma(h)$ is chosen as either
$\gamma(h)=h^{\varepsilon}(0<\varepsilon<1)$, or $\gamma(h)=-1/\log(h)$, the
conclusion in Theorem 14 is as follows

$|\lambda-\lambda_{j,h}|\lesssim h^{2k-2\varepsilon},$

or

$|\lambda-\lambda_{j,h}|\lesssim{\log(h)}^{2}h^{2k}.$

##  5 Asymptotic lower bound

In this section, we show that the numerical eigenvalues are the lower bound of
the exact eigenvalues.

It only need to verify (A7): Suppose $(\lambda,\mathbf{u})$ is an eigenpair of
$(\ref{weak form})$, $(\lambda_{h},\mathbf{u}_{h})$ is an eigenpair of
$(\ref{WG-scheme})$, there holds
$\varepsilon_{h,\mathbf{u}}\geq\lambda_{h}\|\mathbf{u}-\mathbf{u}_{h}\|^{2}.$

The following estimates are crucial, and the proof can be found in [13,
Theorem 2.1].

######  Lemma 15.

The exact eigenfunction $\mathbf{u}$ of the Steklov eigenvalue problem
$(\ref{weak form})$ exhibits lower bounds for its convergence rates as follows

$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}-\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}\gtrsim
h^{2k}.$

######  Lemma 16.

Let $(\lambda,\mathbf{u})$ be an eigenpair of $(\ref{weak form})$,
$(\lambda_{h},\mathbf{u}_{h})$ be an eigenpair of $(\ref{WG-scheme})$. For
sufficiently small $h$, $\gamma(h)\ll 1$, the following estimate holds

$\varepsilon_{h,\mathbf{u}}\geq\lambda_{h}\|\mathbf{u}-\mathbf{u}_{h}\|_{X}^{2}.$

###### Proof.

It follows from Lemma 3 that

$\displaystyle\varepsilon_{h,\mathbf{u}}$ $\displaystyle=$ $\displaystyle
a(\mathbf{u},\mathbf{u})-a_{w}(\mathbf{Q}_{h}\mathbf{u},\mathbf{Q}_{h}\mathbf{u})$
$\displaystyle=$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}\|_{T}^{2}-\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla_{w}\times\mathbf{Q}_{h}\mathbf{u}\|_{T}^{2}-s(Q_{h}u,Q_{h}u)$
$\displaystyle=$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}-\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}-\sum\limits_{T\in\mathcal{T}_{h}}\gamma(h)h_{T}^{-1}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{Q}_{b}\mathbf{u}\|_{\partial
T}^{2}.$

Combining Lemma 15 with the trace inequality [24], there holds

$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\|\nabla\times\mathbf{u}-\mathbb{Q}_{h}\nabla\times\mathbf{u}\|_{T}^{2}\gtrsim
h^{2k},$
$\displaystyle\sum\limits_{T\in\mathcal{T}_{h}}\gamma(h)h_{T}^{-1}\|\mathbf{Q}_{0}\mathbf{u}-\mathbf{Q}_{b}\mathbf{u}\|_{\partial
T}^{2}\lesssim\gamma(h)h^{2k}.$

Notice that $h$ is sufficiently small, $\gamma(h)\ll 1$, then

$\varepsilon_{h,\mathbf{u}}\gtrsim h^{2k}.$

It follows from Theorem 12 that

$\lambda_{h}\|\mathbf{u}-\mathbf{u}_{h}\|^{2}\lesssim\gamma(h)^{-2}h^{2(k+1)}.$

Therefore, for sufficiently small $h$, the following inequality holds

$\varepsilon_{h,\mathbf{u}}\geq\lambda_{h}\|\mathbf{u}-\mathbf{u}_{h}\|^{2}.$

Then we complete the proof. ∎

From [27, Theorem 2.4], we have the following theorem.

######  Theorem 17.

Let $(\lambda,\mathbf{u})$ be an eigenpair of $(\ref{weak form})$,
$(\lambda_{h},\mathbf{u}_{h})$ be an eigenpair of $(\ref{WG-scheme})$, for
sufficiently small $h$, we have

$\lambda\geq\lambda_{h}.$

Combining Theorems 14 and 17, the asymptotic lower bound approximations of the
exact eigenvalues are obtained.

##  6 Numerical Experiments

In this section, we present some numerical experiments.

###  6.1 Square domain

In first example, we consider the Maxwell eigenvalue problem $(\ref{weak
form})$ on the square domain $\Omega=(0,\pi)^{2}$. Let $\gamma(h)=h^{0.1}$,
$\varepsilon_{r}=1$ and $\mu_{r}=1$. The degree $k$ of the Wg finite element
space is selected as 1. The first five exact eigenvalues is: $\lambda_{1}=1$,
$\lambda_{2}=1$, $\lambda_{3}=2$, $\lambda_{4}=4$ and $\lambda_{5}=4$.

The numerical results are presented in Table 1. We can find that the first
five eigenvalues all reach the optimal convergence orders and are the
asymptotic lower bounds of the exact eigenvalues. The images and vectorgraph
of the third eigenfunction are shown in Figures 3-3.

Table 1: $\Omega=(0,\pi)^{2}$, $k=1$, $\gamma(h)=h^{0.1}$. $h$ | 1/8 | 1/16 | 1/32 | 1/64
---|---|---|---|---
$\lambda_{1}-\lambda_{1,h}$ | 1.3857e-01 | 4.0828e-02 | 1.1269e-02 | 3.0525e-03
order |  | 1.7629 | 1.8572 | 1.8842
$\lambda_{2}-\lambda_{2,h}$ | 6.4984e-02 | 1.8187e-02 | 4.9488e-03 | 1.3368e-03
order |  | 1.8371 | 1.8777 | 1.8882
$\lambda_{3}-\lambda_{3,h}$ | 4.1608e-01 | 1.2117e-01 | 3.3223e-02 | 8.9721e-03
order |  | 1.7797 | 1.8668 | 1.8886
$\lambda_{4}-\lambda_{4,h}$ | 1.3155e+00 | 4.4141e-01 | 1.2730e-01 | 3.4933e-02
order |  | 1.5754 | 1.7938 | 1.8655
$\lambda_{5}-\lambda_{5,h}$ | 1.2921e+00 | 4.3273e-01 | 1.2657e-01 | 3.4878e-02
order |  | 1.5782 | 1.7734 | 1.8595

Fig. 1: the first component of the third eigenfunction.

Fig. 2: the second component of the third eigenfunction.

Fig. 3: the vectorgraph of the third eigenfunction.

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# ADIMA: Abuse Detection in Multilingual Audio

###### Abstract

Abusive content detection in spoken text can be addressed by performing
Automatic Speech Recognition (ASR) and leveraging advancements in natural
language processing. However, ASR models introduce latency and often perform
sub-optimally for profane words as they are underrepresented in training
corpora and not spoken clearly or completely. Exploration of this problem
entirely in the audio domain has largely been limited by the lack of audio
datasets. Building on these challenges, we propose ADIMA, a novel,
linguistically diverse, ethically sourced, expert annotated and well-balanced
multilingual profanity detection audio dataset comprising of 11,775 audio
samples in 10 Indic languages spanning 65 hours and spoken by 6,446 unique
users. Through quantitative experiments across monolingual and cross-lingual
zero-shot settings, we take the first step in democratizing audio based
content moderation in Indic languages and set forth our dataset to pave future
work. Dataset and code are available at: https://github.com/ShareChatAI/Adima

Index Terms—  Abusive Content Detection, Multilingual Audio Analysis, Indic
Dataset, Crosslingual Audio Analysis

##  1 Introduction

Detecting abusive content in online content has gained a lot of attention due
to the widespread adoption of social media platforms. Use of profane language,
cyber-bullying, racial slur, hate speech etc. are common examples of abusive
behaviour demanding robust content moderation algorithms to ensure healthy and
safe communication. Majority of the existing work has focused on detecting
abusive behaviour in textual data [1, 2, 3, 4, 5, 6]. Abusive content
detection on images and videos has been accelerated with the contribution of
multimedia datasets [7, 8, 9, 10, 11]. However, abusive content detection in
audio has been underexplored primarily due to the absence of audio datasets.
Profanity detection in audio can also be addressed by transcribing audio into
text using automatic speech recognition (ASR) followed by textual search over
the transcriptions. However, this requires accurate ASR systems which require
large amount of expensive training data, especially in multilingual setups.
Moreover, accuracy of ASR on profane words can be low as they are under-
represented in the training corpora. Another paradigm is to formulate this as
keyword spotting task by using a dictionary of audio exemplars of abusive
words and then use template matching approach. However, this does not exploit
underlying cues and overall context which can be helpful for identifying
profanity. Template matching approaches also fail for novel words and
continuously updating the dictionary is time-consuming. Moreover, these
approaches have high time complexity and require collection of significant
number of reference templates that capture the variations in
style/accent/dialect and environmental conditions. Abusive words are usually
not spoken clearly and completely, further limiting the effectiveness of
keyword spotting.

To tackle these challenges, we contribute a novel and highly diverse
multilingual profanity detection audio dataset - Abuse detection In
Multilingual Audio (ADIMA). ADIMA contains 11,775 audio recordings from
ShareChat chatrooms with a total duration of 65 hours for 10 Indic languages -
Hindi (Hi), Bengali (Be), Punjabi (Pu), Haryanvi (Ha), Kannada (Ka), Odia
(Od), Bhojpuri (Bh), Gujarati (Gu), Tamil (Ta) and Malyalam (Ma). The dataset
is balanced across the languages and has recordings spoken by 6446 different
users making it a highly diverse multilingual and multi-user dataset. The
recordings have been extracted from real-life conversations and capture
natural and in-the-wild conversations. We also formulate a profanity detection
task, where the objective is to classify the audio as Abusive or Non-Abusive.
Since the classifier analyzes the complete audio, it is able to effectively
leverage the context and underlying audio properties like pitch, intensity,
tone and emotion for robust profanity detection. We setup baselines for
monolingual and zero-shot cross-lingual setting for encouraging further
research in this direction. ADIMA presents promise in supervised settings such
as automatic moderation of live/recorded audio/video content, social media
chatrooms etc. to enable safer interactions. In unsupervised settings also,
ADIMA can be used for large-scale pretraining of models for Indic languages.
Competitive crosslingual performance also showcases the strength of ADIMA to
address more languages. Our contributions can be summarized as:

  * •

We release ADIMA, a highly diverse, multilingual, expert annotated audio
dataset for profanity detection in 10 Indic languages spanning a total of 65
hours comprising 11,775 total samples.

  * •

We introduce profanity detection task and report baseline monolingual results
capturing nuances of different languages and architectures.

  * •

Competitive crosslingual and joint training results exhibit the potential of
ADIMA for profanity detection in other languages and possibility of having a
single unified model for all the languages.

##  2 Related Work

Abuse detection in textual data under multilingual and monolingual settings
has received lot of attention from the community [1, 2, 3, 12, 5, 6]. Video
datasets for identifying offensive videos and a weakly annotated dataset of
videos from YouTube labelled for profanity detection are contributed by [8]
and [9] respectively. Datasets to identify specific cases such as pornography
and child abuse in images and videos [10] and multimodal hateful memes
identification using visual and textual features [7] are also present. A
YouTube video dataset to identify racist, sexist and normal videos is also
contributed by [11]. While the above mentioned datasets exist for abusive
content detection in text, image and video modality, audio modality has been
rather under explored. Recently, [13] explored self-attentive networks for
toxic language classification. Our dataset ADIMA is an attempt to reduce this
gap. Audio classification is a well studied area and has been accelerated with
the presence of large scale datasets [14, 15, 8]. Audio classification has
been performed using Gaussian Mixture Models, Support Vector Machines over Mel
Frequency Cepstrum Coefficients, Convolutional Neural Network (CNN) [14] and
Recurrent Neural Networks (RNN) [16]. Finetuning and extracting
representations from transformer based models [17] which are trained using
unlabelled raw audios has also gained significant interest and we leverage
these methods for our task.

(a) Dataset Distribution

(b) User Distribution

(c) Unique abusive words

(d) Abusive words per recording

Fig. 1: Analyzing ADIMA across multiple dimensions: (a) Number of samples for Abusive and Non-Abusive categories for all the languages (b) Unique users for all the languages (c) Frequency of profane words in the dataset (words with more than 5 instances) (d) Frequency distribution of the number of profane words in each sample. (Best viewed in color) Table 1: ADIMA statistics across sample distribution, linguistic diversity and audio duration. Data Description | Value
---|---
# Languages | 10
# Total Samples | 11,775
# Abusive Samples | 5,108
# Non-Abusive Samples | 6,667
# Unique Users | 6,446
Total Duration | 65 hours
Average Duration |  20 $(\pm 3)$ seconds
Min/Max Duration | 5/58 seconds

##  3 ADIMA Dataset

###  3.1 Data Collection

Recordings are collected from public audio chatrooms of
ShareChat111https://sharechat.com/. ShareChat is a leading, Indian social
media application supporting over 10 Indic languages with penetration across
India. ShareChat’s public chatrooms are open for anyone to join and informed
consent of the users is requested for recording and broadcasting the
discussions. The data was collected for a period of 6 months (January-June,
2021) from audio chatrooms pertaining to 10 Indic languages. These chatrooms
provide an interactive, audio-only platform for users to speak in regional
dialect. To build ADIMA, we sampled audio from conversations which were
reported as abusive by the users in the chatroom. Focusing on creating a well-
balanced and diverse dataset, we select audio samples across 10 languages in
similar proportions from a total of 6,446 users of the ShareChat platform.

###  3.2 Data Annotation

Specific to each language, an independent set of three annotators per language
were employed on a contract basis and fairly compensated to annotate each data
sample as abusive or non-abusive. We considered the presence of swear, cuss
and abusive words/phrases for annotating an audio as abusive. The abusive
words/phrases were catalogued and reviewed to ensure consensus among the three
reviewers. On average, the inter-annotator agreement measured by Cohen’s Kappa
$\kappa=0.88$ was observed to indicate a high degree of agreement amongst the
annotators for each language. The Cohen’s Kappa varied from $\kappa=0.77$ to
$\kappa=1.0$ across different languages, indicating the variation in
annotation complexity and annotator diversity across languages for the same
task. In the case of disagreements, the final label was selected based on
review by a fourth, expert annotator. Further, we removed low quality
recordings after which the final dataset comprises 11,775 audio recordings
spanning over 65 hours of audio across 10 languages.

###  3.3 Dataset Analysis

We briefly summarize the key statistics pertaining to ADIMA in Table 1. The
dataset is well balanced (43.38%) with 6,667 non-abusive and 5,108 abusive
recordings. We now analyze ADIMA across four different dimensions.

Language Distribution: Figure 1(a) shows the statistics for each language. We
note that on average, each language is almost equally represented in ADIMA.
Further, while ADIMA is well balanced on average, there exist class imbalance-
related variations across languages.

User Distribution: Figure 1(b) depicts the frequency distribution of the
number of samples in the dataset spoken by each individual user. On average,
there are around 1,500 ( 23.7%) users that have spoken more than one audio
recording in the dataset, indicating a diverse set of users while also
presenting a promising opportunity of potential user profiling and leveraging
similarity across the samples spoken by the same user for improved
performance.

Vocabulary Distribution: In Figure 1(c), we plot the frequency of profane
words which occur more than 5 times in the dataset. We note that Hindi has
around 50 dominant profane words, followed by Tamil and Kannada with 35 and 25
words. Overall, there are 1059, 820 and 690 unique profane words in Hindi,
Tamil and Kannada respectively. The distribution of unique words showcases the
diversity of the vocabulary.

Profanity Distribution: In Figure 1 (d), we plot the distribution of instances
of profane words present in the dataset for three languages. Lot of recordings
have lesser than 5 profane words making it challenging to spot the profanity
while some recordings are majorly abusive. Each abusive recording contains
atleast one profane word.

ADIMA is highly diverse across languages, users, vocabulary and density. The
recordings are sampled at 16kHz, mono-channel and range from 5-60 seconds with
an average duration of 20 seconds. We randomly split the dataset in 70:30
ratio for each language to form the train and test set.

##  4 Formulation and Methodology

###  4.1 Problem Formulation

We consider the task of classifying audio recording $x$ into $c$ $\in$
{abusive, non-abusive} categories. We extract features using VGG [14]
(pretrained over audio dataset) and Wav2Vec2 [17] (pretrained over speech
datasets) as backbones. The features are then aggregated across temporal
dimension and are passed through a fully connected classifier for
classification.

###  4.2 Feature Representations

VGG: Log-mel spectrograms are extracted from raw audios with the window and
hop length as 25ms and 10 ms for short-time fourier transform following [14].
We use 64 mel-spaced frequency bins and transform the magnitude using log to
arrive at the log-mel spectogram features for the recordings. Following [14],
we train VGG network over AudioSet dataset [18] for extracting features from
the spectograms.

Wav2Vec2: Wav2Vec2 models [17] are transformer based models and are trained in
a semi-supervised way on unlabelled raw audios. The models can be finetuned
for downstream tasks with task-specific labelled data. We explore XLSR-53
model (trained over 53 languages with little overlap with Indic languages),
CLSRIL-23 [19] (trained on Indic languages) and Him-4200
222https://huggingface.co/Harveenchadha/vakyansh-wav2vec2-hindi-him-4200 which
was finetuned using 4200 hours of labelled Hindi data over CLSRIL-23. For
extracting Wav2Vec2 features, we pass the raw recordings as input.

###  4.3 Model Architecture

We experiment with Mean-Pool, Max-Pool, and recurrent networks (GRU, LSTM). We
represent audio recordings by taking average and maximum of the features
across temporal dimension in Mean-Pool and Max-Pool, respectively. The
accumulated features are passed into a fully connected classifier
$(512\rightarrow 256\rightarrow 128\rightarrow 2)$ with ReLU activation and
0.1 dropout. For RNNs, audio features are processed through single-layer
bidirectional Gated Recurrent Units (GRU) and Long Short-Term Memory (LSTM).
The output of the final time step is used as input to the classifier.

###  4.4 Training Setup and Evaluation

We train the networks using cross entropy loss with Adam optimizer with
learning rate of 0.001 and batch size of 16 for 50 epochs. 0.1 is used as
dropout for the classification layers. The recordings are normalized by zero-
padding shorter audios. We augment data by applying temporal jittering of 0.1
and mask the temporal and frequency/feature dimension randomly between 0 and
10%. We report Macro F1 (MaF1), Accuracy (Acc), area under the ROC (AUC) and
precision-recall curve (AUCpr) on the test set.

##  5 Results

###  5.1 Monolingual Experiments

From Table 2, we note that CLSRIL-23 outperforms other backbones which can be
attributed to the pretraining of CLSRIL-23 in Indic languages. Surprisingly,
VGG which has been trained for identifying different sounds instead of spoken
text demonstrates superior performance than XLSR-53 model for Hindi. However,
Him-4200 which has been finetuned for Hindi outperforms XLSR-53 showing the
advantage of language specific finetuning. The RNN baselines overfit during
the training and do not improve the results. We evaluate the baselines on
other languages in Table 3 and note that Wav2Vec2 models work better for
majority of the languages. This can be attributed to the pretraining of these
models on speech data.

Table 2: Accuracy (Acc), Macro F1 (MaF1), Area under ROC curve (AUC) and Area under Precision-Recall curve (AUCpr) for different architectures and backbones for Hindi. Backbone | Model | Acc | MaF1 | AUC | AUCpr
---|---|---|---|---|---
VGG | Max-Pool | 78.05 | 78.01 | 0.84 | 0.85
Mean-Pool | 78.59 | 78.57 | 0.85 | 0.86
LSTM | 77.77 | 77.71 | 0.84 | 0.86
GRU | 78.57 | 78.56 | 0.85 | 0.86
XLSR-53 | Max-Pool | 76.96 | 76.90 | 0.84 | 0.84
Mean-Pool | 77.51 | 77.34 | 0.83 | 0.85
Him-4200 | Max-Pool | 79.13 | 79.03 | 0.85 | 0.86
Mean-Pool | 78.86 | 78.69 | 0.85 | 0.85
CLSRIL-23 | Max-Pool | 79.67 | 79.48 | 0.86 | 0.86
Mean-Pool | 78.59 | 78.59 | 0.86 | 0.84
LSTM | 70.19 | 69.53 | 0.78 | 0.79
GRU | 75.34 | 75.23 | 0.82 | 0.83
Table 3: Accuracy (Acc) and F1 (MaF1) for languages for VGG and XLSR-53 (Mean-Pool) and CLSRIL-23 features with Max-Pool aggregation. | VGG | XLSR-53 | CLSRIL-23
---|---|---|---
Lang | Acc | MaF1 | Acc | MaF1 | Acc | MaF1
Hi | 78.59 | 78.57 | 77.51 | 77.34 | 79.67 | 79.48
Be | 78.65 | 77.63 | 81.08 | 79.46 | 79.73 | 77.95
Pu | 82.01 | 81.97 | 82.01 | 81.99 | 82.01 | 82.01
Ha | 81.15 | 81.12 | 80.05 | 79.91 | 79.23 | 79.10
Ka | 82.91 | 80.06 | 82.92 | 80.15 | 79.67 | 75.39
Od | 81.64 | 81.46 | 83.29 | 82.21 | 81.64 | 80.21
Bh | 76.19 | 71.85 | 76.48 | 71.10 | 75.89 | 72.30
Gu | 79.56 | 74.11 | 79.28 | 69.21 | 80.94 | 76.38
Ta | 79.78 | 70.77 | 80.59 | 75.04 | 80.59 | 73.39
Ma | 81.72 | 77.31 | 81.70 | 75.45 | 86.29 | 83.41

###  5.2 Cross-lingual Experiments

In Table 4, we train zero-shot models on the source language and evaluate the
performance on the target language using CLSRIL-23 and Max-Pool. The cross-
lingual performance is competitive and even better for some languages showing
strong cross learning among languages for this task. We hypothesize that
models are able to leverage audio properties like pitch, emotions, intensity
etc. for this task instead of relying on the actual words, which is highly
encouraging. We also combine the data (All) for all these languages together
for training and evaluate on each language separately. We note that
combination of all the languages shows improvement for majority of the
languages paving path for truly multilingual models for profanity detection.

Table 4: Macro F1 score across languages using CLSRIL-23 model with Max-Pool aggregation. Bold represent the best combinations. source/target | Hi | Be | Pu | Ka | Ta
---|---|---|---|---|---
Hi | 79.5 | 77.0 | 81.5 | 79.1 | 74.7
Be | 78.3 | 77.9 | 82.0 | 75.2 | 74.1
Pu | 78.5 | 77.1 | 82.0 | 77.6 | 71.0
Ka | 78.3 | 77.4 | 82.4 | 75.4 | 74.1
Ta | 77.1 | 76.0 | 83.4 | 77.1 | 73.3
All | 80.7 | 79.1 | 83.4 | 78.4 | 75.2

##  6 Conclusion and Future Work

Detection of abusive content in spoken text is an important problem.
Performing ASR followed by a NLP layer for processing the transcription
introduces complexity and cost of developing ASR models. In this paper, we
contribute a novel and diverse multilingual audio dataset - ADIMA for tackling
this problem entirely in the audio domain. The dataset covers 10 Indic
languages with 11,775 samples (65 hours) spoken by 6446 unique users and
annotated by expert team of reviewers. We also perform comprehensive
experiments and report baselines for encouraging further exploration in this
direction.

Ethical Considerations: Keeping in mind the sensistive nature of the task, we
ensure to mandate certain ethical considerations throughout the course of this
research and public release of data. Specifically, ShareChat’s public
chatrooms are open for anyone to join and users’ informed consent is sought
for recording and broadcasting the discussions. Further, we remove any
Personally Identifiable Information (PII) from the dataset and anonymize it.
The raw data is kept on secure servers with strong access restrictions to
prevent any malicious usage.

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