An arbitrary order Reconstructed Discontinuous Approximation to Fourth-order Curl Problem

We are concerned in this paper with the fourth-order curl problem, which has applications in inverse electromagnetic scattering, and magnetohydrodynamics (MHD) when modeling magnetized plasmas. Discretizing the fourth-order curl operator is one of the keys to simulate these models. Additionally, the fourth-order curl operator plays an important role in approximating the Maxwell transmission eigenvalue problem. Therefore, it is important to design highly efficient and accurate numerical methods for fourth-order curl problems [2, 17].

Finite element methods (FEMs) are a widely used numerical scheme for solving partial differential equations. The design of FEMs for fourth-order curl problems is challenging. Nevertheless, in recent years, researchers have proposed and analysed various FEMs for these problems to address this difficulty. According to whether the finite element space is contained within the space of the exact solution, finite element methods can be classified into conforming methods and nonconforming methods. For conforming methods, one needs to construct $H(\mathrm{curl}^{2})$-conforming finite elements, we refer to [22, 6] for some recent works. Due to the high order of the fourth-order curl operator, it is still a difficult task to implement a high-order conforming space. Therefore, many researches focus on nonconforming methods. The discontinuous Galerkin (DG) methods use completely discontinuous piecewise polynomial to approximate the solution of partial differential equations. These methods enforce numerical solutions to be close to the exact solution by adding penalties. DG methods are commonly used in solving complex problems due to their simplicity and flexibility in approximation space. For fourth-order curl problems, we refer to [24, 5, 21, 20, 23] for some DG methods.

A significant drawback of DG methods is the large number of degrees of freedom in DG space, which results in high computational costs. This drawback is a matter of concern. In this paper, we propose an arbitrary order discontinuous Galerkin FEM for solving the fourth-order curl problem. The method is based on a reconstructed approximation space that has only one unknown per dimension in each element. The construction of the approximation space includes creating an element patch per element and solving a local least squares fitting problem to obtain a local high-order polynomial. Methods based on the reconstructed spaces are called reconstructed discontinuous approximation (RDA) methods, which have been successfully applied to a series of classical problems [13, 10, 16, 15, 8, 14]. The reconstructed space is a subspace of the standard DG space, which can approximate functions to high-order accuracy and inherits the flexibility on the mesh partition in the meanwhile. One advantage of this space is that it has very few degrees of freedom, which gives high approximation efficiency of finite element. Using this new space, we design the discrete scheme for the fourth-order curl problem under the symmetric interior penalty discontinuous Galerkin (IPDG) framework. We prove the convergence rates under the energy norm and the $L^{2}$ norm, and numerical experiments are conducted to verify the theoretical analysis ,showing our algorithm is simple to implement and can reach high-order accuracy.

The rest of this paper is organized as follows. In Section 2, we introduce the fourth-order curl problem without div-free condition and give the basic notations about the Sobolev spaces and the partition. We also recall two commonly used inequalities in this section. In Section 3, we establish the reconstruction operator and the corresponding approximation space. Some basic properties of the reconstruction are also proven. In Section 4, we define the discrete variational form for the fourth-order curl problem and analyse the error under the energy norm and the $L^{2}$ norm, and prove that the convergence rate is optimal with respect to the energy norm. In Section 5, we carried out some numerical examples to validate our theoretical results. Finally, a brief conclusion is given in Section 6.

Let $\Omega\subset\mathbb{R}^{d}(d=2,3)$ be a bounded polygonal (polyhedral) domain with a Lipschitz boundary $\partial\Omega$.
Given a bounded domain $D\subset\Omega$, we follow the standard definitions to the space $L^{2}(D),L^{2}(D)^{d}$, the spaces $H^{q}(D),H^{q}(D)^{d}$ with the regular exponent $q\geq 0$. The corresponding norms and seminorms are defined as

$$\|\cdot\|_{H^{q}(D)}:=\left(\|\cdot\|_{H^{q}(D)}^{2}\right)^{1/2},\,\,|\cdot|_{H^{q}(D)}:=\left(|\cdot|_{H^{q}(D)}^{2}\right)^{1/2},$$

respectively.

Throughout this paper, for two vectors $\boldsymbol{a}=(a_{1},\ldots,a_{d})\in\mathbb{R}^{d},\boldsymbol{b}=(b_{1},\ldots,b_{d})\in\mathbb{R}^{d}$, the cross product $\boldsymbol{a}\times\boldsymbol{b}$ is defined as

$$\boldsymbol{a}\times\boldsymbol{b}:=\begin{cases}a_{1}b_{2}-a_{2}b_{1},\quad d=2,\\ (a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1})^{T},\quad d=3.\\ \end{cases}$$

The cross product between a vector and a scalar will be used in two dimensions, and in this case for a vector $\boldsymbol{a}=(a_{1},a_{2})\in\mathbb{R}^{2}$ and a scalar $b\in\mathbb{R}$, $\boldsymbol{a}\times b$ is defined as $\boldsymbol{a}\times b:=(a_{2}b,-a_{1}b)^{T}$. Specifically, for a vector-valued function $\boldsymbol{u}\in\mathbb{R}^{d}$, the curl of $\boldsymbol{u}$ reads

$$\mathrm{curl}\boldsymbol{u}:=\nabla\times\boldsymbol{u}=\begin{cases}\frac{\partial u_{2}}{\partial x}-\frac{\partial u_{1}}{\partial y},\quad d=2,\\ (\frac{\partial u_{3}}{\partial y}-\frac{\partial u_{2}}{\partial z},\frac{\partial u_{1}}{\partial z}-\frac{\partial u_{3}}{\partial x},\frac{\partial u_{2}}{\partial x}-\frac{\partial u_{1}}{\partial y})^{T},\quad d=3,\end{cases}$$

and for the scalar-valued function $q(x,y)$, we let $\nabla\times q$ be the formal adjoint, which reads

$$\nabla\times q=\left(\frac{\partial q}{\partial y},-\frac{\partial q}{\partial x}\right)^{T}.$$

For convenience, we mainly use the notations for the three-dimensional case.

In this paper, we consider the following fourth-order curl problem

(1)

$$\left\{\begin{aligned} &\mathrm{curl}^{4}\boldsymbol{u}+\boldsymbol{u}=\boldsymbol{f},&\quad\text{ in }\Omega,\\ &\boldsymbol{u}\times\boldsymbol{n}=g_{1},&\quad\text{ on }\partial\Omega,\\ &(\nabla\times\boldsymbol{u})\times\boldsymbol{n}=g_{2},&\quad\text{ on }\partial\Omega,\end{aligned}\right.$$

For the problem domain $\Omega$, we define the space

$\displaystyle H(\mathrm{curl}^{s},\Omega)$

$\displaystyle:=\left\{\boldsymbol{v}\in L^{2}(\Omega)^{d}\ |\ \mathrm{curl}^{j}\boldsymbol{v}\in L^{2}(\Omega),1\leq j\leq s\right\},$

and

$\displaystyle H_{0}(\mathrm{curl}^{2},\Omega)$

$\displaystyle:=\left\{\boldsymbol{v}\in L^{2}(\Omega)^{d}\ |\ \mathrm{curl}^{j}\boldsymbol{v}\in L^{2}(\Omega),\ \mathrm{curl}^{j-1}\boldsymbol{v}=0\ \text{on}\ \partial\Omega,1\leq j\leq 2\right\},$

For the weak formulation we are going to find $\boldsymbol{u}\in H(\mathrm{curl}^{2},\Omega)$ such that

$\displaystyle a(\boldsymbol{u},\boldsymbol{v})=(\boldsymbol{f},\boldsymbol{v})_{L^{2}(\Omega)},\ \forall\boldsymbol{v}\in H_{0}(\mathrm{curl}^{2},\Omega),$

where

$$a(\boldsymbol{u},\boldsymbol{v}):=(\mathrm{curl}^{2}\boldsymbol{u},\mathrm{curl}^{2}\boldsymbol{v})_{L^{2}(\Omega)}+(\boldsymbol{u},\boldsymbol{v})_{L^{2}(\Omega)}.$$

We refer to [18, 23] for some regularity results of this problem.

Next, we define some notations about the mesh. Let $\mathcal{T}_{h}$ be a regular and quasi-uniform partition $\Omega$ into disjoint open triangles (tetrahedra). Let $\mathcal{E}_{h}$ denote the set of all $d-1$ dimensional faces of $\mathcal{T}_{h}$, and we decompose $\mathcal{E}_{h}$ into $\mathcal{E}_{h}=\mathcal{E}_{h}^{i}\cup\mathcal{E}_{h}^{b}$, where $\mathcal{E}_{h}^{i}$ and $\mathcal{E}_{h}^{b}$ are the sets of interior faces and boundary faces, respectively. We let

$$h_{K}:=\text{diam}(K),\quad\forall K\in\mathcal{T}_{h},\quad h_{e}:=\text{diam}(e),\quad\forall e\in\mathcal{E}_{h},$$

and define $h:=\max_{K\in\mathcal{T}_{h}}h_{K}$. The quasi-uniformity of the mesh $\mathcal{T}_{h}$ is in the sense that there exists a constant $\nu>0$ such that $h\leq\nu\min_{K\in\mathcal{T}_{h}}\rho_{K}$, where $\rho_{K}$ is the diameter of the largest ball inscribed in $K$. One can get the inverse inequality and the trace inequality from the regularity of the mesh, which are commonly used in the analysis.

We refer to [1] for more details of these inequalities.

Finally, we introduce the trace operators that will be used in our numerical schemes. For $e\in\mathcal{E}_{h}^{i}$, we denote by $K^{+}$ and $K^{-}$ the two neighbouring elements that share the boundary $e$, and $\boldsymbol{\mathrm{n}}^{+},\boldsymbol{\mathrm{n}}^{-}$ the unit out normal vector on $e$, respectively. We define the jump operator $[[\cdot]]$ and the average operator $\{\cdot\}$ as

(4)		$\displaystyle[[\boldsymbol{q}]]$	$\displaystyle:=\boldsymbol{q}|_{K^{+}}\times\boldsymbol{\mathrm{n}}^{+}+\boldsymbol{q}|_{K^{-}}\times\boldsymbol{\mathrm{n}}^{-},\quad\text{ for vector-valued function},$
		$\displaystyle[[\mathrm{curl}\boldsymbol{q}]]$	$\displaystyle:=\mathrm{curl}\boldsymbol{q}|_{K^{+}}\times\boldsymbol{\mathrm{n}}^{+}+\mathrm{curl}\boldsymbol{q}|_{K^{-}}\times\boldsymbol{\mathrm{n}}^{-},\quad\text{ for vector-valued function},$
		$\displaystyle\{v\}$	$\displaystyle:=\frac{1}{2}(v|_{K^{+}}+v|_{K^{-}}),\quad\text{ for scalar-valued function},$
		$\displaystyle\{\boldsymbol{q}\}$	$\displaystyle:=\frac{1}{2}(\boldsymbol{q}|_{K^{+}}+\boldsymbol{q}|_{K^{-}}),\quad\text{ for vector-valued function}.$

For $e\in\mathcal{E}_{h}^{b}$, we let $K\in\mathcal{T}_{h}$ such that $e\in\partial K$ and $\boldsymbol{\mathrm{n}}$ is the unit out normal vector. We define

(5)			$\displaystyle[[\boldsymbol{q}]]:=\boldsymbol{q}|_{K}\times\boldsymbol{\mathrm{n}},\quad[[\mathrm{curl}\boldsymbol{q}]]:=\mathrm{curl}\boldsymbol{q}|_{K}\times\boldsymbol{\mathrm{n}}\text{ for vector-valued function},$
			$\displaystyle\{v\}:=v|_{K}\quad\text{ for scalar-valued function},\qquad\{\boldsymbol{q}\}:=\boldsymbol{q}|_{K}\quad\text{ for vector-valued function}.$

Throughout this paper, $C$ and $C$ with subscripts denote the generic constants that may differ between lines but are independent of the mesh size.

In this section, we will introduce a linear reconstruction operator to obtain a discontinuous approximation space for the given mesh $\mathcal{T}_{h}$. The reconstructed space can achieve a high-order accuracy while the number of degrees of freedom in each dimension remain the same as the number of elements in $\mathcal{T}_{h}$. The construction of the operator includes the following steps.

Step 1. For each $K\in\mathcal{T}_{h}$, we construct an element patch $S(K)$, which consists of $K$ itself and some surrounding elements. The size of the patch is controlled with a given threshold $\#S$. The construction of the element patch $S(K)$ is conducted by a recursive algorithm. We begin by setting $S_{0}(K)=\{K\}$, and define $S_{t}(K)$ recursively:

(6)

$$S_{t}(K)=\bigcup_{K^{\prime}\in S_{t-1}(K)}\bigcup_{K^{\prime\prime}\in\Delta(K^{\prime})}K^{\prime\prime},\quad t=0,1,\dots$$

where $\Delta(K):=\{K^{\prime}\in\mathcal{T}_{h}\ |\ \overline{K}\cap\overline{K^{\prime}}\neq\emptyset\}$. The recursion stops once $t$ meets the condition that the cardinality $\#S_{t}(K)\geq\#S$, and we let the patch $S(K):=S_{t}(K)$. We apply the recursive algorithm (6) to all elements in $\mathcal{T}_{h}$.

Step 2. For each $K\in\mathcal{T}_{h}$, we solve a local least squares fitting problem on the patch $S(K)$. We let $\boldsymbol{x}_{K}$ be the barycenter of the element $K$ and mark barycenters of all elements as collocation points. Let $I(K)$ be the set of collocation points located inside the domain of $S(K)$,

$$I(K):=\{\boldsymbol{x}_{K^{\prime}}\ |\ K^{\prime}\in S(K)\},$$

Let $U_{h}^{0}$ be the piecewise constant space with respect to $\mathcal{T}_{h}$,

$$U_{h}^{0}:=\{v_{h}\in L^{2}(\Omega)\ |\ v_{h}|_{K}\in\mathbb{P}_{0}(K),\ \forall K\in\mathcal{T}_{h}\}.$$

and we denote by $\boldsymbol{\mathrm{U}}_{h}^{0}:=(U_{h}^{0})^{d}$ the vector-valued piecewise constant space. Given a function $\boldsymbol{g}_{h}\in\boldsymbol{\mathrm{U}}_{h}^{0}$ and an integer $m\geq 1$, for the element $K\in\mathcal{T}_{h}$, we consider the following local constrained least squares problem:

(7)

$$\boldsymbol{p}_{S(K)}=\mathop{\arg\min}_{\boldsymbol{q}\in\mathbb{P}_{m}(S(K))^{d}}\sum_{\boldsymbol{x}\in I(K)}\|\boldsymbol{q}(\boldsymbol{x})-\boldsymbol{g}_{h}(\boldsymbol{x})\|_{l^{2}}^{2},\quad\text{s.t. }\boldsymbol{q}(\boldsymbol{x}_{K})=\boldsymbol{g}_{h}(\boldsymbol{x}_{K}),$$

where $I(K):=\{\boldsymbol{x}_{\widetilde{K}}\ |\ \widetilde{K}\in S(K)\cap\mathcal{T}_{h}\}$ denotes the set of collocation points located in $S(K)$, and $\|\cdot\|_{l^{2}}$ denotes the discrete $l^{2}$ norm for vectors. We make the following geometrical assumption on the location of collocation points [11, 10]:

The Assumption 1 excludes the case that the points in $I(K)$ are on an algebraic curve of degree $m$ and requires cardinality $\#I(K)\geq\mathrm{dim}(\mathbb{P}_{m}(\cdot))$. Under this assumption, the fitting problem (7) admits a unique solution. By solving (7) and considering the restriction $(\boldsymbol{p}_{S(K)})|_{K}$, we actually gain a local polynomial defined on $K$. Since $\boldsymbol{p}_{S(K)}$ is sought in the least squares sense, we note that $\boldsymbol{p}_{S(K)}$ depends linearly on the given function $\boldsymbol{g}_{h}$. This property inspires us to define a linear operator from the piecewise constant space $\boldsymbol{\mathrm{U}}_{h}^{0}$ to a piecewise polynomial space with respect to $\mathcal{T}_{h}$.

We define a local reconstruction operator $\mathcal{R}_{K}$ for elements in $\mathcal{T}_{h}$,

$$\begin{aligned} \mathcal{R}_{K}:\boldsymbol{\mathrm{U}}_{h}^{0}&\rightarrow(\mathbb{P}_{m}(K))^{d},\\ \boldsymbol{g}_{h}&\rightarrow(\boldsymbol{p}_{S(K)})|_{K},\\ \end{aligned}\quad\forall K\in\mathcal{T}_{h}.$$

Further, the linear reconstruction operator $\mathcal{R}$ for $\mathcal{T}_{h}$ is defined in a piecewise manner as

$$\begin{aligned} \mathcal{R}:\boldsymbol{\mathrm{U}}_{h}^{0}&\rightarrow\boldsymbol{\mathrm{U}}_{h}^{m},\\ \boldsymbol{g}_{h}&\rightarrow\mathcal{R}\boldsymbol{g}_{h},\\ \end{aligned}\quad(\mathcal{R}\boldsymbol{g}_{h})|_{K}:=\mathcal{R}_{K}\boldsymbol{g}_{h},\quad\forall K\in\mathcal{T}_{h},$$

where $\boldsymbol{\mathrm{U}}_{h}^{m}$ is the image space of the operator $\mathcal{R}$. Clearly, by the operator $\mathcal{R}$, any piecewise constant function $\boldsymbol{g}_{h}$ will be mapped into a piecewise polynomial function $\mathcal{R}\boldsymbol{g}_{h}\in\boldsymbol{\mathrm{U}}_{h}^{m}$. Then, we present more details to the space $\boldsymbol{\mathrm{U}}_{h}^{m}$, which are piecewise polynomial spaces for the partition $\mathcal{T}_{h}$. For any element $K$, we pick up a function $\boldsymbol{w}_{K,j}\in\boldsymbol{\mathrm{U}}_{h}^{0}$ such that

$$\boldsymbol{w}_{K,j}(\boldsymbol{x})=\begin{cases}\boldsymbol{e}_{j},&\boldsymbol{x}\in K,\\ \boldsymbol{0},&\text{otherwise},\\ \end{cases}\quad 1\leq j\leq d,$$

where $\boldsymbol{e}_{j}$ is the unit vector in $\mathbb{R}^{d}$ whose $j$-th entry is $1$. We let $\boldsymbol{\lambda}_{K,j}:=\mathcal{R}\boldsymbol{w}_{K,j}(K\in\mathcal{T}_{h}\leq j\leq d)$ and we state that the space $\boldsymbol{\mathrm{U}}_{h}^{m}$ is spanned by $\{\boldsymbol{\lambda}_{K,j}\}$.

Lemma 3 claims that the basis functions of $\boldsymbol{\mathrm{U}}_{h}^{m}$ are indeed the group of functions $\{\boldsymbol{\lambda}_{K,j}\}$, and for any function $\boldsymbol{g}_{h}\in\boldsymbol{\mathrm{U}}_{h}^{0}$, one can explicitly write $\mathcal{R}\boldsymbol{g}_{h}$ as

(9)

$$\mathcal{R}\boldsymbol{g}_{h}=\sum_{K\in\mathcal{T}_{h}}\sum_{j=1}^{d}g_{h,j}(\boldsymbol{x}_{K})\boldsymbol{\lambda}_{K,j}.$$

From the problem (7), the basis function $\boldsymbol{\lambda}_{K,j}$ vanishes on the element $K^{\prime}$ that $K\not\in S(K^{\prime})$. This fact indicates $\boldsymbol{\lambda}_{K,j}$ has a finite support set that $\text{supp}(\boldsymbol{\lambda}_{K,j})=\{K^{\prime}\in\mathcal{T}_{h}\ |\ K\in S(K^{\prime})\}$, and $\text{supp}(\boldsymbol{\lambda}_{K,j})$ is different from the patch $S(K)$.

We extend the operator $\mathcal{R}$ to act on smooth functions. For any $\boldsymbol{g}\in H^{m+1}(\Omega)^{d}$, we define a piecewise constant function $\boldsymbol{g}_{h}\in\boldsymbol{\mathrm{U}}_{h}^{0}$ as

$$\boldsymbol{g}_{h}(\boldsymbol{x}_{K}):=\boldsymbol{g}(\boldsymbol{x}_{K}),\quad\forall K\in\mathcal{T}_{h},$$

and we directly define $\mathcal{R}\boldsymbol{g}:=\mathcal{R}\boldsymbol{g}_{h}$. In this way, any smooth function in $H^{m+1}(\Omega)^{d}$ is mapped into a piecewise polynomial function with respect to $\mathcal{T}_{h}$ by the operator $\mathcal{R}$. For any $\boldsymbol{g}\in H^{m+1}(\Omega)^{d}$, $\mathcal{R}\boldsymbol{g}$ can also be written as (9).

Next, we will focus on the approximation property of the operator $\mathcal{R}$. We first define a constant $\Lambda_{m,K}$ for every element patch,

$$\Lambda^{2}_{m,K}:=\max_{p\in\mathbb{P}_{m}(S(K))}\frac{\|p\|_{L^{2}}}{h^{d}\sum_{\boldsymbol{x}\in I(K)}p(\boldsymbol{x})^{2}}.$$

Assumption 1 as well as the norm equivalence in finite dimensional spaces actually ensures $\Lambda_{m,K}<\infty$.

From the stability result (10), we can prove the approximation results.

From (12), it can be seen that the operator $\mathcal{R}$ has an approximation error of degree $O(h^{m+1-q})$ under the case that $\Lambda_{m}$ admits an upper bound independent of the mesh size $h$. However, it is not trivial to bound the constant $\Lambda_{m}$, see Remark 1. We can prove that under some mild conditions on the element patch, the constant admits a uniform upper bound.

We define the approximation problem to solve the problem (1) based on the space $\boldsymbol{\mathrm{U}}_{h}^{m}$ constructed in the previous section: Seek $\boldsymbol{u_{h}}\in\boldsymbol{\mathrm{U}}_{h}^{m}$ ($m\geq 2$) such that

(14)

$$B_{h}(\boldsymbol{u_{h}},\boldsymbol{v_{h}})=l_{h}(\boldsymbol{v_{h}}),\quad\forall\boldsymbol{v_{h}}\in\boldsymbol{\mathrm{U}}_{h}^{m},$$

where we define the bilinear form $B_{h}(\cdot,\cdot)$ on $U_{h}:=\boldsymbol{\mathrm{U}}_{h}^{m}+H(\mathrm{curl}^{2},\Omega)\cap\{\boldsymbol{v}:\|\mathrm{curl}^{j}\boldsymbol{v}\|_{1/2+\sigma,\Omega}<\infty,0\leq j\leq 3\}$, as in [4].

	$\displaystyle B_{h}(\boldsymbol{u},\boldsymbol{v})$	$\displaystyle:=\sum_{K\in\mathcal{T}_{h}}\int_{K}\mathrm{curl}^{2}\boldsymbol{u}\cdot\mathrm{curl}^{2}\boldsymbol{v}\ \mathrm{d}\boldsymbol{\boldsymbol{x}}+\sum_{K\in\mathcal{T}_{h}}\int_{K}\boldsymbol{u}\cdot\boldsymbol{v}\ \mathrm{d}\boldsymbol{\boldsymbol{x}}$
		$\displaystyle+\sum_{e\in\mathcal{E}_{h}}\int_{e}\left([[\boldsymbol{u}]]\cdot\{\mathrm{curl}^{3}\boldsymbol{v}\}+[[\mathrm{curl}\boldsymbol{u}]]\cdot\{\mathrm{curl}^{2}\boldsymbol{v}\}\right)\mathrm{d}\boldsymbol{\boldsymbol{s}}+\sum_{e\in\mathcal{E}_{h}}\int_{e}\left([[\boldsymbol{v}]]\cdot\{\mathrm{curl}^{3}\boldsymbol{u}\}+[[\mathrm{curl}\boldsymbol{v}]]\cdot\{\mathrm{curl}^{2}\boldsymbol{u}\}\right)\mathrm{d}\boldsymbol{\boldsymbol{s}}$
		$\displaystyle+\sum_{e\in\mathcal{E}_{h}}\int_{e}\left(\mu_{1}[[\boldsymbol{u}]]\cdot[[\boldsymbol{v}]]+\mu_{2}[[\mathrm{curl}\boldsymbol{u}]]\cdot[[\mathrm{curl}\boldsymbol{v}]]\right)\mathrm{d}\boldsymbol{\boldsymbol{s}}$

The parameter $\mu_{1}$ and $\mu_{2}$ are positive penalties which are set by

$\displaystyle\mu_{1}|_{e}=\frac{\eta}{h_{e}^{3}},\quad\mu_{2}|_{e}=\frac{\eta}{h_{e}}.\quad\text{ for }e\in\mathcal{E}_{h}.$

The linear form $l_{h}(\cdot)$ is defined for $\boldsymbol{v}\in U_{h}$,

$\displaystyle l_{h}(\cdot)$

$\displaystyle:=\sum_{K\in\mathcal{T}_{h}}\int_{K}\boldsymbol{f}\cdot\boldsymbol{v}\mathrm{d}\boldsymbol{\boldsymbol{x}}$

$\displaystyle+\sum_{e\in\mathcal{E}_{h}^{b}}\int_{e}\left(g_{1}\cdot(\mathrm{curl}^{3}\boldsymbol{v_{h}}+\mu_{1}\boldsymbol{v_{h}}\times\boldsymbol{\mathrm{n}})+g_{2}\cdot(\mathrm{curl}^{2}\boldsymbol{v_{h}}+\mu_{2}\mathrm{curl}\boldsymbol{v_{h}}\times\boldsymbol{\mathrm{n}})\right)\mathrm{d}\boldsymbol{\boldsymbol{s}}.$

We introduce two energy norms for the space $U_{h}$,

$\displaystyle\|\boldsymbol{u}\|_{\mathrm{DG}}^{2}$

$\displaystyle:=\sum_{K\in\mathcal{T}_{h}}\int_{K}|\boldsymbol{u}|^{2}\mathrm{d}\boldsymbol{\boldsymbol{x}}+\sum_{K\in\mathcal{T}_{h}}\int_{K}|\mathrm{curl}^{2}\boldsymbol{u}|^{2}\mathrm{d}\boldsymbol{\boldsymbol{x}}+\sum_{e\in\mathcal{E}_{h}}\int_{e}h_{K}^{-3}|[[\boldsymbol{u}]]|^{2}\mathrm{d}\boldsymbol{\boldsymbol{s}}+\sum_{e\in\mathcal{E}_{h}}\int_{e}h_{K}^{-1}|[[\mathrm{curl}\boldsymbol{u}]]|^{2}\mathrm{d}\boldsymbol{\boldsymbol{s}},$

and

$\displaystyle|\!|\!|\boldsymbol{u}|\!|\!|^{2}$

$\displaystyle:=\|\boldsymbol{u}\|_{\mathrm{DG}}^{2}+\sum_{e\in\mathcal{E}_{h}}\int_{e}h_{e}^{3}|\{\mathrm{curl}^{3}\boldsymbol{u}\}|^{2}\mathrm{d}\boldsymbol{\boldsymbol{s}}+\sum_{e\in\mathcal{E}_{h}}\int_{e}h_{e}|\{\mathrm{curl}^{2}\boldsymbol{u}\}|^{2}\mathrm{d}\boldsymbol{\boldsymbol{s}}.$

We claim that the two energy norms are equivalent over the space $\boldsymbol{\mathrm{U}}_{h}^{m}$.