Hodge Decomposition Methods for a Quad-Curl Problem on Planar Domains

Abstract

We develop and analyze $P_k$ Lagrange finite element methods for a quad-curl problem on planar domains that is based on the Hodge decomposition of divergence-free vector fields. Numerical results that illustrate the performance of the finite element methods are also presented.

1 Introduction

Let $\Omega \subset \mathbb {R}^2$ be a bounded polygonal domain. The energy space for the quad-curl problem to be considered in this paper is

$$\begin{aligned} {\mathbb {E}}=\{\varvec{v}\in [L_2(\Omega )]^2: \mathrm {curl}\,\varvec{v}\in H^1_0(\Omega ), \mathrm {div}\,\varvec{v}=0 \;\text {and}\;\varvec{n}\times \varvec{v}=0\;\text {on}\;\partial \Omega \}, \end{aligned}$$

(1.1)

with the norm $\Vert \cdot \Vert _{\mathbb {E}}$ given by

$$\begin{aligned} \Vert \varvec{v}\Vert _{\mathbb {E}}^2=\Vert \varvec{v}\Vert _{L_2(\Omega )}^2+|\mathrm {curl}\,\varvec{v}|_{H^1(\Omega )}^2. \end{aligned}$$

(1.2)

Here and below we will follow standard notation for differential operators, function spaces and norms that can be found for example in [1, 13, 16, 21, 25].

We will consider the following problem: Find $\varvec{u}\in {\mathbb {E}}$ such that

$$\begin{aligned} \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{u}),\mathrm {curl}\,(\mathrm {curl}\,\varvec{v})\big )+\beta (\mathrm {curl}\,\varvec{u},\mathrm {curl}\,\varvec{v})+ \gamma (\varvec{u},\varvec{v}) =(\varvec{f},\varvec{v}) \qquad \forall \,\varvec{v}\in {\mathbb {E}}, \end{aligned}$$

(1.3)

where $(\cdot ,\cdot )$ denotes the inner product for ${L_2(\Omega )}$ (or $[L_2(\Omega )]^2$), $\beta$ and $\gamma$ are nonnegative constants ($\gamma >0$ if $\Omega$ is not simply connected), and $\varvec{f}\in [L_2(\Omega )]^2$. Since the divergence-free condition is included in the definition of ${\mathbb {E}}$, the problem (1.3) provides an elliptic formulation for the quad-curl problem.

Remark 1.1

In two dimensions the curl of the vector field $\varvec{v}=[v_1, v_2]^t$ is the scalar function $\mathrm {curl}\,\varvec{v}=(\partial v_2/\partial x_1)-(\partial v_1/\partial x_2)$, and the curl of a scalar function $\phi$ is the vector field $\mathrm {curl}\,\phi =[\partial \phi /\partial x_2 , -\partial \phi /\partial x_1]^t$. An alternative notation for $\mathrm {curl}\,\phi$ is $\mathrm {rot}\,\phi$.

The quad-curl problem is related to the Maxwell transmission eigenvalue problem (cf. [14, 26] ) and mathematical models for magnetohydrodynamics with hyperresistivity (cf. [7, 15]). Finite element methods for the quad-curl problem (based on a non-elliptic formulation) were recently developed in [23, 29, 31] using a nonconforming finite element method, a discontinuous Galerkin method and a mixed finite element method. In this paper we will use a Hodge decomposition approach to reduce (1.3) to second order elliptic boundary value problems that can be solved by simple $H^1$ conforming finite element methods.

We note that the Hodge decomposition approach to time harmonic Maxwell equations on planar domains was investigated in [9] for the perfectly conducting boundary condition and extended to general boundary conditions in [11] with applications to metamaterials. Adaptive and multigrid methods for these problems based on the Hodge decomposition approach were developed in [10, 12, 17]. Applications of the Hodge decomposition to other electromagnetic problems can also be found in [2, 3, 5].

The rest of the paper is organized as follows. We recall the Hodge decomposition for divergence-free vector fields in Sect. 2, where the well-posedness of (1.3) is also addressed. The reduction of (1.3) to second order elliptic boundary value problems is established in Sect. 3. Based on this reduction, we develop $P_k$ finite element methods for (1.3) in Sect. 4, followed by a convergence analysis in Sect. 5. Numerical results are presented in Sect. 6, and we end with some concluding remarks in Sect. 7.

2 Hodge Decomposition for $H(\mathrm {div}\,^0;\Omega )$

The space $H(\mathrm {div}\,^0;\Omega )$ of divergence-free vector fields is the orthogonal complement of $\,\mathrm {grad}\,H^1_0(\Omega )$, i.e.,

$$\begin{aligned} H(\mathrm {div}\,^0;\Omega )=\{\varvec{v}\in [L_2(\Omega )]^2:\,(\varvec{v},\,\mathrm {grad}\,\eta )=0 \quad \forall \,\eta \in H^1_0(\Omega )\}, \end{aligned}$$

and $L_2^0(\Omega )=\{v\in {L_2(\Omega )}:\,(1,v)=0\}$ is the zero-mean subspace of ${L_2(\Omega )}$. Given any $\varvec{v}\in H(\mathrm {div}\,^0;\Omega )$, we have a unique decomposition (cf. [9, 21]):

$$\begin{aligned} \varvec{v}=\mathrm {curl}\,\psi +\sum _{j=1}^m d_j\,\mathrm {grad}\,\varphi _j, \end{aligned}$$

(2.1)

where $\psi \in H^1(\Omega )\cap L_2^0(\Omega )$, the non-negative integer m is the Betti number for $\Omega$ ($m=0$ if $\Omega$ is simply connected, cf. Fig. 1), $d_j$ ($1\le j\le m$) are real numbers, and the harmonic functions $\varphi _1,\ldots ,\varphi _m$ are defined as follows.

Let the outer boundary of $\Omega$ be denoted by ${\Gamma }_0$ and the m components of the inner boundary be denoted by ${\Gamma }_1,\ldots ,{\Gamma }_m$. Then the harmonic functions $\varphi _j$ are determined by

$$\begin{aligned} (\,\mathrm {grad}\,\varphi _j,\,\mathrm {grad}\,v)= & {} 0 \quad \forall \,v\in H^1_0(\Omega ), \end{aligned}$$

(2.2a)

$$\begin{aligned} \varphi _j\big |_{{\Gamma }_0}= & {} 0,\end{aligned}$$

(2.2b)

$$\begin{aligned} \varphi _j\big |_{{\Gamma }_k}= & {} {\left\{ \begin{array}{ll} 1&{}\quad j=k\\ 0&{}\quad j\ne k \end{array}\right. } \quad \text {for}\quad 1\le k\le m. \end{aligned}$$

(2.2c)

Fig. 1

Betti numbers

Remark 2.1

Note that $\,\mathrm {grad}\,\varphi _j$ belongs to ${\mathbb {E}}$ for $1\le j\le m$ (cf. [9, Corollary 2.5]) and

$$\begin{aligned} (\mathrm {curl}\,\psi ,\,\mathrm {grad}\,\varphi _j)=0 \qquad \forall \,\psi \in H^1(\Omega ) \end{aligned}$$

(2.3)

(cf. [9, Lemma 2.4]).

2.1 Properties of the Space ${\mathbb {E}}$

Let $\varvec{v}\in {\mathbb {E}}$ be represented by (2.1). We have

$$\begin{aligned} (\mathrm {curl}\,\psi ,\mathrm {curl}\,\rho )=(\varvec{v},\mathrm {curl}\,\rho ) \qquad \forall \,\rho \in H^1(\Omega ) \end{aligned}$$

by (2.3). Since ${\mathbb {E}}$ is a subspace of $H_0(\mathrm {curl}\,;\Omega )$, we also have (cf. [21, Theorems 2.2.11 and 2.2.12])

$$\begin{aligned} (\varvec{v},\mathrm {curl}\,\rho )=(\mathrm {curl}\,\varvec{v},\rho ) \qquad \forall \,\rho \in H^1(\Omega ). \end{aligned}$$

It follows that the function $\psi \in H^1(\Omega )\cap L_2^0(\Omega )$ satisfies

$$\begin{aligned} (\mathrm {curl}\,\psi ,\mathrm {curl}\,\rho )=(\mathrm {curl}\,\varvec{v},\rho ) \qquad \forall \,\rho \in H^1(\Omega ). \end{aligned}$$

(2.4)

Remark 2.2

Since $\mathrm {curl}\,\phi =[{\partial \phi }/{\partial x_2}, -{\partial \phi }/{\partial x_1}]^t$, we have

$$\begin{aligned} (\mathrm {curl}\,\phi ,\mathrm {curl}\,\rho )=(\,\mathrm {grad}\,\phi ,\,\mathrm {grad}\,\rho ) \qquad \forall \,\phi ,\rho \in H^1(\Omega ) \end{aligned}$$

and $\Vert \mathrm {curl}\,\rho \Vert _{L_2(\Omega )}=|\rho |_{H^1(\Omega )}$ for all $\rho \in H^1(\Omega )$.

We can deduce properties of $\varvec{v}$ from the following results (cf. [22, Sect. 5.1] and [18, Sect. 2.5]) for elliptic boundary value problems on polygonal domains, where $\omega$ is the largest interior angle at the corners of $\Omega$.

Lemma 2.3

Let $\mu \in H^1_0(\Omega )$ satisfy

$$\begin{aligned} (\,\mathrm {grad}\,\mu ,\,\mathrm {grad}\,\eta )+\beta (\mu ,\eta )=(g,\eta ) \qquad \forall \,\eta \in H^1_0(\Omega ), \end{aligned}$$

where $g\in H^1(\Omega )$. Then we have $\mu \in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$ and

$$\begin{aligned} \Vert \mu \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert g\Vert _{H^1(\Omega )}. \end{aligned}$$

Lemma 2.4

Let $\lambda \in H^1(\Omega )$ satisfy

$$\begin{aligned} (\,\mathrm {grad}\,\lambda ,\,\mathrm {grad}\,\psi )+(\lambda ,1)(\psi ,1)=(g,\psi ) \qquad \forall \,\psi \in H^1(\Omega ), \end{aligned}$$

where $g\in H^1(\Omega )$. Then we have $\lambda \in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$ and

$$\begin{aligned} \Vert \lambda \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert g\Vert _{H^1(\Omega )}. \end{aligned}$$

Since $\psi$ belongs to $H^1(\Omega )\cap L_2^0(\Omega )$ and $\mathrm {curl}\,\varvec{v}$ belongs to $H^1_0(\Omega )$, we can apply Lemma 2.4 to (2.4) and conclude that $\psi$ belongs to $H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$. Note that (2.2a), (2.2b), (2.2c) can be transformed to a problem of the form in Lemma 2.3 where $\beta =0$ and $g\in C^\infty (\bar{\Omega })$. Hence we can apply Lemma 2.3 to conclude that $\varphi _j$ belongs to $H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for $1\le j\le m$ and any $\epsilon >0$. Then (2.1) implies that $\varvec{v}$ belongs to $[H^{{(\pi /\omega )-\epsilon }}(\Omega )]^2$ and we have established the following result.

Theorem 2.5

The space ${\mathbb {E}}$ is a subspace of $[H^{{(\pi /\omega )-\epsilon }}(\Omega )]^2$ for any $\epsilon >0$, where $\omega$ is the largest angle at the corners of $\Omega$.

Remark 2.6

If $\Omega$ is a smooth domain, then we can apply the elliptic regularity theory for such domains [27] to conclude that $\psi$ belongs to $H^3(\Omega )$ and $\varphi _j$ belongs to $C^\infty (\bar{\Omega })$ for $1\le j\le m$. It follows that ${\mathbb {E}}$ is a subspace of $[H^2(\Omega )]^2$.

2.2 Well-Posedness of (1.3)

Since ${\mathbb {E}}$ is compactly embedded in $[L_2(\Omega )]^2$ by Theorem 2.5 and the Rellich–Kondrachov Theorem [1], we can establish the well-posedness of (1.3) by the Fredholm theory [30]. It suffices to show that if $\varvec{v}\in {\mathbb {E}}$ satisfies

$$\begin{aligned} \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{v}),\mathrm {curl}\,(\mathrm {curl}\,\varvec{w})\big )+\beta (\mathrm {curl}\,\varvec{v},\mathrm {curl}\,\varvec{w})+ \gamma (\varvec{v},\varvec{w}) =0\qquad \forall \,\varvec{w}\in {\mathbb {E}}, \end{aligned}$$

(2.5)

then $\varvec{v}=0$.

This is obvious if $\gamma >0$. In the case where $\gamma =0$ and $\Omega$ is simply connected, we deduce from (2.5) that

$$\begin{aligned} \mathrm {curl}\,(\mathrm {curl}\,\varvec{v})=0 \end{aligned}$$

and hence $\mathrm {curl}\,\varvec{v}=0$ (because $\mathrm {curl}\,\varvec{v}\in H^1_0(\Omega )$). It then follows from (2.4) that $\psi =0$ and hence $\varvec{v}=0$ by (2.1).

3 Reduction to Second Order Elliptic Boundary Value Problems

According to (2.1), we can write

$$\begin{aligned} \varvec{u}=\mathrm {curl}\,\phi +\sum _{j=1}^m c_j\,\mathrm {grad}\,\varphi _j, \end{aligned}$$

(3.1)

where $\phi \in H^1(\Omega )\cap L_2^0(\Omega )$ and $c_1,\ldots ,c_m$ are real numbers. The idea of the Hodge decomposition approach is to find $\phi$ and $c_1,\ldots ,c_m$, and then recover $\varvec{u}$ by (3.1).

Below we will find problems that determine the function $\phi$ and the coefficients $c_1,\ldots ,c_m$ in the decomposition (3.1).

3.1 A Problem for $\phi$

It follows from (2.4) and (3.1) that

$$\begin{aligned} (\mathrm {curl}\,\phi ,\mathrm {curl}\,\psi )=(\xi ,\psi ) \qquad \forall \,\psi \in H^1(\Omega ), \end{aligned}$$

(3.2)

where $\xi =\mathrm {curl}\,\varvec{u}\in H^1_0(\Omega )$. Note that $\varvec{n}\times \varvec{u}=0$ on $\partial \Omega$ implies $(1,\xi )=0$ and hence the singular Neumann boundary value problem (3.2) has a unique solution $\phi \in H^1(\Omega )\cap L_2^0(\Omega )$. An equivalent formulation that avoids the zero-mean constraint is to find $\phi \in H^1(\Omega )$ such that

$$\begin{aligned} (\mathrm {curl}\,\phi ,\mathrm {curl}\,\psi )+(\phi ,1)(\psi ,1)=(\xi ,\psi ) \qquad \forall \,\psi \in H^1(\Omega ). \end{aligned}$$

(3.3)

It only remains to find a problem that determines $\xi$.

3.2 A Problem for $\xi$

We begin with a lemma.

Lemma 3.1

The function $\xi =\mathrm {curl}\,\varvec{u}\in H^1_0(\Omega )\cap L_2^0(\Omega )$ satisfies

$$\begin{aligned} \big (\mathrm {curl}\,\xi ,\mathrm {curl}\,(\mathrm {curl}\,\varvec{\zeta })\big )+\beta (\xi ,\mathrm {curl}\,\varvec{\zeta })+\gamma (\varvec{u},\varvec{\zeta })=(Q\varvec{f},\varvec{\zeta }) \qquad \forall \,\varvec{\zeta }\in [C^\infty _c(\Omega )]^2, \end{aligned}$$

(3.4)

where Q is the orthogonal projection from $[L_2(\Omega )]^2$ onto $H(\mathrm {div}\,^0;\Omega )$.

Proof

Since $\varvec{\zeta }-Q\varvec{\zeta }$ belongs to $\,\mathrm {grad}\,(H^1_0(\Omega ))$, we have $\mathrm {curl}\,Q\varvec{\zeta }=\mathrm {curl}\,\varvec{\zeta }\in H^1_0(\Omega )$, $\varvec{n}\times Q\varvec{\zeta }=\varvec{n}\times \varvec{\zeta }=0$ on $\partial \Omega$ (cf. [9, Corollary 2.5]) and hence $Q\varvec{\zeta }\in {\mathbb {E}}$. Therefore (3.4) follows from (1.3):

$$\begin{aligned}&\,\big (\mathrm {curl}\,\xi ,\mathrm {curl}\,(\mathrm {curl}\,\varvec{\zeta })\big )+\beta (\xi ,\mathrm {curl}\,\varvec{\zeta })+\gamma (\varvec{u},\varvec{\zeta })\\&\quad = \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{u}),\mathrm {curl}\,(\mathrm {curl}\,\varvec{\zeta })\big )+\beta (\mathrm {curl}\,\varvec{u},\mathrm {curl}\,\varvec{\zeta })+ \gamma (\varvec{u},\varvec{\zeta })\\&\quad = \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{u}),\mathrm {curl}\,(\mathrm {curl}\,Q\varvec{\zeta })\big )+\beta (\mathrm {curl}\,\varvec{u},\mathrm {curl}\,Q\varvec{\zeta })+ \gamma (\varvec{u},Q\varvec{\zeta })\\&\quad = (Q\varvec{f},\varvec{\zeta }) \end{aligned}$$

$\square$

It follows from (3.4) that

$$\begin{aligned} \mathrm {curl}\,(-\Delta \xi )=-\beta \,\mathrm {curl}\,\xi -\gamma \varvec{u}+Q\varvec{f}\end{aligned}$$

(3.5)

in the sense of distributions.

We will exploit (3.5) through the following lemma due to Nečas [8, 20, 28].

Lemma 3.2

If $\tau$, $\partial \tau /\partial x_1$ and $\partial \tau /\partial x_2$ belong to $H^{-1}(\Omega )$ , then $\tau$ belongs to ${L_2(\Omega )}$.

Since $-\Delta \xi$ belongs to $H^{-1}(\Omega )=[H^1_0(\Omega )]'$ and the right-hand side of (3.5) belongs to $[L_2(\Omega )]^2$, we can apply Lemma 3.2 to conclude that $-\Delta \xi$ belongs to ${L_2(\Omega )}$. Then (3.5) implies

$$\begin{aligned} -\Delta \xi \in H^1(\Omega ). \end{aligned}$$

(3.6)

Let $\rho \in H^1(\Omega )\cap L_2^0(\Omega )$ be defined by the following consistent singular Neumann boundary value problem

$$\begin{aligned} (\mathrm {curl}\,\rho ,\mathrm {curl}\,\psi )= & {} -\gamma (\varvec{u},\mathrm {curl}\,\psi )+(Q\varvec{f},\mathrm {curl}\,\psi )\\= & {} -\gamma (\xi ,\psi )+(\varvec{f},\mathrm {curl}\,\psi ) \qquad \qquad \forall \,\psi \in H^1(\Omega ).\nonumber \end{aligned}$$

(3.7)

Since the relations (3.5)–(3.7) imply $-\Delta \xi +\beta \xi \in H^1(\Omega )$ and

$$\begin{aligned} (\mathrm {curl}\,(-\Delta \xi +\beta \xi ),\mathrm {curl}\,\psi )=(\mathrm {curl}\,\rho ,\mathrm {curl}\,\psi ) \qquad \forall \,\psi \in H^1(\Omega ), \end{aligned}$$

we have

$$\begin{aligned} -\Delta \xi +\beta \xi =\rho +c \end{aligned}$$

for some constant c, and hence

$$\begin{aligned} (\mathrm {curl}\,\xi ,\mathrm {curl}\,\eta )+\beta (\xi ,\eta )=(\rho ,\eta ) \qquad \forall \,\eta \in H^1_0(\Omega )\cap L_2^0(\Omega ). \end{aligned}$$

(3.8)

Below we will show that the function $\xi \in H^1(\Omega )\cap L_2^0(\Omega )$ is determined by (3.7) and (3.8).

3.2.1 The Case Where $\gamma =0$

When $\gamma$ is 0 (and $\Omega$ is simply connected), the two equations (3.7) and (3.8) are decoupled. We can first solve (3.7) for $\rho$ and then solve (3.8) for $\xi$.

In this case (3.7) becomes a consistent singular Neumann boundary value problem: Find $\rho \in H^1(\Omega )\cap L_2^0(\Omega )$ such that

$$\begin{aligned} (\mathrm {curl}\,\rho ,\mathrm {curl}\,\psi )=(\varvec{f},\mathrm {curl}\,\psi ) \qquad \forall \,\psi \in H^1(\Omega ). \end{aligned}$$

(3.9)

An equivalent formulation without the zero-mean constraint is to find $\rho \in H^1(\Omega )$ such that

$$\begin{aligned} (\mathrm {curl}\,\rho ,\mathrm {curl}\,\psi )+(\rho ,1)(\psi ,1)=(\varvec{f},\mathrm {curl}\,\psi ) \qquad \forall \,\psi \in H^1(\Omega ). \end{aligned}$$

(3.10)

Once we have found $\rho \in H^1(\Omega )\cap L_2^0(\Omega )$, $\xi \in H^1_0(\Omega )\cap L_2^0(\Omega )$ is determined by the well-posed (nonstandard) elliptic boundary value problem (3.8). We can also determine $\xi$ through standard boundary value problems that do not involve the zero-mean constraint.

Lemma 3.3

The solution $\xi$ of (3.8) is given by

$$\begin{aligned} \xi =\xi _0-\frac{(1,\xi _0)}{(1,\xi _1)}\xi _1, \end{aligned}$$

(3.11)

where $\xi _0,\xi _1\in H^1_0(\Omega )$ satisfy

$$\begin{aligned} (\mathrm {curl}\,\xi _0,\mathrm {curl}\,\eta )+\beta (\xi _0,\eta )= & {} (\rho ,\eta ) \qquad \forall \,\eta \in H^1_0(\Omega ),\end{aligned}$$

(3.12)

$$\begin{aligned} (\mathrm {curl}\,\xi _1,\mathrm {curl}\,\eta )+\beta (\xi _1,\eta )= & {} (1,\eta ) \qquad \forall \,\eta \in H^1_0(\Omega ). \end{aligned}$$

(3.13)

Proof

First we note that (3.12) and (3.13) are standard elliptic boundary value problems and that (3.13) implies $(1,\xi _1)>0$.

By construction, the function $\xi$ belongs to $H^1_0(\Omega )\cap L_2^0(\Omega )$ and

$$\begin{aligned} (\mathrm {curl}\,\xi ,\mathrm {curl}\,\eta )+\beta (\xi ,\eta )=(\rho ,\eta ) -\frac{(1,\xi _0)}{(1,\xi _1)}(1,\eta ) \qquad \forall \,\eta \in H^1_0(\Omega ), \end{aligned}$$

which implies (3.8). $\square$

3.2.2 The Case Where $\gamma >0$

When $\gamma$ is positive, the problems (3.7) and (3.8) are coupled and we can reformulate them as the following problem:

Find $(\zeta ,\xi )\in [H^1(\Omega )\cap L_2^0(\Omega )]\times [H^1_0(\Omega )\cap L_2^0(\Omega )]$ such that

$$\begin{aligned} (\mathrm {curl}\,\zeta ,\mathrm {curl}\,\psi )+\gamma ^\frac{1}{2}(\psi ,\xi )= & {} \gamma ^{-\frac{1}{2}}(\varvec{f},\mathrm {curl}\,\psi ) \quad \forall \,\psi \in H^1(\Omega )\cap L_2^0(\Omega ), \nonumber \\\end{aligned}$$

(3.14a)

$$\begin{aligned} -\gamma ^\frac{1}{2}(\zeta ,\eta )+ (\mathrm {curl}\,\xi ,\mathrm {curl}\,\eta )+\beta (\xi ,\eta )= & {} 0\quad \forall \,\eta \in H^1_0(\Omega )\cap L_2^0(\Omega ), \end{aligned}$$

(3.14b)

where $\zeta =\gamma ^{-\frac{1}{2}}\rho$.

We can also write (3.14a), (3.14b) concisely as

$$\begin{aligned} \mathcal {A} \big ((\zeta ,\xi ),(\psi ,\eta )\big )=\gamma ^{-\frac{1}{2}}(\varvec{f},\mathrm {curl}\,\psi )\qquad \forall \,\psi \in H^1(\Omega )\cap L_2^0(\Omega ), \eta \in H^1_0(\Omega )\cap L_2^0(\Omega ),\nonumber \\ \end{aligned}$$

(3.15)

where the bilinear form $\mathcal {A}(\cdot ,\cdot )$ on $H^1(\Omega )\times H^1_0(\Omega )$ is defined by

$$\begin{aligned} \mathcal {A}\big ((\zeta ,\xi ),(\psi ,\eta )\big )= (\mathrm {curl}\,\zeta ,\mathrm {curl}\,\psi )+ \gamma ^\frac{1}{2}(\psi ,\xi ) -\gamma ^\frac{1}{2}(\zeta ,\eta )+ (\mathrm {curl}\,\xi ,\mathrm {curl}\,\eta )+\beta (\xi ,\eta ).\nonumber \\ \end{aligned}$$

(3.16)

The bilinear form $\mathcal {A}(\cdot ,\cdot )$ is clearly bounded on $H^1(\Omega )\times H^1_0(\Omega )$, and it follows from the identity

$$\begin{aligned} \mathcal {A}\big ((\psi ,\eta ),(\psi ,\eta )\big )=(\mathrm {curl}\,\psi ,\mathrm {curl}\,\psi )+(\mathrm {curl}\,\eta ,\mathrm {curl}\,\eta )+\beta (\eta ,\eta ) \end{aligned}$$

(3.17)

and standard Poincaré-Friedrichs inequalities [27] that $\mathcal {A}(\cdot ,\cdot )$ is coercive on $[H^1(\Omega )\cap L_2^0(\Omega )]\times H^1_0(\Omega )$. Therefore the problem (3.14a), (3.14b) is well-posed by the Lax–Milgram theorem [24].

We can also determine $(\zeta ,\xi )$ through problems that do not involve the zero-mean constraint.

Lemma 3.4

The solution $(\zeta ,\xi )$ of (3.14a), (3.14b) is given by

$$\begin{aligned} (\zeta ,\xi )=(\zeta _0,\xi _0)-\frac{(1,\xi _0)}{(1,\xi _1)}(\zeta _1,\xi _1), \end{aligned}$$

(3.18)

where $(\zeta _0,\xi _0), (\zeta _1,\xi _1)\in H^1(\Omega )\times H^1_0(\Omega )$ are defined by

$$\begin{aligned} \mathcal {A}\big ((\zeta _0,\xi _0),(\psi ,\eta )\big )+(\zeta _0,1)(\psi ,1)= & {} \gamma ^{-\frac{1}{2}}(\varvec{f},\mathrm {curl}\,\psi )\qquad \forall \, (\psi ,\eta )\in H^1(\Omega )\times H^1_0(\Omega ),\nonumber \\\end{aligned}$$

(3.19)

$$\begin{aligned} \mathcal {A}\big ((\zeta _1,\xi _1),(\psi ,\eta )\big )+(\zeta _1,1)(\psi ,1)= & {} (1,\eta )\qquad \forall \, (\psi ,\eta )\in H^1(\Omega )\times H^1_0(\Omega ). \end{aligned}$$

(3.20)

Proof

First we note that (3.17) implies

$$\begin{aligned}&\mathcal {A}\big (\psi ,\eta ),(\psi ,\eta )\big )+(\psi ,1)(\psi ,1)\nonumber \\&\quad =(\mathrm {curl}\,\psi ,\mathrm {curl}\,\psi )+(\psi ,1)(\psi ,1)+(\mathrm {curl}\,\eta ,\mathrm {curl}\,\eta )+\beta (\eta ,\eta ) \end{aligned}$$

(3.21)

and hence, by standard Poincaré-Friedrichs inequalities, the problems (3.19) and (3.20) are well-posed. Moreover (3.20) and (3.21) imply that $(1,\xi _1)>0$.

By construction, we have $(1,\xi )=0$ and

$$\begin{aligned} \mathcal {A}\big ((\zeta ,\xi ),(\psi ,\eta )\big )+(\zeta ,1)(\psi ,1)=\gamma ^{-\frac{1}{2}}(\varvec{f},\mathrm {curl}\,\psi ) -\frac{(1,\xi _0)}{(1,\xi _1)}(1,\eta ) \end{aligned}$$

(3.22)

for all $(\psi ,\eta )\in H^1(\Omega )\times H^1_0(\Omega )$, which implies (3.15). We can also recover the zero-mean condition $(1,\zeta )=0$ by taking $(\psi ,\eta )=(1,0)$ in (3.22). $\square$

3.3 A Problem for $\varvec{c}_1,\ldots , \varvec{c}_m$

If $\Omega$ is not simply connected, then $\gamma$ is positive. According to Remark 2.1, we can take $\varvec{v}=\,\mathrm {grad}\,\varphi _i$ in (1.3) to obtain

$$\begin{aligned} \gamma (\varvec{u},\,\mathrm {grad}\,\varphi _i)=(\varvec{f},\,\mathrm {grad}\,\varphi _i) \qquad \text {for}\quad 1\le i\le m. \end{aligned}$$

It then follows from (2.3) and (3.1) that the coefficients $c_1,\ldots ,c_m$ in the Hodge decomposition (3.1) are determined by the $m\times m$ system

$$\begin{aligned} \sum _{j=1}^m (\,\mathrm {grad}\,\varphi _i,\,\mathrm {grad}\,\varphi _j)c_j=\frac{1}{\gamma }(\varvec{f},\,\mathrm {grad}\,\varphi _i) \qquad \text {for}\quad 1\le i\le m. \end{aligned}$$

(3.23)

Note that (3.23) is symmetric positive definite because of (2.2b).

3.4 Regularity of $\varvec{u}$

First we observe that $\rho$ belongs to $H^1(\Omega )$ by construction. Then (3.8) and Lemma 2.3 imply that $\xi \in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$, and (3.2) and Lemma 2.4 imply that $\phi$ belongs to $H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$. Moreover the harmonic functions $\varphi _j$ ($1\le j\le m)$ also belong to $H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ by Lemma 2.3. It follows that $\varvec{u}$ belongs to $[H^{{(\pi /\omega )-\epsilon }}(\Omega )]^2$ for any $\epsilon >0$. Thus the regularity of $\varvec{u}$ is better than $H^1$ for a convex polygon and worse than $H^1$ for a nonconvex polygon.

Remark 3.5

Note that the regularity of $\xi =\mathrm {curl}\,\varvec{u}$ is better than $H^1(\Omega )$. But the regularity of $\varvec{u}$ is the same as the one in Theorem 2.5 for ${\mathbb {E}}$. This is due to the presence of singularities at the corners of $\Omega$ that prevents full elliptic regularity for $\varvec{u}$. In contrast, for a smooth $\Omega$, $\rho \in H^1(\Omega )$ implies $\xi \in H^3(\Omega )$ by (3.8), which in turn implies $\phi \in H^5(\Omega )$ by (3.2). Since the harmonic functions $\varphi _j$ for $1\le j\le m$ belong to $C^\infty (\bar{\Omega })$, $\varvec{u}$ belongs to $[H^4(\Omega )]^2$ by (3.1), which is two orders higher than the regularity of the vector fields in ${\mathbb {E}}$ (cf. Remark 2.6).

4 ${P_k}$ Finite Element Methods

The reduction in Sect. 3 leads to the following numerical procedure for (1.3).

• (1) Find an approximation $\tilde{\xi }\in H^1_0(\Omega )\cap L_2^0(\Omega )$ for $\xi$ numerically. In the case where $\gamma =0$ (and $\Omega$ is simply connected), one can first solve (3.9) numerically to find an approximation $\tilde{\rho }\in H^1_0(\Omega )\cap L_2^0(\Omega )$ for $\rho$, and then solve (3.8) (with $\rho$ replaced by $\tilde{\rho }$) numerically to find an approximation $\tilde{\xi }$ for $\xi$ (cf. Sect. 3.2.1). In the case where $\gamma >0$, we can obtain $\tilde{\xi }$ by solving the coupled problem (3.14a), (3.14b) numerically (cf. Sect. 3.2.2).

• (2) Solve (3.2) (with $\xi$ replaced by $\tilde{\xi }$) numerically to find an approximation $\tilde{\phi }\in H^1(\Omega )\cap L_2^0(\Omega )$ for $\phi$.

• (3) In the case where $\Omega$ is not simply connected (and $\gamma >0$), solve numerically the boundary value problems in (2.2a), (2.2b), (2.2c) to find approximations $\tilde{\varphi }_j \in H^1(\Omega )$ for $\varphi _j$ ($1\le j\le m$) and then solve (3.23) (with $\varphi _j$ replaced by $\tilde{\varphi }_j$) numerically to find approximations $\tilde{c}_1,\ldots ,\tilde{c}_m$ for $c_1,\ldots ,c_m$. Note that the computation of $\tilde{\varphi }_j$ ($1\le j\le m$) only involves $\Omega$ and hence can be carried out in advance.

• (4) The approximation $\tilde{\varvec{u}}$ of $\varvec{u}$ is given by

  $$\begin{aligned} \tilde{\varvec{u}}=\mathrm {curl}\,\tilde{\phi }+\sum _{j=1}^m\tilde{c}_j\,\mathrm {grad}\,\tilde{\varphi }_j. \end{aligned}$$

Therefore any numerical method that works for second order elliptic boundary value problems can also be applied to (1.3). Here we will consider $P_k$ Lagrange finite element methods.

Let $\mathcal {T}_h$ be a quasi-uniform simplicial triangulation of $\Omega$. We denote by $V_h(\subset H^1(\Omega ))$ the $P_k$ $(k\ge 1)$ Lagrange finite element space [13, 16] associated with $\mathcal {T}_h$ and by $\mathring{V}_h(\subset H^1_0(\Omega ))$ the subspace of $V_h$ whose members vanish on $\partial \Omega$.

4.1 The $\varvec{P}_k$ Finite Element Method for the Approximation of $\varvec{\xi }$

We consider two separate cases depending on whether $\gamma$ is 0 or positive.

4.1.1 The Case Where $\gamma =0$

Following the discussion in Sect. 3.2.1, we can first compute $\rho$ and then $\xi$.

The $P_k$ finite element method for (3.10) is to find $\rho _h\in V_h$ such that

$$\begin{aligned} (\mathrm {curl}\,\rho _h,\mathrm {curl}\,\psi )+(\rho ,1)(\psi ,1)=(\varvec{f},\mathrm {curl}\,\psi ) \qquad \forall \,\psi \in V_h. \end{aligned}$$

(4.1)

The $P_k$ finite element method for (3.8) [cf. (3.11)–(3.13)] is to find

$$\begin{aligned} \xi _h=\xi _{0,h}-\frac{(1,\xi _{0,h})}{(1,\xi _{1,h})}\xi _{1,h}, \end{aligned}$$

(4.2)

where $\xi _{0,h},\xi _{1,h}\in \mathring{V}_h$ satisfy

$$\begin{aligned} (\mathrm {curl}\,\xi _{0,h},\mathrm {curl}\,\eta )+\beta (\xi _{0,h},\eta )= & {} (\rho _h,\eta ) \qquad \forall \,\eta \in \mathring{V}_h,\end{aligned}$$

(4.3)

$$\begin{aligned} (\mathrm {curl}\,\xi _{1,h},\mathrm {curl}\,\eta )+\beta (\xi _{1,h},\eta )= & {} (1,\eta ) \qquad \forall \,\eta \in \mathring{V}_h. \end{aligned}$$

(4.4)

4.1.2 The Case Where $\gamma >0$

The $P_k$ finite element method for (3.14a), (3.14b) [cf. (3.18)–(3.20)] is to find

$$\begin{aligned} (\zeta _h,\xi _h)=(\zeta _{0,h},\xi _{0,h})-\frac{(1,\xi _{0,h})}{(1,\xi _{1,h})}(\zeta _{1,h},\xi _{1,h}), \end{aligned}$$

(4.5)

where $(\zeta _{0,h},\xi _{0,h}), (\zeta _{1,h},\xi _{1,h})\in V_h\times \mathring{V}_h$ satisfy

$$\begin{aligned} \mathcal {A}\big ((\zeta _{0,h},\xi _{0,h}),(\psi ,\eta )\big )+(\zeta _{0,h},1)(\psi ,1)= & {} \gamma ^{-\frac{1}{2}}(\varvec{f},\mathrm {curl}\,\psi )\qquad \forall \, (\psi ,\eta )\in V_h\times \mathring{V}_h,\nonumber \\\end{aligned}$$

(4.6)

$$\begin{aligned} \mathcal {A}\big ((\zeta _{1,h},\xi _{1,h}),(\psi ,\eta )\big )+(\zeta _{1,h},1)(\psi ,1)= & {} (1,\eta )\qquad \forall \, (\psi ,\eta )\in V_h\times \mathring{V}_h. \end{aligned}$$

(4.7)

Remark 4.1

The function $\xi _h$ is an approximation of $\mathrm {curl}\,\varvec{u}$.

4.2 The $\varvec{P}_k$ Finite Element Method for the Approximation of $\varvec{\phi }$

The finite element method for (3.3) is to find $\phi _h\in V_h\cap L_2^0(\Omega )$ such that

$$\begin{aligned} (\mathrm {curl}\,\phi ,\mathrm {curl}\,\psi )+(\phi ,1)(\psi ,1)=(\xi _h,\psi ) \qquad \forall \,\psi \in V_h. \end{aligned}$$

(4.8)

4.3 The Approximation of $\varvec{u}$

We take

$$\begin{aligned} \varvec{u}_h=\mathrm {curl}\,\phi _h+\sum _{j=1}^m c_{j,h}\varphi _{j,h} \end{aligned}$$

(4.9)

to be the approximation of $\varvec{u}$, where $c_{1,h},\ldots ,c_{m,h}$ are determined by

$$\begin{aligned} \sum _{j=1}^m (\,\mathrm {grad}\,\varphi _{i,h},\,\mathrm {grad}\,\varphi _{j,h})c_{j,h}=\gamma ^{-1}(\varvec{f},\,\mathrm {grad}\,\varphi _{i,h}) \qquad \text {for}\quad 1\le i\le m, \end{aligned}$$

(4.10)

and the discrete harmonic functions $\varphi _{1,h},\ldots ,\varphi _{m,h}$ are determined by (cf. (2.2a), (2.2b), (2.2c))

$$\begin{aligned} (\,\mathrm {grad}\,\varphi _{j,h},\,\mathrm {grad}\,v)= & {} 0\qquad \forall \,v\in \mathring{V}_h, \end{aligned}$$

(4.11a)

$$\begin{aligned} \varphi _{j,h}\big |_{{\Gamma }_0}= & {} 0,\end{aligned}$$

(4.11b)

$$\begin{aligned} \varphi _{j,h}\big |_{{\Gamma }_k}= & {} {\left\{ \begin{array}{ll} 1\quad j=k\\ 0\quad j\ne k \end{array}\right. }\qquad \text {for}\quad 1\le k\le m. \end{aligned}$$

(4.11c)

For a simply connected $\Omega$, the approximation for $\varvec{u}$ is simplified to $\varvec{u}_h=\mathrm {curl}\,\phi _h$.

5 Convergence Analysis

Since the error analysis for the discrete harmonic functions $\varphi _{1,h},\ldots ,\varphi _{m,h}$ and the coefficients $c_{1,h},\ldots ,c_{m,h}$ has already been carried out in [9], we only need to focus on the error analysis for $\xi _h$ and $\phi _h$.

We will use the following standard polynomial approximation result [13, 16, 19].

Lemma 5.1

Given any $\delta >0$, there exists a positive constant C independent of h such that

$$\begin{aligned}&\inf _{\psi \in V_h}\Vert \lambda -\psi \Vert _{H^1(\Omega )}\le Ch^{\min (\delta ,k)}\Vert \lambda \Vert _{H^{1+\delta }(\Omega )} \qquad \forall \,\lambda \in H^{1+\delta }(\Omega ),\\&\inf _{\eta \in \mathring{V}_h}\Vert \mu -\psi \Vert _{H^1(\Omega )}\le Ch^{\min (\delta ,k)}\Vert \mu \Vert _{H^{1+\delta }(\Omega )} \qquad \forall \,\mu \in H^{1+\delta }(\Omega )\cap H^1_0(\Omega ). \end{aligned}$$

From here on we will use C (with or without subscript) to denote a generic positive constant that is independent of h. The error estimates below will depend on $\omega$, the largest interior angle at the corners of $\Omega$.

5.1 Error Analysis for $\varvec{\xi }_{\varvec{h}}$

Our goal is to establish the following result.

Lemma 5.2

For any $\epsilon >0$, there exists a positive constant ${C_\epsilon }$ independent of h such that

$$\begin{aligned} |\xi -\xi _h|_{H^1(\Omega )}\le C_\epsilon h^{{\min ({(\pi /\omega )-\epsilon },k)}}. \end{aligned}$$

(5.1)

We will consider the two cases $\gamma =0$ and $\gamma >0$ separately.

5.1.1 The Case Where ${\gamma }=0$

We first estimate the error for $\rho _h$ in the norm of $[H^1(\Omega )]'$ by a duality argument.

Lemma 5.3

For any $\epsilon >0$, there exists a positive constant ${C_\epsilon }$ independent of h such that

$$\begin{aligned} |(\rho -\rho _h,\chi )|\le {C_\epsilon }h^{\min ((\pi /\omega )-\epsilon ,k)}\Vert \chi \Vert _{H^1(\Omega )} \qquad \forall \,\chi \in H^1(\Omega ). \end{aligned}$$

(5.2)

Proof

In view of (3.10), (4.1) and the fact that $\rho ,\rho _h\in L_2^0(\Omega )$, we have

$$\begin{aligned} \Vert \mathrm {curl}\,\rho \Vert _{L_2(\Omega )}\le \Vert \varvec{f}\Vert _{L_2(\Omega )}, \quad \Vert \mathrm {curl}\,\rho _h\Vert _{L_2(\Omega )}\le \Vert \varvec{f}\Vert _{L_2(\Omega )}, \end{aligned}$$

(5.3)

and a Galerkin orthogonality relation

$$\begin{aligned} \big (\mathrm {curl}\,(\rho -\rho _h),\mathrm {curl}\,\psi )=0 \qquad \forall \,\psi \in V_h. \end{aligned}$$

(5.4)

Let $\chi \in H^1(\Omega )$ be arbitrary and $\lambda \in H^1(\Omega )$ be defined by

$$\begin{aligned} (\mathrm {curl}\,\psi ,\mathrm {curl}\,\lambda )+(\psi ,1)(\lambda ,1)=(\psi ,\chi ) \qquad \forall \,\psi \in H^1(\Omega ). \end{aligned}$$

(5.5)

Then we have

$$\begin{aligned} (\rho -\rho _h,\chi )=\big (\mathrm {curl}\,(\rho -\rho _h),\mathrm {curl}\,\lambda \big )= \big (\mathrm {curl}\,(\rho -\rho _h),\mathrm {curl}\,(\lambda -\psi )\big ) \qquad \forall \,\psi \in V_h \end{aligned}$$

by (5.4), (5.5) and the fact that $\rho ,\rho _h\in L_2^0(\Omega )$, which implies

$$\begin{aligned} |(\rho -\rho _h,\chi )|\le \Vert \mathrm {curl}\,(\rho -\rho _h)\Vert _{L_2(\Omega )}\inf _{\psi \in V_h}|\lambda -\psi |_{H^1(\Omega )}. \end{aligned}$$

(5.6)

According to Lemma 2.4 and (5.5), we have $\lambda \in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$ and also $\Vert \lambda \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert \chi \Vert _{H^1(\Omega )}$. Lemma 5.1 then implies

$$\begin{aligned} \inf _{\psi \in V_h}|\lambda -\psi |_{H^1(\Omega )}\le Ch^{\min ({(\pi /\omega )-\epsilon },k)}\Vert \lambda \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )} \le {C_\epsilon }h^{\min ({(\pi /\omega )-\epsilon },k)}\Vert \chi \Vert _{H^1(\Omega )}.\nonumber \\ \end{aligned}$$

(5.7)

The estimate (5.2) follows from (5.3), (5.6) and (5.7). $\square$

Next we estimate $|\xi -\xi _h|_{H^1(\Omega )}$.

Let $\tilde{\xi }_{0,h}\in \mathring{V}_h$ be defined by

$$\begin{aligned} (\mathrm {curl}\,\tilde{\xi }_{0,h},\mathrm {curl}\,\eta )+\beta (\tilde{\xi }_{0,h},\eta )=(\rho ,\eta ) \qquad \forall \,\eta \,\in \mathring{V}_h. \end{aligned}$$

(5.8)

On one hand we have

$$\begin{aligned} \big (\mathrm {curl}\,(\tilde{\xi }_{0,h}-\xi _{0,h}),\mathrm {curl}\,\eta \big )+\beta \big (\tilde{\xi }_{0,h}-\xi _{0,h},\eta \big )=(\rho -\rho _h,\eta )\qquad \forall \,\eta \in \mathring{V}_h \end{aligned}$$

by comparing (4.3) and (5.8). It follows that

$$\begin{aligned} |\tilde{\xi }_{0,h}-\xi _{0,h}|_{H^1(\Omega )}^2\le (\rho -\rho _h,\tilde{\xi }_{0,h}-\xi _{0,h}), \end{aligned}$$

which together with (5.2) and a standard Poncaré-Friedrichs inequality implies

$$\begin{aligned} |\tilde{\xi }_{0,h}-\xi _{0,h}|_{H^1(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}. \end{aligned}$$

(5.9)

On the other hand $\tilde{\xi }_{0,h}$ is the Galerkin finite element approximation of $\xi _0$ (cf. (3.12) and (5.8)). Therefore we have

$$\begin{aligned} |\xi _0-\tilde{\xi }_{0,h}|_{H^1(\Omega )}\le C\inf _{\eta \in \mathring{V}_h}|\xi _0-\eta |_{H^1(\Omega )} \end{aligned}$$

(5.10)

by Céa’s lemma [13, 16].

According to Lemma 2.3 and (3.12), we have $\xi _0\in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ and $\Vert \xi _0\Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert \rho \Vert _{H^1(\Omega )}$. It then follows from Lemma 5.1 that

$$\begin{aligned} \inf _{\eta \in \mathring{V}_h}|\xi _0-\eta |_{H^1(\Omega )}\le C h^{{\min ({(\pi /\omega )-\epsilon },k)}}\Vert \xi _0\Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}.\quad \end{aligned}$$

(5.11)

Putting (5.9)–(5.11) together we obtain

$$\begin{aligned} |\xi _0-\xi _{0,h}|_{H^1(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}. \end{aligned}$$

(5.12)

Similarly, since $\xi _{1,h}$ is the Galerkin finite element approximation of $\xi _1$ (cf. (3.13) and (4.4)), we have

$$\begin{aligned} |\xi _1-\xi _{1,h}|_{H^1(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}. \end{aligned}$$

(5.13)

The estimate (5.1) follows from (3.11), (4.2), (5.12) and (5.13).

5.1.2 The Case Where ${\gamma }>0$

The error analysis follows the ideas in Sect. 5.1.1 within the setting of the coupled problem (3.14a), (3.14b).

First we use a duality argument to estimate the error for $\zeta _{0,h}$ in the norm of $[H^1(\Omega )]'$. In view of (3.19), (3.21) and (4.6), we have

$$\begin{aligned} \Vert \zeta _0\Vert _{H^1(\Omega )}+ |\xi _0|_{H^1(\Omega )}\le C\Vert \varvec{f}\Vert _{L_2(\Omega )}, \quad \Vert \zeta _{0,h}\Vert _{H^1(\Omega )}+|\xi _{0,h}|_{H^1(\Omega )}\le C\Vert \varvec{f}\Vert _{L_2(\Omega )},\nonumber \\ \end{aligned}$$

(5.14)

and the Galerkin orthogonality relation

$$\begin{aligned} \mathcal {A}\big ((\zeta _0-\zeta _{0,h},\xi _0-\xi _{0,h}),(\psi ,\eta )\big )+(\zeta _0-\zeta _{0,h},1)(\psi ,1) =0 \qquad \forall \,(\psi ,\eta )\in V_h\times \mathring{V}_h.\nonumber \\ \end{aligned}$$

(5.15)

Let $\chi \in H^1(\Omega )$ be arbitrary and $(\lambda ,\mu )\in H^1(\Omega )\times H^1_0(\Omega )$ be defined by

$$\begin{aligned} \mathcal {A}\big ((\psi ,\eta ),(\lambda ,\mu )\big ) +(\psi ,1)(\lambda ,1)=(\psi ,\chi ) \qquad \forall \,(\psi ,\eta )\in H^1(\Omega )\times H^1_0(\Omega ). \end{aligned}$$

(5.16)

Then the function $\zeta _0-\zeta _{0,h}\in H^1(\Omega )$ satisfies, by (5.15) and (5.16),

$$\begin{aligned} (\zeta _0-\zeta _{0,h},\chi )&=\mathcal {A}\big ((\zeta _0-\zeta _{0,h},\xi _0-\xi _{0,h}),(\lambda ,\mu ) \Big ) +(\zeta -\zeta _{0,h},1)(\lambda ,1)\\&=\mathcal {A}\big ((\zeta _0-\zeta _{0,h},\xi _0-\xi _{0,h}),(\lambda -\psi ,\mu -\eta ) \Big )+(\zeta -\zeta _{0,h},1)(\lambda -\psi ,1) \end{aligned}$$

for all $(\psi ,\eta )\in V_h\times \mathring{V}_h$, and hence

$$\begin{aligned} |(\zeta _0-\zeta _{0,h},\chi )|&\le C\big (\Vert \zeta _0-\zeta _{0,h}\Vert _{H^1(\Omega )}+|\xi _0-\xi _{0,h}|_{H^1(\Omega )}\big )\\&\times \inf _{(\psi ,\eta )\in V_h\times \mathring{V}_h}\big (\Vert \lambda -\psi \Vert _{H^1(\Omega )}+|\mu -\eta |_{H^1(\Omega )}\big ). \nonumber \end{aligned}$$

(5.17)

Observe that the well-posedness of (5.16) implies

$$\begin{aligned} \Vert \lambda \Vert _{H^1(\Omega )}+\Vert \mu \Vert _{H^1(\Omega )}\le C\Vert \chi \Vert _{L_2(\Omega )}. \end{aligned}$$

(5.18)

It then follows from Lemmas 2.3, 2.4 and the relations (cf. (3.16) and (5.16))

$$\begin{aligned} (\mathrm {curl}\,\psi ,\mathrm {curl}\,\lambda )+(\psi ,1)(\lambda ,1)= & {} \gamma ^\frac{1}{2}(\psi ,\mu )+(\psi ,\chi ) \qquad \forall \,\psi \in H^1(\Omega ),\\ (\mathrm {curl}\,\eta ,\mathrm {curl}\,\mu )+\beta (\eta ,\mu )= & {} -\gamma ^\frac{1}{2}(\lambda ,\eta )\qquad \forall \,\eta \in H^1_0(\Omega ), \end{aligned}$$

that $(\lambda ,\mu )\in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )\times H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ and, because of (5.18),

$$\begin{aligned} \Vert \lambda \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}+\Vert \mu \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert \chi \Vert _{H^1(\Omega )}. \end{aligned}$$

Hence Lemma 5.1 implies

$$\begin{aligned} \inf _{(\psi ,\eta )\in V_h\times \mathring{V}_h}\big (\Vert \lambda -\psi \Vert _{H^1(\Omega )}+|\mu -\eta |_{H^1(\Omega )}\big ) \le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}} \Vert \chi \Vert _{H^1(\Omega )}. \end{aligned}$$

(5.19)

Putting (5.14), (5.17) and (5.19) together, we see that

$$\begin{aligned} |(\zeta _0-\zeta _{0,h},\chi )|\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}\Vert \chi \Vert _{H^1(\Omega )} \qquad \forall \,\chi \in H^1(\Omega ). \end{aligned}$$

(5.20)

Next we compare the equation

$$\begin{aligned} (\mathrm {curl}\,\xi _0,\mathrm {curl}\,\eta )+\beta (\xi _0,\eta )=\gamma ^\frac{1}{2}(\zeta ,\eta )\qquad \forall \,\eta \in H^1_0(\Omega ), \end{aligned}$$

that is a part of (3.19) with the equation

$$\begin{aligned} (\mathrm {curl}\,\xi _{0,h},\mathrm {curl}\,\eta )+\beta (\xi _{0,h},\eta )=\gamma ^\frac{1}{2}(\zeta _{0,h},\eta )\qquad \forall \,\eta \in \mathring{V}_h \end{aligned}$$

that is a part of (4.6). Using (5.20) and the arguments in the derivation of(5.12), we find

$$\begin{aligned} |\xi _h-\xi _{0,h}|_{H^1(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}. \end{aligned}$$

(5.21)

Similarly, we have

$$\begin{aligned} |\xi _1-\xi _{1,h}|_{H^1(\Omega )}\le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}} \end{aligned}$$

(5.22)

by comparing (3.20) and (4.7).

The estimate (5.1) follows from (3.18), (4.5), (5.21) and (5.22).

5.2 Error Analysis for $\varvec{\phi }_{\varvec{h}}$

The error analysis for $\phi _h$ is similar to the error analysis for $\xi _h$ in Sect. 5.1.1.

Lemma 5.4

For any $\epsilon >0$, there exists a positive constant ${C_\epsilon }$ independent of h such that

$$\begin{aligned} |\phi -\phi _h|_{H^1(\Omega )}\le {C_\epsilon }h^{\min ({(\pi /\omega )-\epsilon },k)}. \end{aligned}$$

(5.23)

Proof

It follows from Lemma 2.4 and (3.3) that $\phi \in H^{1+{(\pi /\omega )-\epsilon }}(\Omega )$ for any $\epsilon >0$ and $\Vert \phi \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )}\le {C_\epsilon }\Vert \xi \Vert _{H^1(\Omega )}$.

Let the function $\tilde{\phi }_h\in V_h$ be defined by

$$\begin{aligned} (\mathrm {curl}\,\tilde{\phi }_h,\mathrm {curl}\,\psi )+(\tilde{\phi }_h,1)(\psi ,1)=(\xi ,\psi )\qquad \forall \,\psi \in V_h. \end{aligned}$$

(5.24)

On one hand we have, by comparing (3.3) and (5.24),

$$\begin{aligned} |\tilde{\phi }_h-\phi _h|_{H^1(\Omega )}\le C\Vert \xi -\xi _h\Vert _{L_2(\Omega )}. \end{aligned}$$

(5.25)

On the other hand $\tilde{\phi }_h\in V_h$ is the Galerkin $P_k$ finite element approximation of $\phi$, and the orthogonality relation

$$\begin{aligned} \big (\mathrm {curl}\,(\phi -\tilde{\phi }_h),\mathrm {curl}\,\psi \big )=0 \qquad \forall \,\psi \in V_h \end{aligned}$$

together with Lemma 5.1 implies that

$$\begin{aligned} |\phi -\tilde{\phi }_h|_{H^1(\Omega )}&\le \inf _{\psi \in V_h}|\phi -\psi |_{H^1(\Omega )}\nonumber \\&\le Ch^{{\min ({(\pi /\omega )-\epsilon },k)}}\Vert \phi \Vert _{H^{1+{(\pi /\omega )-\epsilon }}(\Omega )} \le {C_\epsilon }h^{{\min ({(\pi /\omega )-\epsilon },k)}}\Vert \xi \Vert _{H^1(\Omega )}. \end{aligned}$$

(5.26)

The estimate (5.23) follows from (5.25), (5.26) and the estimate in Lemma 5.2 for $\xi -\xi _h\in H^1_0(\Omega )$. $\square$

5.3 Error Analysis for $\varvec{\varphi }_{\varvec{h}, j}$ and $\varvec{c}_{\varvec{h},j}$

The following result can be found in [9, Lemmas 4.6 and 4.7].

Lemma 5.5

In the case where $\Omega$ is not simply connected, we have

$$\begin{aligned} |\varphi _j-\varphi _{j,h}|_{H^1(\Omega )}+|c_j-c_{j,h}|\le Ch^{\pi /\omega }\qquad \text {for}\;1\le j\le m. \end{aligned}$$

5.4 Convergence Results

In view of Lemmas 5.2, 5.4, 5.5, (3.1) and (4.9), we immediately have the following result.

Theorem 5.6

The approximations $\xi _h$ and $\varvec{u}_h$ obtained by the $P_k$ finite element method satisfy

$$\begin{aligned} \Vert \varvec{u}-\varvec{u}_h\Vert _{L_2(\Omega )}+ |\mathrm {curl}\,\varvec{u}-\xi _h|_{H^1(\Omega )}\le {C_\epsilon }h^{\min ({(\pi /\omega )-\epsilon },k)}, \end{aligned}$$

(5.27)

for any $\epsilon >0$, where $\omega$ is the largest angle at the corners of $\Omega$.

Remark 5.7

In the case where $(\pi /\omega )$ is not an integer, a more detailed analysis that takes into account the nature of the singularities at the corners of $\Omega$ (cf. [6]) shows that the $\epsilon$ in (5.27) can be removed.

6 Numerical Results

In this section we report the results of numerical experiments for three different domains: the unit square, a nonconvex but simply connected domain and a domain whose Betti number is 1. We use quasi-uniform meshes in all the experiments.

Experiment 6.1

In the first experiment the domain $\Omega$ is the unit square $(0,1)\times (0,1)$. We take $\beta =\gamma =0$ and the exact solution to be $\varvec{u}=\mathrm {curl}\,\phi$ where

$$\begin{aligned} \phi (x)=\sin ^3(\pi x_1)\sin ^3(\pi x_2). \end{aligned}$$

We solve (1.3) by the $P_1$ and $P_2$ finite element methods. The results are presented in Tables 1 and 2. They agree with Theorem 5.6 with $\omega =\pi /2$.

Table 1 Results for the $P_1$ finite element method for Experiment 6.1

Table 2 Results for the $P_2$ finite element method for Experiment 6.1

Experiment 6.2

The domain $\Omega$ for the second experiment is also the unit square $(0,1)\times (0,1)$. We take $\beta =\gamma =0$ and

$$\begin{aligned} \varvec{f}= \begin{bmatrix} (x_1^2+1)\sin x_1+x_1 x_2^3 + 2\\ (x_2^2+1)\cos x_1+x_1^3x_2^2 - 1 \end{bmatrix}. \end{aligned}$$

We solve (1.3) by the $P_1$ and $P_2$ finite element methods and report the results in Tables 3 and 4. Since the exact solution is not known, the relative errors are estimated by comparing the numerical solutions on consecutive refinement levels. The results agree with Theorem 5.6 with $\omega =\pi /2$.

Table 3 Results for the $P_1$ finite element method for Experiment 6.2

Table 4 Results for the $P_2$ finite element method for Experiment 6.2

Experiment 6.3

In the third experiment we solve (1.3) on the nonconvex domain (cf. Fig. 2) whose vertices are $(0, 0), (.5, 0), (.5, .7), (1, .7), (1, 1), (0, 1), (1, .75), (.25, .75), (.25, .625) \;\text {and}\; (0, .625).$

Fig. 2

Domain for Experiment 6.3

We take $\beta =\gamma =0$ and use a piecewise constant vector field $\varvec{f}$ defined by

$$\begin{aligned} {\varvec{f}}(x)= {\left\{ \begin{array}{ll} \left[ \frac{1}{4}, \frac{5}{4}\right] ^\mathrm{t} &{}\qquad |x|<2^{-1/2}\\ \left[ \frac{1}{2} , \frac{3}{2}\right] ^\mathrm{t} &{}\qquad 2^{-1/2}\le |x|<1\\ \left[ 1,2\right] ^\mathrm{t}&{}\qquad |x|\ge 1 \end{array}\right. }. \end{aligned}$$

The estimated relative errors for the $P_1$ and $P_2$ finite element methods are displayed in Tables 5 and 6.

Table 5 Results for the $P_1$ finite element method for Experiment 6.3

Table 6 Results for the $P_2$ finite element method for Experiment 6.3

For this problem the order of convergence predicted by Theorem 5.6 is 2 / 3 (since $\omega =3\pi /2$). This is observed in Table 6 for the $P_2$ finite element method, and also in Table 5 for the $P_1$ finite element method with respect to the convergence of $\varvec{u}_h$. On the other hand the convergence observed in Table 5 for $\xi _h$ is pre-asymptotic.

Experiment 6.4

The domain for the fourth experiment is

$$\begin{aligned} \Omega = (0, 1) \times (0, 1) \setminus [1/4, 3/4] \times [1/4, 3/4] \end{aligned}$$

whose Betti number is 1. We take $\beta =\gamma =1$ and use the same $\varvec{f}$ in Experiment 6.2.

Since the domain is not simply connected, the solution $\varvec{u}$ of (1.3) is given by

$$\begin{aligned} \varvec{u}=\mathrm {curl}\,\phi +c\,\mathrm {grad}\,\varphi , \end{aligned}$$

(6.1)

where $\varphi$ is the harmonic function that vanishes on the outer boundary of $\Omega$ and equals 1 on the inner boundary of $\Omega$. The approximation for $\varvec{u}$ is given by

$$\begin{aligned} \varvec{u}_h=\mathrm {curl}\,\phi _h+c_h\,\mathrm {grad}\,\varphi _h, \end{aligned}$$

(6.2)

where $\varphi _h$ is the discrete analog of $\varphi$.

We solve (1.3) by the $P_1$ finite element method and report the results in Table 7. The order of convergence for $\varvec{u}_h$ is observed to be 2 / 3, which agrees with Theorem 5.6 with $\omega =3\pi /2$. The convergence of $\xi _h$ is pre-asymptotic.

Table 7 Results for the $P_1$ finite element method for Experiment 6.4

Note that the order of convergence for $c_h$ is better than 2 / 3, which is due to the fact that $\varvec{f}$ is smooth (cf. [9, Remark 4.8 and Table 4.3]).

7 Concluding Remarks

We have designed and analyzed $P_k$ Lagrange finite element methods for a quad-curl problem that are based on the Hodge decomposition approach. For simplicity we only considered quasi-uniform meshes and the performance of the methods suffer from the existence of reentrant corners. But optimal convergence rates can be recovered if we would use properly graded meshes (cf. [17] for the case of the Maxwell equations).

Our convergence analysis does not require $H^2$ (or higher) regularity for the exact solution that is assumed in [23, 29, 31]. The computational cost of our approach also compares favorably to those for the methods in [23, 29, 31].

Below are some related topics that can also benefit from the Hodge decomposition approach.

• As in the case of the Maxwell equations, the Hodge decomposition approach lends itself naturally to the development of fast solvers for the quad-curl problem.

• The Hodge decomposition approach can also be applied to the following eigenvalue problems on two dimensional domains. The first eigenvalue problem is to find $(\varvec{u},\lambda )\in {\mathbb {E}}\times \mathbb {R}$ such that

  $$\begin{aligned} \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{u}),\mathrm {curl}\,(\mathrm {curl}\,\varvec{v})\big )=\lambda (\varvec{u},\varvec{v}) \qquad \forall \,\varvec{v}\in {\mathbb {E}}\quad \text {and}\quad \varvec{u}\ne \mathbf{0}. \end{aligned}$$

  The second eigenvalue problem is to find $(\varvec{u},\lambda )\in {\mathbb {E}}\times \mathbb {R}$ such that

  $$\begin{aligned} \big (\mathrm {curl}\,(\mathrm {curl}\,\varvec{u}),\mathrm {curl}\,(\mathrm {curl}\,\varvec{v})\big )=\lambda (\mathrm {curl}\,\varvec{u},\mathrm {curl}\,\varvec{v}) \qquad \forall \,\varvec{v}\in {\mathbb {E}}\quad \text {and}\quad \varvec{u}\ne \mathbf{0}. \end{aligned}$$

  For both problems the Hodge decomposition approach reduces an elliptic eigenvalue problem for vector fields to an elliptic eigenvalue problem for scalar functions that can be solved by standard $H^1$ conforming finite elements.

• For three dimensional domains, the Hodge decomposition approach reduces the quad-curl problem to problems that can be solved numerically by standard $H(\mathrm {curl})$, $H(\mathrm {div})$ and $H^1$ conforming finite elements. Moreover these problems have already been analyzed in [4].

These topics are being investigated in our ongoing projects.