Overlapping Schwarz method is one of the most important methods for computing the large-scale discrete problems arising from partial differential equations (PDE). This domain decomposition (DD) method is essentially parallel and has been extensively studied in the literature (see, e.g., [8, 6, 5, 4, 17, 10, 2, 12, 13] and the references therein). Generally speaking, the iterative convergence rate (e.g., PCG, preconditioned GMRES) depends on the condition number of the discrete system. Therefore, it is very important to obtain the sharp estimate of the preconditioned algebraic systems resulting from overlapping Schwarz methods.

For a long time, the best bound of the condition number of the overlapping domain decomposition method is $C\left(1+\frac{H^{2}}{\delta^{2}}\right)$ for the second order elliptic boundary value problems, cf. [7, 18], where $\delta$ is the overlapping size and $H$ is the diameter of subdomains. In 1994, Dryja and Widlund [8] first improved the bound to $C\left(1+\frac{H}{\delta}\right)$. Subsequently, Brenner [6] proved the best bound is $C\left(1+\frac{H}{\delta}\right)$. The same techniques used are applied to the fourth order elliptic boundary value problems and high-frequency Helmholtz problems (see [6, 5, 4, 10]).

The same situation happens to the analysis of the two-level overlapping domain decomposition method in $\bm{H}(\bm{{\rm curl}};\Omega)$. Toselli [17] proved an upper bound of $C\left(1+\frac{H^{2}}{\delta^{2}}\right)$ for the condition number. For many years, people wonder if the best bound should be $C\left(1+\frac{H}{\delta}\right)$. Numerical results [17] indicate that the best bound should be $C\left(1+\frac{H}{\delta}\right)$. Bonazzoli et al. [2] posed this open problem if the best bound is $C\left(1+\frac{H}{\delta}\right)$. In this paper, we close this open problem and prove that $C\left(1+\frac{H}{\delta}\right)$ is the best bound.

The key ideas in obtaining the sharp estimate of overlapping Schwarz methods in $\bm{H}(\bm{{\rm curl}};\Omega)$ and $\bm{H}({\rm div};\Omega)$ are as follows. Firstly, the functions are limited to one element of the coarse triangulation where all functions are $\bm{H}^{1}$ locally. This way get sharper estimates than the results in [17]. Secondly, by the Helmholtz decomposition, we get a solenoidal subspace in $\bm{H}(\bm{{\rm curl}};\Omega)$ and an irrotational subspace in $\bm{H}({\rm div};\Omega)$ which is also in $\bm{H}^{1}(\Omega)$ when $\Omega$ is convex. After the decomposition we can utilize the techniques from the domain decomposition method for $H^{1}$ problems.

The rest of this paper is organized as follows: In section 2, we introduce model problems and some preliminaries. We introduce the overlapping Schwarz method and give a stable spacial decomposition for $\bm{H}(\bm{{\rm curl}};\Omega)$-elliptic problems in Section 3. An extension to $\bm{H}({\rm div};\Omega)$-elliptic problems is introduced in Section 4. Finally, we present a conclusion in Section 5.

Throughout this paper, we use standard notations for Sobolev spaces $H^{m}(D)$ and $H_{0}^{m}(D)$ with their associated norms $\|\cdot\|_{m,D}$ and semi-norms $|\cdot|_{m,D}$. We denote by $L^{2}(D):=H^{0}(D)$, and use $(\cdot,\cdot)_{0,D}$ to represents the $L^{2}$-inner product and $||\cdot||_{0,D}$ represents the corresponding $L^{2}$-norm. For vector field space, we use bold font $\bm{L}^{2}(D)$ and $\bm{H}^{m}(D)$ to represent $[L^{2}(D)]^{d}$ and $[H^{m}(D)]^{d}$, respectively ($d=2,3$), and still use the notations of the norms $\|\cdot\|_{m,D}$, $|\cdot|_{m,D}$ and $||\cdot||_{0,D}$ with its inner product $(\cdot,\cdot)_{0,D}$. If $D=\Omega$, we drop the subscript $\Omega$ in associated norms or semi-norms or inner products. Let

$\displaystyle\bm{H}(\bm{{\rm curl}};\Omega):=\{\bm{u}\in\bm{L}^{2}(\Omega)\ |\ \bm{{\rm curl}}\bm{u}\in\bm{L}^{2}(\Omega)\ \}$

equipped with the norm $||\cdot||^{2}_{\bm{{\rm curl}}}=||\cdot||_{0}^{2}+||\bm{{\rm curl}}\cdot||_{0}^{2}$. $\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ represents a subspace where the vector valued functions have a vanishing tangential trace at domain boundary. $\bm{H}(\bm{{\rm curl}}_{0};\Omega)$ represents another supspace where the vector valued functions have a vanishing $\bm{{\rm curl}}$. We also let

$\displaystyle\bm{H}({\rm div};\Omega):=\{\bm{u}\in\bm{L}^{2}(\Omega)\ |\ {\rm div}\bm{u}\in\bm{L}^{2}(\Omega)\ \}$

equipped with the norm $||\cdot||^{2}_{{\rm div}}=||\cdot||_{0}^{2}+||{\rm div}\cdot||_{0}^{2}$. Let $\bm{H}({\rm div}_{0};\Omega)$ be a subspace where the vector valued functions are divergence-free. For convenience of notations in this paper, we define

$\displaystyle\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega):=\bm{H}_{0}(\bm{{\rm curl}};\Omega)\cap\bm{H}({\rm div}_{0};\Omega).$

(2.1)

In this paper, we assume $\Omega$ is convex, bounded and simple connected. Then we know that the kernel of $\bm{{\rm curl}}$ operator in $\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ is $\nabla{H_{0}^{1}(\Omega)}$. Moreover, the Helmholtz decomposition holds

$\displaystyle\bm{H}_{0}(\bm{{\rm curl}};\Omega)=\nabla{H_{0}^{1}(\Omega)}\oplus\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega),$

where $\oplus$ is an orthogonal decomposition under $\bm{L}^{2}(\Omega)$. For the theoretical analysis in the following, we define an operator (called as Hodge operator in [11]) for the above decomposition,

$\displaystyle\begin{aligned} &\Theta^{\perp}\ :\ \bm{H}_{0}(\bm{{\rm curl}};\Omega)\to\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega),\\ &\Theta^{\perp}(\bm{w})=\Theta^{\perp}(\nabla s+\bm{w}^{\perp})=\bm{w}^{\perp},\end{aligned}$

(2.2)

where $s\in H_{0}^{1}(\Omega)$ and $\bm{w}^{\perp}\in\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)$. Therefore, $\Theta^{\perp}$ is ${\bf curl}$-preserving that

$\displaystyle{\bf curl}\Theta^{\perp}\bm{w}$

$\displaystyle={\bf curl}\bm{w}.$

(2.3)

Using the Sobolev embedding theorem (see, e.g., [1, 11, 9]), we known that

$\displaystyle\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)=\bm{H}_{0}(\bm{{\rm curl}};\Omega)\cap\bm{H}({\rm div}_{0};\Omega)\hookrightarrow\bm{H}^{1}(\Omega).$

(2.4)

Consider the model problem

$$\begin{cases}\bm{{\rm curl}}\;\bm{{\rm curl}}\bm{u}+\bm{u}=\bm{f}\ \ &\text{in $\Omega,$}\\ \ \ \ \ \ \bm{n}\times\bm{u}=\bm{0}\ \ &\text{on $\partial\Omega,$}\end{cases}$$

(2.5)

where $\bm{n}$ represents the outward unit normal vector on $\partial{\Omega}$. The variational form for (2.5) is as follows:

$$\begin{cases}\text{Given $\bm{f}\in\bm{L}^{2}(\Omega)$, find $\bm{u}\in\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ such that}\\ \text{$a_{\bm{{\rm curl}}}(\bm{u},\bm{v})=(\bm{f},\bm{v})_{0}\ \ \ \ \forall\ \bm{v}\in\bm{H}_{0}(\bm{{\rm curl}};\Omega)$},\end{cases}$$

(2.6)

where

$$a_{\bm{{\rm curl}}}(\bm{w},\bm{v}):=\int_{\Omega}\big{(}\bm{{\rm curl}}\bm{w}\cdot\bm{{\rm curl}}\bm{v}+\bm{w}\cdot\bm{v}\big{)}dx$$

(2.7)

for all $\bm{w},\bm{v}\in\bm{H}(\bm{{\rm curl}};\Omega)$. By the Lax-Milgram theorem, it is easy to see that the solutions with respect to (2.6) are well-posed.

Let $\mathcal{T}_{h}$ be a shape-regular and quasi-uniform triangulation. We consider the $k$-th Nédélec element in $\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ as follows

$$\mathcal{ND}_{h,0}:=\{\bm{v}\in\bm{H}_{0}(\bm{{\rm curl}};\Omega)\ |\ \bm{v}|_{\tau}\in\mathcal{ND}_{k}(\tau),\ \forall\ \tau\in\mathcal{T}_{h}\ \},$$

(2.8)

where $\mathcal{ND}_{k}(\tau)$ is the $k$-th order local Nédélec space on element $\tau$ (see, e.g., [14, 15, 11]). The discrete variational form of (2.6) may be written as:

$$\begin{cases}\text{Given $\bm{f}\in\bm{L}^{2}(\Omega)$, find $\bm{u}_{h}\in\mathcal{ND}_{h,0}$ such that }\\ a_{\bm{{\rm curl}}}(\bm{u}_{h},\bm{v}_{h})=(\bm{f},\bm{v}_{h})_{0}\ \ \ \ \forall\ \bm{v}_{h}\in\mathcal{ND}_{h,0}.\end{cases}$$

(2.9)

Define an operator $A_{h}:\mathcal{ND}_{h,0}\to\mathcal{ND}_{h,0}$ such that $(A_{h}\bm{w}_{h},\bm{v}_{h})=a_{\bm{{\rm curl}}}(\bm{w}_{h},\bm{v}_{h})$ for all $\bm{w}_{h},\bm{v}_{h}\in\mathcal{ND}_{h,0}.$ We denote the discrete divergence-free space by

$\displaystyle\mathcal{ND}_{h,0}^{\perp}=\{\bm{v}_{h}\in\mathcal{ND}_{h,0}\ |\ (\bm{v}_{h},\nabla{p}_{h})_{0}=0,\ \ \forall\ p_{h}\in S_{h,0}\ \},$

(2.10)

with $S_{h,0}$ being a continuous and piecewise $P_{k+1}$ polynomial space on $\mathcal{T}_{h}$ with vanishing trace on $\partial{\Omega}$. Therefore, we have the discrete Helmholtz decomposition

$\displaystyle\mathcal{ND}_{h,0}=\nabla{S_{h,0}}\oplus\mathcal{ND}_{h,0}^{\perp}.$

(2.11)

It is easy to see that $\mathcal{ND}_{h,0}^{\perp}\not\subset\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)$, where $\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)$ is defined in (2.1). Let a subspace be

$\displaystyle\bm{V}^{+}:=\Theta^{\perp}\mathcal{ND}_{h,0}^{\perp}\subset\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega),$

where $\Theta^{\perp}$ is defined in (2.2). Define another operator

$\displaystyle P_{h}:\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)\to\bm{V}^{+}$

such that

$\displaystyle(\bm{{\rm curl}}P_{h}\bm{w},\bm{{\rm curl}}\bm{v})_{0}=(\bm{{\rm curl}}\bm{w},\bm{{\rm curl}}\bm{v})_{0}\ \ \ \ \forall\ \bm{w}\in\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega),\ \bm{v}\in\bm{V}^{+}.$

(2.12)

Due to the fact that the Poincaré inequality holds in $\bm{H}^{\perp}_{0}(\bm{{\rm curl}};\Omega)$, we know that the operator $P_{h}$ is well-defined in (2.12). Further, we extend the operator $P_{h}$ to $\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ by

$\displaystyle\begin{aligned} &P_{h}\nabla s:=0,\quad s\in H^{1}_{0}(\Omega),\\ &P_{h}\bm{w}=P_{h}(\nabla s+\bm{w}^{\perp})=P_{h}\bm{w}^{\perp},\quad\ \bm{w}^{\perp}\in\bm{H}_{0}^{\perp}(\bm{{\rm curl}};\Omega).\end{aligned}$

(2.13)

It holds that for any $\bm{w}_{h}^{\perp}\in\mathcal{ND}_{h,0}^{\perp}$, we have

$\displaystyle P_{h}\bm{w}_{h}^{\perp}$

$\displaystyle=P_{h}(\nabla s+\Theta^{\perp}\bm{w}_{h}^{\perp})=P_{h}\Theta^{\perp}\bm{w}_{h}^{\perp}=\Theta^{\perp}\bm{w}_{h}^{\perp}$

(2.14)

for some $s\in H^{1}_{0}(\Omega)$. By (2.3), we have

$\displaystyle\bm{{\rm curl}}\;P_{h}\bm{w}_{h}^{\perp}=\bm{{\rm curl}}\;\Theta^{\perp}\bm{w}_{h}^{\perp}$

$\displaystyle=\bm{{\rm curl}}\;\bm{w}_{h}^{\perp}.$

(2.15)

The following lemma holds (see Lemma 10.6 in [18]).

Let $\mathcal{T}_{H}:=\{K_{i}\}_{i=1}^{N}$ be a shape-regular and quasi-uniform coarse triangular or tetrahedral mesh on $\Omega$, where $H:=\max\{H_{i}\ |\ i=1,2,...,N\}$. We let domain be subdivided into $N$ subdomains where

$\displaystyle\Omega_{i}=K_{i},\quad i=1,\dots,N.$

The fine shape-regular and quasi-uniform triangulation $\{\tau\}$ is obtained by subdividing $\mathcal{T}_{H}$ and we denote it by $\mathcal{T}_{h}=\{\tau\}$. We may construct the edge element spaces $\mathcal{ND}_{H,0}\subset\mathcal{ND}_{h,0}$ on $\mathcal{T}_{H}$ and $\mathcal{T}_{h}$ but it is well-known that $\mathcal{ND}_{h,0}^{\perp}\not\subset\mathcal{ND}_{h,0}^{\perp}$. To get overlapping subdomains $(\Omega_{i}^{{}^{\prime}},\ 1\leq i\leq N)$, we enlarge a subdomain $\Omega_{i}$ by adding a size-$\delta$ layer of fine elements, where $\delta=O({\rm dist}(\partial\Omega_{i}\setminus\partial\Omega,\partial\Omega_{i}^{{}^{\prime}}\setminus\partial\Omega))$. We define, see the gray region in Figure 1 in 2D (The 3D case is same), the overlapping region inside $\Omega_{i}^{\prime}$ by

$\displaystyle\begin{aligned} \Omega_{i,j,\delta}&=\bigcup_{\tau\in\mathcal{T}_{h},\ \tau\subset(\Omega_{i}^{\prime}\cap\Omega_{j}^{\prime})}\tau,\\ \Omega_{i,\delta}:&=\bigcup_{\Omega_{j}^{\prime}\cap\Omega_{i}^{\prime}\neq\emptyset}\Omega_{i,j,\delta}.\end{aligned}$

(3.1)

We decompose the finite element space $\mathcal{ND}_{h,0}$ in (2.8) into overlapping subspaces,

$\displaystyle\bm{V}_{i}:=\mathcal{ND}_{h,0}\cap\bm{H}_{0}(\bm{\mbox{curl}};\Omega_{i}^{{}^{\prime}}),\ \ \ \ i=1,2,...,N.$

For describing the overlapping domain decomposition preconditioner for $A_{h}$, we define $A_{H}:\mathcal{ND}_{H,0}\to\mathcal{ND}_{H,0}$ such that

$\displaystyle(A_{H}\bm{w}_{H},\bm{v}_{H})_{0}=a_{\bm{{\rm curl}}}(\bm{w}_{H},\bm{v}_{H})\ \ \ \ \forall\ \bm{w}_{H},\bm{v}_{H}\in\mathcal{ND}_{H,0}.$

Similarly, we also define $A_{i}:\bm{V}_{i}\to\bm{V}_{i}$ such that

$\displaystyle(A_{i}\bm{w}_{i},\bm{v}_{i})_{0}=a_{\bm{{\rm curl}}}(\bm{w}_{i},\bm{v}_{i})\ \ \ \ \forall\ \bm{w}_{i},\bm{v}_{i}\in\bm{V}_{i}.$

We denote by $Q_{H}:\bm{L}^{2}(\Omega)\to\mathcal{ND}_{H,0}$ a $\bm{L}^{2}$-orthogonal projector and $Q_{i}:\bm{L}^{2}(\Omega)\to\bm{V}_{i},\ (i=1,2,...,N)$ $\bm{L}^{2}$-orthogonal projectors. So the preconditioner is

$\displaystyle B_{h}^{-1}=A_{H}^{-1}Q_{H}+\sum_{i=1}^{N}A_{i}^{-1}Q_{i}.$

(3.2)

Next, we introduce an assumption on overlapping domain decomposition (see [18]).

According to Assumption 1, we obtain a partition of unity, and then there exists a family continuous and piecewise linear polynomials $\{\theta_{i}\}_{i=1}^{N}$, which satisfy the following properties

$\displaystyle\begin{aligned} &{\rm supp}(\theta_{i})\subset\overline{\Omega_{i}^{{}^{\prime}}},\ \ \ \ 0\leq\theta_{i}\leq 1,\\ &\sum_{i=1}^{N}\theta_{i}\equiv 1,\ \ \ \ x\in\Omega,\\ &\big{(}\nabla{\theta_{i}}\big{)}|_{\Omega_{i}^{\circ}}=0,\ \ \ \ ||\nabla{\theta_{i}}||_{0,\infty,\Omega_{i,\delta}}\leq\frac{C}{\delta},\end{aligned}$

(3.3)

where $\Omega_{i}^{\circ}=\Omega_{i}^{{}^{\prime}}\backslash\overline{\Omega_{i,\delta}}$.

Next, we first present the main result in this paper, and delay its proof.

For convenience of theoretical analysis, using the “local” argument (see [3]), we may denote by $Q_{0,\Omega_{i}}:\bm{L}^{2}(\Omega_{i})\to\bm{P}_{0}(\Omega_{i}),\ \ \Omega_{i}\in\mathcal{T}_{H}$, a local $\bm{L}^{2}$-orthogonal projector. We have

$\displaystyle||\bm{w}-Q_{0,\Omega_{i}}\bm{w}||_{0,\Omega_{i}}\leq CH|\bm{w}|_{1,\Omega_{i}}\ \ \ \ \forall\ \bm{w}\in\bm{H}^{1}(\Omega_{i}).$

(3.5)

In order to prove the main result, we first give some technical lemmas. In the following theoretical analysis, we also take advantage of the global $\bm{L}^{2}$-orthogonal projector $Q_{H}$. The following lemma holds (see [17, 18]).

We claim all triangles in $\Omega_{i_{0},\delta}$ belong to at least one of the following stripes, cf. Figures 2 and 3 where $I_{0}=12$,

$\displaystyle\begin{aligned} \Omega^{\prime}_{i_{1}}\cap\Omega_{i_{0}},\ \;\Omega_{i_{5}}^{\prime}\cap\Omega_{i_{0}},\ \;\Omega_{i_{9}}^{\prime}\cap\Omega_{i_{0}},\ \;\Omega_{i_{1}}^{\prime}\cap\Omega_{i_{2}},\ \dots,\ \;\Omega_{i_{I_{0}}}^{\prime}\cap\Omega_{i_{1}},\\ \Omega_{i_{1}}\cap\Omega_{i_{0}}^{\prime},\ \;\Omega_{i_{5}}\cap\Omega_{i_{0}}^{\prime},\ \;\Omega_{i_{9}}\cap\Omega_{i_{0}}^{\prime},\ \;\Omega_{i_{1}}\cap\Omega_{i_{2}}^{\prime},\ \dots,\ \;\Omega_{i_{I_{0}}}\cap\Omega_{i_{1}}^{\prime}.\end{aligned}$

(3.8)

We note that inside each stripe $\bm{w}$ is $\bm{H}^{1}$. We will prove (3.7) on one stripe first, then get (3.7) by summing over all strips in (3.8).

We separate one stripe $\Omega_{i_{0}}^{\prime}\cap\Omega_{i_{1}}$ into finite patches $\{s_{l}\}_{l=1}^{n_{i_{0}}}$ of triangles $\tau_{j}$ from $\mathcal{T}_{h}$, cf. Figure 2,

$\displaystyle s_{l}=\cup_{j=1}^{m_{l}}\tau_{j},\quad\ \ |s_{l}|=C\delta^{2},\quad|\partial s_{l}\cap\partial\Omega_{i_{1}}|=C\delta.$

By using the Ponicaré-Friedrichs inequality on $s_{l}$, because $\bm{w}\in\bm{H}^{1}(\Omega_{i})$, we have

$\displaystyle||\bm{w}||_{0,s_{l}}^{2}\leq C\delta^{2}|\bm{w}|_{1,s_{l}}^{2}+C\delta||\bm{w}||^{2}_{0,\partial{s_{l}}\cap\partial{\Omega_{i_{1}}}}.$

Summing over the patches $\{s_{l}\}_{1\leq l\leq n_{i_{0}}}$, we get, using the trace theorem in $\Omega_{i_{1}}$

	$\displaystyle||\bm{w}||_{0,\Omega_{i_{0}}^{\prime}\cap\Omega_{i_{1}}}^{2}$	$\displaystyle\leq C\delta^{2}|\bm{w}|_{1,\Omega_{i_{1}}}^{2}+C\delta||\bm{w}||^{2}_{0,\partial{\Omega_{i_{0}}}\cap\partial{\Omega_{i_{1}}}}$
		$\displaystyle\leq C\delta^{2}|\bm{w}|_{1,\Omega_{i_{1}}}^{2}+C\delta H|\bm{w}|^{2}_{1,\Omega_{i_{1}}}+C\delta H^{-1}||\bm{w}||^{2}_{0,\Omega_{i_{1}}}$
		$\displaystyle\leq C\delta^{2}\{\big{(}1+\frac{H}{\delta}\big{)}|\bm{w}|_{1,\Omega_{i_{1}}}^{2}+\frac{1}{\delta H}||\bm{w}||^{2}_{0,\Omega_{i_{1}}}\}.$

Summing over all strips $\Omega_{i_{j}}^{\prime}\cap\Omega_{i_{k}}$ in (3.8), by finite covering, we get (3.7). This completes the proof of the lemma. $\Box$

For any $\bm{u}_{h}\in\mathcal{ND}_{h,0}$, we decompose it as $\bm{u}_{h}=\sum_{i=0}^{N}\bm{u}_{i}$, where

$\displaystyle\begin{aligned} \bm{u}_{0}&:=\nabla{q_{0}}+\bm{w}_{0}\in\mathcal{ND}_{H,0},\\ \bm{u}_{i}&:=\nabla{q_{i}}+\bm{w}_{i}\in\bm{V}_{i},\ \ \ \ i=1,2,...,N,\\ q_{0}&=\tilde{I}_{H}q_{h},\\ q_{i}&=I_{h}(\theta_{i}(q_{h}-q_{0})),\\ \bm{w}_{0}&=Q_{H}P_{h}\bm{w}_{h}^{\perp},\\ \bm{w}_{i}&=\Pi_{E,h}(\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0})),\\ \bm{u}_{h}&=\nabla q_{h}+\bm{w}_{h}^{\perp},\end{aligned}$

(3.9)

where $q_{h}$ and $\bm{w}_{h}^{\perp}$ are defined in (2.11), $\Pi_{E,h}$ is the H-curl interpolation operator to $\mathcal{ND}_{h,0}$, $\theta_{i}$ is defined in (3.3), $P_{h}$ is defined in (2.12), $Q_{H}$ is defined in (3.6), $I_{h}$ is the nodal value interpolation to $S_{h,0}$ defined in (2.10) and $\tilde{I}_{H}$ is the Scott-Zhang interpolation operator (see [16]) to $S_{H,0}$. In (3.9), the operation $\Pi_{E,h}(\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0}))$ is well-defined, because $\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0})\in\bm{H}_{0}(\bm{{\rm curl}};\Omega)$ and $\bm{w}_{0}\in\mathcal{ND}_{H,0}$. We check the decomposition,

	$\displaystyle\sum_{i=0}^{N}\bm{u}_{i}$	$\displaystyle=\nabla{q_{0}}+\bm{w}_{0}+\sum_{i=1}^{N}(\nabla{q_{i}}+\bm{w}_{i})$
		$\displaystyle=\nabla{q_{0}}+\bm{w}_{0}+\sum_{i=1}^{N}\left(\nabla{I_{h}(\theta_{i}(q_{h}-q_{0}))}+\Pi_{E,h}(\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0}))\right)$
		$\displaystyle=\nabla{q_{0}}+\bm{w}_{0}+\nabla{I_{h}\left(\sum_{i=1}^{N}\theta_{i}(q_{h}-q_{0})\right)}+\Pi_{E,h}\left(\sum_{i=1}^{N}\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0})\right)$
		$\displaystyle=\nabla{q_{0}}+\bm{w}_{0}+\nabla{I_{h}\left(q_{h}-q_{0}\right)}+\Pi_{E,h}\left(\bm{w}_{h}^{\perp}-\bm{w}_{0}\right)$
		$\displaystyle=\nabla{q_{0}}+\bm{w}_{0}+\nabla{\left(q_{h}-q_{0}\right)}+\left(\bm{w}_{h}^{\perp}-\bm{w}_{0}\right)$
		$\displaystyle=\nabla q_{h}+\bm{w}_{h}^{\perp}=\bm{u}_{h}.$

In order to prove the lower bound in (3.4), we only need to give a stable decomposition (see Chapter 2 in [18]), and then prove that the stable parameter can be bounded by $C(1+\frac{H}{\delta})$.

For the terms $\nabla{q_{i}}$ of $\bm{u}_{h}$, using the result for continuous finite element spaces conforming in $H_{0}^{1}(\Omega)$ given in the proof of lemma 3.12 in [18] and (3.9), we have

$\displaystyle\begin{aligned} \sum_{i=0}^{N}a_{\bm{{\rm curl}}}(\nabla{q_{i}},\nabla{q_{i}})&=\sum_{i=0}^{N}|q_{i}|^{2}_{1,\Omega_{i}^{{}^{\prime}}}\leq C(1+\frac{H}{\delta})|q_{h}|^{2}_{1}\\ &=C(1+\frac{H}{\delta})a_{\bm{{\rm curl}}}(\nabla{q_{h}},\nabla{q_{h}}).\end{aligned}$

(3.11)

For the terms $\bm{w}_{i}$ of $\bm{u}_{h}$, we will prove the following by the four steps below.

$\displaystyle\begin{aligned} \sum_{i=0}^{N}a_{\bm{{\rm curl}}}(\bm{w}_{i},\bm{w}_{i})&=||\bm{{\rm curl}}\bm{w}_{0}||_{0}^{2}+||\bm{w}_{0}||_{0}^{2}+\sum_{i=1}^{N}||\bm{{\rm curl}}\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}^{2}+\sum_{i=1}^{N}||\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}^{2}\\ &\leq C(1+\frac{H}{\delta})a_{\bm{{\rm curl}}}(\bm{w}^{\perp}_{h},\bm{w}^{\perp}_{h}).\end{aligned}$

(3.12)

• (1)

  For the term $||\bm{{\rm curl}}\bm{w}_{0}||_{0}$ in (3.12), by (3.9), (3.6), (2.4) and (2.15), we get

  $\displaystyle||\bm{{\rm curl}}\bm{w}_{0}||_{0}=||\bm{{\rm curl}}Q_{H}P_{h}\bm{w}_{h}^{\perp}||_{0}\leq C|P_{h}\bm{w}_{h}^{\perp}|_{1}\leq C||\bm{{\rm curl}}P_{h}\bm{w}_{h}^{\perp}||_{0}=C||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}.$

  (3.13)

• (2)

  For the term $||\bm{w}_{0}||_{0}$ in (3.12), by (3.9), $\bm{L}^{2}$-orthogonal projector $Q_{H}$ and (2.14), we have

  $\displaystyle||\bm{w}_{0}||_{0}=||Q_{H}P_{h}\bm{w}_{h}^{\perp}||_{0}\leq||P_{h}\bm{w}_{h}^{\perp}||_{0}=||\Theta^{\perp}\bm{w}_{h}^{\perp}||_{0}\leq||\bm{w}_{h}^{\perp}||_{0}.$

  (3.14)

  Here the last inequality follows the argument

  $\displaystyle(\Theta^{\perp}\bm{w}_{h}^{\perp},\bm{g})_{0}=(\Theta^{\perp}(\nabla s+\tilde{\bm{w}}^{\perp}),\bm{g})_{0}=(\tilde{\bm{w}}^{\perp},\bm{g})_{0}=(\nabla s+\tilde{\bm{w}}^{\perp},\bm{g})_{0}=(\bm{w}_{h}^{\perp},\bm{g})_{0}$

  with $\bm{g}=\Theta^{\perp}\bm{w}_{h}^{\perp}$ and the Cauchy-Schwarz inequality.

• (3)

  For the terms $||\bm{{\rm curl}}\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}$ in (3.12), by the properties of $\{\theta_{i}\}_{i=1}^{N}$ and $\Pi_{E,h}$ (see Lemma 10.8 in [18]), we have

  $\displaystyle\sum_{i=1}^{N}||\bm{{\rm curl}}\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}^{2}\leq C\sum_{i=1}^{N}\delta^{-2}||\bm{v}||_{0,\Omega_{i,\delta}}^{2}+C||\bm{{\rm curl}}\bm{v}||_{0}^{2},$

  where

  $\displaystyle\bm{v}=\bm{w}_{h}^{\perp}-\bm{w}_{0}=\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}.$

  Denoting by $\widetilde{\bm{v}}:=P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}$ and using the triangle inequality, we obtain from above inequality that

  $\displaystyle\begin{aligned} \sum_{i=1}^{N}||\bm{{\rm curl}}\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}^{2}&\leq C\sum_{i=1}^{N}\delta^{-2}||\widetilde{\bm{v}}||_{0,\Omega_{i,\delta}}^{2}+C\sum_{i=1}^{N}\delta^{-2}||\widetilde{\bm{v}}-\bm{v}||_{0,\Omega_{i,\delta}}^{2}+C||\bm{{\rm curl}}\bm{v}||_{0}^{2}\\ &=:I_{1}+I_{2}+I_{3}.\end{aligned}$

  (3.15)

  For the first term $I_{1}$ in (3.15), by Lemma 3.3 and the fact that $\widetilde{\bm{v}}\in\bm{H}^{1}(\Omega_{i})$, we get, because of finite overlapping,

  $\displaystyle I_{1}=C\sum_{i=1}^{N}\delta^{-2}||\widetilde{\bm{v}}||_{0,\Omega_{i,\delta}}^{2}$

  $\displaystyle\leq C(1+\frac{H}{\delta})\sum_{i=1}^{N}|\widetilde{\bm{v}}|^{2}_{1,\Omega_{i}}+C\frac{1}{\delta H}||\widetilde{\bm{v}}||^{2}_{0,\Omega}.$

  (3.16)

  For the first term in (3.16), by the triangle inequality, the inverse estimate and (3.5), we obtain

  	$\displaystyle|\widetilde{\bm{v}}|_{1,\Omega_{i}}$	$\displaystyle=|P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}\leq|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}+|Q_{H}P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}$
  		$\displaystyle=|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}+|Q_{H}P_{h}\bm{w}_{h}^{\perp}-Q_{0,\Omega_{i}}P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}$
  		$\displaystyle\leq|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}+CH^{-1}||Q_{H}P_{h}\bm{w}_{h}^{\perp}-Q_{0,\Omega_{i}}P_{h}\bm{w}_{h}^{\perp}||_{0,\Omega_{i}}$
  		$\displaystyle\leq|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}+CH^{-1}||Q_{H}P_{h}\bm{w}_{h}^{\perp}-P_{h}\bm{w}_{h}^{\perp}||_{0,\Omega_{i}}$
  		$\displaystyle\ \ \ \ +CH^{-1}||P_{h}\bm{w}_{h}^{\perp}-Q_{0,\Omega_{i}}P_{h}\bm{w}_{h}^{\perp}||_{0,\Omega_{i}}$
  		$\displaystyle\leq|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}}+CH^{-1}||P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}||_{0,\Omega_{i}}+C|P_{h}\bm{w}_{h}^{\perp}|_{1,\Omega_{i}},$

  which, together with (3.6), (2.4) and (2.15), yields

  $\displaystyle\begin{aligned} \sum_{i=1}^{N}|\widetilde{\bm{v}}|^{2}_{1,\Omega_{i}}&\leq C\sum_{i=1}^{N}|P_{h}\bm{w}_{h}^{\perp}|^{2}_{1,\Omega_{i}}+CH^{-2}\sum_{i=1}^{N}||P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}||^{2}_{0,\Omega_{i}}\\ &\leq C|P_{h}\bm{w}_{h}^{\perp}|^{2}_{1}+CH^{-2}||P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}||_{0}^{2}\\ &\leq C|P_{h}\bm{w}_{h}^{\perp}|^{2}_{1}\leq C||\bm{{\rm curl}}P_{h}\bm{w}_{h}^{\perp}||^{2}_{0}=C||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||^{2}_{0}.\end{aligned}$

  (3.17)

  For the second term in (3.16), by (3.6), (2.4) and (2.15), we have

  $\displaystyle\begin{aligned} \frac{C}{\delta H}||\widetilde{\bm{v}}||_{0}^{2}&=\frac{C}{\delta H}||P_{h}\bm{w}_{h}^{\perp}-Q_{H}P_{h}\bm{w}_{h}^{\perp}||_{0}^{2}\\ &\leq\frac{C}{\delta H}CH^{2}|P_{h}\bm{w}_{h}^{\perp}|^{2}_{1}\leq C\frac{H}{\delta}||\bm{{\rm curl}}P_{h}\bm{w}_{h}^{\perp}||^{2}_{0}=C\frac{H}{\delta}||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||^{2}_{0}.\end{aligned}$

  (3.18)

  Combining (3.15), (3.16), (3.17) and (3.18), we get

  $\displaystyle I_{1}=C\sum_{i=1}^{N}\delta^{-2}||\widetilde{\bm{v}}||_{0,\Omega_{i,\delta}}^{2}\leq C(1+\frac{H}{\delta})||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}.$

  For the second term $I_{2}$ in (3.15), by Lemma 2.1, we deduce

  	$\displaystyle I_{2}$	$\displaystyle=C\sum_{i=1}^{N}\delta^{-2}||\widetilde{\bm{v}}-\bm{v}||_{0,\Omega_{i,\delta}}^{2}=C\delta^{-2}\sum_{i=1}^{N}||\bm{w}_{h}^{\perp}-P_{h}\bm{w}_{h}^{\perp}||_{0,\Omega_{i,\delta}}^{2}$
  		$\displaystyle\leq C\delta^{-2}||\bm{w}_{h}^{\perp}-P_{h}\bm{w}_{h}^{\perp}||_{0}^{2}\leq Ch^{2}\delta^{-2}||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}$
  		$\displaystyle\leq C||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}.$

  For the third term $I_{3}$ in (3.15), we have, because of (3.13),

  $\displaystyle I_{3}=C||\bm{{\rm curl}}\bm{v}||_{0}^{2}\leq C\{||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}+||\bm{{\rm curl}}\bm{w}_{0}||_{0}^{2}\}\leq C||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}.$

  By using (3.15) and the estimates of three terms $I_{1},I_{2}$ and $I_{3}$, we obtain

  $\displaystyle\sum_{i=1}^{N}||\bm{{\rm curl}}\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}^{2}\leq I_{1}+I_{2}+I_{3}\leq C(1+\frac{H}{\delta})||\bm{{\rm curl}}\bm{w}_{h}^{\perp}||_{0}^{2}.$

  (3.19)

• (4)

  For the terms $||\bm{w}_{i}||_{0,\Omega_{i}^{{}^{\prime}}}$ in (3.12), we have, by (3.9), the fact $|\theta_{i}|\leq 1$, triangle inequality, finite overlapping and (3.14),

  $\displaystyle\begin{aligned} \sum_{i=1}^{N}||\bm{w}_{i}||^{2}_{0,\Omega_{i}^{{}^{\prime}}}&\leq C\sum_{i=1}^{N}||\theta_{i}(\bm{w}_{h}^{\perp}-\bm{w}_{0})||^{2}_{0,\Omega_{i}^{{}^{\prime}}}\leq C\sum_{i=1}^{N}\{||\bm{w}_{h}^{\perp}||^{2}_{0,\Omega_{i}^{{}^{\prime}}}+||\bm{w}_{0}||^{2}_{0,\Omega_{i}^{{}^{\prime}}}\}\\ &\leq C||\bm{w}_{h}^{\perp}||_{0}^{2}+C||\bm{w}_{0}||_{0}^{2}\leq C||\bm{w}_{h}^{\perp}||_{0}^{2}.\end{aligned}$

  (3.20)