Fully $H$(GRADCURL)-nonconforming finite element method for the singularly perturbed quad-curl problem on cubical meshes

The quad-curl equation appears in various models, such as the inverse electromagnetic scattering theory [3, 13, 16], couple stress theory in linear elasticity [11, 14], and magnetohydrodynamics [24]. The corresponding quad-curl eigenvalue problem plays a fundamental role in the analysis and computation of the electromagnetic interior transmission eigenvalues [15]. In this paper, we consider the following quad-curl singular perturbation problem on a bounded polyhedral domain $\Omega\subset\mathbb{R}^{3}$: find $\bm{u}$ such that

Here $\bm{f}\in H(\text{div}^{0};\Omega):=\{\bm{u}\in[L^{2}(\Omega)]^{3}:\;\operatorname{div}\bm{u}=0\}$, $\alpha>0$ and $\beta\geq 0$ are constants of moderate size, $\epsilon>0$ is a constant that can approach 0, and $\bm{n}$ is the unit outward normal vector to $\partial\Omega$.

Standard conforming finite element methods to solve problem (1.1) require function spaces to be subspaces of $H(\operatorname{grad}\operatorname{curl};\Omega)$, see Section 2 for its precise definition. Some $H$(gradcurl)-conforming finite elements have been constructed in the past years. So far, the construction in 2 dimensions(2D) is relatively complete [20, 7, 18], however, that is not the case for 3 dimensions (3D). In [22], one of the authors and Z. Zhang developed a tetrahedral $\operatorname{grad}\operatorname{curl}$-conforming element, which has $315$ degrees of freedom (DOFs) on each element. To reduce the number of DOFs, they, together with their collaborators, enriched the shape function space on each tetrahedron with piecewise-polynomial bubbles [8], and the resulting element has only 18 DOFs. However, it is challenging to extend the idea of enriching bubbles to cubical element. The only grad curl-conforming element on cubical meshes constructed in [17] has $144$ DOFs on each cube.

Therefore, we resort to nonconforming elements to reduce the number of DOFs. On tetrahedral meshes, Zheng et al. constructed the first nonconforming finite element in [24], which has $20$ DOFs on each element. Recently, Huang [9] proposed a nonconforming finite element Stokes complex which includes two $\operatorname{grad}\operatorname{curl}$-nonconforming finite elements: one coincides with the element in [24] and the other one has only 14 DOFs. However, the convergence rates of the two nonconforming finite elements in [9, 24] will deteriorate when $\epsilon\to 0$ in problem (1.1). Hence, Huang and Zhang in [10] developed two families of $H(\operatorname{grad}\operatorname{curl})$-nonconforming but $H$(curl)-conforming finite elements on tetrahedral meshes, which can solve problem (1.1) when $\epsilon$ tends to zero, see also [19]. On cubical meshes, one of the authors and her collaborators proposed a family of nonconforming finite elements with at least $48$ DOFs in [21]. This family is also $H$(curl)-conforming, and hence can solve problem (1.1) as $\epsilon$ vanishes. To further reduce the number of DOFs, in this paper, we will construct both $H(\operatorname{grad}\operatorname{curl})$-nonconforming and $H(\operatorname{curl})$-nonconforming finite elements. Different with the tetrahedral elements, the fully nonconforming elements that we will construct still work for problem (1.1) when $\epsilon\rightarrow 0$.

To this end, we consider the following Stokes complex:

We will develop the fully $\operatorname{grad}\operatorname{curl}$-nonconforming elements by constructing a discrete nonconforming Stokes complex on cubical meshes:

where $S_{h}^{r}(\mathcal{T}_{h})$, $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$, $\bm{W}_{h}(\mathcal{T}_{h})$, and $Q_{h}(\mathcal{T}_{h})$ are conforming or nonconforming finite element spaces for $H^{1}(\Omega)$, $H(\operatorname{grad}\operatorname{curl};\Omega)$, ${[H^{1}(\Omega)]}^{3}$, and $L^{2}(\Omega)$, respectively. To make the number of DOFs as minimal as possible, we use the $r$-th order serendipity finite element space with $r=1,2$ for $S_{h}^{r}(\mathcal{T}_{h})$. We choose the nonconforming Stokes pair developed in [23] for $\bm{W}_{h}(\mathcal{T}_{h})$ and $Q_{h}(\mathcal{T}_{h})$. Then the $\operatorname{grad}\operatorname{curl}$-nonconforming finite element space $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$ is obtained as the gradient of $S_{h}^{r}(\mathcal{T}_{h})$ and a complementary part whose $\operatorname{curl}_{h}$ falls into $\bm{W}_{h}(\mathcal{T}_{h})$. To be precise, the shape function space $\bm{V}_{h}^{r-1}(K)$ of $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$ is

where $S_{h}^{r}(K)$ and $\bm{W}_{h}(K)$ are shape function spaces of $S_{h}^{r}(\mathcal{T}_{h})$ and $\bm{W}_{h}(\mathcal{T}_{h})$ and $\mathfrak{p}$ is the Poincaré operator with the precise definition given in Sections 3. The DOFs of $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$ can be obtained from the DOFs of $S_{h}^{r}(\mathcal{T}_{h})$ and $\bm{W}_{h}(\mathcal{T}_{h})$. By taking $r=1,2$, we will get two versions of $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$. The number of DOFs on each element is 24 for $r=1$ and 36 for $r=2$. By constructing $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$ in this way, we can show that the complex (1.3) is exact on contractible domains.

Based on the newly proposed $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$ and $S_{h}^{r}(\mathcal{T}_{h})$, we develop a robust mixed finite element method for solving the quad-curl singular perturbation problem (1.1). The wellposedness of the numerical scheme holds by proving the discrete Poincaré inequality, the discrete inf-sup condition, and the coercivity condition. Moreover, we prove a special property of functions in $\bm{V}_{h}^{r-1}(\mathcal{T}_{h})$:

where $\bm{I}_{h}^{r-1}$ is defined in Section 4. With the interpolation results, projection error estimates, and property (1.4), we obtain an optimal convergence order $\mathcal{O}(h)$ for any fixed $\epsilon$. As $\epsilon$ approaches $0$, the right-hand side of the optimal estimate will blow up, which makes the estimate useless. Therefore, we also provide a uniform convergence order $\mathcal{O}(h^{1/2})$ for any value of $\epsilon$ in the sense of the energy norm. We note that the optimal estimate for a moderate $\epsilon$ can be obtained without using property (1.4), while the uniform estimate can not. The fully nonconforming finite elements on tetrahedral meshes [9] do not possess this property, which is the reason that those elements can not work for the quad-curl singular perturbation problem.

The rest of the paper is organized as follows. In section 2 we list some notation that will be used throughout the paper. In section 3 we define the fully $\operatorname{grad}\operatorname{curl}$-nonconforming finite element on a cube and estimate the interpolation errors. Nonconforming and exact finite element complexes are constructed in section 4. In section 5 we use our proposed elements to solve the quad-curl singularly perturbed problem and obtain the optimal convergence order of the energy error for any fixed $\epsilon$. In section 6 we provide a uniform error estimate with respect to $\epsilon$. In section 7, numerical examples are shown to verify the correctness and efficiency of our method. Finally, some concluding remarks are given in section 8.

We assume that $\Omega\subset\mathbb{R}^{3}$ is a contractible Lipschitz domain throughout the paper. We adopt conventional notation for Sobolev spaces such as $W^{s,p}(D)$ or $W_{0}^{s,p}(D)$ on a sub-domain $D\subset\Omega$ furnished with the norm $\left\|\cdot\right\|_{W^{s,p}(D)}$ and the semi-norm $\left|\cdot\right|_{W^{s,p}(D)}$. In the case of $p=2$, we use notation $H^{s}(D)$ or $H_{0}^{s}(D)$ for spaces $W^{s,p}(D)$ or $W_{0}^{s,p}(D)$. The norm and semi-norm are $\|\cdot\|_{s,D}$ and $|\cdot|_{s,D}$, respectively. In the case of $s=0$, the space $H^{0}(D)$ coincides with $L^{2}(D)$ which is equipped with the inner product $(\cdot,\cdot)_{D}$ and the norm $\left\|\cdot\right\|_{D}$. When $D=\Omega$, we drop the subscript $D$.

In addition to the standard Sobolev spaces, we also define

For a subdomain $D$, we use $P_{r}(D)$, or simply $P_{r}$ when there is no possible confusion, to denote the space of polynomials with degree at most $r$ on $D$. We denote $Q_{i,j,k}$ the space of polynomials in three variables $(x,y,z)$ whose maximum degrees are $i$ in $x$, $j$ in $y$, and $k$ in $z$, respectively.

For a scalar function space $S$, we use $\bm{S}$ or $[S]^{3}$ to denote the $S\otimes\mathbb{R}^{3}$.

Let  $\mathcal{T}_{h}\,$ be a partition of the domain $\Omega$ consisting of shape-regular cuboids. For $K=(x_{1}^{c}-h_{1},x_{1}^{c}+h_{1})\times(x_{2}^{c}-h_{2},x_{2}^{c}+h_{2})\times(x_{3}^{c}-h_{3},x_{3}^{c}+h_{3})$, we denote $h_{K}=\sqrt{h_{1}^{2}+h_{2}^{2}+h_{3}^{2}}$ as the diameter of $K$ and $h=\max_{K\in\mathcal{T}_{h}}h_{K}$ as the mesh size of $\mathcal{T}_{h}$. Denote by $\mathcal{V}_{h}$, $\mathcal{E}_{h}$, and $\mathcal{F}_{h}$ the sets of vertices, edges, and faces in the partition, and $\mathcal{V}_{h}^{\text{int}}$, $\mathcal{E}_{h}^{\text{int}}$, and $\mathcal{F}_{h}^{\text{int}}$ the sets of interior vertices, edges, and faces. We also denote by $\mathcal{V}_{h}(K)$, $\mathcal{E}_{h}(K)$, and $\mathcal{F}_{h}(K)$ the sets of vertices, edges, and faces related to the element $K\in\mathcal{T}_{h}$. We use $\bm{\tau}_{e}$ and $\bm{n}_{f}$ to denote the unit tangential vector and the unit normal vector to $e\in\mathcal{E}_{h}$ and $f\in\mathcal{F}_{h}$, respectively. Suppose $f=K_{1}\cap K_{2}\in\mathcal{F}_{h}$. For a function $v$ defined on $K_{1}\cup K_{2}$, we define $[\![v]\!]_{f}=f|_{K_{1}}-f|_{K_{2}}$ to be the jump across $f$. When $f\subset\partial\Omega$, $[\![v]\!]_{f}=v$.

We use $C$ to denote a generic positive constant that is independent of $h$.

In this section, we will construct $H(\operatorname{grad}\operatorname{curl})$-nonconforming finite elements on cubical meshes. To this end, recall the Poincaré operator $\mathfrak{p}:[C^{\infty}]^{3}\to[C^{\infty}]^{3}$ [4] defined by

The Poincaré operator satisfies

where $\mathfrak{p}^{3}v=\bm{x}\int_{0}^{1}t^{2}v(t\bm{x})\text{d}t$ and $\mathfrak{p}^{1}\bm{v}=\int_{0}^{1}\bm{v}(t\bm{x})\cdot\bm{x}\text{d}t$. It holds

For each element $K\in\mathcal{T}_{h}$, we define the following shape function space:

where

and for $r=1,2$, the shape function space $S_{h}^{r}(K)$ contains all polynomials of superlinear degree at most $r$. The superlinear degree of a polynomial is the degree with ignoring variables that enter linearly. For example, $x^{2}yz^{3}$ is a polynomial with superlinear degree $5$. To be specific, the space $S_{h}^{2}(K)$ is spanned by the monomials in $P_{2}(K)$ and

We have $\dim S_{h}^{1}(K)=8$ and $\dim S_{h}^{2}(K)=20$. The space $\bm{W}_{h}(K)$ is the same as the space $\bm{V}_{T}^{(3)}$ defined in [23] and $\dim\bm{W}_{h}(K)=18$.

The right-hand side of (3.4) is a direct sum. In fact, if $\bm{v}\in\operatorname{grad}S_{h}^{r}(K)\cap\mathfrak{p}\bm{W}_{h}(K)$, then $\operatorname{curl}\bm{v}=0$ and $\mathfrak{p}^{1}\bm{v}=0$. According to (3.2), we have $\bm{v}=0$. From the definition of $\bm{V}^{r-1}_{h}(K)$, we have $\bm{P}_{r-1}(K)\subset\bm{V}^{r-1}_{h}(K)$ and

We define the following DOFs for $\bm{v}\in\bm{V}^{r-1}_{h}(K)$: