1. Introduction
The quadcurl equation appears in various applications, such as the inverse electromagnetic scattering theory [10, 11, 30] or magnetohydrodynamics [36]. The corresponding quadcurl eigenvalue problem plays a fundamental role in the analysis and computation of the electromagnetic interior transmission eigenvalues [20, 29]. To compute eigenvalues, one usually starts with the corresponding source problem [3, 7, 28]. Some methods have been proposed for the source problem, i.e., the quadcurl problem, in [33, 36, 30, 19, 9, 27, 34, 12, 35, 31, 32, 8]. Recently, a family of conforming finite elements using incomplete th order polynomials is proposed in [33] for the qualcurl problem. In this paper, we construct a new family of elements by using the complete
th order polynomials. Due to the large kernel space of the curl operator, the Helmholtz decomposition of splitting an arbitrary vector field into the irrotational and solenoidal components plays an important role in the analysis. However, in general, the irrotational component is not
regular when is nonconvex. Therefore, we propose a new decomposition for , which further splits the irrotational component into a function in and a function in the kernel space of curl operator.There exist a few results on the numerical methods for the quadcurl eigenvalue problem. The problem was first proposed in [30] by Sun, who applied a mixed finite element method and proved an a priori error estimate. Two multigrid methods based on the Rayleigh quotient iteration and the inverse iteration with fixed shift were proposed and analyzed in [18]. In the first part of the paper, we apply the classical framework of Babuška and Osborn [3, 24] to prove an a priori estimate.
At reentrant corners or material interfaces, the eigenvectors feature strong singularities [23]. For more efficient computation, adaptive local refinements are considered. A posteriori error estimators are essential for the adaptive finite element methods. We refer to [14, 4, 21, 25] for the a posteriori estimates of source problems and [15, 6, 5] for eigenvalue problems. In terms of the quadcurl eigenvalue problem, to the authors’ knowledge, no work on a posteriori error estimations has been done so far. To this end, we start by relating the eigenvalue problem to a source problem. An a posteriori error estimator for the source problem is constructed, The proof uses the new decomposition and makes no additional regularity assumption. Then we apply the idea of [15] to obtain an a posteriori error estimate for the eigenvalue problem.
The rest of this paper is organized as follows. In Section 2, we present some notations, the new elements, the new decomposition, and an
Clément interpolation. In Section 3, we derive an a priori error estimate for the quadcurl eigenvalue problem. In Section 4, we prove an a posteriori error estimate. Finally, in Section 5, we show some numerical experiments.
2. Notations and basis tools
2.1. Notations
Let be a simplyconnected Lipschitz domain. For any subdomain , denotes the space of square integrable functions on with norm . If is a positive integer, denotes the space of scalar functions in whose derivatives up to order are also in . If , . When , we omit the subscript in the notations of norms. For vector functions, and .
Let and , where the superscript denotes the transpose. Then and . For a scalar function , . We now define a space concerning the curl operator
whose norm is given by
The spaces , , and are defined, respectively, as
Let be a triangular partition of . Denote by and the sets of vertices and edges. Let be the tangent vector of an edge . We refer to and as the sets of vertices and edges in the interior of , respectively. Let and be the sets of vertices and edges on the element . Denote by the diameter of and . In the following, we introduce some subdomains called patches:

: the union of elements sharing a common edge with , ;

: the union of elements sharing as an edge, ;

: the union of elements sharing as a vertex, .
We use to represent the space of polynomials on an edge or on a subdomain with degrees at most and .
2.2. A decomposition of
We mimic the proof of [16, Prop. 5.1] to obtain a decomposition of the space , which plays a critical role in the analysis.
Lemma 2.1.
Let be the set of gradients of functions in . Then is a closed subspace of and
(2.1) 
where Namely, for , with and Furthermore, admits the splitting
(2.2) 
where and satisfying
(2.3)  
(2.4) 
Proof.
The proof of (2.1) can be found in [33]. We only need to prove (2.2). Let be a bounded, smooth, simplyconnected open set with . For any , we can extend in the following way:
Obviously, and . Now, we consider the following problem: Find defined in such that
(2.5)  
(2.6) 
Since and has a smooth boundary, there exists a function satisfying (2.5) and (2.6) and
(2.7) 
In addition, (2.5) can be rewritten as Based on [17, Thm. 2.9], there exists a unique function of such that
(2.8) 
Now, we restrict (2.8) to the domain and obtain
(2.9) 
Using the extension theorem [13], we can extend to defined on satisfying
(2.10) 
where we have used PoincaréFriedrichs inequality for since we can choose for which Restricting on , we have
Note that since is the extension of . Therefore, (2.2) is proved. Combining (2.7) and (2.10), we obtain
and
2.3. A new family of conforming elements
In this subsection, we propose a new family of conforming finite elements. The new elements can lead to one order higher accuracy than the elements in [33] when the solution is smooth enough.
Definition 2.1.
For an integer , an conforming element is given by the triple:
where
is the set of DOFs (degree of freedom) defined as follows.

is the set of DOFs on all vertex nodes and edge nodes :
(2.11) with the points chosen at vertex nodes and distinct nodes on each edge.

is the set of DOFs given on all edges of with the unit tangential vector :
(2.12) 
is the set of DOFs on the element :
(2.13) where when and when . Here is the space of a homogeneous polynomial of degree .
Lemma 2.2.
The above finite elements are unisolvent and conforming.
Using the above Lemma, the global finite element space on is given by
Provided and with , define an interpolation , whose restriction on , denoted by , is such that
(2.14) 
Theorem 2.1.
If , with , then the following error estimate for the interpolation holds:
2.4. An (curl)type Clément interpolation
Let be a patch on a vertex and be the projection of on , i.e., such that
Similarly, we can define an projection on a patch for an edge .
For , the lowestorder interpolation can be rewritten as
where
and the functions , , , and are the corresponding Lagrange basis functions. Now we define a new Clément interpolation for :
where and . The interpolation is welldefined and the following error estimate holds.
Theorem 2.2.
For any , let . Then, for , it holds that
(2.15) 
The theorem can be obtained using the similar arguments for Theorem 2.1 and the boundedness of the operators .
3. An a priori error estimate for the eigenvalue problem
Following [30], the quadcurl eigenvalue problem is to seek and such that
(3.1) 
where is the unit outward normal to . The assumption that is simplyconnected implies . The variational form of the quadcurl eigenvalue problem is to find and such that
(3.2) 
In addition to defined in Section 3, we need more discrete spaces. Define
The discrete problem for (3.2) is to find and such that
(3.3) 
3.1. The source problem
We start with the associated source problem. Given , find and such that
(3.4) 
Note that for .
The weak formulation is to find such that
(3.5) 
where
The wellposedness of (3.5) is proved in Thm. 1.3.2 of [28]. Consequently, we can define an solution operator such that . In fact, is compact due to the following result.
Lemma 3.1.
processes the continuous compactness property.
Proof.
Since [22], then .
The conforming FEM seeks and such that
(3.6) 
The wellposedness of problems (3.6) is due to the discrete compactness of with , which is stated in the following theorem. Its proof is similar to that of Theorem 7.17 in [22] and thus is omitted.
Theorem 3.1.
processes the discrete compactness property.
Consequently, we can define a discrete solution operator such that is the solution of (3.6). It is straightforward to use the standard finite element framework and the approximation property of the interpolation to show the following theorem.
Theorem 3.2.
Assume that , , and . It holds that
3.2. An a priori error estimate of the eigenvalue problem
We first rewrite the eigenvalue problem as follows. Find and such that
(3.7) 
Due to the fact that , we have . Then (3.7) can be written as an operator eigenvalue problem of finding and such that
(3.8) 
The discrete eigenvalue problem is to find and such that
(3.9) 
Using the operator , the eigenvalue problem is to find and such that
(3.10) 
where .
(1) is collectively compact, and
(2) is pointwise convergent, i.e., for strongly in as .
Theorem 3.3.
Let be an eigenvalue of with multiplicity and
be the associated eigenspace. Let
be an orthonormal basis for . Assume that for . Then, for small enough, there exist exactly discrete eigenvaluesand the associated eigenfunctions
of such that(3.11)  
(3.12) 
4. A posteriori error estimates for the eigenvalue problem
Assume that is a simple eigenpair of (3.2) with and is the associated finite element eigenpair of (3.3) with . According to Theorem 3.3 and [2, (3.28a)], the following inequalities hold:
(4.1)  
(4.2) 
where and
It is obvious that as .
Define two projection operators as follows. For and , find , such that
According to the orthogonality and the uniqueness of the discrete eigenvalue problem,
Let be the solution of (3.5) with . Then
(4.3) 
The following theorem relates the eigenvalue problem to a source problem with .
Theorem 4.1.
Let . It holds that
(4.4) 
Furthermore, for small enough, there exist two constants and such that
(4.5) 
Proof.
Since , by the triangle inequality, we have that
Using and (4.3), we obtain that
(4.6) 
Due to the wellposedness of (3.5), it holds that
which, together with (4.1) and (4.2), leads to
(4.7) 
Then (4.4) follows immediately. Note that as . For small enough, (4.4) implies (4.5).
We first derive an a posteriori error estimate when (a) or (b) is a vector polynomial for which , . Note that for (a) and for (b). Hence holds for both cases.
Denote the total errors by . Then satisfy the defect equations
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