The quad-curl equation appears in various applications, such as the inverse electromagnetic scattering theory [10, 11, 30] or magnetohydrodynamics . The corresponding quad-curl 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 quad-curl 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  for the qual-curl 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 non-convex. 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 quad-curl eigenvalue problem. The problem was first proposed in  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 . 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 . 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 quad-curl 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  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 quad-curl 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
Let be a simply-connected 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.
Let be the set of gradients of functions in . Then is a closed subspace of and
where Namely, for , with and Furthermore, admits the splitting
where and satisfying
Obviously, and . Now, we consider the following problem: Find defined in such that
Now, we restrict (2.8) to the domain and obtain
Using the extension theorem , we can extend to defined on satisfying
where we have used Poincaré-Friedrichs inequality for since we can choose for which Restricting on , we have
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  when the solution is smooth enough.
For an integer , an -conforming element is given by the triple:
is the set of DOFs (degree of freedom) defined as follows.
is the set of DOFs on all vertex nodes and edge nodes :
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 :
is the set of DOFs on the element :
where when and when . Here is the space of a homogeneous polynomial of degree .
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
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 lowest-order interpolation can be rewritten as
and the functions , , , and are the corresponding Lagrange basis functions. Now we define a new Clément interpolation for :
where and . The interpolation is well-defined and the following error estimate holds.
For any , let . Then, for , it holds that
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 , the quad-curl eigenvalue problem is to seek and such that
where is the unit outward normal to . The assumption that is simply-connected implies . The variational form of the quad-curl eigenvalue problem is to find and such that
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
3.1. The source problem
We start with the associated source problem. Given , find and such that
Note that for .
The weak formulation is to find such that
processes the continuous compactness property.
Since , then .
The -conforming FEM seeks and such that
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.
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
Due to the fact that , we have . Then (3.7) can be written as an operator eigenvalue problem of finding and such that
The discrete eigenvalue problem is to find and such that
Using the operator , the eigenvalue problem is to find and such that
(1) is collectively compact, and
(2) is point-wise convergent, i.e., for strongly in as .
4. A posteriori error estimates for the eigenvalue problem
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
The following theorem relates the eigenvalue problem to a source problem with .
Let . It holds that
Furthermore, for small enough, there exist two constants and such that
Since , by the triangle inequality, we have that
Using and (4.3), we obtain that
Due to the well-posedness of (3.5), it holds that
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