A priori and a posteriori error estimates for the quad-curl eigenvalue problem

In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a priori estimate of the eigenvalue with a convergence order 2(s-1) is obtained if the eigenvector u\in H^{s+1}(\Omega). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the eigenvector in an energy norm and an upper bound for the eigenvalues. Numerical examples are presented for validation.


Introduction
The quad-curl equation appears in various applications, such as the inverse electromagnetic scattering theory [10,11,28] or magnetohydrodynamics [36].The corresponding quad-curl eigenvalue problem plays a fundamental role in the analysis and computation of the electromagnetic interior transmission eigenvalues [22,27].To compute eigenvalues, one usually starts with the corresponding source problem [3,5,30].Some methods have been proposed for the source problem, i.e., the quad-curl problem, in [8,9,12,19,28,29,[31][32][33][34][35][36].Recently, a family of H(curl 2 )conforming finite elements using incomplete k-th order polynomials is proposed in [33] for the qual-curl problem.In this paper, we construct a new family of elements by using the complete k-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 H 2 -regular when Ω is non-convex.Therefore, we propose a new decomposition for H 0 (curl 2 ; Ω), which further splits the irrotational component into a function in H 2 (Ω) 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 [28] 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 [4,14,20,25] for the a posteriori estimates of source problems and [6,7,15] 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 [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 H(curl 2 ) 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.Let u = (u 1 , u 2 ) t and w = (w 1 , w 2 ) t , where the superscript t denotes the transpose.Then

Notations and basis tools
(∂v/∂x 2 , −∂v/∂x 1 ) t .We now define a space concerning the curl operator The spaces H 0 (curl 2 ; D), H 1 0 (D), and H(div 0 ; D) are defined, respectively, as Let T h be a triangular partition of Ω. Denote by N h and E h the sets of vertices and edges.
Let τ e be the tangent vector of an edge e ∈ E h .We refer to N int h and E int h as the sets of vertices and edges in the interior of Ω, respectively.Let N h (T ) and E h (T ) be the sets of vertices and edges on the element T .Denote by h T the diameter of T ∈ T h and h = max In the following, we introduce some subdomains called patches: • ω T : the union of elements sharing a common edge with T , T ∈ T h ; • ω e : the union of elements sharing e as an edge, e ∈ E h ; • ω v : the union of elements sharing v as a vertex, v ∈ N h .We use P k to represent the space of polynomials on an edge or on a subdomain D ⊂ Ω with degrees at most k and P k (D) = (P k (D)) 2 .

2.3.
A new family of H(curl 2 )-conforming elements.In this subsection, we propose a new family of H(curl 2 )-conforming finite elements.The new elements can lead to one order higher accuracy than the elements in [33] when the solution u is smooth enough.
Definition 2.1.For an integer k ≥ 4, an H(curl 2 )-conforming element is given by the triple: T is a triangle, where Σ T is the set of DOFs (degree of freedom) defined as follows.
• M p (u) is the set of DOFs on all vertex nodes and edge nodes p i : with the points p i chosen at 3 vertex nodes and (k − 1) distinct nodes on each edge.
• M e (u) is the set of DOFs given on all edges e i of T with the unit tangential vector τ e i : M e (u) = e i u • τ e i qds, ∀q ∈ P k (e i ), i = 1, 2, 3 . (2.12) • M T (u) is the set of DOFs on the element T : where x when k ≥ 5 and D = P 0 x ⊕ P 1 x ⊕ P 2 x when k = 4.Here P k is the space of a homogeneous polynomial of degree k.
Using the above Lemma, the global finite element space V h on T h is given by Provided u ∈ H 1/2+δ (Ω) and ∇ × u ∈ H 1+δ (Ω) with δ > 0, define an H(curl 2 ; Ω) interpolation Π h u ∈ V h , whose restriction on T , denoted by Π T u, is such that where M p , M e , and M T are the sets of DOFs in (2.11)-(2.13).
Theorem 2.1.If u ∈ H s+1 (Ω), 1 + δ ≤ s ≤ k with δ > 0, then the following error estimate for the interpolation Π h holds: Since the proofs of Lemma 2.2 and Theorem 2.1 are similar to those in [33], we omit them.
Similarly, we can define an L 2 projection R k e on a patch ω e for an edge e.For u ∈ H 0 (curl 2 ; Ω), the lowest-order H(curl 2 ; Ω) interpolation Π h u can be rewritten as where ) for any node v e,i on an edge e, α i e (u) = e u • τ e q i ds for any q i ∈ P 4 (e), and the functions φ v , φ ve , φ i e , and φ i T are the corresponding Lagrange basis functions.Now we define a new where αv (u ).The interpolation is well-defined and the following error estimate holds.

Theorem 2.2. For any T
The theorem can be obtained using the similar arguments for Theorem 2.1 and the bounded- 3. An a priori error estimate for the eigenvalue problem Following [28], the quad-curl eigenvalue problem is to seek λ and u such that where n is the unit outward normal to ∂Ω.The assumption that Ω is simply-connected implies λ = 0.The variational form of the quad-curl eigenvalue problem is to find λ ∈ R and u ∈ X such that In addition to V h defined in Section 3, we need more discrete spaces.Define 3.1.The source problem.We start with the associated source problem.Given f ∈ L 2 (Ω), find u ∈ H 0 (curl 2 ; Ω) and p ∈ H 1 0 (Ω) such that Note that p = 0 for f ∈ H(div 0 ; Ω).
The weak formulation is to find (u; p) ∈ H 0 (curl where The well-posedness of (3.5) is proved in Thm.1.3.2 of [30].Consequently, we can define an solution operator A : In fact, A is compact due to the following result.
The well-posedness of problems (3.6) is due to the discrete compactness of {X h } h∈Λ with Λ = h n , n = 0, 1, 2, • • •, which is stated in the following theorem.Its proof is similar to that of Theorem 7.17 in [21] and thus is omitted.
Theorem 3.1.X h processes the discrete compactness property.6).It is straightforward to use the standard finite element framework and the approximation property of the interpolation to show the following theorem.

Consequently, we can define a discrete solution operator
2. An a priori error estimate of the eigenvalue problem.We first rewrite the eigenvalue problem as follows.Find λ ∈ R and (u; p) ∈ H 0 (curl Due to the fact that ∇ • u = 0, we have p = 0. Then (3.7) can be written as an operator eigenvalue problem of finding µ := 1/(λ + 1) ∈ R and u ∈ X such that The discrete eigenvalue problem is to find Using the operator A h , the eigenvalue problem is to find µ h ∈ R and u h ∈ X h such that where µ h = 1/(λ h + 1).

Define a collection of operators,
Due to Theorem 3.1 and Theorem 3.2, (1) A is collectively compact, and (2) A is point-wise convergent, i.e., for f ∈ L 2 (Ω), A hn f → Af strongly in L 2 (Ω) as n → ∞.
Theorem 3.3.Let µ be an eigenvalue of A with multiplicity m and E(µ) be the associated eigenspace.Let {φ j } m j=1 be an orthonormal basis for E(µ).Assume that φ ∈ H s+1 (Ω) for φ ∈ E(µ).Then, for h small enough, there exist exactly m discrete eigenvalues µ j,h and the associated eigenfunctions Proof.Note that A and A h are self-adjoint.We have that Due to [21, Thm 2.52], it holds that In addition, we have that Since E(µ) is finite dimensional, we obtain (3.11) and which proves (3.12).
Define two projection operators R h , Q h as follows.For u ∈ H 0 (curl 2 ; Ω) and p ∈ H 1 0 (Ω), find According to the orthogonality and the uniqueness of the discrete eigenvalue problem, Let ω h ; p h be the solution of (3.5) with f = (λ h + 1)u h .Then The following theorem relates the eigenvalue problem to a source problem with f = (λ h + 1)u h .
Furthermore, for h small enough, there exist two constants c and C such that Proof.Since u h = R h ω h , by the triangle inequality, we have that Using u = (λ + 1)Au and (4.3), we obtain that Due to the well-posedness of (3.5), it holds that which, together with (4.1) and (4.2), leads to Then (4.4) follows immediately.Note that r(h) → 0 as h → 0. For h small enough, (4.4) implies (4.5).
We first derive an a posteriori error estimate when (a) f ∈ H(div 0 , Ω) or (b) f is a vector polynomial for which (f , ∇q h ) = 0, ∀q h ∈ S 0 h .Note that p = p h = 0 for (a) and p h = 0 for (b).Hence p h = 0 holds for both cases.
Denote the total errors by e := u − u h and ε := p − p h = p.Then e ∈ H 0 (curl 2 ; Ω) and ε ∈ b(e, q) = r 2 (∇q), ∀q ∈ H 1 0 (Ω), ( where and r 2 (∇q) = −(u h , ∇q).We have the following Galerkin orthogonality The error estimator will be constructed by employing Lemma 2.1.Writing e = e 0 + e ⊥ and v = v 0 + v ⊥ with e 0 , v 0 ∈ ∇H 1 0 (Ω) and e ⊥ , v ⊥ ∈ X, we obtain that (∇×) 2 e ⊥ , (∇×) (e 0 , ∇q) = r 2 (∇q), ∀q ∈ H 1 0 (Ω).(4.14) The estimators for the irrotational part e 0 , the solenoidal part e ⊥ , and ∇ε will be derived separately.Firstly, consider the irrotational part e 0 and ∇ε.For a ϑ ∈ H 1 0 (Ω), we have where the jump with E ∈ E int h the common edge of two adjacent elements T 1 , T 2 ∈ T h and n E the unit normal vector of E directed towards the interior of T 1 .We also have We introduce the error terms which are related to the upper and lower bounds for e 0 and ∇ε: where Next, we consider the bounds for e ⊥ .For w ∈ X, the residual r 1 (w) can be expressed as where [[(∇×) 2 u h × n E ]] E stands for the jump of the tangential component of (∇×) 2 u h and [[(∇×) 3 u h ]] E stands for the jump of (∇×) 3 u h .The bounds for |||e ⊥ ||| contain the error terms , (4.17) where and π h f denotes the L 2 -projection of f onto P k (T ).
Now we state the a posteriori estimate for e and ε in the energy norm.
Proof.Since e = e 0 + e ⊥ , the proof is split into two parts.Note that r 2 (∇q h ) = 0, ∀q h ∈ S 0 h and e 0 = ∇ϕ for some ϕ.Define a projection operator P k h : H 1 0 (Ω) −→ S 0 h such that (see, e.g., [4,24,26]) Due to (4.14) and the orthogonal property (4.11), we have that Using integration by parts, (4.21), and (4.22), we obtain that Therefore, we have Similarly, we can obtain the upper bounds of ∇ε .Due to (4.12) and (4.14) , we have Therefore, we have that

.27)
We now derive the lower bounds of e 0 and ∇ε using the bubble functions.Denote by Using the technique in [1], we have the following norm equivalences. ) Using (4.28), integration by parts, the inverse inequality, and the fact that b which implies that Extend The estimate of the local upper bound for η E 3 can be obtained similarly: where we have used the fact that Consequently, Now collecting (4.25), (4.31), and (4.33), we have that Similarly, ) , we obtain that (ii) Estimation of the solenoidal part e ⊥ .We start with proving the upper bound for η T 1 by using b T again.Employing the similar technique in [1], we have the following estimates for any v in finite dimensional spaces: Setting Due to the inverse inequality and (4.39), it holds that Thus we obtain that Dividing the above inequality by φ h T and multiplying by h 2 T , we obtain Next we estimate the upper bound for η E 1;1 by using the bubble functions b T , b E .Let T 1 and T 2 be two elements sharing the edge E. We extend the jump Similar to (4.38) and (4.39), the following inequalities hold Now we are ready to construct the upper bound for η E 1;1 : By applying the inverse inequality, (4.41), and (4.43), we get which, together with (4.40), leads to The upper bound for η E 1;2 can be constructed in a similar way.Extend [[(∇×) Dividing the above inequality by [[(∇×) 3 u h ]] E E and applying (4.40) and (4.44), we obtain Due to the Galerkin orthogonality (4.10), for any .
Let w h = Π C w.According to the trace inequality and Theorem 2.2, we obtain Furthermore, we use (2.2), (2.3), and the Poincaré inequality to obtain Similar to the proof of (4.27), using (2.4), it holds that Hence, When f = (λ h + 1)u h , according to the definition of η 0 , η 2 , and η 3 , we have that η 0 = λ h η 3 and η 2 = 0.The following error estimator is a direct consequence of Theorem 4.2 and (4.2).
Theorem 4.3.For h small enough, there exist constants c 1 , C 1 , and C 2 such that where η 1 and η 3 are respectively defined in (4.17

Conclusion
A H(curl 2 )-conforming element is proposed for the quad-curl problem in 2D.We construct a priori and robust a posteriori error estimates for the eigenvalue problem.Due to a new decomposition for the solution for the quad-curl problem, the theory assumes no extra regularity of the eigenfunctions.In future, we plan to use the estimator to develop adaptive finite element methods.The 3D counterpart is anther interesting but challenging topic.

2. 1 .
Notations.Let Ω ∈ R 2 be a simply-connected Lipschitz domain.For any subdomain D ⊂ Ω, L 2 (D) denotes the space of square integrable functions on D with norm • D .If s is a positive integer, H s (D) denotes the space of scalar functions in L 2 (D) whose derivatives up to order s are also in L 2 (D).If s = 0, H 0 (D) = L 2 (D).When D = Ω, we omit the subscript Ω in the notations of norms.For vector functions, L 2 (D) = (L 2 (D)) 2 and H s (D) = (H s (D)) 2 .

λ T 3
the barycentric coordinates of T ∈ T h and define the bubble function b T by b

1 and T 2
a common edge of T 1 and T 2 , let ω E = T 1 ∪ T 2 and enumerate the vertices of T such that the vertices of E are numbered first.Define the edge-bubble function b E by b

Figure 5 . 1 .
In Tables 5.1, 5.3, and 5.5, we list the first five eigenvalues.Tables 5.2, 5.4, and 5.6 show the convergence rates of the relative errors for the first eigenvalues, which agree with the theory.

Figure 5 .
2 shows global error estimators and the relative errors of some simple eigenvalues for the three domains.It can be observed that both the relative errors and the estimators have the same convergence rates.Figure5.3 shows the distribution of the local estimators.The estimators are large at corners and catch the singularities effectively.