# 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∈H^s+1(Ω). 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.

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## 1. Introduction

The quad-curl equation appears in various applications, such as the inverse electromagnetic scattering theory [10, 11, 30] or magnetohydrodynamics [36]. 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 [33] 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 [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 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

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

### 2.1. Notations

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

 H(curl2;D) :={u∈L2(D):∇×u∈L2(D),(∇×)2u∈L2(D)},

whose norm is given by

 ∥u∥H(curl2;D)=√(u,u)+(∇×u,∇×u)+((∇×)2u,(∇×)2u).

The spaces , , and are defined, respectively, as

 H0(curl2;D):={u∈H(curl2;D):n×u=0and∇×u=0on ∂D}, H10(D):={u∈H1(D):u=0on ∂D}, H(div0;D):={u∈L2(D):∇⋅u=0}.

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 H0(curl2;Ω)

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

 H0(curl2;Ω)=X⊕∇H10(Ω), (2.1)

where Namely, for , with and Furthermore, admits the splitting

 u⊥=∇ϕ+v, (2.2)

where and satisfying

 ∥v∥2≤C∥∇×u⊥∥1. (2.3) ∥∇ϕ∥≤C(∥∇×u⊥∥1+∥u⊥∥). (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, simply-connected open set with . For any , we can extend in the following way:

 ~u ={u⊥,Ω,0,O−¯Ω.

Obviously, and . Now, we consider the following problem: Find defined in such that

 −△ψ =∇×~u, in O, (2.5) ψ =0, on ∂O. (2.6)

Since and has a smooth boundary, there exists a function satisfying (2.5) and (2.6) and

 ∥ψ∥3,O≤C∥∇×~u∥1,O. (2.7)

In addition, (2.5) can be rewritten as Based on [17, Thm. 2.9], there exists a unique function of such that

 −∇×ψ+~u=∇p. (2.8)

Now, we restrict (2.8) to the domain and obtain

 ∇p=−∇×ψ∈H2(O/¯Ω). (2.9)

Using the extension theorem [13], we can extend to defined on satisfying

 ∥~p∥3,O≤C∥p∥3,O/¯Ω≤C∥∇p∥2,O/¯Ω≤C∥∇×ψ∥2,O/¯Ω, (2.10)

where we have used Poincaré-Friedrichs inequality for since we can choose for which Restricting on , we have

 u⊥=∇×ψ+∇~p∈H2(Ω)+∇(p−~p)∈H1(Ω)≜v+∇ϕ.

Note that since is the extension of . Therefore, (2.2) is proved. Combining (2.7) and (2.10), we obtain

 ∥v∥2,Ω=∥∇×ψ+∇~p∥2,Ω≤∥∇×ψ+∇~p∥2,O≤C∥∇×ψ∥2,O≤C∥∇×~u∥1,O=C∥∇×u⊥∥1,Ω

and

 ∥∇ϕ∥Ω=∥u⊥−v∥Ω≤∥u⊥∥Ω+∥v∥Ω≤∥u⊥∥Ω+∥v∥2,Ω≤C(∥u⊥∥Ω+∥∇×u⊥∥1,Ω).

### 2.3. A new family of H(curl2)-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:

 Tis a triangle,PT=Pk(T),ΣT=Mp(u)∪Me(u)∪MK(u),

where

is the set of DOFs (degree of freedom) defined as follows.

• is the set of DOFs on all vertex nodes and edge nodes :

 Mp(u)={∇×u(pi),i=1,2,⋯,3k} (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 :

 Me(u)={∫eiu⋅τeiqds, ∀q∈Pk(ei),i=1,2,3}. (2.12)
• is the set of DOFs on the element :

 MT(u)={∫Tu⋅qdV, ∀q∈D}, (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

 Vh={vh∈H(curl2;Ω): vh|T∈Pk(T) ∀T∈Th}.

Provided and with , define an interpolation , whose restriction on , denoted by , is such that

 Mp(u−ΠTu)=0, Me(u−ΠTu)=0, and MT(u−ΠTu)=0, (2.14)

where , and are the sets of DOFs in (2.11)-(2.13).

###### Theorem 2.1.

If , with , then the following error estimate for the interpolation holds:

 ∥u−Πhu∥T+hT∥∇×(u−Πhu)∥T+h2T∥∥(∇×)2(u−Πhu)∥∥T≤Chs+1∥u∥s+1,T.

Since the proofs of Lemma 2.2 and Theorem 2.1 are similar to those in [33], we omit them.

### 2.4. An H(curl2)-type Clément interpolation

Let be a patch on a vertex and be the projection of on , i.e., such that

 ∫ωv(ϕ−Rkvϕ)pdV=0,∀p∈Pk(ωv).

Similarly, we can define an projection on a patch for an edge .

For , the lowest-order interpolation can be rewritten as

 Πhu=∑v∈Ninthαv(u)ϕv+∑e∈Einth∑i(αcurl,ie(u)ϕcurl,ie+αie(u)ϕie)+∑T∈Th∑iαiT(u)ϕiT,

where

 αv(u) =∇×u(v) for any vertex v, αcurl,ie(u) =∇×u(ve,i) for any node ve,i on an % edge e, αie(u) =∫eu⋅τeqids % for any qi∈P4(e), αiT(u) =∫Tu⋅qidV for any qi∈D,

and the functions , , , and are the corresponding Lagrange basis functions. Now we define a new Clément interpolation for :

 ΠCu=∑v∈Ninth~αv(u)ϕv+∑e∈Einth∑i(~αcurl,ie(u)ϕcurl,ie+αie(u)ϕie)+∑T∈Th∑iαiT(u)ϕiT,

where and . The interpolation is well-defined and the following error estimate holds.

###### Theorem 2.2.

For any , let . Then, for , it holds that

 ∥u−ΠCu∥T+hT∥∇(u−ΠCu)∥T+h2T∥∇(∇×(u−ΠCu))∥T≤Ch2∥u∥2,ω. (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 quad-curl eigenvalue problem is to seek and such that

 (∇×)4u=λu inΩ,∇⋅u=0 inΩ,u×n=0on∂Ω,∇×u=0 on∂Ω, (3.1)

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

 ((∇×)2u,(∇×)2v)=λ(u,v),∀v∈X. (3.2)

In addition to defined in Section 3, we need more discrete spaces. Define

 V0h={vh∈Vh: n×vh=0 and ∇×vh=0 on ∂Ω}, Sh={wh∈H1(Ω): wh|T∈Pk}, S0h={wh∈Wh,wh|∂Ω=0}, Xh={uh∈V0h | (uh,∇qh)=0,  for all  qh∈S0h}.

The discrete problem for (3.2) is to find and such that

 ((∇×)2uh,(∇×)2vh)=λh(uh,vh),∀v∈Xh. (3.3)

### 3.1. The source problem

We start with the associated source problem. Given , find and such that

 (∇×)4u+u+∇p=f inΩ,∇⋅u=0 inΩ,u×n=0on∂Ω,∇×u=0 on∂Ω. (3.4)

Note that for .

The weak formulation is to find such that

 a(u,v)+b(v,p)=(f,v),∀v∈H0(curl2;Ω),b(u,q)=0,∀q∈H10(Ω), (3.5)

where

 a(u,v) =((∇×)2u,(∇×)2v)+(u,v), b(v,p) =(v,∇p).

The well-posedness 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

 a(uh,vh)+b(vh,ph)=(f,vh),∀vh∈V0h,b(uh,qh)=0,∀qh∈S0h. (3.6)

The well-posedness 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

 ∥Af−Ahf∥H(curl2;Ω)≤Chs−1(∥Af∥s+∥∇×Af∥s+∥p∥s).

### 3.2. An a priori error estimate of the eigenvalue problem

We first rewrite the eigenvalue problem as follows. Find and such that

 a(u,v)+b(v,p)=(λ+1)(u,v),∀v∈H0(curl2;Ω),b(u,q)=0,∀q∈H10(Ω). (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

 Au=μu. (3.8)

The discrete eigenvalue problem is to find and such that

 a(uh,vh)+b(vh,ph)=(λh+1)(uh,vh),∀vh∈V0h,b(uh,qh)=0,∀qh∈S0h. (3.9)

Using the operator , the eigenvalue problem is to find and such that

 Ahuh=μhuh, (3.10)

where .

Define a collection of operators,

 A={Ah:L2(Ω)→L2(Ω), h∈Λ}.

Due to Theorem 3.1 and Theorem 3.2,

(1) is collectively compact, and

(2) is point-wise 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 eigenvalues

and the associated eigenfunctions

of such that

 |μ−μj,h| ≤Cmax1≤i≤ma(ϕi−ϕi,h,ϕi−ϕi,h), 1≤j≤m, (3.11) |μ−μj,h| =O(h2(s−1)), 1≤j≤m. (3.12)
###### Proof.

Note that and are self-adjoint. We have that

 ((A−Ah)ϕi,ϕj) =(∇×∇×(A−Ah)ϕi,∇×∇×Aϕj) =(∇×∇×(A−Ah)ϕi,∇×∇×(A−Ah)ϕj).

Due to [22, Thm 2.52], it holds that

 |μ−μj,h|≤C{maxi∥∇×∇×(A−Ah)ϕi∥2+∥(A−Ah)|E(μ)∥2}. (3.13)

Let ,

 ∥(A−Ah)ϕ∥H(curl2;Ω)≤infvh∈Xh∥Aϕ−vh∥H(curl2;Ω)≤Chs−1∥Aϕ∥s+1≤Cμhs−1∥ϕ∥s+1.

 ∥(A−Ah)ϕ∥H(curl2;Ω) ≤infvh∈Xh∥Aϕ−vh∥H(% curl2;Ω)=μinfvh∈Xh∥ϕ−(1/μ)vh∥H(curl2;Ω) ≤μ∥ϕ−ϕh∥H(curl2;Ω)≤μ√a(ϕ−ϕh,ϕ−ϕh).

Since is finite dimensional, we obtain (3.11) and

 ∥(A−Ah)|E(μ)∥H(curl2) ≤Cμhs−1,

which proves (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:

 ∥u−uh∥≤CρΩ(h)|||u−uh|||, (4.1) |λh−λ|≤C|||u−uh|||2, (4.2)

where and

 ρΩ(h)=supf∈L2(Ω),∥f∥=1infv∈V0h∥Af−v∥H(curl2;Ω).

It is obvious that as .

Define two projection operators as follows. For and , find , such that

 a(u−Rhu,vh)+b(vh,p−Qhp) =0,∀vh∈V0h, b(u−Rhu,qh) =0,∀qh∈S0h.

According to the orthogonality and the uniqueness of the discrete eigenvalue problem,

 uh=(λh+1)RhAuh.

Let be the solution of (3.5) with . Then

 ωh=(λh+1)Auhanduh=Rhωh. (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

 ≤|||u−ωh|||+|||ωh−Rhωh|||.

Using and (4.3), we obtain that

 =|||(λ+1)Au−(λh+1)Auh||| (4.6)

Due to the well-posedness 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

 a(e,v)+b(v,ε) =r1(v), ∀v∈