CONVERGENCE ANALYSIS OF TWO FINITE ELEMENT METHODS FOR THE MODIFIED MAXWELL’S STEKLOV EIGENVALUE PROBLEM

. The modified Maxwell’s Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.


Introduction
The Steklov eigenvalue problem is governed by the elliptic equation with the eigen-parameter in the boundary condition.It has many applications in physics, e.g., surface waves [4] and stability of mechanical oscillators immersed in a viscous fluid [13].Various numerical methods for the Steklov eigenvalue problem have been developed and analyzed [1,6,19,22,24,26].Recently, a new application was considered in [10] using the Steklov eigenvalues as a target signature in nondestructive testing (see e.g., [12,17] for different choices of target signatures).The associated non-selfadjoint Steklov eigenvalue problem for inhomogeneous absorbing media has drawn significant attention in the numerical analysis community [20,25,27].
Most earlier papers on Steklov eigenvalues focused on the Laplace or the Helmholtz equation.For the Maxwell's equation, the so-called modified Steklov eigenvalues was studied in [11] for an electromagnetic inverse scattering problem.In the same paper a finite element method was proposed for computing the eigenvalues.The term "modified" refers to the insertion of a boundary-to-boundary operator  into the standard Steklov eigenvalue problem.Through this modification the authors showed the compactness of the corresponding solution operator and the existence of the eigenvalues.For numerical analysis about this eigenvalue problem, to our knowledge, there exist only two papers [15,16].Halla [16] provided a general framework considering the original and the modified Maxwell's Steklov eigenvalue problem, which guaranteed that the Galerkin approximation is convergent as long as certain commuting projection operator exists.While in [15], a specific finite element method was considered and a convergence order of the corresponding discrete eigenvalues were obtained.However, neither of these two results cover the method proposed in [11].
The difference between the two finite element methods, the one used in [11] and the one considered in [15], is how  is discretized.The boundary-to-boundary operator  projects vectors into the surface-divergence-free B. GONG space.It has two equivalent representations on the continuous level, but the corresponding finite element discretizations differ substantially.We denote by  ℎ the discretization used in [11], and by ̃︀  ℎ the one in [15].What is crucial of ̃︀  ℎ is that it maintains the property of  to map vectors into the surface-divergence-free space.In contrast,  ℎ does not, i.e., it has a range not surface-divergence-free.Despite of this difference between ̃︀  ℎ and  ℎ , they display similar numerical behaviour (see [15]).Both the eigenvalues  ℎ and ̃︀  ℎ computed by the method using  ℎ and ̃︀  ℎ respectively, converge to the exact values.Moreover,  ℎ seems to be more accurate than ̃︀  ℎ .This motivates us to analyze the finite element method that uses  ℎ , which is the main goal of the current paper.
As mentioned above, the major difficulty in the analysis lies in the fact that the range of  ℎ is not surfacedivergence-free.To this end, we define a solution operator slightly different than that from [11] so that its domain and range are the  2 space other than the surface-divergence-free space.Both finite element methods (with ̃︀  ℎ and  ℎ ) under this  2 -to- 2 framework are analyzed.The main tools used here are the Helmholtz decomposition and the Babuška-Osborn theory for eigenvalue problems [3] (see [5,23] for some recent developments).In addition, we prove under some conditions the monotonic convergence of ̃︀  ℎ and an inequality between  ℎ and The rest of the paper is arranged as follows.In Section 2, we introduce the notations and present some useful estimations and identities.Sections 3 and 4 contain the error analysis for the finite element methods using ̃︀  ℎ and  ℎ , respectively.Convergence in norm of the discrete solution operators and the convergence order of the associated eigenvalues are obtained.In Section 5, we prove some other properties for the discrete eigenvalues.

Preliminary
Let Ω ⊂ R 3 be a simply connected bounded Lipschitz polyhedron with a connected boundary Γ.Let  be the unit outward normal to Γ. Denote by   (Ω) and   (Γ) the standard Sobolev spaces for  ∈ R and  ∈ [−1, 1], respectively.Define We denote the norm of   (Ω),  2  (Γ) and   (Γ) as (Γ) be the trace operators that maps  to  × | Γ and ( × | Γ ) × , respectively.The operators   and   can be continuously extended to (curl; Ω).Denote by   =    the tangential component of  on the boundary.
Let ∇ Γ and curl Γ denote, respectively, the surface gradient and surface vector curl, which can be defined on  1/2 (Γ).The surface divergence and surface scalar curl, denoted by div Γ and curl Γ , are respectively the duals of −∇ Γ and curl Γ , i.e.,

⟨𝜑, ∇
Define the surface-divergence-free space as For more details on these operators and spaces, we refer the readers to [7,9].We also use ⟨•, •⟩ to denote certain duality of spaces on the boundary.There are two equivalent ways to define the surface-divergence-free projection operator .One is The above definition can be applied on, say,  ∈   (curl; Ω) or  ∈  2  (Γ).The other is  =  + ∇ Γ  with  ∈  1 (Γ)/R being the solution of which can be applied on, say,  ∈   (curl; Ω) or  ∈  2  (Γ).Let  ℎ be a regular tetrahedral mesh for polyhedron Ω with size ℎ.The faces of  ℎ on Γ induce a triangular mesh for Γ.We use the notations in Chapter 5 of [21] to denote by  ℎ ⊂ (div, Ω) the divergence-conforming finite element space of degree , by  ℎ ⊂ (curl; Ω) the curl-conforming finite element space of degree , and by  ℎ ⊂  1 (Ω) the Lagrange element space of degree .In addition, denote by  Γ ℎ ⊂  1 (Γ) the Lagrange element space of degree  on the boundary.We shall mainly discuss the case when  = 1.
Based on the definitions of  by (2.2) and (2.3), the two finite element approximations ̃︀  ℎ and  ℎ are defined as follows.̃︀ And
From now on we use  to represent some positive number which is less than  Ω .
To obtain regularity of the solution of (2.2) and (2.3), we shall apply the following result.
A direct use of the definitions of , ̃︀  ℎ and  ℎ gives one the following.
Lemma 2.10.The operators , ̃︀  ℎ and  ℎ are linear and bounded from  2  (Γ) to  2  (Γ).Moreover, they are orthogonal projection operators under the  2  (Γ) inner product, and given  ,  ∈  2  (Γ), it holds that which implies that on  2  (Γ) we have Then by the definition of  and  we have the following Hence we have the first line of equalities verified, by which we can show that which means  ℎ =  on  2  (Γ).The rest can be proved similarly.
In [11] the solution operator of (2.1) is defined from  (︀ div 0 Γ ; Γ )︀ to itself by   ℒ, which is inconvenient for the current paper.Instead, we define for (2.1) another solution operator  such that We show in the following that  is compact and indeed represents the spectrum of (2.1).
Lemma 2.11. is compact and self-adjoint.There is a bijection between the eigenpairs of  and those of (2.1) for nonzero eigenvalues. Proof.
. Then due to the wellposedness of ℒ, it holds  := ℒ ∈ .Using the remark of [14], we have that curl Γ   ∈  2 (Γ) and (2.13) Therefore, the regularity result of Lemma 2.9 indicates that for the solution Hence  is self-adjoint.Note that we have used the assumption that   is real.

Finite element method using ̃︀ 𝒮 ℎ
For the completeness of the theory, also as a preparation for the next section, we first analyze the finite element method that uses ̃︀  ℎ as the discretization (see (2.5)).Instead of defining the corresponding solution operator ̃︀  ℎ (as is done in [15]) from , we shall define it from  2  (Γ) to  2  (Γ).One consequence is that, the operator ̃︀  ℎ thus defined is self-adjoint on  2  (Γ), while ̃︀  ℎ defined in [15] is not on Let the solution operator of (3.1) be defined as That ̃︀  ℎ is self-adjoint and the correspondence of the eigenpairs between ̃︀  ℎ and (3.1) can be shown in the same way as in Lemma 2.11.
Next we prove the convergence of ̃︀  ℎ to  .

Finite element method using 𝒮 ℎ
Now consider the finite element eigenvalue problem of finding ( ℎ ,  ℎ ) ∈ R ×  ℎ such that where  ℎ is defined in (2.6).We start with the well-posedness of the corresponding source problem: for  ∈  2  (Γ), find  ℎ ∈  ℎ such that In other words, to find  ℎ = ℒ ℎ  ℎ  .Formally, we characterize  ℎ as the solution of the operator equation Using the discrete decomposition In what follows, we deal with the existence and (uniform) continuous dependence on  ∈  2  (Γ) of  ℎ .In (4.3), the uniform boundedness of ( +  ℎ ) −1 from  2 (Ω) to  2 (Ω) is guaranteed by the collective compactness of { ℎ } ℎ .Consequently, the operator ( +  ℎ ) −1 =  −  ℎ ( +  ℎ ) −1 confined on  ℎ is also uniformly bounded from  ℎ to  ℎ .Hence it requires the uniform boundedness of ℒ + ℎ  ℎ on  2  (Γ) to show the well-posedness of  ℎ .However, what's in hand now is only the well-posedness of ℒ + ℎ on  Proof.Note that On one hand, by Lemma 2.3, we have that On the other hand, using the identity where  is a face of an element on the boundary.Let  be the element containing  .Noting that  ℎ,0 ∈  1/2+ (Ω) and curl  ℎ,0 ∈  ℎ , we map )︀  ℎ,0 from the element  to the reference element K then map back to see that Altogether we obtain that ) Using the coercivity and boundedness of  + (•, •), we obtain the well-posedness of Due to Lemmas 4.1 and 2.7, (4.3) and (4.2) are well-posed.Moreover, and ‖ℒ ℎ  ℎ ‖ .The proof is complete.
We now can define the solution operator  ℎ :  2  (Γ) →  2  (Γ) for (4.1) by Lemma 4.3. ℎ are uniformly bounded and self-adjoint.There is a bijection between the eigenpairs of (4.1) and those of  ℎ for nonzero eigenvalues.
Proof.Due to Lemma 4.2,  ℎ are uniformly bounded with The self-adjoint property and the correspondence between the eigenpairs of  ℎ and (4.1) can be shown the same way as in Lemma 2.11.
The proof of  ℎ converging to  and  ℎ converging to  follows similar steps in that of Theorems 3.2 and 3.3, respectively.However, there are several technical differences.
Let   ,  ℎ, , ̃︀  ℎ, and  0 ℎ, be the -th eigenvalues in descending order of ( the set of all subspaces of  ℎ with dimension .Then, for   we have the min-max property Similarly, the min-max property holds for

Conclusion
In this work, we propose new definitions of the solution operator  and two finite element approximations ︀  ℎ and  ℎ for the modified Maxwell's Steklov eigenvalue problem.With the help of these operators, we are able to prove the convergence order of eigenvalues for both finite element methods.Moreover, the ordering between the eigenvalues  ℎ and ̃︀  ℎ , and the monotonic converging property of ̃︀  ℎ are proved.Besides, it is observed in numerical results of [15] that  ℎ has the same monotonic converging property as ̃︀  ℎ .This property of  ℎ is interesting to investigate, since, if  ℎ truly converge from below, we can claim using Lemma 5.1 that the finite element method (4.1) is a better choice than (3.1).
Here  is the wavenumber which is real and positive and   is the relative permittivity.Assume that the media is isotropic and dielectric, i.e.,   is a real scalar function.In addition, we require that   is smooth, bounded and away from zero.More precisely, there exist constants  > 0 and  > 0 such that   ∈  1 (Ω) and    .
3.1.̃︀  ℎ are uniformly bounded and self-adjoint.There is a bijection between the eigenpairs of (3.1) and those of ̃︀  ℎ for nonzero eigenvalues.