Compact Equivalent Inverse of the Electric Field Integral Operator on Screens

We construct inverses of the variational electric field boundary integral operator up to compact perturbations on orientable topologically simple screens. We describe them as solution operators of variational problems set in low-regularity standard trace spaces. On flat disks these variational problems do not involve the inversion of any non-local operators. This result lays the foundation for operator preconditioning for the discretized electric field integral equation.


"Simple" Screens
A simple screen in the sense of this article is a compact orientable twodimensional manifold Γ ⊂ R 3 with boundary ∂Γ, which is the image of the unit disk D := {x ∈ R 3 : x 3 = 0 and x < 1} under a bi-Lipschitz mapping. In particular, Γ need not be smooth; shapes with corners and kinks are admitted. Nevertheless, Γ has a tangent plane and an unit normal vector n almost everywhere. We point out that simple screens are a special case of the Lipschitz screens considered in [7], and, of course, of the even more general class of screens introduced in [14].

Motivation and Objectives
In this paper, we pursue the construction of bounded linear operators which provide compact-equivalent inverses of the EFIE operator on simple screens in the sense that where A k : H −1/2 (div Γ , Γ) → ( H −1/2 (div Γ , Γ)) is the EFIE operator induced by the bilinear form a k , and C k : H −1/2 (div Γ , Γ) → H −1/2 (div Γ , Γ) is a compact operator.
In addition, we demand that the evaluation of N k g for any g ∈ ( H −1/2 (div Γ , Γ)) (A) does not entail solving any integral equation, but merely the evaluation of integral operators on Γ, and (B) entirely relies on solving variational equations in low-regularity trace spaces.
Remark 1.2. The rationale behind (A) and (B) above is the use of (1.2) as basis for operator preconditioning of the linear systems of equations arising from low-order boundary element discretization of (1.1). This approach, harnessing the Calderón identity (1.3), has been successfully applied on closed surfaces [3] and scalar boundary integral equations on screens, and yields methods that are robust with respect to mesh refinement.

Related Work, Novelty and Outline
Our main new contribution is the explicit construction of a suitable operator N k complying with (1.2) and (A) and (B) under the assumption that (compact-equivalent) inverses of the single-layer and hypersingular boundary integral operators (BIOs) on Γ for the Laplacian −Δ are available in the form of concrete BIOs. In [17] we verified this assumption for the disk D. Thus, for this particular simple screen we have fully achieved the goals advertised above, but we hope that such inverses will be discovered for more general shapes in the future.
Therefore, we have decided to elaborate the construction of N k in Sect. 3 for general simple screens. The key tool is the Hodge decomposition of the trace space H −1/2 (div Γ , Γ), which we recall in Sect. 2.2. The proper realization of N k through variational equations is presented in Sect. 4.
Another important feature of the operator N k is uniform stability in the low-frequency limit k → 0, as will be shown in Sect. 3.
The idea to tackle the EFIE by means of Hodge decompositions is well established, see [12,Sect. 6] also for screen problems [4,Sect. 3]. In these works, it was used as an analysis tool. In other works, most prominently [10] and [16], the Hodge decomposition served to convert the EFIE into boundary equations for scalar traces. Our policy for constructing N k also draws on this trick. Similar ideas, though in a BEM setting, have recently been proposed for the construction of preconditioners in [1].

Trace Operators and Trace Spaces
From [19,Ch. 3] we adopt standard notations and definitions for Sobolev spaces H s (Γ) and H s (Γ), −1 ≤ s ≤ 1, on the simple screen Γ. Bold font will mark corresponding Sobolev spaces H s (Γ) and H s (Γ) of vector fields on Γ.
We point out that in the case of screens the vector Sobolev spaces satisfy duality relations analogous to the scalar case, i.e.
with L 2 (Γ) as pivot space. The variational EFIE (1.1) is set in a jump trace space for H(curl, R 3 \ Γ). Theoretical investigations of these traces spaces started with [8] and [9] and were further developed in [11] and, for screens, in [7, Sect. 2] and [14]. For a very brief review, let us introduce the space of tangential square-integrable vector fields on the simple screen Γ L 2 t (Γ) := {u ∈ L 2 (Γ) | u · n = 0 a.e. on Γ}, (2.2) endowed with the L 2 -inner product. We define the tangential trace γ t as the operator that suitably extends We will make use of the following tangential trace space together with its dual space (relying on L 2 t (Γ) as pivot space) ) . Next, we recall the space of div Γ -conforming tangential surface vector fields with vanishing in-Γ normal component on ∂Γdefined in [7, Sect. 2,Def. 1] (there denoted as X) and its dual space (with respect to In addition, we define the spaces that are dual to each other.

Proposition 2.1. The following duality relation holds
Proof. For any v ∈ H 1/2 (Γ), let us consider the linear map v → Γ vdS and define which is a subset of H −1/2 (Γ). We note that H Alternatively, one may arrive to (2.10) by using the definition of H As homeomorphic image of the disk D the screen Γ is connected and has trivial co-homology; it has no holes. As a consequence we have the following result about surface differential operators and related spaces.
Theorem 2.2. The surface differential operators curl Γ and div Γ generate the following deRham exact sequence of Hilbert spaces: Proof. This theorem is the essence of results for polyhedral surfaces from [9,Sect. 6], in particular [9, Proposition 4.7] and [9, Theorem 6.1]. Alternatively, one can pull back everything to the unit disk D and there use the smoothed Poincaré lifting invented in [15].
(2.12) In addition, we learn that the surface divergence operator div Γ : is continuous and surjective.

Hodge Decomposition
with closed subspaces Thanks to the trivial topology of Γ, the exact sequence of Theorem 2.2 guarantees the existence of scalar potentials Since the mapping curl Γ : H 1/2 (Γ) → X z (Γ) is bijective, we can view this as a parameterization of X z (Γ) over H 1/2 (Γ). In order to find a parameterization of X ⊥ (Γ), based on Theorem 2.2 let us introduce a divergence lifting L : H −1/2 * (Γ) → X ⊥ (Γ) as a right inverse of div Γ in the sense that div Γ •L = Id, through By means of the lifting operator L, we find the following representation

Compact-Equivalent Inverses
As explained in Sect. 1.3, we aim to find an operator N k such that with a compact operator C k that may also depend on the wave number k. We begin by considering the scaled Hodge decompositions ξ = ξ z +k ξ ⊥ and η = η z + k η ⊥ with (ξ z , ξ ⊥ ), (η z , η ⊥ ) ∈ X z (Γ) × X ⊥ (Γ), and plug it into the EFIE variational problem We split the terms and get where the three "cross-terms" in braces are compact due to Lemma 2.4 and behave like O(k) when k → 0. Now, let us recall the following result from literature.  Then, we exploit the fact that V k − V 0 is compact and rewrite (3.2) as where the operator C k contains all the compact terms from (3.2) plus some containing V k − V 0 and V k − V 0 . We see that the final expression involves only two terms that are not compact and that they only act in either X z (Γ) or X ⊥ (Γ). This motivates that we define the operators S z : X z (Γ) → (X z (Γ)) and S ⊥ : X ⊥ (Γ) → (X ⊥ (Γ)) induced by them, (3.4) and consider the following variational problem: For g ∈ (H −1/2 (curl Γ , Γ)) , find ξ z ∈ X z (Γ) and ξ ⊥ ∈ X ⊥ (Γ) such that As we want N k to be a compact-equivalent inverse of A k , we point out that it suffices to solve (3.5) and (3.6). Let us denote the associated inverses by N z := S −1 z and N ⊥ := S −1 ⊥ ; their existence will be established below. Then, we define In other words, given g ∈ H −1/2 (curl Γ , Γ), we can compute ξ = N k g = (N z − k 2 N ⊥ )g as follows: (I) To compute N z g we find ξ z ∈ X z (Γ) such that (3.8) Note that unique solvability of (3.8) is ensured by the H −1/2 (Γ)ellipticity of V 0 [20, Theorem 3.5.9]. Equivalently, we can use the scalar potential representation (2.20) of X z (Γ) and solve: Find u ∈ H 1/2 (Γ) such that where curl * Γ : ( H −1/2 (div Γ , Γ) → ( H 1/2 (Γ)) = H −1/2 (Γ). Finally, we conclude that Page 8 of 14 R. Hiptmair, C. Urzúa-Torres IEOT (II) The evaluation of N ⊥ boils down to solving: Find ξ ⊥ ∈ X ⊥ (Γ) such that We again point out that existence and uniqueness of solutions of (3.12) follow by the H −1/2 (Γ)-ellipticity of V 0 and the bijectivity of div Γ : Unfortunately, the space X ⊥ (Γ) is not a low-regularity trace space and thus (3.12) violates (B). Nevertheless, (2.23) permits us to write ξ ⊥ = Lψ, ψ ∈ H −1/2 * (Γ) and recast (3.12) as which reduces to when using div Γ •L = Id and the adjoint operator Theorem 3.2. For any k > 0, the continuous operators

16)
with compact operators C k that are uniformly bounded as k → 0.
Proof. For g ∈ H −1/2 (curl Γ , Γ) and η ∈ H −1/2 (div Γ , Γ) we have where the last two terms are compact due to Lemma 3.1. For short, we gather these terms and write Now, let us plug in N k = N z − k 2 N ⊥ and obtain where we have already used the fact that N z maps to X z (Γ) and that X z (Γ) = ker div Γ in H −1/2 (div Γ , Γ). Then, by also plugging in the Hodge decomposition η = η z + η ⊥ , we arrive to IEOT Compact Equivalent Inverse Page 9 of 14 9 where we have again employed X z (Γ) = ker div Γ . Finally, let us re-order the right hand side and plug in the definition of N ⊥ : With this it becomes clear that the first line gives us the identity due to the definitions of N z and N ⊥ . On the other hand, we have that the expressions on the last line are compact as a consequence of Lemma 2.4. Hence, collecting all compact terms as C k , we find and therefore the desired identity plus a compact operator.
The compact terms in C k are from where it is clear that C k remains bounded for k → 0.

Mixed Variational Formulation for N ⊥
This section is devoted to derive a formulation of N k that complies with (B). Difficulties arise specifically from N ⊥ and we start by briefly discussing why one cannot use straightforward variational formulations.
Remark 4.1. From (3.15), one is tempted to compute N ⊥ g, with g ∈ ( H −1/2 (div Γ , Γ)) , through the four following steps: Nevertheless, this is not possible. Problematic is the right hand side of the variational problem (4.1). It is well-defined only if grad Γ v ∈ H −1/2 (div Γ , Γ), and thus we need v ∈ H(Γ), with H(Γ) as defined in (2.16). However, H(Γ) is not a low-regularity trace space and thus violates (B).