Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the $L^2$-norm and $L^{\infty}$-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).


Introduction
This paper is concerned with the time-domain electromagnetic scattering by a perfectly conducting obstacle which is modeled by the exterior boundary value problem: Here, E and H denote the electric and magnetic fields, respectively, Ω ⊂ R 3 is a bounded Lipschitz domain with boundary Γ and n is the unit outer normal vector to Γ. Throughout this paper, the electric permittivity ε and the magnetic permeability µ are assumed to be positive constants. Equation (1.1e) is the well-known Silver-Müller radiation condition in the time domain withx := x/|x|. Time-domain scattering problems have been widely studied recently due to their capability of capturing wide-band signals and modeling more general materials and nonlinearity, including their mathematical analysis (see, e.g., [1,11,[26][27][28]31,33,39,40] and the references quoted there). The well-posedness and stability of solutions to the problem (1.1a)-(1.1e) have been proved in [16] by employing an exact transparent boundary condition (TBC) on a large sphere. Recently, a spherical PML method has been proposed in [42] to solve the problem (1.1a)-(1.1e) efficiently, based on the real coordinate stretching technique associated with [Re(s)] −1 in the Laplace transform domain with the Laplace transform variable s ∈ C + := {s = s 1 +is 2 ∈ C : s 1 > 0, s 2 ∈ R}, and its exponential convergence has also been established in terms of the thickness and absorbing parameters of the PML layer.
In this paper, we continue our previous study in [42] and propose and study the uniaxial PML method for the problem (1.1a)-(1.1e), based on the real coordinate stretching technique introduced in [42], which uses a cubic domain to define the PML problem and thus is of great advantage over the spherical one in dealing with problems involving anisotropic scatterers. We first establish the existence, uniqueness and stability estimates of the PML problem by the Laplace transform technique and the energy argument and then prove the exponential convergence in both the L 2 -norm and the L ∞ -norm in time of the time-domain uniaxial PML method. Our proof for the L 2 -norm convergence follows naturally from the error estimate between the EtM operators for the original scattering problem and its truncated PML problem established also in the paper, which is different from [42]. The L ∞ -norm convergence is obtained directly from the time-domain variational formulation of the original scattering problem and its truncated PML problem with using special test functions.
The PML method was first introduced in the pioneering work [3] of Bérenger in 1994 for efficiently solving the time-dependent Maxwell's equations. Its idea is to surround the computational domain with a specially designed medium layer of finite thickness in which the scattered waves decay rapidly regardless of the wave incident angle, thereby greatly reducing the computational complexity of the scattering problem. Since then, various PML methods have been developed and studied in the literature (see, e.g., [4, 10, 23-25, 29, 35] and the references quoted there). Convergence analysis of the PML method has also been widely studied for time-harmonic acoustic, electromagnetic, and elastic wave scattering problems. For example, the exponential convergence has been established in terms of the thickness of the PML layer in [2,4,8,13,15,21,30,32] for the circular or spherical PML method and in [5-7, 14, 17, 19, 20] for the uniaxial (or Cartesian) PML method. Among them, the proof in [2] is based on the error estimate between the electric-to-magnetic (EtM) operators for the original electromagnetic scattering problem and its truncated PML problem, while the key ingredient of the proof in [13] and [14] is the decay property of the PML extensions defined by the series solution and the integral representation solution, respectively. On the other hand, there are also several works on convergence analysis of the time-domain PML method for transient scattering problems. For two-dimensional transient acoustic scattering problems, the exponential convergence was proved in [12] for the circular PML method and in [18] for the uniaxial PML method, based on the complex coordinate stretching technique. For the 3D time-domain electromagnetic scattering problem (1.1a)-(1.1e), the spherical PML method was proposed in [42] based on the real coordinate stretching technique associated with [Re(s)] −1 in the Laplace transform domain with the Laplace transform variable s ∈ C + , and its exponential convergence was established by means of the energy argument and the exponential decay estimates of the stretched dyadic Green's function for the Maxwell equations in the free space. In addition, we refer to [1] for the well-posedness and stability estimates of the time-domain PML method for the two-dimensional acoustic-elastic interaction problem, and to [41] for the convergence analysis of the PML method for the fluid-solid interaction problem above an unbounded rough surface.
The remaining part of this paper is as follows. In Section 2, we introduce some basic Sobolev spaces needed in this paper. In Section 3, the well-posedness of the time-domain electromagnetic scattering problem is presented, and some important properties are given for the transparent boundary condition (TBC) in the Cartesian coordinate. In Section 4, we propose the uniaxial PML method in the Cartesian coordinate, study the well-posedness of the truncated PML problem and establish its exponential convergence. Some conclusions are given in Section 5.

Functional spaces
We briefly introduce the Sobolev space H(curl, ·) and its related trace spaces which are used in this paper. For a bounded domain D ⊂ R 3 with Lipschitz continuous boundary Σ, the Sobolev space H(curl, D) is defined by Denote by u Σ = n × (u × n)| Σ the tangential component of u on Σ, where n denotes the unit outward normal vector on Σ. By [9] we have the following bounded and surjective trace operators: γ : H 1 (D) → H 1/2 (Σ), γϕ = ϕ on Σ, where γ t and γ T are known as the tangential trace and tangential components trace operators, and Div and Curl denote the surface divergence and surface scalar curl operators, respectively (for the detailed definition of H −1/2 (Div, Σ) and H −1/2 (Curl, Σ), we refer to [9]). By [9] again we know that H −1/2 (Div, Σ) and H −1/2 (Curl, Σ) form a dual pairing satisfying the integration by parts formula where (·, ·) D and ·, · Σ denote the L 2 -inner product on D and the dual product between H −1/2 (Div, Σ) and H −1/2 (Curl, Σ), respectively. For any S ⊂ Σ, the subspace with zero tangential trace on S is denoted as

The well-posedness of the scattering problem
Let Ω be contained in the interior of the cuboid B 1 := {x = (x 1 , x 2 , x 3 ) ⊤ ∈ R 3 : |x j | < L j /2, j = 1, 2, 3} with boundary Γ 1 = ∂B 1 . Denote by n 1 the unit outward normal to Γ 1 . The computational domain B 1 \Ω is denoted by Ω 1 . In this section, we assume that the current density J is compactly supported in B 1 with and that J is extended so that Define the following time-domain transparent boundary condition (TBC) on Γ 1 : which is essentially an electric-to-magnetic (EtM) Calderón operator. Then the original scattering problem (1.1a)-(1.1e) can be equivalently reduced into the initial boundary value problem in a bounded domain Ω 1 × (0, T ): The well-posedness of the original scattering problem (1.1a)-(1.1e) has been established in [16] by using the transparent boundary condition on a sphere. Thus the problem (3.6) is also well-posed since it is equivalent to the problem (1.1a)-(1.1e). However, for convenience of the subsequent use in the following sections, we study the problem (3.6) directly by studying the property of the EtM operator T . For any s ∈ C + := {s = s 1 + is 2 ∈ C : be the Laplace transform of E and H with respect to time t, respectively (for extensive studies on the Laplace transform, the reader is referred to [22]). Let B : whereĚ andȞ satisfy the exterior Maxwell's equation in the Laplace domain     as |x| → ∞.
It is obvious that T = L −1 • B • L . For each s ∈ C + it is known that, by the Lax-Milgram theorem the problem (3.8) has a unique solution (Ě,Ȟ) ∈ H(curl, R 3 \B 1 ) . Thus the operator B is a well-defined, continuous linear operator.
where L(X, Y ) denotes the standard space of bounded linear operators from the Hilbert space X to the Hilbert space Y . Further, we have where · Γ 1 denotes the dual product between H −1/2 (Div, Γ 1 ) and H −1/2 (Curl, Γ 1 ).
Proof. First, eliminatingȞ from (3.8) and multiplying both sides of the resulting equation with (3.8) and integrating by parts the resulting equation multiplied withĚ over B R \B 1 , we obtain that Taking the real part of (3.11) and noting that (3.12) By the Silver-Müller radiation condition (1.1e) in the s-domain, it is known that the right-hand side of (3.12) tends to zero as R → ∞. This implies that Re Bω, ω Γ 1 ≥ 0. The proof is thus complete.
By using Lemma 3.1 and [40, Lemmas 4.5-4.6], the time-domain EtM operator T has the following positive properties which will be used in the error analysis of the time-domain PML solution.
Lemma 3.2. Given ξ ≥ 0 and ω(·, t) ∈ L 2 (0, ξ; H −1/2 (Curl, Γ 1 )) it holds that We now introduce the equivalent variational formulation in the Laplace transform domain to the problem (3.6). To this end, eliminate the magnetic field H and take the Laplace transform of (3.6) to get The variational formulation of (3.13) is then as follows: find a solutionĚ ∈ H Γ (curl, where the sesquilinear form a(·, ·) is defined as By Lemma 3.1 it is easy to see that a(·, ·) is uniformly coercive, that is, Then, by the Lax-Milgram theorem the problem (3.13) is well-posed for each s ∈ C + . Thus, and by the energy argument in conjunction with the inversion theorem of the Laplace transform (cf. [16]) the well-posedness of the problem (3.6) follows. In particular,

The uniaxial PML method
In practical applications, the scattering problems may involve anisotropic scatterers. In this case, the uniaxial PML method has a big advantage over the circular or spherical PML method as it provides greater flexibility and efficiency in solving such problems. Thus, in this section, we propose and study the uniaxial PML method for solving the time-domain electromagnetic scattering problem (1.1a)-(1.1e).

The PML equation in the Cartesian coordinates
In this subsection, we derive the PML equation in the Cartesian coordinates. To this end, define Let Ω PML = B 2 \B 1 be the PML layer and let Ω 2 = B 2 \Ω be the truncated PML domain. See Figure 1 for the uniaxial PML geometry.
, let s 1 > 0 be an arbitrarily fixed parameter and let us define the PML medium property as with positive constants σ j , j = 1, 2, 3, and integer m ≥ 1. In what follows, we will take the real part of the Laplace transform variable s ∈ C + to be s 1 , that is, Re(s) = s 1 .
In the rest of this paper, we always make the following assumptions on the thickness of the PML layer and the parameters σ j , which are reasonable in our model: for a fixed generic constant C 0 . Under the assumptions (4.18) and (4.19) we have We remark that the constant assumption on d j and σ j in (4.18)-(4.19) is only to simplify the convergence analysis but not mandatory. We now introduce the real stretched Cartesian Noting that the solution of the exterior problem (3.8) in R 3 \B 1 can be derived as the integral representation [34, Theorem 12.2], we can derive the PML extension under the stretched coordinates x by following [42]. For any p ∈ H −1/2 (Div, Γ 1 ) and q ∈ H −1/2 (Div, Γ 1 ), define where the stretched single-and double-layer potentials are defined as Here, the stretched dyadic Green's function is given by with the stretched fundamental solution and the complex distance Introduce the stretched curl operator acting on vector u = (u 1 , u 2 , u 3 ) ⊤ : The PML extension in the s-domain in R 3 \B 1 of γ t (Ě)| Γ 1 and γ t (curlĚ)| Γ 1 is then defined aš Then it is easy to see that (ˇ E,ˇ H) satisfies the Maxwell equation in the s-domain: Then (E PML , H PML ) can be viewed as the extension in the region R 3 \B 1 of the solution of the problem (1.1a)-(1.1e) since, by the fact that α j = 1 on (4.28) The truncated PML problem in the time domain is to find (E p , H p ), which is an approximation (4.29)

Well-posedness of the truncated PML problem
We now study the well-posedness of the truncated PML problem (4.29), employing the Laplace transform technique and a variational method. Eliminate H p and take the Laplace transform of (4.29) to obtain that (4.30) The variational formulation of (4.30) can be derived as follows: find a solutionĚ p ∈ H 0 (curl, Ω 2 ) such that where the sesquilinear form a p (·, ·) is defined as We have the following result on the well-posedness of the variational problem (4.31).
Re a PML (ω, ω) This, together with the definition of ω and the Cauchy-Schwartz inequality, implies that The desired estimate (4.44) then follows from the trace theorem. Now, by using B the truncated PML problem (4.30) for the electric fieldĚ p can be equivalently reduced to the boundary value problem in Ω 1 :

(4.46)
Similarly, for the problem (4.46) we can derive its equivalent variational formulation: findĚ p ∈ H Γ 1 (curl, Ω 1 ) such that where the sesquilinear form a(·, ·) is defined as By using B and the Laplace and inverse Laplace transform, the truncated PML problem (4.29) is equivalent to the initial boundary value problem in Ω 1 :

Exponential convergence of the uniaxial PML method
In this subsection, we prove the exponential convergence of the uniaxial PML method. We begin with the following lemma which is useful in the proof of the exponential decay property of the stretched fundamental solution Φ s (x, y).
Lemma 4.5. Let s = s 1 + is 2 with s 1 > 0, s 2 ∈ R. Then, for any x ∈ Γ 2 and y ∈ Γ 1 the complex distance ρ s defined by (4.24) satisfies Then, by the definition of the complex distance ρ s ( x, y) (see (4.24)) we have where we have used the fact that x jσj (x j )(x j − y j ) ≥ 0 for x ∈ Γ 2 and y ∈ Γ 1 . In addition, and |curl x E(p, q)(x)| (4.51) We now establish the L 2 -norm and L ∞ -norm error estimates in time between solutions to the original scattering problem and the truncated PML problem (4.29) in the computational domain Ω 1 .
We now estimate the norm ( By [34,Theorem 12.2] it is easy to see thatˇ E p has the integral representatioň Defineˇ H p := −(µs) −1 curlˇ E p . Then (ˇ E p ,ˇ H p ) satisfies the stretched Maxwell equations in (4.27) in R 3 \B 1 . It is worth noting thatˇ H p is not the extension ofȞ p | Γ 1 .
Noting that ∇ × v = A∇ × Bv, we know that Bˇ E p satisfies the problem where we have used the fact thatˇ E p is the extension ofĚ p | Γ 1 and B = diag{1, 1, 1} on Γ 1 . By the definition of B, and since A = diag{1, 1, 1} on Γ 1 , it is easy to see that By the definition of B in (4.38), we obtain that where ω satisfies ∇ × (µs) −1 BA∇ × ω + εs(BA) −1 ω = 0 in Ω PML , By Lemma 4.4 and the estimate for BA and (BA) −1 in (4.34)-(4.35), we have Since ∇ × v = A∇ × Bv and |A −1 | ≤ (1 + σ 0 ) 2 in Ω PML , we have by the boundedness of the trace operator γ t that By Lemma 4.6 and the boundedness of γ T and γ t it is derived that This, together with the Parseval identity for the Laplace transform (see [22, (2.46 for all s 1 > λ, where λ is the abscissa of convergence forǔ andv, gives U 2 L 2 (0,T ;H(curl,Ω 1 )) + V 2 L 2 (0,T ;H(curl,Ω 1 )) where we have used the assumptions (3.3) and (3.4) to get the last inequality. It is obvious that m should be chosen small enough to ensure rapid convergence (thus we need to take m = 1). Since s −1 1 = T in (4.62), we obtain the required estimate (4.52) by using the Cauchy-Schwartz inequality.
We now prove (4.53). Since (E, H) and (E p , H p ) satisfy the equations (3.6) and (4.49), respectively, it is easy to verify that (U , V ) satisfies the problem where C = L −1 • sB • L . The variational problem of (4.64) is to find U ∈ H Γ (curl, Ω 1 ) for all t > 0 such that For 0 < ξ < T , introduce the auxiliary function Then it is easy to verify that For any φ(x, t) ∈ L 2 0, ξ; L 2 (Ω 1 ) 3 , using integration by parts and condition (4.66), we have Taking the test function ω = Ψ 1 in (4.65) and using (4.66) give By (4.67) we have the estimate which implies that Integrating (4.65) from t = 0 to t = ξ and taking the real parts yield First, using (4.67) and Lemma 3.2, we have Then, and by (4.67) we deduce the estimate where we have used the trace theorem to get the last inequality. The right-hand of (4.72) contains the term which cannot be controlled by the left-hand of (4.72). To address this issue, we consider the new problem which is obtained by differentiating each equation of (4.64) with respect to t. By a similar argument as in deriving (4.65), we obtain the variational formulation of (4.73): find u such that for all ω ∈ H Γ (curl, Ω 1 ), Define the auxiliary function Similarly as in the derivation of (4.68)-(4.69), we conclude by integration by parts that Choosing the test function ω = Ψ 2 in (4.74), integrating the resulting equation with respect to t from t = 0 to t = ξ and taking the real parts yield Similarly to (4.71), it follows from (4.67) and Lemma 3.3 that Thus, and by (4.77) we have Combining (4.72) and (4.78) gives Taking the L ∞ -norm of both sides of (4.79) with respect to ξ and using the Young inequality yield L 1 (0,T ;H −1/2 (Div,Γ 1 )) , which, together with the Cauchy-Schwartz inequality, implies that U L ∞ (0,T ;L 2 (Ω 1 ) 3 ) + ∂ t U L ∞ (0,T ;L 2 (Ω 1 ) 3 ) + ∇ × U L ∞ (0,T ;L 2 (Ω 1 ) 3 ) (4.80) T 3/2 (T − T )[∂ t E p Γ 1 ] L 2 (0,T ;H −1/2 (Div,Γ 1 )) + T 1/2 (T − T )[∂ 2 t E p Γ 1 ] L 2 (0,T ;H −1/2 (Div,Γ 1 )) .
We now only need to estimate the right-hand term of (4.80). By (4.57) and the definition ofT (see (4.49)) we know that (T − T )[∂ t E p From this, the definition of U and Maxwell's system (4.63) the required estimate (4.53) then follows. The proof is thus complete.
Remark 4.8. The L 2 -norm error estimate (4.52) can also be obtained by integrating (4.79) with respect to ξ from 0 to T . The idea of using the uniform coercivity of the variational form in our proof of the L 2 -norm error estimate (4.52) is also known for the time-harmonic PML method. This builds a connection between our proposed time-domain PML method with the real coordinate stretching technique and the time-harmonic PML method in some sense.

Conclusions
In this paper, by using the real coordinate stretching technique we proposed a uniaxial PML method in the Cartesian coordinates for 3D time-domain electromagnetic scattering problems, which is of advantage over the spherical one in dealing with scattering problems involving anisotropic scatterers. The well-posedness and stability estimates of the truncated uniaxial PML problem in the time domain were established by employing the Laplace transform technique and the energy argument. The exponential convergence of the uniaxial PML method was also proved in terms of the thickness and absorbing parameters of the PML layer, based on the error estimate between the EtM operators for the original scattering problem and the truncated PML problem established in this paper via the decay estimate of the dyadic Green's function. Our method can be extended to other electromagnetic scattering problems such as scattering by inhomogeneous media or bounded elastic bodies as well as scattering in a two-layered medium. It is also interesting to study the spherical and Cartesian PML methods for time-domain elastic scattering problems, which is more challenging due to the existence of shear and compressional waves with different wave speeds. We hope to report such results in the near future.