Time domain analysis for electromagnetic scattering by an elastic obstacle in a two-layered medium
aa r X i v : . [ m a t h . A P ] F e b Time domain analysis for electromagnetic scatteringby an elastic obstacle in a two-layered medium
Changkun Wei ∗ Jiaqing Yang † Bo Zhang ‡ Abstract
In this paper, we consider the scattering of a time-dependent electromagnetic wave by anelastic body immersed in the lower half-space of a two-layered background medium which isseparated by an unbounded rough surface. By proposing two exact transparent boundaryconditions (TBCs) on the artificial planes, we reformulate the unbounded scattering probleminto an equivalent initial-boundary value problem in a strip domain with the well-posednessand stability proved using the Laplace transform, variational method and energy method.A perfectly matched layer (PML) is then introduced to truncate the interaction problemwith two finite layers containing the elastic body, leading to a PML problem in a finite stripdomain. We further verify the existence, uniqueness and stability estimate of solution for thePML problem. Finally, we establish the exponential convergence in terms of the thicknessand parameters of the PML layers via an error estimate on the electric-to-magnetic (EtM)capacity operators between the original problem and the PML problem.
Keywords:
Electromagnetic wave equation, elastic wave equation, two-layered medium,time-domain, well-posedness, perfectly matched layer, exponential convergence.
Let us consider the interaction scattering of a time-dependent electromagnetic field by an elasticbody embedded in a two-layered medium in three dimensions. This problem can be categorizedinto the class of the unbounded rough surface scattering problems, which are the subject ofintensive studies in the engineering and mathematics. In the problem setting, the whole space isdivided into two parts by an unbounded rough surface Γ f with the elastic body Ω immersed in thelower half-space. We assume that the electromagnetic field initiated by an electric current densityproduces a tangential stress on the interface Γ := ∂ Ω which excites an elastic displacement of theelastomer. Following the Voigt’s model (cf. [37, 10, 4, 28]), we assume that the electromagneticfield does not considerably penetrate inside the elastomer. Several important works have beendone on this typical electromagnetic-elastic interaction problem, which is confined to the time-harmonic setting. It was shown in [10] that Cakoni & Hsiao established a mathematical model, ∗ Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea( [email protected] ) † School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China( [email protected] ) ‡ NCMIS, LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049,China ( [email protected] ) • Prove the well-posedness and stability for the interaction problem; • Propose a time-domain PML method and show the well-posedness and stability; • Establish the exponential convergence of the PML method in terms of thickness and pa-rameters of the PML layer.Due to the unbounded interface, the usual Silver-M¨ u ller radiation condition is not valid any-more to describe the asymptotic behavior of scattered waves away from the rough surface. More-over, the classical Fredholm alternative theorem may not be applied into this kind of problemsdue to the lack of compactness result. These make the studies of interface scattering problemsquite challenging. For the time-harmonic setting, there exists lots of works for the mathematicalanalysis with using either the boundary integral equations method or the variational method;see, e.g. [15, 11, 12, 14, 47] for the acoustic wave and [29, 35, 36] for the electromagnetic wave.Recently, the time-domain scattering problems have attracted much attention due to their ca-pability of capturing wide-band signals and modeling more general material and nonlinearity[16, 33, 41, 42, 48]. Precisely, the mathematical analysis can be found in [17, 41] for time-dependent scattering problems in the full acoustic wave cases, and [18, 34, 25, 26] in the fullelectromagnetic wave cases. In addition, the time-dependent fluid-solid interaction problemshas been also studied for the bounded elastomer [1], local rough surfaces [43], and unboundedlayered structures in the three-dimensional case [27]. To the best of our knowledge, the math-ematical analysis is quite rare for the electromagnetic-elastic interaction problems in the timedomain. Here, we refer to a recently related work [45] for a bounded obstacle embedded in thehomogeneous background medium.As is known, the perfectly matched layer (PML) method is a fast and effective methodfor solving unbounded scattering problems which was originally proposed by B´erenger in 1994for Maxwell’s equations [3]. The purpose of the PML method is to surround the computationaldomain with a specially designed medium in a finite thickness layer in which the scattered wavesdecay rapidly regardless of the wave incident angle, thereby greatly reducing the computationalcomplexity of the scattering problem. Since then, various PML formulations have been widelycreated and studied for solving the wave scattering problems (see, e.g., [40, 24, 32, 38, 19, 13, 17]).The broad applications of the PML method attract great interests for mathematicians to studythe convergence analysis for the time-harmonic scattering problems; see, e.g. [32, 30, 21, 8, 2, 5,6, 7, 23] for the acoustic and electromagnetic obstacle scattering problems. However, the PMLtechnique is much less studied for unbounded rough surface scattering problems. A general linearconvergence was proved in [13] for the acoustic scattering problem depending on the thicknessand composition of the layer. Moreover, an exponential convergence was also established in [35]for the electromagnetic scattering problems. 2ompared with the time-harmonic setting, very few results are available for the mathemat-ical analysis of the time-domain PML method, which is challenged by the dependence of theabsorbing medium on all frequencies. For the 2D time-domain acoustic scattering problem, theexponential convergence of both a circular PML method [17] and a uniaxial PML method [20]were established in terms of the thickness and absorbing parameters. For the 3D time domainelectromagnetic scattering problem, the exponential convergence of a spherical PML methodwas very recently shown in [46] in terms of the thickness and parameter of the PML layer, basedon a real coordinate stretching technique associated with [Re( s )] − in the Laplace domain, where s ∈ C + is the Laplace transform variable. It is also noticed that for the acoustic-elastic interac-tion problem, the well-posedness and stability estimates of the time-domain PML method wasproved in [1], but no convergence analysis was provided. We also remark that an exponentialconvergence of the PML method was recently established in [44] for the fluid-solid interactionproblem above an unbounded rough surface, which generalized our previous idea [46] with thereal coordinate stretching technique.In this paper, we intend to study the time-dependent electromagnetic-elastic interactionproblem in a two-layered medium associated with a bounded elastic body immersed in the lowerhalf-space. With the aid of the factorization on the interface conditions [45] and two exact timedomain TBCs, we establish the well-posedness and stability of the interaction problem based onthe variational method and the Laplace transform and its inversion. Further, we propose a time-domain PML method along x direction by using the real coordinate stretching technique in [46]associated with [Re( s )] − in the frequency domain. The well-posedness and stability estimateof the truncated PML problem are proved by the Laplace transform and energy method. Anexponential convergence is then proved in terms of the thickness and parameters of the PMLlayer, through an error estimate on the EtM operators between the original problem and thePML problem.The outline of this paper is as follows. In section 2, we introduce some basic notations andgive a brief description of our model problem. In section 3, the original interaction scatteringproblem is firstly reduced into an equivalent initial-boundary value problem in a strip domain.Then we study the well-posedness and stability for the reduced problem by the variationalmethod and the energy method. In section 4, a time-domain PML method is introduced totruncate the interaction problem with two finite layers containing the elastic body, leading to atruncated PML problem in a finite strip domain. The well-posedness and stability estimate forthe truncated PML problem is further verified. An exponential convergence of the PML methodis finally established. Some conclusions are given in section 5. Consider the propagation of an electromagnetic wave which is excited by an electric currentdensity in a two-layered medium with a bounded elastic body immersed in the lower half-space;see the problem geometry in Figure 1. For x = ( x , x , x ) ⊤ ∈ R , let e x = ( x , x ) ⊤ ∈ R andΓ f := { x ∈ R : x = f ( e x ) } be the unbounded rough surface with f ∈ C ( R ), which separates the whole space into atwo-layered domainΩ + f := { x ∈ R : x > f ( e x ) } and Ω − f := { x ∈ R : x < f ( e x ) } . h fh h source Figure 1: Geometric configuration of the scattering problemHere, the electromagnetic medium fills with distinct parameters ε, µ . We assume that Ω is abounded domain with Lipschitz-continuous boundary Γ := ∂ Ω representing a homogeneousand isotropic elastic body immersed in the lower medium Ω − f and the exterior Ω c = R \ Ω ofΩ is simply connected. Furthermore, we assume Ω to be with a constant mass density ρ i > λ i , µ i satisfying the condition that µ i > λ i + 2 µ i >
0. Define twoartificial planar surfaces Γ h := { x ∈ R : x = h } , where h > sup e x ∈ R f ( e x ) is a constantand Γ h := { x ∈ R : x = h } , where h < h .Let Ω h := { x ∈ R : f ( e x ) < x < h } and Ω h := { x ∈ R : h < x < f ( e x ) } ∩ Ω c , andΩ h = Ω h ∪ Ω h ∪ Γ f . In what follows, we denote by n the unit outward normal vector both on Γand Γ f as well as n = (0 , , ⊤ , n = (0 , , − ⊤ the unit outward normal vectors on Γ h andΓ h , respectively. To the end, we define C + := { s = s + is ∈ C with s , s ∈ R and s > } and remark hereafter that the index j is always valued in the set { , } except special statement. Elastic wave equation . In the elastic body Ω, the elastic displacement u is governed bythe linear elastodynamic equation: ρ i ∂ u ∂t − ∆ ∗ u = , in Ω × (0 , T ) (2.1)where ∆ ∗ is the Lam´e operator defined as∆ ∗ u := µ i ∆ u + ( λ i + µ i ) ∇ div u = div σ ( u ) . In above, σ ( u ) and ε ( u ) are called stress and strain tensors respectively, which are given by σ ( u ) = ( λ i div u ) I + 2 µ i ε ( u ) and ε ( u ) = 12 ( ∇ u + ( ∇ u ) ⊤ ) . Furthermore, the homogeneous initial conditions are imposed for the elastic wave equation u ( x,
0) = ∂ u ∂t ( x,
0) = 0 , x ∈ Ω . (2.2)4 axwell’s equations . In the electromagnetic domain Ω c , the electric field E and magneticfield H satisfy the time-domain Maxwell equations ∇ × E + µ ∂ H ∂t = , ∇ × H − ε ∂ E ∂t = J , in Ω c × (0 , T ) (2.3)where J is the electric current density which is assumed to be compactly supported in Ω h and J | t =0 = , the electric permittivity ε and magnetic permeability µ are both positive andpiece-wise constants: ε ( x ) = ( ε , x ∈ Ω + f ,ε , x ∈ Ω − f \ Ω , µ = ( µ , x ∈ Ω + f ,µ , x ∈ Ω − f \ Ω . (2.4)On the interface Γ f between the two-layered medium, we have the jump conditions n × [ E ] = n × [ µ − ∇ × E ] = , on Γ f × (0 , T ) (2.5)where [ · ] stands for the jump of a function across the interface Γ f . In addition, the homogeneousinitial conditions are also imposed for the Maxwell’s equations: E ( x,
0) = H ( x,
0) = , x ∈ Ω c . (2.6)Using the Maxwell’s system (2.3), it is obvious that ∂ t E ( x,
0) = ε − ( ∇ × H )( x, − ε − J ( x,
0) = , x ∈ Ω c , (2.7) ∂ t H ( x,
0) = − µ − ( ∇ × E )( x,
0) = , x ∈ Ω c . (2.8)and ∇ · E = ∇ · H = 0 , in Ω c × (0 , T ) . (2.9)Due to the unbounded structure of the medium, it is no longer valid to impose the usualSilver-M¨uler radiation condition. Instead, we employ the following radiation condition: theelectromagnetic fields ( E , H ) consist of bounded outgoing waves in Ω + h and Ω − h , whereΩ + h = { x ∈ R : x > h } and Ω − h = { x ∈ R : x < h } . Interface conditions . The two medium are coupled by the interface condition (cf. [10]): H ( x, t ) × E ( x, t ) · n = T u ( x, t ) · u t ( x, t ) , on Γ × (0 , T ) (2.10)where T u := 2 µ i n · ∇ u + λ i n ∇ · u + µ i n × ( ∇ × u ) denotes the elastic surface traction operator.There are infinite many decomposition of above interface condition (2.10). According to theVoigt’s model [37], the stress tensor is proportional to the magnetic field which leads to thefollowing decomposition (see [45]) n × H = T u and n × E = n × u t , on Γ × (0 , T ) . (2.11)Note that it is easily checked out that (2.11) implies (2.10).5 The well-posedness of scattering problem
In this section, we firstly introduce two exact time-domain transparent boundary conditions(TBCs) on the artificial plane surfaces to reformulate the scattering problem into an initial-boundary value problem in a finite strip domain. Then, we will show the well-posedness forthe reduced problem in s -domain by the method of Laplace transform and the Lax-Milgramlemma. To the end, the existence, uniqueness, and stability for the reduced problem in the timedomain shall be verified by using the abstract inversion theorem of the Laplace transform, andthe energy argument. In this subsection, we start by introducing two transparent boundary conditions (TBCs) on theartificial planar surfaces (cf. [26]): T j [ E Γ hj ] = H × n j , on Γ h j × (0 , T ) , j = 1 , , (3.1)which maps the tangential component of electric field E to the tangential trace of magnetic field H on Γ h j . Then the time-dependent electromagnetic-elastic wave interaction problem can bereduced to an equivalent initial boundary value problem in the strip domain Ω h : ρ i ∂ u ∂t − ∆ ∗ u = , in Ω × (0 , T ) ∇ × E + µ∂ t H = , in Ω h × (0 , T ) ∇ × H − ε∂ t E = J , in Ω h × (0 , T ) u ( x,
0) = ∂ t u ( x,
0) = , in Ω E ( x,
0) = H ( x,
0) = , in Ω h n × [ E ] = n × [ µ − ∇ × E ] = , on Γ f × (0 , T ) n × H = T u , n × E = n × u t , on Γ × (0 , T ) T j [ E Γ hj ] = H × n j , on Γ h j × (0 , T ) , j = 1 , . (3.2)Taking the Laplace transform of (3 .
2) and employing (A.2) together with initial conditions (2.2)and (2.6), we obtain the time harmonic electromagnetic-elastic interaction problem in s-domain: ∆ ∗ ˇ u − ρ i s ˇ u = , in Ω ∇ × ˇ E + µs ˇ H = , in Ω h ∇ × ˇ H − εs ˇ E = ˇ J , in Ω h n × [ ˇ E ] = n × [ µ − ∇ × ˇ E ] = , on Γ f n × ˇ H = T ˇ u , n × ˇ E = n × s ˇ u on Γ B j [ ˇ E Γ hj ] = ˇ H × n j , on Γ h j , j = 1 , , (3.3)where s ∈ C + , and B j is the electric-to-magnetic (EtM) capacity operators on Γ h j in s-domainsatisfying T j = L − ◦ B j ◦ L .In [26], Y. Gao and P. Li derived the formulation of the EtM operators B j and showed someof important properties including boundness and coercivity. Here, we present the main results6f TBCs in [26] without detailed proof. The explicit representations of EtM operators B j takethe following form: for any tangential vector ω = ( ω , ω , ⊤ on Γ h j , B j [ ω ] = ( v , v , ⊤ , (3.4)where b v = 1 µ j sβ j ( ξ ) (cid:2) ε j µ j s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , b v = 1 µ j sβ j ( ξ ) (cid:2) ε j µ j s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , where b v j denotes the Fourier transform of v j with respect to e x (see Appendix A for the definitionof Fourier transform), and β j ( ξ ) = ( ε j µ j s + | ξ | ) / , with Re[ β j ( ξ )] > . (3.5)For convenience, we eliminate the magnetic field ˇ H and get the TBCs for electric field ˇ E in thes-domain and time domain, respectively:( µ j s ) − ( ∇ × ˇ E ) × n j + B j [ ˇ E Γ hj ] = , on Γ h j , (3.6)( µ j ) − ( ∇ × E ) × n j + C j [ E Γ hj ] = , on Γ h j , (3.7)where C j = L − ◦ s B j ◦ L .The following lemma on the boundedness and coercivity of B j plays a key role in the proofof the well-posedness which has been shown in [26]. Lemma 3.1.
For j = 1 , , B j is continuous from H − / (curl , Γ h j ) to H − / (div , Γ h j ) (seeAppendix B for the definition of the trace spaces). Moreover, for any ω ∈ H − / (curl , Γ h j ) , wehave Re h B j ω , ω i Γ hj ≥ . Eliminating the magnetic field ˇ H in (3.3), we consider the reduced vector boundary valueproblem ∆ ∗ ˇ u − ρ i s ˇ u = , in Ω (3.8a) ∇ × (( sµ ) − ∇ × ˇ E ) + sε ˇ E = − ˇ J , in Ω h (3.8b) n × [ ˇ E ] = n × [ µ − ∇ × ˇ E ] = , on Γ f (3.8c) − ( µ s ) − n × ∇ × ˇ E = T ˇ u , on Γ (3.8d) n × ˇ E = n × s ˇ u , on Γ (3.8e)( µ j s ) − ( ∇ × ˇ E ) × n j + B j [ ˇ E Γ hj ] = , on Γ h j , j = 1 , , (3.8f)in the Hilbert space X s := (cid:8) ( V , v ) ∈ H (curl , Ω h ) × H (Ω) , n × V = n × s v , on Γ (cid:9) under thenorm k ( V , v ) k X s := (cid:16) k V k H (curl , Ω h ) + k v k H (Ω) (cid:17) / . (3.9)7e shall prove the well-posedness of problem (3.8a)-(3.8f) in X s by the Lax-Milgram lemma. Tothis end, we derive the variational formulation of (3.8a)-(3.8f) by multiplying (3.8b) and (3.8a)with the complex conjugates of a pair of test functions ( V , v ) ∈ X s , respectively, and applyingintegration by part, coupling interface condition (3.8d), and TBCs (3.8f). Hence, the variationalformulation of (3.8a)-(3.8f) reads as follows: find a solution ( ˇ E , ˇ u ) ∈ X s such that Z Ω h ( sµ ) − ( ∇ × ˇ E ) · ( ∇ × V ) dx + Z Ω h sε ˇ E · V dx (3.10)+ Z Γ ( sµ ) − ∇ × ˇ E × n · V dγ + X j =1 h B j [ ˇ E Γ hj ] , V Γ hj i Γ hj = − Z Ω h ˇ J · V dx, and Z Ω h s (cid:16) λ i ( ∇ · ˇ u )( ∇ · v ) + 2 µ i ε ( ˇ u ) : ε ( v ) (cid:17) + ρ i s | s | ˇ u · v i dx − Z Γ ( sµ ) − ∇ × ˇ E × n · s v dγ = 0 , (3.11)where A : B = tr ( AB ⊤ ) denotes the Frobenius inner product of square matrices A and B.Adding (3.11) to (3.10) gives the final variational form: a (cid:0) ( ˇ E , ˇ u ) , ( V , v ) (cid:1) = − Z Ω h ˇ J · V dx, (3.12)where the sesquilinear form a ( · , · ) is defined as a (cid:0) ( ˇ E , ˇ u ) , ( V , v ) (cid:1) = Z Ω h (cid:0) ( sµ ) − ( ∇ × ˇ E ) · ( ∇ × V ) dx + sε ˇ E · V (cid:1) dx (3.13)+ Z Ω h s E ( ˇ u , v ) + ρ i s | s | ˇ u · v i dx + X j =1 h B j [ ˇ E Γ hj ] , V Γ hj i Γ hj . Here, the bilinear form E ( u , v ) is defined by E ( u , v ) := λ i (div u )(div v ) + 2 µ i ε ( u ) : ε ( v ) (3.14)= 2 µ i (cid:16) X i,j =1 ∂ i u j ∂ i v j (cid:17) + λ i (div u )(div v ) − µ i curl u · curl v . Under our assumptions on the Lam´e constants: µ > , λ + 2 µ >
0, we have the estimate (see[31, Chap. 5.4]) Z Ω E ( u , u ) dx ≥ C Ω k ε ( u ) k F (Ω) , (3.15)where the positive constant C Ω only depends on Ω, and k ε ( u ) k F (Ω) denotes the Frobenius normdefined by k ε ( u ) k F (Ω) := (cid:16) X i,j =1 k ε ij ( u ) k L (Ω) (cid:17) / . emma 3.2. For each s ∈ C + , the variational problem (3.12) has a unique solution ( ˇ E , ˇ u ) ∈ X s which satisfies the following estimates: k∇ × ˇ E k L (Ω h ) + k s ˇ E k L (Ω h ) . s − k s ˇ J k L (Ω h ) , (3.16) k∇ ˇ u k F (Ω) + k∇ · ˇ u k L (Ω) + k s ˇ u k L (Ω) . s − max { , s − }k ˇ J k L (Ω h ) . (3.17) Hereafter, the expression a . b or a & b stands for a ≤ Cb or a ≥ Cb , where C is a positiveconstant and its specific value is not required but should be always clear from the context.Proof. i) By Cauchy-Schwartz inequality, the boundness of B j and Lemma B.3, it follows that (cid:12)(cid:12) a (cid:0) ( ˇ E , ˇ u ) , ( V , v ) (cid:1)(cid:12)(cid:12) . | s | − k∇ × ˇ E k L (Ω h ) k∇ × V k L (Ω h ) + | s |k ˇ E k L (Ω h ) k V k L (Ω h ) + | s |k∇ · ˇ u k L (Ω) k∇ · v k L (Ω) + | s | k ˇ u k L (Ω) k v k L (Ω) + | s |k∇ ˇ u k F (Ω) k∇ v k F (Ω) + X j =1 k B j [ ˇ E Γ hj ] k H − / (div , Γ hj ) k V Γ hj k H − / (curl , Γ hj ) . k ˇ E k H (curl , Ω h ) k V k H (curl , Ω h ) + k ˇ u k H (Ω) k v k H (Ω) , which yields that a ( · , · ) is continuous in the product space X s × X s .ii) a ( · , · ) is uniformly coercive. In fact, setting ( V , v ) := ( ˇ E , ˇ u ) in (3.13) yields a (cid:0) ( ˇ E , ˇ u ) , ( ˇ E , ˇ u ) (cid:1) = Z Ω h (cid:0) ( sµ ) − |∇ × ˇ E | dx + sε | ˇ E | (cid:1) dx (3.18)+ Z Ω h s E ( ˇ u , ˇ u ) + ρ i s | s ˇ u | i dx + X j =1 h B j [ ˇ E Γ hj ] , ˇ E Γ hj i Γ hj . Define µ max := max { µ , µ } , ε min := min { ε , ε } . Combining the estimate (3.15) and the well-known Korn’s inequality [31, Lemma 5.4.4] k ε ( v ) k F (Ω) + k v k L (Ω) ≥ C Ω k v k H (Ω) , ∀ v ∈ H (Ω) (3.19)then taking the real part of (3.18) and using Lemma 3.1, we haveRe[ a (( ˇ E , ˇ u ) , ( ˇ E , ˇ u ))] ≥ s | s | ( µ − max k∇ × ˇ E k L (Ω h ) + ε min k s ˇ E k L (Ω h ) )+ s ( C Ω k ε ( ˇ u ) k F (Ω) + ρ i k s ˇ u k L (Ω) ) ≥ s | s | C k ˇ E k H (curl , Ω h ) + s C k ˇ u k H (Ω) ≥ C k ( ˇ E , ˇ u ) k X s , (3.20)where C is defined as C := min { s | s | C , s C } , C = min { µ − max , ε min | s | } , C = C Ω min { C Ω , ρ i | s | } .
9t follows from the Lax-Milgram lemma that the variational problem (3.12) has a uniquesolution ( ˇ E , ˇ u ) ∈ X s for each s ∈ C + . Moreover, using (3.12), we clearly have a (cid:0) ( ˇ E , ˇ u ) , ( ˇ E , ˇ u ) (cid:1) . | s | k ˇ J k L (Ω h ) k s ˇ E k L (Ω h ) ≤ ǫ | s | k ˇ J k L (Ω h ) + ǫ k s ˇ E k L (Ω h ) , (3.21)where we have used ǫ -inequality in the last inequality.Choosing ǫ sufficiently small such that ǫ < s | s | , e.g., ǫ = s | s | , combining (3.20) with (3.21),we obtain s | s | (cid:16) k∇ × ˇ E k L (Ω h ) + k s ˇ E k L (Ω h ) (cid:17) + s min { , s } (cid:16) k∇ ˇ u k F (Ω) + k∇ · ˇ u k L (Ω) + k s ˇ u k L (Ω) (cid:17) . s − k ˇ J k L (Ω h ) , (3.22)we arrive at (3.16) and (3.17) after using Cauchy-Schwartz inequality for (3.22). For 0 ≤ t ≤ T , to show the well-posedness of the reduced problem (3.2) and the convergence ofthe PML method, we make the following assumptions on the source term J : J ∈ H (0 , T ; L (Ω h ) ) , ∂ lt J | t =0 = , l = 0 , , , , . (3.23)Furthermore, in the rest of the paper, we will always assume that J can be extended to ∞ withrespect to t such that J ∈ H (0 , ∞ ; L (Ω h ) ) , k J k H (0 , ∞ ; L (Ω h ) ) . k J k H (0 ,T ; L (Ω h ) ) . (3.24) Theorem 3.3.
The reduced initial-boundary value problem (3.2) has a unique solution ( E ( x, t ) , H ( x, t ) , u ( x, t )) satisfying E ( x, t ) ∈ L (0 , T ; H (curl , Ω h )) ∩ H (cid:0) , T ; L (Ω h ) (cid:1) , H ( x, t ) ∈ L (0 , T ; H (curl , Ω h )) ∩ H (cid:0) , T ; L (Ω h ) (cid:1) , u ( x, t ) ∈ L (cid:0) , T ; H (Ω) (cid:1) ∩ H (cid:0) , T ; L (Ω) (cid:1) , with the stability estimate max t ∈ [0 ,T ] (cid:0) k ∂ t E k L (Ω h ) + k∇ × E k L (Ω h ) (3.25)+ k ∂ t H k L (Ω h ) + k∇ × H k L (Ω h ) (cid:1) . k J k H (0 ,T ; L (Ω h ) ) , max t ∈ [0 ,T ] (cid:0) k ∂ t u k L (Ω) + k∇ · u k L (Ω) + k∇ u k F (Ω) (cid:1) . k J k L (0 ,T ; L (Ω h ) ) . (3.26)10 roof. Simple calculations yields the following estimate Z T ( k∇ × E k L (Ω h ) + k ∂ t E k L (Ω h ) + k∇ u k F (Ω) + k ∂ t u k L (Ω) ) dt ≤ Z T e − s ( t − T ) ( k∇ × E k L (Ω h ) + k ∂ t E k L (Ω h ) + k∇ u k F (Ω) + k ∂ t u k L (Ω) ) dt . Z ∞ e − s t ( k∇ × E k L (Ω h ) + k ∂ t E k L (Ω h ) + k∇ u k F (Ω) + k ∂ t u k L (Ω) ) dt. It is therefore sufficient to estimate the integral Z ∞ e − s t ( k∇ × E k L (Ω h ) + k ∂ t E k L (Ω h ) + k∇ u k F (Ω) + k ∂ t u k L (Ω) ) dt. Recalling the s -domain reduced system (3.3), by estimates (3.16) and (3.17) in Lemma 3.2,it follows from [39, Lemma 44.1] that ( ˇ E , ˇ u ) are holomorphic functions of s on the half plane s > γ > , where γ is any positive constant. Hence we have from Lemma A.2 that the inverseLaplace transform of ˇ E and ˇ u exist and are supported in [0 , ∞ ].Denote by E = L − ( ˇ E ) and u = L − ( ˇ u ). It follows that using the Parseval identity (A.5)and estimate (3.16) Z ∞ e − s t ( k∇ × E k L (Ω h ) + k ∂ t E k L (Ω h ) ) dt = 12 π Z ∞−∞ ( k∇ × ˇ E k L (Ω h ) + k s ˇ E k L (Ω h ) ) ds . Z ∞−∞ s − k s ˇ J k L (Ω h ) ds . s − Z ∞ e − s t k ∂ t J k L (Ω h ) dt, which shows that E ( x, t ) ∈ L (0 , T ; H (curl , Ω h )) ∩ H (cid:0) , T ; L (Ω h ) (cid:1) , thanks to the Maxwell system in (3.2), we also have H ( x, t ) ∈ L (0 , T ; H (curl , Ω h )) ∩ H (cid:0) , T ; L (Ω h ) (cid:1) . For elastic wave, combining Parseval identity (A.5) with estimate (3.17), we similarly have Z ∞ e − s t ( k ∂ t u k L (Ω) + k∇ u k F (Ω) ) dt = 12 π Z ∞−∞ ( k s ˇ u k L (Ω) + k∇ ˇ u k F (Ω) ) ds . Z ∞−∞ s − max { , s − }k s ˇ J k L (Ω h ) ds . s − max { , s − } Z ∞ e − s t k ∂ t J k L (Ω h ) dt, which means that u ( x, t ) ∈ L (cid:0) , T ; H (Ω) (cid:1) ∩ H (cid:0) , T ; L (Ω) (cid:1) .
11n what follows, we shall prove the stability of solution in (3.2) by means of the initialconditions. We start by defining an energy function ε ( t ) = e ( t ) + e ( t ) , for t ∈ (0 , T )with e ( t ) = k ε / E ( · , t ) k L (Ω h ) + k µ / H ( · , t ) k L (Ω h ) ,e ( t ) = k ρ / i ∂ t u k L (Ω) + Z Ω E ( u , u ) dx. Observe that ε ( · ) can be equivalently written as ε ( t ) − ε (0) = Z t ε ′ ( τ ) dτ = Z t (cid:16) e ′ ( τ ) + e ′ ( τ ) (cid:17) dτ. (3.27)By simple calculations using the system (3.2) and integration by parts, we have Z t e ′ ( τ ) dτ = 2Re Z t Z Ω h (cid:0) ε∂ τ E · E + µ∂ τ H · H (cid:1) dxdτ = 2Re Z t Z Ω h (cid:0) ( ∇ × E ) · H − ( ∇ × E ) · H (cid:1) dxdτ − X j =1 Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ − Z t Z Ω h J · E dxdτ + 2Re Z t Z Γ ( H × n ) · E dγdτ = − X j =1 Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ − Z t Z Ω h J · E dxdτ + 2Re Z t Z Γ ( H × n ) · E dγdτ. (3.28)Noting the definition of E ( u , v ) (see (3.14)), by the elastic wave equation (2.1) and using theintegration by parts, it can be similarly shown that Z t e ′ ( τ ) dτ = 2Re Z t Z Ω (cid:16) ρ e ∂ τ u · ∂ τ u + E ( ∂ τ u , u ) (cid:17) dxdτ = Z t Z Ω (cid:16) − E ( u , ∂ τ u ) + E ( ∂ τ u , u ) (cid:17) dxdτ + 2Re Z t Z Γ T u · ∂ τ u dγdτ = 2Re Z t Z Γ T u · ∂ τ u dγdτ. (3.29)12ombining (3.27)-(3.29) with ε (0) = 0 and the interface condition (2.11), we obtain ε ( t ) = − X j =1 Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ − Z t Z Ω h J · E dxdτ + 2Re Z t Z Γ ( H × n ) · E dγdτ + 2Re Z t Z Γ T u · ∂ τ u dγdτ = − X j =1 Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ − Z t Z Ω h J · E dxdτ + 2Re Z t Z Γ H · ( n × E − n × ∂ τ u ) dγdτ = − X j =1 Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ − Z t Z Ω h J · E dxdτ. (3.30)By [26, equation (4.11)], it holds thatRe Z t Z Γ hj T j [ E Γ hj ] · E Γ hj dγdτ ≥ . This, combining the ǫ -inequality and (3.15) one has the following estimate k E ( · , t ) k L (Ω h ) + k H ( · , t ) k L (Ω h ) + k ∂ t u k L (Ω) + k ε ( u ) k F (Ω) . ε ( t ) ≤ − Z t Z Ω h J · E dxdτ . Z t k J k L (Ω h ) · k E k L (Ω h ) dτ . ǫ t ∈ [0 ,T ] k E ( · , t ) k L (Ω h ) + 12 ǫ k J k L (0 ,T ; L (Ω h ) ) . (3.31)Finally, letting ǫ > ǫ = 1 and applying Cauchy-Schwartz inequalityyields max t ∈ [0 ,T ] (cid:16) k E ( · , t ) k L (Ω h ) + k H ( · , t ) k L (Ω h ) + k ∂ t u k L (Ω) + k ε ( u ) k F (Ω) (cid:17) . k J k L (0 ,T ; L (Ω h ) ) . k J k L (0 ,T ; L (Ω h ) ) . (3.32)Now, by using the Cauchy-Schwartz inequality again, we have for any 0 ≤ ξ ≤ T k u ( · , ξ ) k L (Ω) = Z ξ ∂ t k u ( · , t ) k L (Ω) dt = 2Re Z ξ Z Ω ∂ t u ( x, t ) · u ( x, t ) dxdt ≤ Z ξ (cid:16) ǫ k ∂ t u ( · , t ) k L (Ω) + 14 ǫ k u ( · , t ) k L (Ω) (cid:17) dt . T ǫ k ∂ t u ( · , ξ ) k L (Ω) + T ǫ k u ( · , ξ ) k L (Ω) . (3.33)Choosing ǫ = T in (3.33) gives k u ( · , ξ ) k L (Ω) . T k ∂ t u ( · , ξ ) k L (Ω) . (3.34)13pplying Korn’s inequality (3.19) and using (3.34) gives k ∂ t u k L (Ω) + k ε ( u ) k F (Ω) & k u k H (Ω) & k∇ · u k L (Ω) + k∇ u k F (Ω) , This, combining (3.32) leads to the stability estimate (3.26).Taking the derivative of (3.2) with respect to t , observing that ( ∂ t E , ∂ t H ) satisfy the sameset of equations with the source J replaced by ∂ t J , and the initial conditions replaced by ∂ t E = ∂ t H = using (2.7)-(2.8) and ∂ t u also satisfies elastodynamic equation with ∂ t u ( x,
0) = ρ − i ∆ ∗ u ( x,
0) = , therefore we can follow the same steps as deriving (3.32) for ( ∂ t E , ∂ t H )which leads to max t ∈ [0 ,T ] (cid:16) k ∂ t E ( · , t ) k L (Ω h ) + k ∂ t H ( · , t ) k L (Ω h ) + k ∂ t u k L (Ω) + k ε ( ∂ t u ) k F (Ω) (cid:17) . k ∂ t J k L (0 ,T ; L (Ω h ) ) . (3.35)This, combining (3.32) with the Maxwell’s equations completes our proof of (3.25). In this section, we shall derive the time domain PML formulation of the electromagnetic-elasticinteraction scattering problem. The well-posedness and stability of the PML problem is estab-lished based on the variational method and the energy method which is adopt in section 3. Inthe end, we shall show the exponential convergence analysis of the time domain PML methodapplying a novel technique to construct the PML layer. h h fh h h L h L h +L h -L source Figure 2: Geometric configuration of the truncated PML problem14 .1 The PML equations and Well-posedness
We firstly introduce geometry of the PML problem as shown in Figure 2. Let Ω L h := { x ∈ R : h < x < h + L } and Ω L h := { x ∈ R : h − L < x < h } denote the PMLlayers with thickness L and L which surround the strip domain Ω h . Denote by Ω h + L := { x ∈ R : h − L < x < h + L } ∩ Ω c the truncated PML domain with boundariesΓ h + L := { x ∈ R : x = h + L } and Γ h − L := { x ∈ R : x = h − L } . Now, let s > σ = σ ( x ): σ ( x ) = , if h ≤ x ≤ h , s − σ ( x − h L ) m , if h < x < h + L , s − σ ( h − x L ) m , if h − L < x < h , (4.1)where σ j are two positive constants and m ≥ s ∈ C + to be s , that is, Re( s ) = s .Next, we shall derive the PML equations by the change of variables technique, starting byintroducing the real stretched coordinate ˆ x ˆ x = x , ˆ x = x , ˆ x = Z x σ ( τ ) dτ. Since supp J ⊂ Ω h , taking the Laplace transform of the original Maxwell’s equation (2.3) withrespect to t , we have for j = 1 , ( ∇ × ˇ E + µ j s ˇ H = , in Ω L j h j ∇ × ˇ H − ε j s ˇ E = , in Ω L j h j (4.2)Let ˇ E (ˆ x ) and ˇ H (ˆ x ) be the PML extensions of the electromagnetic field ˇ E and ˇ H satisfying (4.2).To be more precise, the change of variables technique is to require ˇ E (ˆ x ) and ˇ H (ˆ x ) satisfying ( ∇ p × ˇ E (ˆ x ) + µ j s ˇ H (ˆ x ) = , in Ω L j h j ∇ p × ˇ H (ˆ x ) − ε j s ˇ E (ˆ x ) = , in Ω L j h j (4.3)where ∇ p × u := ( ∂ x u − σ − ∂ x u , σ − ∂ x u − ∂ x u , ∂ x u − ∂ x u ) ⊤ for any vector u =( u , u , u ) ⊤ . Observing that ∇ × diag(1 , , σ ) u = diag( σ, σ, ∇ p × u , we introduce the PML solutions ( ˇ e E , ˇ f H ) byˇ e E ( x ) = diag(1 , , σ ) ˇ E (ˆ x ) , (4.4)ˇ f H ( x ) = diag(1 , , σ ) ˇ H (ˆ x ) . (4.5)Inserting (4.4) and (4.5) into (4.3) and combining the elastic wave equations, we obtain the15runcated PML equations of ˇ e E , ˇ f H and ˇ e u ∇ × ˇ e E + e µs ˇ f H = , in Ω h + L ∇ × ˇ f H − e εs ˇ e E = ˇ J , in Ω h + L ∆ ∗ ˇ e u − ρ i s ˇ e u = , in Ω n × [ ˇ e E ] = n × [ µ − ∇ × ˇ e E ] = , on Γ f n × ˇ f H = T ˇ e u , n × ˇ e E = n × s ˇ e u on Γˇ e E × n j = , on Γ h j ± L j , j = 1 , , (4.6)where e µ := diag( σ, σ, σ − ) µ and e ε := diag( σ, σ, σ − ) ε , respectively, and the perfect electricconductor (PEC) boundary conditions have been imposed on the PML boundary Γ h + L andΓ h − L (Hereafter, we always take the sign + when j = 1, and − when j = 2 in Γ h j ± L j ).Eliminating the magnetic field ˇ f H from (4.6) yields the equation of ( ˇ e E , ˇ e u ) ∇ × (( s e µ ) − ∇ × ˇ e E ) + s e ε ˇ e E = − ˇ J , in Ω h + L ∆ ∗ ˇ e u − ρ i s ˇ e u = , in Ω n × [ ˇ e E ] = n × [ µ − ∇ × ˇ e E ] = , on Γ f − n × ( s e µ ) − ∇ × ˇ e E = T ˇ e u , on Γ n × ˇ e E = n × s ˇ e u , on Γˇ e E × n j = , on Γ h j ± L j , j = 1 , . (4.7)In the following, we shall show the well-posedness of (4.7) by the variational method in theHilbert space f X s := (cid:8) ( V , v ) ∈ H (curl , Ω h + L ) × H (Ω) , n × V = n × s v , on Γ (cid:9) where H (curl , Ω h + L ) := { u ∈ H (curl , Ω h + L ) : u × n = , on Γ h + L and u × n = , on Γ h − L } . And the norm on f X s is defined as (3.9) with Ω h replaced by Ω h + L . To this end,we introduce the variational formulation of (4.7): to find a solution ( ˇ e E , ˇ e u ) ∈ f X s such that e a (cid:0) ( ˇ e E , ˇ e u ) , ( V , v ) (cid:1) = − Z Ω h ˇ J · V dx for all ( V , v ) ∈ f X s , (4.8)where the sesquilinear form e a ( · , · ) is defined as e a (cid:0) ( ˇ e E , ˇ e u ) , ( V , v ) (cid:1) = Z Ω h + L (cid:16) ( s e µ ) − ( ∇ × ˇ e E ) · ( ∇ × V ) dx + s e ε ˇ e E · V (cid:17) dx + Z Ω h s E (ˇ e u , v ) + ρ i s | s | ˇ e u · v i dx. Noting that 1 ≤ σ ≤ s − σ , for x ∈ Ω h + L , combining the boundness of ε , µ , Korn’s inequality(3.19) and (3.15), we haveRe e a (cid:0) ( ˇ e E , ˇ e u ) , ( ˇ e E , ˇ e u ) (cid:1) &
11 + s − σ s | s | (cid:16) k∇ × ˇ e E k L (Ω h + L ) + k s ˇ e E k L (Ω h + L ) (cid:17) + s min { , s } (cid:16) k∇ ˇ e u k F (Ω) + k∇ · ˇ e u k L (Ω) + k s ˇ e u k L (Ω) (cid:17) . σ := max { σ , σ } , which implies the uniform coercivity of e a ( · , · ).Arguing similarly as in the proof of Lemma 3.2 (noting that the TBC in the s-domain is nowreplaced with the PEC boundary condition), we can obtain the following lemma. Lemma 4.1.
The truncated PML variational problem (4.8) has a unique solution ( ˇ e E , ˇ e u ) ∈ f X s for each s ∈ C + with Re( s ) = s > . Further, it holds that k∇ × ˇ e E k L (Ω h + L ) + k s ˇ e E k L (Ω h + L ) . s − (1 + s − σ ) k s ˇ J k L (Ω h ) , (4.9) k∇ ˇ e u k F (Ω) + k∇ · ˇ e u k L (Ω) + k s ˇ e u k L (Ω) . q s − σ s min { , s } k ˇ J k L (Ω h ) . (4.10)Taking the inverse Laplace transform of system (4.6), we obtain the truncated PML problemin the time domain ∇ × e E + e µ∂ t f H = , in Ω h + L × (0 , T ) ∇ × f H − e ε∂ t e E = J , in Ω h + L × (0 , T ) ρ i ∂ e u ∂t − ∆ ∗ e u = , in Ω × (0 , T ) e E | t =0 = f H | t =0 = , in Ω h + L e u ( x,
0) = ∂ t e u ( x,
0) = , in Ω n × [ e E ] = n × [ µ − ∇ × e E ] = , on Γ f × (0 , T ) e E × n j = , on Γ h j ± L j × (0 , T ) , j = 1 , . (4.11)Note that s appearing in the matrix e µ and e ε is an arbitrarily fixed, positive parameter, asmentioned earlier at the beginning of this subsection. In the Laplace transform domain, thetransform variable s ∈ C + is taken so that Re( s ) = s >
0, and in the subsequent study of thewell-posedness and convergence of the truncated PML problem (4.11), we take s = 1 /T .The well-posedness and stability of the truncated PML problem in the time domain (4.11)can be obtained similarly as Theorem 3.3 with using the estimate (4.9)-(4.10) in Lemma 4.1 aswell as the energy method. Theorem 4.2.
Let s = 1 /T . The truncated initial-boundary value problem (4.11) has a uniquesolution (cid:0) e E ( x, t ) , f H ( x, t ) , e u ( x, t ) (cid:1) satisfying e E ( x, t ) ∈ L (cid:0) , T ; H (curl , Ω h + L ) (cid:1) ∩ H (cid:0) , T ; L (Ω h + L ) (cid:1) , f H ( x, t ) ∈ L (cid:0) , T ; H (curl , Ω h + L ) (cid:1) ∩ H (cid:0) , T ; L (Ω h + L ) (cid:1) , e u ( x, t ) ∈ L (cid:0) , T ; H (Ω) (cid:1) ∩ H (cid:0) , T ; L (Ω) (cid:1) , with the stability estimate max t ∈ [0 ,T ] (cid:16) k ∂ t e E k L (Ω h + L ) + k∇ × e E k L (Ω h + L ) + k ∂ t f H k L (Ω h + L ) + k∇ × f H k L (Ω h + L ) (cid:17) . (1 + σ T ) k J k H (0 ,T ; L (Ω h ) ) , and max t ∈ [0 ,T ] ( k ∂ t e u k L (Ω) + k∇ · e u k L (Ω) + k∇ e u k F (Ω) ) . p σ T k J k L (0 ,T ; L (Ω h ) ) . .2 EtM operators for the PML problem Recalling the truncated PML problem (4.6) in s -domain, let ˇ e E = ( ˇ e E , ˇ e E , ˇ e E ) ⊤ and ˇ f H =( ˇ e H , ˇ e H , ˇ e H ) ⊤ , denote by ˇ e E Γ hj = − n j × ( n j × ˇ e E | Γ hj ) = ( ˇ e E ( e x, h j ) , ˇ e E ( e x, h j ) , ⊤ and ˇ f H × n j =( ˇ e H ( e x, h j ) , − ˇ e H ( e x, h j ) , ⊤ the tangential component of the electric field and the tangential traceof the magnetic field on Γ h j , respectively. We start by introducing the EtM operators for thePML problem (4.6) e B j : H − / (curl , Γ h j ) → H − / (div , Γ h j )ˇ e E Γ hj → ˇ f H × n j where ˇ e E and ˇ f H satisfy the following equations in the PML layer Ω L j h j ∇ × ˇ e E + e µ j s ˇ f H = , in Ω L j h j ∇ × ˇ f H − e ε j s ˇ e E = , in Ω L j h j ˇ e E ( e x, x ) = ˇ e E ( e x, h j ) , on Γ h j ˇ e E × n j = , on Γ h j ± L j . (4.12)Using the Maxwell’s equations in (4.12), we easily haveˇ e H ( e x, h j ) = ( µ j s ) − ( ∂ x ˇ e E − ∂ x ˇ e E ) , (4.13) − ˇ e H ( e x, h j ) = ( µ j s ) − ( ∂ x ˇ e E − ∂ x ˇ e E ) . (4.14)Eliminating magnetic field f H from (4.12) and writing it into component form, we obtain σ − ∂ x ( σ − ∂ x ˇ e E ) + ∂ x ˇ e E − ∂ x [ ∂ x ˇ e E + σ − ∂ x ( σ − ˇ e E )] − s µ j ε j ˇ e E = 0 , (4.15a) σ − ∂ x ( σ − ∂ x ˇ e E ) + ∂ x ˇ e E − ∂ x [ ∂ x ˇ e E + σ − ∂ x ( σ − ˇ e E )] − s µ j ε j ˇ e E = 0 , (4.15b) ∂ x ( ∂ x ˇ e E + ∂ x ˇ e E ) − ∂ x ˇ e E − ∂ x ˇ e E + s µ j ε j ˇ e E = 0 . (4.15c)Noting that ∇ · ( e ε j ˇ e E ) = ε j ( σ∂ x ˇ e E + σ∂ x ˇ e E + ∂ x ( σ − ˇ e E )) = 0 , (4.16)then inserting (4.16) into (4.15) yields σ − ∂ x ( σ − ∂ x ˇ e E ) + ∂ x ˇ e E + ∂ x ˇ e E − s µ j ε j ˇ e E = 0 , (4.17a) σ − ∂ x ( σ − ∂ x ˇ e E ) + ∂ x ˇ e E + ∂ x ˇ e E − s µ j ε j ˇ e E = 0 , (4.17b) ∂ x [ σ − ∂ x ( σ − ˇ e E )] + ∂ x ˇ e E + ∂ x ˇ e E − s µ j ε j ˇ e E = 0 . (4.17c)18or convenience, we only consider the derivation of EtM operator e B on Γ h . To do this,taking the Fourier transform of (4.17a) and (4.17b) with respect to e x leads to the ODEs ∂ x b ˇ e E j ( ξ, x ) − ( µ ε s + | ξ | ) b ˇ e E j ( ξ, x ) = 0 , in Ω L h . b ˇ e E j ( ξ, x ) = b ˇ e E j ( ξ, h ) , on Γ h b ˇ e E j ( ξ, x ) = 0 , on Γ h + L . (4.18)The general solutions of ODEs (4.18) can be easily represented as b ˇ e E j ( ξ, x ) = A j e β ( ξ )(ˆ x − h ) + B j e − β ( ξ )(ˆ x − h ) , h < x < h + L . (4.19)Letting x = h and x = h + L and applying the boundary conditions in (4.19), respectivelyyields A j = − e − β ( ξ ) e L b ˇ e E j ( ξ,h ) e β ( ξ ) e L − e − β ( ξ ) e L , B j = e β ( ξ ) e L b ˇ e E j ( ξ,h ) e β ( ξ ) e L − e − β ( ξ ) e L , where e L := Z h + L h σ ( τ ) dτ = L + s − m + 1 L σ . (4.20)Hence, the solution of (4.18) is described as b ˇ e E j ( ξ, x ) = e − β ( ξ )(ˆ x − h − e L ) − e β ( ξ )(ˆ x − h − e L ) e β ( ξ ) e L − e − β ( ξ ) e L b ˇ e E j ( ξ, h ) , h < x < h + L . (4.21)Taking the normal derivative of (4.21) and evaluate the value on Γ h , we obtain ∂ b ˇ e E j ( ξ, h ) ∂x = − β ( ξ ) coth[ β ( ξ ) e L ] b ˇ e E j ( ξ, h ) , (4.22)where coth( t ) := e t + e − t e t − e − t denotes the hyperbolic cotangent function and the fact that σ = 1 onΓ h has been used.Next, we consider the equation (4.17c). Let P = σ − ˇ e E , by divergence free condition (4.16)and PEC boundary condition on Γ h + L , we have ∂ x P ( e x, x ) = 0 , on Γ h + L . Taking the Fourier transform of (4.17c) with respect to e x , we obtain ∂ x b P ( ξ, x ) − ( µ ε s + | ξ | ) b P ( ξ, x ) = 0 , in Ω L h . b P ( ξ, x ) = b ˇ e E ( ξ, h ) , on Γ h ∂ ˆ x b P ( ξ, x ) = 0 . on Γ h + L (4.23)Similarly, we get the general solution of (4.23) that b P ( ξ, x ) = e − β ( ξ )(ˆ x − h − e L ) + e β ( ξ )(ˆ x − h − e L ) e β ( ξ ) e L + e − β ( ξ ) e L b ˇ e E ( ξ, h ) , h < x < h + L . (4.24)19aking the normal derivative of (4.24) and evaluate the value on Γ h , we obtain ∂ b P ( ξ, h ) ∂x = − β ( ξ )coth[ β ( ξ ) e L ] b ˇ e E ( ξ, h ) , It follows from (4.16) and σ = 1 on Γ h that b ˇ e E ( ξ, h ) = − coth[ β ( ξ ) e L ] ∂ x b P ( ξ, h ) β ( ξ )= coth[ β ( ξ ) e L ] iβ ( ξ ) (cid:18) ξ b ˇ e E ( ξ, h ) + ξ b ˇ e E ( ξ, h ) (cid:19) . This, combining (4.13)-(4.14) and (4.22) leads to b ˇ e H ( ξ, h ) = 1 µ s (cid:20) iξ b ˇ e E ( ξ, h ) − ∂ x b ˇ e E ( ξ, h ) (cid:21) = coth[ β ( ξ ) e L ] µ s (cid:20) − ξ β ( ξ ) (cid:18) ξ b ˇ e E ( ξ, h ) + ξ b ˇ e E ( ξ, h ) (cid:19) + β ( ξ ) b ˇ e E ( ξ, h ) (cid:21) = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:20) ε µ s b ˇ e E ( ξ, h ) + ξ (cid:18) ξ b ˇ e E ( ξ, h ) − ξ b ˇ e E ( ξ, h ) (cid:19)(cid:21) , and − b ˇ e H ( ξ, h ) = 1 µ s (cid:20) iξ b ˇ e E ( ξ, h ) − ∂ x b ˇ e E ( ξ, h ) (cid:21) = coth[ β ( ξ ) e L ] µ s (cid:20) − ξ β ( ξ ) (cid:18) ξ b ˇ e E ( ξ, h ) + ξ b ˇ e E ( ξ, h ) (cid:19) + β ( ξ ) b ˇ e E ( ξ, h ) (cid:21) = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:20) ε µ s b ˇ e E ( ξ, h ) + ξ (cid:18) ξ b ˇ e E ( ξ, h ) − ξ b ˇ e E ( ξ, h ) (cid:19)(cid:21) . Now, for any tangential vector ω = ( ω , ω , ⊤ defined on Γ h , we obtain the explicitrepresentation of the EtM operator e B e B ω = ( v , v , ⊤ , (4.25)where b v = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:2) ε µ s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , b v = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:2) ε µ s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , with e L := Z h + L h σ ( τ ) dτ = L + s − m + 1 L σ . (4.26)Similarly, for any tangential vector ω = ( ω , ω , ⊤ defined on Γ h , the EtM operator e B hasthe following form e B ω = ( v , v , ⊤ , (4.27)20here b v = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:2) ε µ s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , b v = coth[ β ( ξ ) e L ] µ sβ ( ξ ) (cid:2) ε µ s b ω + ξ ( ξ b ω − ξ b ω ) (cid:3) , with e L := Z h h − L σ ( τ ) dτ = L + s − m + 1 L σ . (4.28)We now find that the truncated PML problem (4.7) is equivalently reduced to the followingboundary value problem ∇ × (( s e µ ) − ∇ × ˇ e E ) + s e ε ˇ e E = − ˇ J , in Ω h + L ∆ ∗ ˇ e u − ρ i s ˇ e u = , in Ω n × [ ˇ e E ] = n × [ µ − ∇ × ˇ e E ] = , on Γ f − n × ( s e µ ) − ∇ × ˇ e E = T ˇ e u , on Γ n × ˇ e E = n × s ˇ e u , on Γ( sµ j ) − ∇ × ˇ e E × n j + e B j [ ˇ e E Γ hj ] = , on Γ h j , j = 1 , . (4.29)The variational formulation of (4.29) can be obtained: to find ( ˇ e E , ˇ e u ) ∈ X s such that a p (cid:0) ( ˇ e E , ˇ e u ) , ( V , v ) (cid:1) = − Z Ω h ˇ J · V dx for all ( V , v ) ∈ X s , (4.30)where the sesquilinear form a p ( · , · ) is defined as a p (cid:0) ( ˇ e E , ˇ e u ) , ( V , v ) (cid:1) = Z Ω h (cid:0) ( s e µ ) − ( ∇ × ˇ e E ) · ( ∇ × V ) dx + s e ε ˇ e E · V (cid:1) dx (4.31)+ Z Ω h s E (ˇ e u , v ) + ρ i s | s | ˇ e u · v i dx + X j =1 h e B j [ ˇ e E Γ hj ] , V Γ hj i Γ hj . In this section, we shall give an error estimate between the solution ( E , u ) of the originalequations (3.2) and the solution ( e E , e u ) of the truncated PML problem (4.11). The followingfundamental Lemma on the error estimate between the EtM operators B j and the EtM operators e B j is essential to the exponential convergence of the PML method. Lemma 4.3.
For j = 1 , , denote L j = L j σ j m +1 . Then for s = s + is with s > , we have thefollowing estimate k B j − e B j k L ( H − / (curl , Γ hj ) ,H − / (div , Γ hj )) ≤ Γ j e − √ ε j µ j L j − e − √ ε j µ j L j := M j , where Γ j is defined in (4.37) , and L ( X, Y ) denotes the standard space of the bounded linearoperators from the Hilbert space X to the Hilbert space Y . roof. Given u = ( u , u , ⊤ , v = ( v , v , ⊤ ∈ H − / (curl , Γ h j ), we have from the definitionsof B j (see (3.4)) and e B j (see (4.25) and (4.27)) that h ( B j − e B j ) u , v i Γ hj = Z R (1 + | ξ | ) / µ j sβ j ( ξ ) (1 − coth[ β j ( ξ ) e L j ])(1 + | ξ | ) − / h ε j µ j s ( b u b v + b u b v ) + ( ξ b u − ξ b u ) · ( ξ b v − ξ b v ) i dξ. (4.32)Hence we need to estimate the term(1 + | ξ | ) / | β j ( ξ ) | (cid:12)(cid:12)(cid:12) − coth[ β j ( ξ ) e L j ] (cid:12)(cid:12)(cid:12) . Firstly, we denote ε j µ j s = a j + ib j , with a j = ε j µ j ( s − s ) , b j = 2 ε j µ j s s , and β j = ε j µ j s + | ξ | = φ j + ib j , with φ j = Re( ε j µ j s ) + | ξ | = a j + | ξ | . Noting that (1 + | ξ | ) / | β j ( ξ ) | = h (1 + φ j − a j ) φ j + b j i / , we define an auxiliary function F j ( t ) = (1 + t − a j ) t + b j , t ≥ a j . Simple calculations gives the derivative F ′ j ( t ) = 2( t − a j + 1)[( a j − t + b j ]( t + b j ) . We consider the following two cases:(I) If s ≥ s , then a j ≤
0. Setting K j := b j − a j , it can be verified that F j ( t ) increases in[ a j , K j ], and decreases in [ K j , + ∞ ). Hence F j ( t ) reaches its maximum (1 − a j ) + b j b j at K j .(II) If s < s , then a j >
0. We have another three possibilities.(II.a) 1 − a j <
0, then F j ( t ) increases in [ a j , + ∞ ), hence F j ( t ) ≤ lim t → + ∞ F j ( t ) = 1 . (II.b) 1 − a j = 0, it can be easily verified that F j ( t ) = t t + b j ≤ . − a j >
0, that is 1 − ε j µ j s + ε j µ j s >
0. In this case, we need to compare the size of a j and K j . Note that K j ≤ a j is equivalent to s + (cid:18) s + 1 ε j µ j (cid:19) s + s (cid:18) s − ε j µ j (cid:19) ≤ . Thus define ε ( s ) := − ( s + 12 ε j µ j ) + s s ε j µ j + 14 ε j µ j . (4.33)We further have three cases:(II.c.i) 1 − ε j µ j s <
0, then s > ε j µ j s − ε j µ j > ε ( s ) <
0, then a j < K j . Hence F j ( t ) ≤ F j ( K j ) = (1 − a j ) + b j b j . (II.c.ii) 1 − ε j µ j s = 0, then s > ε ( s ) = 0, it holds that F j ( t ) ≤ F j ( K j ) = 1 + s s .(II.c.iii) 1 − ε j µ j s >
0, then we have the following two cases:(II.c.iii.1) If s ≤ ε ( s ), then K j ≤ a j , therefore F j ( t ) decreases in [ a j , + ∞ ), then F j ( t ) ≤ F j ( a j ) = 1 a j + b j . (II.c.iii.2) If s > ε ( s ), then K j > a j . Hence F j ( t ) ≤ F j ( K j ) = (1 − a j ) + b j b j . Recalling the definitions of a j and b j , by the above discussions, we arrive at(1 + | ξ | ) / | β j ( ξ ) | ≤ Λ j ( s , s ) , (4.34)where Λ j ( s , s ) is defined as:(1) when 1 − ε j µ j s < j ( s , s ) = , ≤ s ≤ s − ε j µ j , h (cid:0) − ε j µ j ( s − s ) (cid:1) ε j µ j s s i / , s > s − ε j µ j . (2) when 1 − ε j µ j s = 0, Λ j ( s , s ) = (cid:0) s s (cid:1) / . (3) when 1 − ε j µ j s > j ( s , s ) = √ ε j µ j | s | , ≤ s ≤ ε ( s ) , h (cid:0) − ε j µ j ( s − s ) (cid:1) ε j µ j s s i / , s > ε ( s ) .
23n the following, we further estimatesup ξ ∈ R (cid:12)(cid:12)(cid:12) − coth[ β j ( ξ ) e L j ] (cid:12)(cid:12)(cid:12) = sup ξ ∈ R (cid:12)(cid:12)(cid:12) e − β jr ( ξ ) e L j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e − β jr ( ξ )+ iβ ji ( ξ )) e L j (cid:12)(cid:12)(cid:12) ≤ sup ξ ∈ R e − β jr ( ξ ) e L j − e − β jr ( ξ ) e L j , (4.35)where β j r ( ξ ) = Re[ β j ( ξ )], and β j i ( ξ ) = Im[ β j ( ξ )]. By the formulas z / = r | z | + z i sgn( z ) r | z | − z , for z = z + iz , Re[ z / ] > , we have β j r ( ξ ) = s | β j ( ξ ) | + Re[ β j ( ξ )]2= " [( ε j µ j ( s − s ) + | ξ | ) + 4 ε j µ j s s ] / + ε j µ j ( s − s ) + | ξ | / . Note that e − βjr ( ξ ) e Lj − e − βjr ( ξ ) e Lj is monotonically decreasing with respect to β j r ( ξ ). Hence, we need toseek the maximum of β j r ( ξ ) in R . Simple calculations yields that ξ = 0 is the unique extremepoint of the function β j r ( ξ ), and β j r (0) = √ ε j µ j s , e − β jr ( ξ ) e L j − e − β jr ( ξ ) e L j (cid:12)(cid:12)(cid:12) ξ =0 = 2 e − √ ε j µ j s e L j − e − √ ε j µ j s e L j . Besides, β j r ( ξ ) → + ∞ , thereby, e − βjr ( ξ ) e Lj − e − βjr ( ξ ) e Lj →
0, as ξ → ∞ .By the definitions of e L and e L (see (4.26) and (4.28)), we therefore conclude thatsup ξ ∈ R e − β jr ( ξ ) e L j − e − β jr ( ξ ) e L j = 2 e − √ ε j µ j s e L j − e − √ ε j µ j s e L j ≤ e − √ ε j µ j L j − e − √ ε j µ j L j . (4.36)Combining (4.34) and(4.36) as well as Cauchy-Schwartz inequality for (4.32) yields |h ( B j − e B j ) u , v i Γ hj | ≤ Γ j e − √ ε j µ j L j − e − √ ε j µ j L j k u k H − / (curl , Γ hj ) k v k H − / (curl , Γ hj ) , where Γ j = 1 µ j | s | Λ j ( s , s ) max { ε j µ j | s | , } . (4.37)This completes the proof. 24et ω = ( ˇ E , ˇ u ) and ω p = ( ˇ e E , ˇ e u ) be the solutions of the variational problems (3.12) and(4.30), respectively. By the definitions of variational formulations of a ( · , · ) and a p ( · , · ), we obtain | a ( ω − ω p , ω − ω p ) | = | a ( ω , ω − ω p ) − a ( ω p , ω − ω p ) | = | a p ( ω p , ω − ω p ) − a ( ω p , ω − ω p ) | = (cid:12)(cid:12)(cid:12) X j =1 h ( B j − e B j )[ ˇ e E Γ hj ] , ( ˇ E − ˇ e E ) Γ hj i Γ hj (cid:12)(cid:12)(cid:12) ≤ η X j =1 k B j − e B j k L ( H − / (curl , Γ hj ) ,H − / (div , Γ hj )) k ω p k X s k ω − ω p k X s , (4.38)where the constant η = max { p h − h ) − , √ } is defined in Lemma B.3. Now we arriveat our main theorem by concluding the above argument. Theorem 4.4.
Let ( E , u ) be the solution of problem (3.2) , and ( e E , e u ) be the solution of problem (4.11) with s = 1 /T , σ = max { σ , σ } , then under the assumptions (3.23) and (3.24) we havethe following error estimate Z T ( k E − e E k H (curl , Ω h ) + k u − e u k H (Ω) ) dt (4.39) . max { , T } ( T + 2 T )( γ + γ )(1 + σ T ) (cid:16) X j =1 e −√ ε j µ j σ j L j − e −√ ε j µ j σ j L j (cid:17) k J k H (0 ,T ; L (Ω h ) ) , where γ and γ are positive constants independent of ( E , u ) and ( e E , e u ) , but that may dependon T .Proof. Combining (4.38) with Lemma 4.3 and the uniform coercivity (3.20) of a ( · , · ), we have k ω − ω p k X s ≤ C − η ( M + M ) k ω p k X s . By the Parseval identity (A.5) and the definitions of M , M in Lemma 4.3, we get Z ∞ e − s t k L − ( ω − ω p ) k X s dt = 12 π Z ∞−∞ k ω − ω p k X s ds ≤ π Z ∞−∞ C − η (cid:16) X j =1 Γ j e − √ ε j µ j L j − e − √ ε j µ j L j (cid:17) k ω p k X s ds . This implies that Z T ( k E − e E k H (curl , Ω h ) + k u − e u k H (Ω) ) dt ≤ e s T Z ∞ e − s t ( k E − e E k H (curl , Ω h ) + k u − e u k H (Ω) ) dt = e s T Z ∞ e − s t k L − ( ω − ω p ) k X s dt ≤ η e s T π Z ∞ C − (cid:16) X j =1 Γ j e − √ ε j µ j L j − e − √ ε j µ j L j (cid:17) k ω p k X s ds . (4.40)25ince s > C in (3.20) and Γ j in (4.37), thereexists a sufficiently large positive constant M , such that C − Γ j , C − Γ Γ ≤ γ | s | , (4.41)when s ≥ M , where γ is a constant independence of s . On the other hand, it’s clear that C − Γ j , C − Γ Γ ≤ γ , (4.42)when 0 ≤ s ≤ M , where γ is a constant independence of s . Thus the last inequality in (4.40)becomes Z ∞ C − (cid:16) X j =1 Γ j e − √ ε j µ j L j − e − √ ε j µ j L j (cid:17) k ω p k X s ds ≤ (cid:16) X j =1 e − √ ε j µ j L j − e − √ ε j µ j L j (cid:17) (cid:18)Z M γ k ω p k X s ds + Z ∞ M γ k s ω p k X s ds (cid:19) . (4.43)Now, only the right-hand integral in (4.43) remains to be estimated. Combining Lemma 4.1with Parseval identity (A.5) and the assumptions (3.23)-(3.24) yields Z M γ k ω p k X s ds + Z ∞ M γ k s ω p k X s ds ≤ (1 + s − σ ) (cid:16) Z M γ (cid:2) s s min { , s } k ˇ J k L (Ω h ) + s − k s ˇ J k L (Ω h ) (cid:3) ds + Z ∞ M γ (cid:2) s s min { , s } k s ˇ J k L (Ω h ) + s − k s ˇ J k L (Ω h ) (cid:3) ds (cid:17) ≤ s s min { , s } ( γ + γ )(1 + s − σ ) Z ∞ X l =0 k s l ˇ J k L (Ω h ) ds = π s s min { , s } ( γ + γ )(1 + s − σ ) Z ∞ X l =0 k ∂ lt J k L (Ω h ) dt. By this inequality and (4.40), (4.43) the required estimate (4.39) follows easily on taking s = T − and using the assumption (3.24) again, where integer m ≥ m = 1) noting the definition of L j = σ j L j / ( m + 1). The proof is thus complete. Remark 4.5.
Theorem 4.4 implies that, for large T the exponential convergence of the PMLmethod can be achieved by enlarging the thickness L j or the PML absorbing parameter σ j whichincreases as ln T . In this paper, the scattering of a time-dependent electromagnetic wave by an an elastic bodyimmersed in the lower half-space of a two-layered background medium is studied. The well-posedness and stability estimate is verified by using the Laplace transform, the variational26ethod and the energy method. In addition, we propose an effective PML method to solve thisinteraction problem, based on a real coordinate stretching technique associated with [Re( s )] − in the frequency domain, where s is the Laplace transform variable. The well-posedness andstability of the truncated PML problem are proved by using the Laplace transform and energymethod. At last, through the error estimate between the EtM operators of the original prob-lem and the EtM operators for the PML problem, we establish the exponential convergencedepending on the thickness and parameters of the PML layers.In practical computation, the PML medium must be truncated along the lateral directionwhich may be achieved by constructing the rectangular or cylindrical PML. Further, the idea ofreal coordinate stretching could be extended to other time-dependent scattering problems, suchas diffraction gratings, elastic rough surface scattering problems. We hope to report such resultsin the future. A Laplace transform
For each s ∈ C + , the Laplace transform of the vector field u ( t ) is defined as:ˇ u ( s ) = L ( u )( s ) = Z ∞ e − st u ( t ) dt. The Fourier transform of φ ( e x, x ) is normalized as follows: b φ ( ξ, x ) = F ( φ )( ξ, x ) = 12 π Z R e − i e x · ξ φ ( e x, x ) d e x, ξ ∈ R and the inverse Fourier transform of b φ ( ξ ) is φ ( e x, x ) = F − ( b φ )( e x, x ) = 12 π Z R e i e x · ξ b φ ( ξ, x ) dξ. Some related properties on the Laplace transform and its inversion are summarized as L ( d u dt )( s ) = s L ( u )( s ) − u (0) , (A.1) L ( d u dt )( s ) = s L ( u )( s ) − s u (0) − d u dt (0) , (A.2) L (cid:16) Z t u ( τ ) dτ (cid:17) ( s ) = s − L ( u )( s ) , (A.3)which can be easily verified from the integration by parts.Next, we present the relation between Laplace and Fourier transform. According to thedefinition on the Fourier transform, it holds √ π F ( u ( · ) e − s · ) = Z + ∞−∞ u ( t ) e − s t e − is t dt = Z ∞ u ( t ) e − ( s + is ) t dt = L ( u )( s + is ) . We can verify from the formula of the inverse Fourier transform that u ( t ) e − s t = 1 √ π F − { F ( u ( · ) e − s · ) } = 1 √ π F − (cid:16) L ( u ( s + is )) (cid:17) , u ( t ) = 1 √ π F − (cid:16) e s t L ( u ( s + is )) (cid:17) . (A.4)where F − denotes the inverse Fourier transform with respect to s .By (A.4), the Plancherel or Parseval identity for the Laplace transform can be obtained (see[22, (2.46)]). Lemma A.1 (Parseval identity) . If ˇ u = L ( u ) and ˇ v = L ( v ) , then π Z ∞−∞ ˇ u ( s ) · ˇ v ( s ) ds = Z ∞ e − s t u ( t ) · v ( t ) dt. (A.5) for all s > λ where λ is the abscissa of convergence for the Laplace transform of u and v . Lemma A.2. ([39, Theorem 43.1])
Let ˇ ω ( s ) denotes a holomorphic function in the half plane s > σ , valued in the Banach space E . The following statements are equivalent:1. there is a distribution ω ∈ D ′ + ( E ) whose Laplace transform is equal to ˇ ω ( s ) , where D ′ + ( E ) isthe space of distributions on the real line which vanish identically in the open negative halfline;2. there is a σ with σ ≤ σ < ∞ and an integer m ≥ such that for all complex numbers s with s > σ , it holds that k ˇ ω ( s ) k E . (1 + | s | ) m . B Functional spaces
In this subsection, we give a brief summary of some fundamental functional spaces. For abounded Lipschitz domain D ∈ R with unit outward normal vector ν on its boundary Σ, weset H (curl , D ) := { ω ∈ L ( D ) : ∇ × ω ∈ L ( D ) } , which is clearly a Hilbert space equipped with the norm k ω k H (curl ,D ) = (cid:16) k ω k L ( D ) + k∇ × ω k L ( D ) (cid:17) / . From [9], we define the bounded surjective trace operator γ , tangential trace operator γ t andtangential projection operator γ T by γ : H ( D ) → H / (Σ) , γϕ = ϕ on Σ ,γ t : H ( D ) → L t (Σ) , γ t ω = ω × ν on Σ ,γ T : H ( D ) → L t (Σ) , γ T ω = ν × ( ω × ν ) on Σ , where L t (Σ) := { ω ∈ L (Σ) : ω · ν = 0 } and denote by ω Σ = ν × ( ω × ν ) the tangentialcomponent of ω on Σ. In fact, the range of γ t and γ T H / k (Σ) := (cid:8) ξ ∈ L t (Σ) : ξ = γ T ω , ω ∈ H ( D ) } ,H / ⊥ (Σ) := (cid:8) ξ ∈ L t (Σ) : ξ = γ t ω , ω ∈ H ( D ) } , L t (Σ) , and γ t : H ( D ) → H / ⊥ (Σ), γ T : H ( D ) → H / k (Σ) are bounded andsurjective operators. The dual spaces of H / ⊥ (Σ) and H / k (Σ) with respect to the pivot space L t (Σ) are denoted by H − / ⊥ (Σ) and H − / k (Σ), respectively. In this paper, we will also usethe notion γ t φ (or γ T φ ) for the composite operator γ t ◦ γ − φ (or γ T ◦ γ − φ ). According to [9,Theorem 4.1], the definitions of γ t and γ T can be extended into H (curl , D ). Lemma B.1. H − / (Div , Σ) := n λ ∈ H − / k (Σ) : Div λ ∈ H − / (Σ) o and H − / (Curl , Σ) := n λ ∈ H − / ⊥ (Σ) : Curl λ ∈ H − / (Σ) o . The operators γ t : H (curl , D ) → H − / (Div , Σ) and γ T : H (curl , D ) → H − / (Curl , Σ) arelinear, continuous, and surjective. Moreover, the L t (Σ) -inner product can be extended to definea duality product h· , ·i Σ between the spaces H − / (Div , Σ) and H − / (Curl , Σ) . We refer to [9] for the detailed definitions of the surface divergence and surface scalar curl op-erators Div and Curl in lemmaB.1. In addition, the dual pair H − / (Div , Σ) and H − / (Curl , Σ)satisfy the following vector integration by parts Z D ( ω · ∇ × v − ∇ × ω · v ) dx = h γ t ω , γ T v i Σ ∀ ω , v ∈ H (curl , D ) . (B.1)For a finite strip domain Ω h , the definition of Sobolev space H (curl , Ω h ) can be found in[26, 35]. Denote by C ∞ e x the linear space of infinitely differentiable functions with compactsupport with respect to the variable e x on Ω h . According to the dense argument of C ∞ e x (Ω h ) in H (curl , Ω h ) (see [35, Lemma 2.2]), one may only need to consider the proof in C ∞ e x (Ω h ) andthen extend them by limiting argument to more general functions in H (curl , Ω h ). Therefore, theboundary integrals only on Γ h j and Γ need to be considered when formulating the variationalproblems in Ω h .For a smooth vector ω = ( ω , ω , ω ) ⊤ defined on Γ h j , denote bydiv Γ hj ω = ∂ x ω + ∂ x ω , curl Γ hj ω = ∂ x ω − ∂ x ω the surface divergence and the surface scalar curl, respectively. Now we introduce two vectortrace spaces on the planar surface: H − / (curl , Γ h j ) := (cid:8) ω ∈ H − / (Γ h j ) : ω = 0 , curl Γ hj ω ∈ H − / (Γ h j ) (cid:9) ,H − / (div , Γ h j ) := (cid:8) ω ∈ H − / (Γ h j ) : ω = 0 , div Γ hj ω ∈ H − / (Γ h j ) (cid:9) , which are equipped with the norm defined by the Fourier transform: k ω k H − / (curl , Γ hj ) = (cid:16) Z R (1 + | ξ | ) − / ( | b ω | + | b ω | + | ξ b ω − ξ b ω | ) dξ (cid:17) / , k ω k H − / (div , Γ hj ) = (cid:16) Z R (1 + | ξ | ) − / ( | b ω | + | b ω | + | ξ b ω + ξ b ω | ) dξ (cid:17) / . The following two lemmas about the duality between the spaces H − / (curl , Γ h j ) and H − / (div , Γ h j ) and the trace regularity in H (curl , Ω h ) can be found the proofs in [35, Lemma2.3, Lemma 2.4]. 29 emma B.2. The spaces H − / (div , Γ h j ) and H − / (curl , Γ h j ) are mutually adjoint with respectto the scalar product in L (Γ h j ) defined by h ω , v i Γ hj = Z Γ hj ω · v dγ = Z R ( b ω b v + b ω b v ) dξ. Lemma B.3.
Let η = max { p h − h ) − , √ } . We have the estimate k ω k H − / (curl , Γ hj ) ≤ η k ω k H (curl , Ω) , ∀ ω ∈ H (curl , Ω h ) . Acknowledgements
The work was partially supported by the National Natural Science Foundation of Chinagrants 11771349 and 91630309, and the National Research Foundation of Korea (NRF-2020R1I1A1A01073356).
References [1] G. Bao, Y. Gao and P. Li, Time-domain analysis of an acoustic-elastic interaction problem,
Arch. Rational Mech. Anal. (2018), 835-884.[2] G. Bao and H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations,
SIAM J. Numer. Anal. (2005), 2121-2143.[3] J.P. B´erenger, A perfectly matched layer for the absorption of electromagnetic waves, J.Comput. Phys. (1994), 185-200.[4] A. Bernardo, A. Marquez and S. Meddahi, Analysis of an interaction problem between anelectromagnetic field and an elastic body,
Int. J. Numer. Anal. Model. (2010), 749-765.[5] J.H. Bramble, J.E. Pasciak, Analysis of a finite PML approximation for the three dimen-sional time-harmonic Maxwell and acoustic scattering problems, Math. Comp. (2007),597-614.[6] J.H. Bramble and J.E. Pasciak, Analysis of a finite element PML approximation for thethree dimensional time-harmonic Maxwell problem, Math. Comp. (2008), 1-10.[7] J.H. Bramble and J.E. Pasciak, Analysis of a Cartesian PML approximation to the threedimensional electromagnetic wave scattering problem, Int. J. Numer. Anal. Model. (2012),543-561.[8] J.H. Bramble and J.E. Pasciak, Analysis of a Cartesian PML approximation to acousticscattering problems in R and R , Math. Comp. (2013), 209-230.[9] A. Buffa, M. Costabel and D. Sheen, On traces for H (curl , Ω) in Lipschitz domains,
J.Math. Anal. Appl. (2002), 845-867.[10] F. Cakoni and G. C. Hsiao, Mathematical model of the interaction problem between elec-tromagnetic field and elastic body, in:
Acoustics, Mechanics, and the Related Topics ofMathematical Analysis , (2002), 48-54. 3011] S.N. Chandler-Wilde, E. Heinemeyer and R. Potthast, Existence, A well-posed integralequation formulation for three-dimensional rough surface scattering,
Proc. Roy. Soc. LondonA (2006), 3683-3705.[12] S.N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods forscattering by unbounded rough surfaces,
SIAM J. Math. Anal. (2005), 598-618.[13] S.N. Chandler-Wilde and P. Monk, The PML for rough surface scattering, Appl. Numer.Math. (2009), 2131-2154.[14] S.N. Chandler-Wilde, C.R. Ross, and B. Zhang, Scattering by infinite one-dimensionalrough surfaces, Proc. Roy. Soc. London A (1999), 3767-3787.[15] S.N. Chandler-Widle and B. Zhang, A uniqueness result for scattering by infinite roughsurfaces,
SIAM J. Appl. Math. (1998), 1774-1790.[16] Q. Chen and P. Monk, Discretization of the time domain CFIE for acoustic scatteringproblems using convolution quadrature, SIAM J. Math. Anal. (2014), 3107-3130.[17] Z. Chen, Convergence of the time-domain perfectly matched layer method for acousticscattering problems, Int. J. Numer. Anal. Model. (2009), 124-146.[18] Z. Chen and J.C. N´ed´elec, On Maxwell equations with the transparent boundary condition, J. Comput. Math. (2008), 284-296.[19] Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbinglayers for the wave scattering by periodic structures, SIAM J. Numer. Anal. (2003),799-826.[20] Z. Chen and X. Wu, Long-time stability and convergence of the uniaxial perfectly matchedlayer method for time-domain acoustic scattering problems, SIAM J. Numer. Anal. (2012), 2632-2655.[21] Z. Chen and W. Zheng, Convergence of the uniaxial perfectly matched layer method fortime-harmonic scattering problems in two-layered medium, SIAM J. Numer. Anal. (2010), 2158-2185.[22] A.M. Cohen, Numerical Methods for Laplace Transform Inversion , Springer, 2007.[23] Z. Chen and W. Zheng, PML method for electromagnetic scattering problem in a two-layermedium,
SIAM J. Numer. Anal. (2017), 2050-2084.[24] F. Collino, P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci.Comput. (1998), 2061-2090.[25] Y. Gao and P. Li, Analysis of time-domain scattering by periodic structures, J. Differ.Equations. (2016), 5094-5118.[26] Y. Gao and P. Li, Electromagnetic scattering for time-domain Maxwell’s equations in anunbounded structure,
Math. Models Methods Appl. Sci. (2017), 1843-1870.3127] Y. Gao, P. Li and B. Zhang, Analysis of transient acoustic-elastic interaction in an un-bounded structure, SIAM J. Math. Anal. (2017), 3951-3972.[28] G.N. Gatica, G.C. Hsiao and S. Meddahi, A coupled mixed finite element method for theinteraction problem between an electromagnetic field and an elastic body, SIAM J. Numer.Anal. (2010), 1338-1368.[29] H. Haddar and A. Lechleiter, Electromagnetic wave scattering from rough penetrable layers, SIAM J. Math. Anal. (2011), 2418-2443.[30] T. Hohage, F. Schmidt, and L. Zschiedrich, Solving time-harmonic scattering problemsbased on the pole condition II: Convergence of the PML method, SIAM J. Math. Anal. (2003), 547-560.[31] G.C. Hsiao and W.L. Wendland, Boundary Integral Equations , Springer, Berlin, 2008.[32] M. Lassas and E. Somersalo, On the existence and convergence of the solution of PMLequations,
Computing. (1998), 229-241.[33] J. Li and Y. Huang, Time-Domain Finite Element Methods for Maxwell’s Equations inMetamaterials , Springer, New York, 2012.[34] P. Li, L. Wang and A. Wood, Analysis of transient electromagnetic scattering from a three-dimensional open cavity,
SIAM J. Appl. Math. (2015), 1675-1699.[35] P. Li, H. Wu and W. Zheng, Electromagnetic scattering by unbounded rough surfaces, SIAM J. Math. Anal. (2011), 1205-1231.[36] P. Li, G. Zheng and W. Zheng, Maxwell’s equations in an unbounded structure, Math.Meth. Appl. Sci. (2017), 573-588.[37] G.A. Maugin, Continuum Mechanics of Electromagnetic Solids , North-Holland, Amster-dam, 1988.[38] F. Teixeira and W. Chew, Advances in the theory of perfectly matched layers,
Fast EfficientAlgorithms in Computational Electromagnetics (2001), 409-433.[39] F. Tr`eves, Basic Linear Partial Differential Equations , Academic Press, New York, 1975.[40] E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations,
Appl.Numer. Math. (1998), 533-557.[41] B. Wang and L. Wang, On L -stability analysis of time-domain acoustic scattering problemswith exact nonreflecting boundary conditions, J. Math. Study (2014), 65-84.[42] L. Wang, B. Wang and X. Zhao, Fast and accurate computation of time-domain acousticscattering problems with exact nonreflecting boundary conditions, SIAM J. Appl. Math. (2012), 1869-1898.[43] C. Wei and J. Yang, Analysis of a time-dependent fluid-solid interaction problem above alocal rough surface, Sci. China Math. (2020), 887-906.3244] C. Wei, J. Yang and B. Zhang, Convergence of the perfectly matched layer method fortransient acoustic-elastic interaction above an unbounded rough surface, arXiv:1907.09703,2019.[45] C. Wei, J. Yang and B. Zhang, A time-dependent interaction problem between an electro-magnetic field and an elastic body, Acta Math. Appl. Sin. Engl. Ser. (2020), 95-118.[46] C. Wei, J. Yang and B. Zhang, Convergence analysis of the PML method for time-domainelectromagnetic scattering problems, SIAM J. Numer. Anal. (2020), 1918-1940.[47] B. Zhang and S.N. Chandler-Wilde, Integral equation methods for scattering by infiniterough surfaces, Math. Methods Appl. Sci. (2003), 463-488.[48] X. Zhao and L. Wang, Efficient Spectral-Galerkin method for waveguide problem in infinitedomain, Commun. Appl. Math. Comput27