Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants

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1 Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants A.S. Bonnet-Bendhia, P. Ciarlet Jr., C. M. Zwölf CNRS-INRIA-ENSTA UMR 706 3, Boulevard Victor, Paris Cedex 5, France Abstract We consider a simplified scalar model problem related to Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic constants. We build two variational formulations equivalent to the model problem. We show that, under some suitable conditions, both formulations are well-posed since they fit into the coercive plus compact framework. Advantages over previous studies is the validity of the formulations in the general case of Lipschitz interface between the two media and L dielectric and magnetic constants. An interesting feature of these formulations is that they allow a simple finite element numerical implementation. Key words: wave transmission problem, opposite sign dielectric constants, augmented variational formulation Introduction Physical models describing the electromagnetic properties of some metamaterials, semiconductors near plasmon resonance [6], plasmas under cyclotron frequency and superconductors according to London s phenomenologic approach) lead to negative dielectric constant ɛ. In recent years [8] metamaterials, modelled with simultaneously negative dielectric constant ɛ and magnetic permeability µ, have been thorougly studied, due to their specific electromagnetic behaviour and their wide application range in modern electronics. For practical applications, it is therefore important to be able to capture numerically the electromagnetic field near interfaces between classical dielectric media ɛ > 0, µ > 0), and superconductors ɛ < 0, µ > 0) or metamaterials ɛ < 0, µ < 0). address: Carlo-Maria.Zwolf@ensta.fr C. M. Zwölf). URL: zwolf C. M. Zwölf). Preprint submitted to Elsevier Science 6 January 006

2 Mathematically however, due to the dielectric constant sign shift at the interface, the natural variational formulation of such problems is neither coercive nor coercive plus compact, so it seems not possible to fit straightforwardly the model into a framework leading to a well-posed problem. In this paper we focus on a simplified scalar model problem related to Maxwell equations, which involves similar interface discontinuities. Assume the domain is split in two parts Ω and Ω, with dielectric constant called ɛ, positive on Ω and ɛ negative on Ω, and consider the following equation ) div ɛ u + ω µu = f. ) This simplified model problem has already been studied in the case of a piecewise constant ɛ, such that ɛ R + and ɛ R. In [4] it has been shown, using integral equations, that for a smooth interface Σ = Ω Ω, the model problem fits into the Fredholm framework if the contrast κ := ɛ /ɛ is not equal to. In [7], using Dirichlet to Neumann operators, it has been shown that the model fits into the Fredholm framework if κ >> or κ << no required regularity of the interface Σ). The effect of a geometrical singularity on the interface id est, a non-smooth interface has been investigated more precisely in []. It has been proved there that, for an interface with a right angle, the operator associated with the non-coercive transmission problem is selfadjoint and has compact resolvent if κ / [ 3, /3]; similar results can be derived for any angle. Since it is very hard to generalize integral equation methods in the case of a nonsmooth interface geometry or in the case of non-constant ɛ i ) i=,, a variational approach is adopted here. By extending a method employed in [], we introduce two variational formulations allowing both a Lipschitz interface and variable ɛ and ɛ. To that aim, we introduce a new unknown, which is equal to the gradient of u in one of the subdomains. The two-field formulation is valid for interfaces between superconductor and dielectric medium. The three-field formulation is more general: it is valid also for interfaces between metamaterial and dielectric medium, and further it allows to consider a vanishing frequency ω. For each formulation we derive conditions on ɛ, µ and on the geometry, ensuring that the considered formulation fits into the coercive plus compact framework. One of the main interests of these formulations is that they can be solved numerically with a standard finite element method. As we have mentioned above, the three-field formulation is more general, however it is also more expensive computationally. Regularity assumptions and the model problem Let Ω be an open, bounded domain of R d d=,3) with Lipschitz boundary Ω. Due to the limited total number of pages, we present here only the 3D case. Nevertheless, the results and proofs can be derived in the same way in D.

3 It is assumed that the domain Ω can be split in two simply connected sub-domains Ω and Ω with Lipschitz boundaries: Ω = Ω Ω, Ω Ω =. Moreover, if we let Σ = Ω Ω be the interface, and define Γ i = Ω i \ Σ, it is assumed that Γ and Γ are connected. Hereafter we adopt the notation, for all quantities v defined on Ω, v i := v Ωi, for i =,. Furthermore, we use the notations If v i > 0 a. e. in Ω i : vi max = sup x Ωi v i x), vi min = inf x Ωi v i x). If v i < 0 a. e. in Ω i : v i + = sup x Ωi v i x), vi = inf x Ωi v i x). Finally, the outgoing normal from Ω i i=,) is called n i. From now on, we assume that ɛ belongs to L Ω), that it is strictly positive on Ω and strictly negative on Ω and that ɛ L Ω). Also, we assume that µ belongs to L Ω). By setting α := /ɛ and β := ω µ the scalar model problem ) for any given f L Ω), may be rewritten as: find u H Ω) such that div α u) + βu = f in Ω. ) We choose to apply, with no loss of generality, an homogeneous Dirichlet condition on Ω: in other words, u Ω = 0. In this case, the natural variational formulation of ) supplemented with this boundary condition is: find u H 0Ω) such that v H 0Ω), α u, v) L Ω) β u, v) L Ω) = f, v) L Ω). As α u, v) L Ω) has no specific sign, its coercivity does not hold. It is easy to prove that problem ) is equivalent to: find u, u ) X X such that +div α u ) + β u = f in Ω div α u ) + β u = f in Ω u i Γi = 0, i =, u Σ = u Σ α n u + α n u ) Σ = 0 weakly, 3) with X i := {p H Ω i ) p Γi = 0}. 3

4 In what follows, we will also use the following Sobolev spaces: H 0,Γi curl; Ω i ), ih 0,Σ curl; Ω i ) and X i, for i =,, respectively defined by H 0,Γi curl; Ω i ) := {p curl p L Ω i ), p n Γi = 0} H 0,Σ curl; Ω i ) := {p curl p L Ω i ), p n Σ = 0} X i := {p Hdiv ; Ω i ) curlp/ α i ) L Ω i ), p/ α i n) Γi = 0}. The spaces H 0,Γi curl; Ω i ), and H 0,Σ curl; Ω i ) are endowed with the usual norm of Hcurl; Ω i ), whereas X i is endowed with the graph norm. And finally,, ) i,j resp. i,j ) denotes the usual scalar product resp. norm) on H i Ω j ). Duality brackets on Σ are understood in the sense of the duality H / 00 Σ)) H / 00 Σ). 3 Two-field variational formulation 3. Derivation of the formulation As we will see in theorem 3.3, the formulation we are going to derive fits into the coercive plus compact framework when at least in one of two subdomains of Ω, the ratio α/β is negative. The main idea behind the construction of a suitable two-field formulation is to replace, in the subdomain where α k /β k < 0, the scalar unknown u k by the vector unknown u k := α k u k. In order to illustrate our approach, assume β > 0, almost everywhere in Ω, and set u := α u an equivalent choice would be u := α u provided that β < 0). Note that the condition α /β < 0 is needed only for the well-posedness of the formulation and not for its derivation. To build the two-field formulation 4) let us successively take the L -scalar product of the first equation of 3) with a test function v X, integrate by parts, and use the second equality of traces in 3): α u, v ) 0, + u n, v Σ β u, v ) 0, = f, v ) 0,. divide the second equation of 3) by β ; take the L -scalar product between the result of the previous operation and the divergence of a vector test function v X, integrate by parts, and use the first equality of traces in 3): ) ) ) u div α, v u f +, div v + v n, u Σ =, div v. β β Since β := ω µ, this corresponds exactly to ω 0 and µ > 0 a. e. 4

5 sum the results obtained at points one and two. Finally, to recover an augmented variational formulation as proposed in [3], we may add the term curl u α, curl v ), since curl u = 0 by construction. α α L Ω ) The overall result is the two-field formulation below: find U = u, u ) X X such that V = v, v ) X X, AU, V ) = LV ), 4) where the forms A and L are respectively defined by AU, V ) := and ) ) u div α, v u +, div v + curl u β α, curl v ) α + v n, u Σ + u n, v Σ + α u, v ) 0, β u, v ) 0,, LV ) := f, v ) 0, ) f, div v. β It is important to note that in the definition of the bilinear form A, the two boundary terms v n, u Σ and u n, v Σ are homogeneous, i. e. without any scaling factor between the two. N.B. We propose an augmented variational formulation, so that the vector fields can be discretized with the help of a continuous Galerkin method see [9]). 3. Equivalence with the initial problem Proposition 3.: The two-field formulation 4) is equivalent to problem 3). Proof: To begin with, one finds that u Γi = 0 i =, ), according to the definition of X and X. Then, let us take in 4) successively v = 0 and v = 0: it is straightforward to show that u, u ) satisfy 5) and 6). ) div u v X,, div v + curl u β α, curl v ) + α ) ) u α, v f + v n, u Σ =, div v. β v X, α u, v ) 0, β u, v ) 0, + u n, v Σ = f, v ) 0,. 6) 5) 5

6 First, we show that 5) implies curlu / α ) = 0. To that aim, we choose divergencefree test functions in 5). For this, given f L Ω ), we introduce the auxiliary problem 7): find χ H 0,Σ curl; Ω ) such that ) curl χ ϕ H 0,Σ curl; Ω ), χ, ϕ) +, curl ϕ α = f, ϕ). 7) Note that such χ is unique and satisfies according to the Proposition 3.6 of [5] for the trace equality) both ) ) curlχ curlχ curl = f χ in Ω, n Γ = 0. α α Thus, we can choose v = curl χ in 5) and integrate by parts to reach: curl u ) α, f = 0. In other words, we get the desired property curlu / α ) = 0 in L Ω ). From 5), we now recover the second and fourth equations of 3): Ω being simply connected, there exists u H Ω ) such that u = u / α. Recall that u / α n) Γ = 0 and α is strictly negative. Thus, Γ being connected, we obtain that u Γ is actually a constant. The scalar potential u being defined up to a constant, let us choose u Γ = 0. If we integrate 5) by parts, we reach div α u ) u + f ), div v + v n, u u Σ = 0 8) β β We want to use in 8) ad hoc test functions v. Given p, s) L Ω ) H / 00 Σ)), let us introduce the auxiliary problem 9): find χ X such that z X, α χ, z) = p, z) + s, z Σ. 9) Problem 9) is well-posed. Its solution χ is such that div α χ = p and α χ n Σ = s. So we can choose v = α χ in 8). Let us take first s = 0, to recover the second equation of 3), and then p = 0, to recover the fourth equation of 3). In order to conclude the proof, we have to recover the first and the last equations of 3). One chooses simply in 6) test functions which span DΩ ), and then functions, which span X the trace mapping X H / 00 Σ) is onto.) 6

7 3.3 Finding a well-posed variational setting for the form A. Below, we build a splitting of the bilinear form A in a two term sum, so that the first term is coercive over {X X }, and the second one is a compact perturbation of the first one. Let us write A = A coer + A comp, with A comp = β + α min )u, v ) 0,. Thanks to the compact imbedding of H Ω ) into L Ω ), A comp is indeed a compact perturbation of A coer. We then prove that the form A coer is coercive under some suitable conditions cf. theorem 3.3). We introduce some constant, related to the lifting of the trace of scalar fields and of the normal trace of vector fields. Let the constant c R + be such that 0) holds optimally i. e. the constant c takes the smallest admissible value.) v, v ) X X, v n, v Σ c v Hdiv ;Ω ) v,. 0) Theorem 3.3: Assume that α /β < 0 a. e., and that α min α + > c {min holds. Then, the form A coer is coercive over {X X }., α + )} ) β max Proof: Since α /β < 0 and α < 0, we have β > 0 in Ω. Thus, A coer can be bounded from below by A coer V, V ) α min v, + min + curl v α α +, ) v β max Hdiv ;Ω ) c v Hdiv ;Ω ) v,. We have to control the term c v Hdiv ;Ω ) v, with a fraction of) the others. Let us recall that, given m; p R + and x; y R the following equality is true. mx + y pxy = m + p x p ) m + p y + m p x + m p. ) m + p y We find that, if we identify x := v,, y := v Hdiv ;Ω ) and set m := α min [ min α +, )] [ )], and p := c min β max α +,, β max 7

8 the form A coer is coercive as soon as m > p. By rewriting m and p as defined above, this last inequality leads to ). Remark 3.3: The dependencies on geometrical conditions shape and regularity of Σ) are implicitly included in the definition of the constant c. Corollary 3.3: Assume that α /β < 0 a. e., and that ) holds. Then, the variational formulation 4) fits into the coercive plus compact framework. Evidently, the knowledge of the vector field u which solves 4) is enough to recover the scalar field u, both theoretically and numerically. If one goes back to the original dielectric and magnetic parameters, ) corresponds precisely to ɛ ɛ max > c {min, )} ω µ max ɛ. 3) Inequality 3) is a sufficient condition. Moreover, it implies ɛ /ɛ max > c. In accordance with the litterature we find that the model problem fits into the coercive plus compact framework in the case of small contrasts recall that κ := ɛ /ɛ.) To recover a similar result in the case of large contrasts provided that β < 0 one could alternatively build a two-field formulation by choosing u := α u and using vector test functions in X. 4 Three field variational formulation As we have already seen, for the two-field formulation to be valid, we had to assume that at least over one of the two subdomains, we have α k /β k < 0. Moreover, in the case of vanishing β we cannot build the two-field formulation. In order to relax those constraints on β, we derive a three-field variational formulation, which allows to handle any β L Ω). 4. Derivation of the formulation In this paragraph we propose a more general formulation, which allows to handle a wider set of conditions on the parameters ɛ, µ and ω: the three-field formulation. This time, we keep both scalar unknowns u and u, and we add the vector unknown Albeit this assumption may not be optimal. 8

9 u. To begin with, keeping both u and u leads to a reformulated definition of H0 Ω); we introduce Now, let us X := {v, w) X X v Σ = w Σ }. take the L -scalar product of the first equation of 3) with a test function v X, integrate by parts, and use the second equality of traces in 3): α u, v ) 0, + u n, v Σ β u, v ) 0, = f, v ) 0,. take the L -scalar product between the second equation of 3) and the divergence of a vector test function v X ; multiply the resulting equality by a constant factor ρ > 0. ρdiv u, div v ) ρβ u, div v ) = ρf, div v ). consider, for v, v ) X X, the two identities recall that u = α u ) α u, v ) + div u, v ) + u n, v Σ = 0; 4) u, div v ) + v, u ) + v n, u Σ = 0. 5) α replace v Σ by v Σ and sum these two equalities with the results obtained following the first two items. Finally add the term curl u α, curl v ) α to build an augmented variational formulation. To summarize, we introduce the variational formulation 6): find U = u, u ), u ) X X such that cf. [3] or paragraph 3.) in order V = v, v ), v ) X X A ρ U, V ) = L ρ V ). 6) The forms A ρ and L ρ are respectively defined by A ρ U, V ) := ρdiv u, div v ) + and curl u α, curl v ) α + ) u α, v + α u, v ) + div u, v ) + u, div v ) ρβ u, div v ) +α u, v ) 0, β u, v ) 0, + u n, v Σ + v n, u Σ, L ρ V ) := f, v ) 0, ρf, div v ). 9

10 Again, it is important to note that in the definition of the bilinear form A ρ, the two boundary terms v n, u Σ and u n, v Σ remain homogeneous. In addition, we remark that this is true for any choice of the factor ρ, which we will fit to some optimal value when we establish the coercivity of A ρ. N.B. Again, the use of an augmented variational formulation allows to use a continuous Galerkin discretisation of the vector fields cf. [9]). 4. Equivalence with the initial problem Proposition 4.: The three-field formulation 6) is equivalent to problem 3). Proof: To begin with, one finds that u Σ = u Σ and u Γi = 0 i =, ), according to the definition of X. Then, one recovers the first equation 3), by choosing in 6) test functions v which span DΩ ), and v, v ) = 0, 0). Next, we have that curlu / α ) = 0: this is achieved as in Proposition 3., by taking v, v ) = 0, 0) and v = curl χ, χ being the solution to 7). From there, we establish that u = α u and that the second equation of 3) is recovered. Unfortunately, it does not seem possible to carry out the proof sequentially, so we proceed in parallel... We introduce τ = u u / α, and η = div u + β u f, and prove that both fields vanish over Ω. To start with, we know that τ H 0,Γ curl; Ω ), curlτ = 0, and that η L Ω ). Choose first in 6) v, v ) = 0, 0), and v DΩ ) 3, to reach ρ η = τ in the sense of distributions over Ω. Therefore, η belongs to H Ω ) and, in addition since Γ is connected), η Γ = c R. Then, let us prove that the trace of η is actually equal to c over the whole boundary Ω. For that, choose in 6) v, v ) = 0, 0), and v X, and integrate by parts. This yields v X, v n, η Ω = 0. 7) Consider then an ad hoc test function v, built in the following way: solve problem 9), with p = 0 and s = c η Σ, which belongs to L Σ), and set v = α χ. One gets div v = 0 and v n Σ = c η Σ. We note that since div v = 0, there holds in particular v n, Ω = 0. Using this vector field in 7) leads to 0 = v n, η Ω = v n, η c Ω = v n, η c Σ = η c L Σ). Therefore, η Ω = c, so that τ H 0 curl; Ω ) since τ = ρ η.) 0

11 Next, choose in 6) v, v ) = 0, 0), and v DΩ ). One finds div α τ = 0 in the sense of distributions over Ω. Thus η belongs to H Ω ), and it satisfies div α η = 0 since ρ η = τ) with a constant trace =c ) over Ω. In other words, η = c over Ω, and τ = 0, so that u = α u holds. There remains to prove that c = 0 to recover the second equation of 3). We choose again in 6) v, v ) = 0, 0), and v X, without integrating by parts, to reach v X, c, div v ) = 0. 8) Since the range of the divergence from X is exactly L Ω ), there follows c = 0, our intended target. In order to conclude the proof, we consider in 6) v, v ) X and v = 0. By integrating by parts and using the previous results, one reaches easily v X, α n u + α n u ), v Σ = 0. The last equation of 3) follows. 4.3 Finding a well-posed variational setting for the form A ρ As for the two-field formulation, we split A ρ as A ρ = A ρ coer + A ρ comp, with A ρ comp := ρ u, v ) ρβ u, div v ) + div u, v ) +u, div v ) ) ) β + α min u, v Thanks to the compact imbedding of H Ω i ) into L Ω i ), i =,, A ρ comp is a compact perturbation of A ρ coer. Let us prove that the form A ρ coer is coercive under some suitable conditions. 0, Theorem 4.3: Assume that α min α + > c 9) holds, with c defined by 0). Then, for ρ /α +, the form A ρ coer is coercive over {X X }.

12 Proof: Let us first compute the value of A ρ coer V, V ), A ρ coerv, V ) = ρ div v + curl v α + v, ) v + α v, v ) α + ρ v + α v, v ) 0, + α min v, v ) 0, + 3 v n, v Σ. Thus, introducing the real parameter η [0, 3], A ρ coer V, V ) may be bounded from below by A ρ coer V, V ) ρ div v + curl v α + v, + ρ v + α v, v ) 0, + α min v, v ) 0, 3 η) v n, v Σ η v n, v Σ. ) v + α v, v ) α The term v n, v Σ is bounded as in 0), whereas v n, v Σ is bounded by v n, v Σ ρ / div v, ρ / v ) + α / v, α / v ) [ ρ div v + ρ v ] + α v, v ) + α v, v ). To get coercivity, it is advised to restrict η to [0, [... We deduce A ρ coerv, V ) α min v, + η/) min α +, ρ ) v Hdiv ;Ω ) 3 η)c v Hdiv ;Ω ) v, 0) + curl v [ + η/) α α / v + ] ρ v. Since ρ /α +, one has actually min/α +, ρ) = /α +. In equation ) let us identify x = v,, y = v X and set m := α min α + The form A ρ coer is coercive when m > p, i. e. η, p := c 3 η α+ η. α min α + > c 3 η) 4 η. )

13 Now, f : η 3 η) /4 η) takes is minimal value over [0, [ at η =, and f) =. For this optimal value, ) reduces to 9). Remark 4.3: If we consider ρ in ]0, /α + [, the right-hand side of 9) changes to c /α + ρ). Corollary 4.3: Assume that 9) holds. Then, the variational formulation 6) fits into the coercive plus compact framework, for ρ /α +. In the sudomain Ω, the numerical approximation is overdetermined, in the sense that both the scalar field u and the vector field u are computed. If one goes back to the original dielectric and magnetic parameters, 9) corresponds precisely to ɛ ɛ max > c. ) The model problem thus fits into the coercive plus compact framework in the case of small contrasts. To derive a similar result in the case of large contrasts one simply builds a three-field formulation by choosing u := α u and using vector test functions in X. 5 Conclusion In this paper we focused on solving a scalar wave transmission problem between media with opposite sign dielectric and/or magnetic constants. For this, we derived two- and three-field variational formulations. The following table summarizes, for all possible transitions between the two media, which simplest formulation can be chosen for solving this problem. Below, N. F., F. and 3 F. denote respectively the natural, the two- and the three-field variational formulations. ɛ < 0 ɛ < 0 ɛ > 0 ɛ > 0 µ < 0 µ > 0 µ < 0 µ > 0 ɛ < 0 ; µ < 0 N.F. N.F. F. 3 F. ɛ < 0 ; µ > 0 N.F. N.F. F. ) F. ɛ > 0 ; µ < 0 F. F. ) N.F. N.F. ɛ > 0 ; µ > 0 3 F. F. N.F. N.F. Nota Bene: As we saw in paragraph 3., the two-field formulation is valid when at least over one of two subdomains Ω, Ω we have ɛµ < 0. In the cases ) both 3

14 ɛ µ and ɛ µ are negative: we can build the two-field formulation by arbitrarily choosing where to introduce the vector unknown. One possible continuation of the present work is to deal with the numerical implementation of the formulations and their comparison. Also of interest is to try and replace the volume vector unknown by an interface unknown in the three-field formulation, and to derive a suitable Domain Decomposition Method to solve the original scalar problem. Finally, one can try and extend the approach followed here to the static and/or harmonic Maxwell equations. References [] A.S. Bonnet-Ben Dhia, M. Dauge, K. Ramdani, Analyse spectrale et singularités d un problème de transmission non coercif, C. R. Acad. Sci. Paris, Ser. I, 38, 77-70, 999. [] F. Cakoni, D. Colton, H. Haddar, The linear sampling method for anisotropic media, J. Comput. Appl. Math., 46, 85-99, 00. [3] P. Ciarlet Jr., Augmented formulations for solving Maxwell equations, Comput. Methods. Appl. Mech. Engrg.,94, , 005. [4] M. Costabel, E. Stephan, A direct boundary integral method for transmission problems, J. of Math. Anal. and Appl., 06, , 985. [5] P. Fernandes, G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci., 7, , 997. [6] B. Prade, J. Y. Vinet, A. Mysyrowicz, Guided optical waves in planar heterostructures with negative dielectric constant, Phys. Rev. B, 44, , 99. [7] K. Ramdani, Lignes supraconductrices : analyse mathématique et numérique, PhD thesis, Paris 6 University, 999. [8] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schutz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., 84, , 000. [9] E. Jamelot, Numerical solution of Maxwell equations with Continuous Galerkin Finite Element Methods in French), Thèse de Doctorat de l École Polytechnique,