Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section

Transcription

1 Error estimates for 1D asymptotic moels in coaxial cables with non-homogeneous cross-section ébastien Imperiale, Patrick Joly o cite this version: ébastien Imperiale, Patrick Joly. Error estimates for 1D asymptotic moels in coaxial cables with non-homogeneous cross-section. Avances in Applie Mechanics, New York ; Lonon ; Paris [etc] : Acaemic Press, 2012, xx, <10.420/aamm >. <hal-00750v2> HAL I: hal ubmitte on 21 Nov 201 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. he ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech., Vol. xx, No. x, pp. xx-xx DOI: xxxx xxx 200x Error estimates for 1D asymptotic moels in coaxial cables with non-homogeneous cross-section ébastien Imperiale 1,2 an Patrick Joly 2, 1 Department of Applie Physics an Applie Mathematics, Columbia University, New York NY, INRIA Rocquencourt, POems, omaine e Voluceau, 715 Le Chesnay, FRANCE Abstract. his paper is the first contribution towars the rigorous justification of asymptotic 1D moels for the time-omain simulation of the propagation of electromagnetic waves in coaxial cables. Our general objective is to erive error estimates between the exact solution of the full D moel an the approximate solution of the 1D moel known as the elegraphist s equation. AM subject classifications: 5L05, 5A5, 7R05, 5A40 Key wors: Coaxial cables, elegraphist s equation, Error estimates, Non-homogeneous crosssection 1 Introuction his work is a continuation of a previous article [4] evote to the asymptotic moeling of electromagnetic waves propagations in a thin co-axial cable. By thin cable, we mean a D elongate (infinitely long in this paper) cylinrical omain whose transverse imensions are small with respect to the consiere wavelengths. By co-axial cable, we mean that each transverse cross-section of the cable is not simply connecte, which is essential. Of course, as a cable is a thin structure whose transverse imensions are much smaller than the longituinal one, one woul like to use a simplifie 1D moel : this is even necessary for the effective efficiency of the computational tool (one wants in particular to avoi using a D mesh for the thin cable). In such a situation, electrical engineers use the well-known elegraphist s equations for perfect coaxial cables (homogeneous with circular cross-section) where the electric unknowns are reuce to an electric potential V(x, t) an an electric current I(x, t), where x Corresponing author. si2245@columbia.eu (ébastien Imperiale), patrick.joly@inria.fr (Patrick Joly) 1 c 200x Global cience Press

3 2 ébastien Imperiale, Patrick Joly enotes the abscissa along the cable, t is time an in absence of source >< C V + GV L I + RI I = 0, V = 0, (1.1) where the capacitance C, the inuctance L, the conuctance G an the resistance R can be expresse it terms of the geometry of the cross-section. In [4], thanks to a formal asymptotic expansions with respect to the small parameter := iameter of the cable / unit reference length, we erive a simplifie 1D effective moel uner quite general assumptions: the cross section is heterogeneous, slowly variable in the longituinal irection an possibly mae of lossy meia (i.e. with electric or magnetic conuctivities). o erive this effective moel, we consiere a family of problems pose in omains that epen on a small geometric parameter > 0. Of course, a given cable correspons to a given value of but the effective moel will be constructe by an asymptotic analysis in. he resulting moel appears as an extension of the elegraphist s equation (1.1) currently use in the engineering community [], [] (in particular, we show that the presence of lossy meia inuces the apparition of time convolution terms in the limit moel). he coefficients of the homogenize moel are given explicitly as the solutions of two 2D scalar elliptic problems pose in the cable cross-section. uch moels can be use as an efficient tool for the time-omain numerical simulation of the propagation of electromagnetic waves in coaxial cables, which is neee in many inustrial applications. In our case, we were motivate by the simulation of non-estructive testing experiments by ultra-souns [5], where coaxial cables are use for the electric supply process for piezo-electric transucers [9]. he present paper is the first contribution towars the rigorous justification of the results of [4]. More than a simple convergence theorem, the general objective is to erive error estimates (in a sense that will be explaine later) between the exact solution of the full D moel an the approximate solution of the 1D moel. We focus in this first paper on the (moel) situation of a perfectly cylinrical cable (invariant uner translation in the longituinal irection) whose cross-section is heterogeneous (constitutive coefficients epen on transverse variables) but mae of non-lossy meia. A more general situation will be consiere in a future work. he paper is organize as follows. In section 2, we present the consiere moel problem an more precisely the family of epenent problems that we wish to analyze. In section, we recall the main results of [4] in the simplifie situation consiere in this paper. hen we give the main results of this work (theorem.2), that provie various error estimates uner the only assumption that the ata of the problem (the source terms) are aequately well-prepare. Finally, in section 4, we give a etaile proof of theorem.2, that relies on appropriate vector fiel ecompositions, energy estimates an aequate versions of Poincaré-Frierichs inequalities (see Appenix 5).

4 2 he homogenize 1D moel in non-conuctive cylinrical coaxial cables We consier a family of problems pose in cylinrical omains (cables) that epen on a small geometric parameter > 0 measuring the transverse imensions of the cables. he omain associate to the cables are enote W : W = R, where is a connecte boune an Lipschitz reference omain (of unit iameter) in R 2 (see Figure 1). In what follows, we shall enote by x =(x, x ) the D space variable where x =(x 1, x 2 ) represents the transverse coorinates. An essential assumption is that is not simply connecte: x + Figure 1: Geometry of the coaxial cables. = O\, O where O an (the hole) are simply connecte, Lipschitz, open sets of R 2. his correspons to the case where the cable contains only one metallic (perfectly conucting) wire. However, the extension to several holes (or several metallic wires) is rather straightforwar [4]. In this case, the bounary of has two connecte components, the exterior one ( + ) an the interior one ( ): + := O, :=. o efine the effective moel, we shall introuce an artificial cut in the cross-section, namely a line G joining + to so that the omain: G := \ G, (2.1) is simply connecte. he family of (thin) omains W is relate to the reference omain W = R by W = G W,

5 4 ébastien Imperiale, Patrick Joly with the transverse scaling transformation G : (x 1, x 2, x ) W, the outwar unitary normal vector n satisfies! (x 1, x 2, x ). Along a. e. G (x) 2 W, n = n 1 (x), n 2 (x),0 t, n =(n 1, n 2 ) t :! R 2, (2.2) where n is the (2D) outwar unitary normal vector to. Next, we assume that the material properties, namely the electric permittivity # (x) an magnetic permeability µ (x), o not epen on x an are obtaine by a scaling in x = (x 1, x 2 ) of fixe istributions over the reference omain W: # = # G 1, µ = µ G 1, (2.) an where (#, µ) are ientifie to (measurable) functions efine on that satisfy the usual positivity properties: 0 < # apple #(x) apple # +, 0 < µ apple µ(x) apple µ +, a. e. x 2. (2.4) he equations governing the electric fiel E (x, t) an the magnetic fiel H (x, t) are Maxwell s equations (r enoting the usual D curl operator) >< # E µ H with perfectly conucting bounary conitions the system being consiere at rest at t = 0: r H = J, in W, t > 0, + r E = 0, in W, t > 0, (2.5) E n = 0 on W, t > 0, (2.6) E (x,0)=h (x,0)=0, a. e. x 2 W. (2.7) o procee in our analysis we shall assume that the source term, namely the current ensity J (x, t), has no longituinal component, is ivergence free, vanishes at time t = 0 an is obtaine by scaling in (x 1, x 2 ) of a fixe current ensity in W (this correspons to what we call well-prepare ata): J = J G 1, J =(J 1, J 2,0) t, J J 2 2 = 0, J(x,0)=0, a. e. x 2 W. (2.) For the analysis, it appears juicious to introuce the tangential components E =(E 1, E 2) t, H =(H 1, H 2) t, of the electric an magnetic fiels as well as the longituinal components of these fiels: E an H. We can rewrite the equations (2.5) with these new unknowns, using

6 the following notations: for all scalar functions u an 2D transverse vector fiels v an w with two components v 1 an v 2 or w 1 an w 2, we efine: u ru, u t,! u u t, v rot u, iv v 1 + v 2, rot v v 2 v Moreover, for any v =(v 1, v 2 ) an w =(w 1, w 2 ), we shall set v w v 1 w 1 + v 2 w 2, v w v 1 w 2 v 2 w 1, e v ( v 2, v 1 ) t. We will also use the following properties,! rot u reu =! rot eu ru, e ru =! rot u, ru v = v 5! rot u. (2.9) Remark 2.1. In the sequel, we shall enote the L 2 scalar prouct in a omain D R 2 (with n the unit outgoing normal vector to D) of two scalar functions u an eu in L 2 (D) or two vector fiels v an ev inl 2 (D) 2 as (u, eu) L2 (D) = u eu x, (v, ev) L2 (D) = v ev x D an k k L2 (D) this associate norm. We shall also use the following Green s formula: hv n, ui W =(v, D! rot u) L2 (W) ( rot v, u) L2 (W), (2.10) vali for any (u, v) 2 L 2 (W) L 2 (W) 2 such that! rot u 2 L 2 (W) 2 an v 2 L 2 (W) an where h, i represents the uality prouct between H 1/2 ( W) an H 1/2 ( W). he equations (2.5) can be rewritten as (with J =(J 1, J 2 )t ) # E e H! rot H = J, in W, t > 0, >< # E µ H rot H = 0, in W, t > 0, + e E +! rot E = 0, in W, t > 0, (2.11) µ H + rot E = 0, in W, t > 0. Using (2.2), the bounary conitions (2.6) become E n = 0, E = 0, on W, t > 0. (2.12) Moreover, taking the ivergence of the equations (2.11) an using (2.), we get, after time integration, the hien ivergence equations: >< iv # E + # E = 0, in W, t > 0, iv µ H + µ H = 0, in W, t > 0. (2.1)

7 6 ébastien Imperiale, Patrick Joly Finally, from the Maxwell s equations (2.11), the ivergence equations (2.1) an the bounary conitions (2.12), it is classical to erive an aitional hien bounary conition for the magnetic fiel (see for instance [2]): H n = 0, on W, t > 0. (2.14) We want to escribe the behavior of (E, H ) when tens to 0. For this, it is useful to apply a change of variables in orer to work in a fixe geometry. Doing so, the parameter appears only in the coefficients of the governing equations. We introuce the rescale fiels (ee, ee, eh, eh ) efine by: E = ee G 1, E = ee G 1, H = eh G 1, H = eh G 1. (2.15) We can write from (2.11, 2.12) the equations for (ee, ee, eh, eh ) in the fixe omain W (the 1 terms simply comes from erivatives in (x 1, x 2 ) an J =(J 1, J 2 ) t ): # ee e eh 1! rot eh = J, in W, t > 0, >< # ee µ eh µ eh while from the ivergence equations (2.1), >< 1 rot eh = 0, in W, t > 0, + e ee + 1! rot ee = 0, in W, t > 0, + 1 rot ee = 0, in W, t > 0, 1 iv # ee + # ee = 0, in W, t > 0, 1 iv µ eh + µ eh = 0, in W, t > 0. he equations (2.16) are complete by zero initial conitions (2.16) (2.17) ee (x,0)=0, ee (x,0)=0, eh (x,0)=0, eh (x,0)=0, a. e. x 2 W, (2.1) an bounary conitions easily euce from (2.12, 2.14) ee n = 0, ee = 0, eh n = 0, on W, t > 0. (2.19) o conclue this section, we recall (without proof) the stanar existence, uniqueness an regularity results (see [6] for instance) for the evolution problem (2.16, 2.1, 2.19) together with a priori estimates that are obtaine via stanar energy techniques. It is useful to introuce some notation. For integers (p, q), we set p+q D p.q := p q

8 7 an for m a strictly positive integer, we introuce the Banach spaces W m (W [0, ]) = J 2 L 1 0, ; L 2 (W) 2 / D p.q J 2 L 1 0, ; L 2 (W) 2, p + q apple m}, W m 0 (W [0, ]) = J 2Wm (W [0, ]) / D 0,q J(,0)=0, q apple m 1}. heorem 2.1. Assuming that J satisfies J 2W 0 (W [0, ]) := L 1 [0, ], L 2 (W) 2, (2.20) the problem (2.16, 2.1, 2.19) amits a unique solution ee =(ee, ee ), eh =(eh, eh ) with ( ee, eh ) 2 C 0 [0, ], L 2 (W) 2, which satisfies the a priori estimate (with C > 0 epening only on # an µ) t apple, k ee (, t)k L2 (W) + k eh (, t)k L2 (W) apple C Moreover if J 2W0 m (W [0, ]) then, t 0 kj (, s)k L2 (W) s. (2.21) (D p.q ee, D p.q eh ) 2 C 0 [0, ], L 2 (W) 2, p + q apple m an for any t apple kd p.q ee (, t)k L2 (W) + kd p.q eh (, t)k L2 (W) apple C t 0 kd p.q J (, s)k L2 (W) s. (2.22) Main results Before stating the main results of this article, we briefly recap the results from [4] in the particular context of section 2. o characterize the limit behavior of the electric an magnetic fiels (E, H ), we nee to introuce j s (x) an y s (x) solutions of particular 2D electro-static an magneto-static problems pose in. More precisely, we efine j s 2 H 1 () as the unique solution of the problem < iv # rj s = 0 in, : j s = 0 on +, j s = 1 on. (.1) an y s 2 H 1 ( G ) as the unique solution of the problem ( [ ] G enoting the jump of a quantity through G ) >< iv µ ry s = 0 in G, y s x = 0, h G (.2) µ ry s n ig = 0 ys G = 1 on G, ry s n = 0 on.

9 ébastien Imperiale, Patrick Joly Remark.1. As alreay emphasize in [4], y s epens on the cut G but not its graient. More precisely, even though y s is not in H 1 (), its graient ry s efine in the sense of istribution in G efines a vector fiel in L 2 () 2 which oes not epen on G. Remark.2. In the forthcoming analysis we will use the following property proven in [4]: (! rot y s, rj s ) L2 () = (! rot j s, ry s ) L2 () = 1. (.) he limit moel corresponing to the D equations (2.16) will be a 1D wave equation with coefficients, calle homogenize coefficients, that are obtaine by some kin of weighte averages of the original physical coefficients. More precisely we efine the capacitance C an the inuctance L of the cable as coefficients C, L as C := #(x) rj s (x) 2 x, L := µ(x) ry s (x) 2 x. (.4) At the (formal) limit! 0, the electromagnetic fiel becomes purely transverse: t t, E (x, x, t) E 0 (x, x, t) E 0 (x, x, t),0 E e 0 (x/, x, t),0 (! 0), t t, H (x, x, t) H 0 (x, x, t) H 0 (x, x, t),0 H e 0 (x/, x, t),0 (! 0). (.5) where the limit transverse electric fiel ee 0 an the limit transverse magnetic fiel eh 0 are given by (this means in particular that one has asymptotically separation of variables between x an x ) < ee 0(x, x, t) =V(x, t) rj s (x), (.6) : eh 0(x, x, t) =I(x, t) ry s (x), where V(x, t) an I(x, t) are the solutions of >< C V (x, t)+ I (x, t) =I (x, t), in R, t > 0, L I (x, t)+ V (x, t) =0, in R, t > 0, (.7) with (C, L) efine by (.4) an where I is efine by I (x, t) = J (x, t) rj s (x) x. (.) he equations (.7) are naturally complete with zero initial conitions V(x,0)=I(x,0)=0, x 2 R. (.9) We now state for the limit 1D problem, the equivalent of theorem (2.1) (existence, uniqueness, regularity).

10 9 heorem.1. Assuming that J satisfies J 2W 0 (W [0, ]), then the problem (.7,.,.9) amits a unique solution V 2 C 0 0, ; L 2 (R), I 2 C 0 0, ; L 2 (R). which satisfies the a priori estimate (C > 0 epens only on (, #, µ)) kv(, t)k L2 (R) + ki(, t)k L2 (R) apple C t 0 kj (, s)k L2 (W) s, (.10) If, moreover, J satisfies J 2W 1 (W [0, ]), 2 then (V, I) 2 C 1 0, : L 2 (R) \ C 0 0, : L (R) 2 an for any t apple : We set V (, t) L 2 (R) + V (, t) L2 (R) + I (, t) L 2 (R) apple C t J (, t) L2 (R) apple C t J s 1,1, := J (, s) L 1 (0,;L 2 (W)) + an more generally, for arbitrary integer p 1 an q J (, s) s, (.11) L 2 (W) J (, s) s. (.12) L2 (W) J (, s) L1 (0,;L 2 (W)), J s p,q, := D p 1.q J L1 (0,;L 2 (W)) + Dp.q 1 J k L1 (0,;L 2 (W)). We are now in position to give the main results of this article. heorem.2. Assuming that J 2W 1 0 (W [0, ]) (cf. 2.20), then an for the transverse fiels k ee k L (0,;H 1 (W)) + k eh k L (0,;H 1 (W)) apple C J s 1,1, (.1) k ee ee 0 k L (0,;L 2 (W)) + k eh eh 0 k L (0,;L 2 (W)) apple C 1 2 ( ) Js 1,1,. (.14) If, in aition, J 2W0 2 (W [0, ]) then k ee ee 0 k L (0,;L 2 (W)) + k eh eh 0 k L (0,;L 2 (W)) apple C ( J s 1,1, + J s 2,1, ). (.15) Finally, if J 2W0 (W [0, ]) then k ee ee 0 k L (0,;L 2 (W)) + k eh eh 0 k L (0,;L 2 (W)) apple C 2 J s 1,2, + J s 2,1, + J s 2,2, + J s,1,. (.16)

11 10 ébastien Imperiale, Patrick Joly 4 Proof of the main theorem We ecompose the proof in several steps. As usual, in what follows, C will enote a generic positive scalar that may change from one line to the other. Unless specifically mentione, C epens only on (, #, µ). Note that, with the regularity assumption on the source term J, we euce from theorem 2.1 an the equations (2.5) that the solution of the full D problem has the regularity: ( ee, eh ) 2 C 1 0, ; L 2 (W) 2, (r ee, r eh ) 2 C 0 0, ; L 2 (W) 2. tep 1 : Proof of the estimate (.1) for the longituinal fiels. his step is quite immeiate. Using the first an the thir equations of (2.16), we immeiately get kr eh (, t)k L2 (W) apple C kree (, t)k L2 (W) apple C ee (, t) L 2 (W) + eh (, t) L 2 (W) + eh (, t) ee (, t) L2 (W) L2 (W),, so that, using the stability estimates (2.22) of theorem 2.1 kr eh k L (0,;L 2 (W)) + kree k L (0,;L 2 (W)) apple C J s 1,1,. (4.1) o conclue, it suffices to use Poincaré s type inequalities. For the electric fiel, (.1) results from the classical Poincaré s inequality since by (2.19) (secon equation) E (, x, t) belongs to H0 1 () for each t > 0 an almost every x 2 R. For the magnetic fiel, we can use a (generalize) Poincaré-Wirtinger s inequality since : µ(x) eh (x, x, t) x = 0, 0 apple t apple, a. e. x 2 R. (4.2) his is obtaine by integrating over W the fourth equation of (2.16), after multiplication by a smooth 1D test function with compact support j(x ). his gives, using Green s formula, µ eh t j x W = 1 = 1 R R rot ee (x, x, t) x j(x ) x h E n, j(x ) i x = 0, thanks to the bounary conition (2.19) (first equation). (4.2) follows easily since H vanishes at time t = 0.

12 It is then easy to obtaine analogous estimates for erivatives in x an t of these longituinal fiels. Inee, since all coefficients in equations (2.16) are inepenent of x an t, it is clear that the fiels D p,q ee an D p,q eh are relate to D p,q ej by the same partial ifferential equations that the ones which link ee an eh to ej. Moreover, since the omain W is a cyliner, they satisfy the same homogeneous bounary conition. Finally, provie that time erivatives of J or orer less or equal to q 1 vanish, these fiels vanish at time t = 0. From this remarks, the reaer will easily check that, if then for any p + q apple m J 2W m 0 (W [0, ]), kd p,q ee k L (0,;H 1 (W)) + kd p,q eh k L (0,;H 1 (W)) apple C D p,q J s 1,1,. (4.) tep 2 : ecomposition of the transverse fiels an relate Poincaré-Frierichs inequalities. he ecomposition we shall use is relate to the following orthogonal ecomposition of spaces of square integrable 2D vector fiels in. Concerning the transverse electric fiel, we first efine the Hilbert spaces: V(#) L 2 () 2 equippe with the inner prouct (u, v) # := # u vx, (4.4) W(#) = u 2 L 2 () 2 / iv # u 2 L 2 (), rot u 2 L 2 (), u n = 0 on. V(#) can be ecompose as (the ecomposition is orthogonal with respect to (u, v) # ) V(#) =U(#) U(#)?, where U(#) := span[rj s ]. (4.5) In [4], it has been shown that U(#) is characterize by U(#) = u 2 W(#) / iv # u = 0, rot u = 0, (4.6) a result which is relate to the following Poincaré-Frierichs inequality (see the appenix for the proof), is: Proposition 4.1. here exists C > 0 epening only on (, #) such that, u 2 W(#), kuk L2 () apple C krot uk L2 () + kiv # uk L2 () + u, rj s #. (4.7) In the same way, for the transverse magnetic fiel, we first efine the Hilbert spaces: V(µ) L 2 () 2 equippe with the inner prouct (u, v) µ := µ u vx, (4.) W(µ) = u 2 L 2 () 2 / iv µ u 2 L 2 (), rot u 2 L 2 (), u n = 0 on. V(µ) can be ecompose as (the ecomposition is orthogonal with respect to (u, v) µ ) V(µ) =U(µ) U(µ)?, where U(µ) := span[ry s ]. (4.9) 11

13 12 ébastien Imperiale, Patrick Joly In [4], it has been shown that U(µ) is characterize by U(µ) = u 2 W(µ) / iv µ u = 0, rot u = 0. It is possible to show the following Poincaré-Frierichs inequality (that we assume here but can be rigorously proven) Proposition 4.2. here exist C > 0 epening only on (, µ) such that, u 2 W(µ), kuk L2 () apple C krot uk L2 () + kiv µ uk L2 () + u, ry s µ. (4.10) Accoring to the orthogonal ecompositions (4.5) an (4.9), for each (x, t), the transverse fiels ee (, x, t) an eh (, x, t) will be splitte as follows : ee (, x, t) =V (x, t) rj s + ee R, (, x, t), ee R, (, x, t) 2 U(#)?, eh (, x, t) =I (x, t) ry s + eh R, (, x, t), eh R, (, x, t) 2 U(µ)?. (4.11) where, from the orthogonality of the ecompositions, the efinitions (.4) an equalities (4.11), the scalar quantities I (x, t) an V (x, t) are given by V (x, t) =C 1 ( ee (, x, t), rj s ) #, I (x, t) =L 1 ( eh (, x, t), ry s ) µ. (4.12) Accoring to (.5,.6), we expect that the resiual transverse fiels ee R, an eh R, converge to 0 when! 0 while V (x, t) an I (x, t) converge to V(x, t) an I(x, t) (the solutions of (.7,.,.9)). his is exactly the way the error estimates (.14) an (.15) will be proven in the next two steps, using the triangular inequality: ke kh ee 0 k L (0,;L 2 (W)) applekee R, k L (0,;L 2 (W)) + k(v V )rj s k L (0,;L 2 (W)), eh 0 k L (0,;L 2 (W)) applekeh R, k L (0,;L 2 (W)) + k(i I )ry s k L (0,;L 2 (W)), (4.1) which yiels by a straightforwar calculation exploiting the separation of variables: ke kh ee 0 k L (0,;L 2 (W)) applekee R, k L (0,;L 2 (W)) + C k(v V )k L (0,;L 2 (R)), eh 0 k L (0,;L 2 (W)) applekeh R, k L (0,;L 2 (W)) + C k(i I )k L (0,;L 2 (R)). (4.14) tep : estimates of the resiual transverse fiels. First note that, from the bounary conitions (2.12, 2.14) an the efinitions of (j s, y s ) we euce the bounary equations E R, n = 0, eh R, n = 0.

14 hus as consequences of Poincaré-Frieriches inequalities (4.7) an (4.10), we euce that k ee R, k L 2 (W) apple C krot ee R, k L 2 (W) + kiv # ee R, k L 2 (W), (4.15) k eh R, k L 2 (W) apple C krot eh R, k L 2 (W) + kiv µ eh R, k L 2 (W) 1. (4.16) Moreover, from the efinitions of ee R, an eh R, given by (4.11), as well as from the efinition of bj s an by s given by (.1,.2) we have rot ee R, = rot ee, iv # ee R, = iv # ee, rot eh R, = rot eh, iv µ eh R, = iv µ eh. Using the secon an the fourth equations of (2.16), this implies that for any t apple, krot eh R, (, t)k L 2 (W) = k# E e t (, t)k L2 (W) apple C 0 k J (, s)k L 2 (W) s, krot ee R, (, t)k L 2 (W) = kµ H e t (, t)k L2 (W) apple C k J 0 (, s)k L 2 (W) s. (4.17) Moreover, using the hien equation (2.17): kiv # ee R, k L 2 (W) = k# ee t k L2 (W) apple C k J (, s)k 0 L2 (W) s, kiv µ eh R, k L 2 (W) = kµ eh t k L2 (W) apple C k J (, s)k 0 L2 (W) s. (4.1) ubstituting (4.17) an (4.1) into (4.15), we easily euce that k ee R, k L (0,;L 2 (W)) + k eh R, k L (0,;L 2 (W)) apple C J 1,1,. (4.19) Moreover, with one more egree of regularity on the source, namely if one assumes that J 2W0 2 (W [0, ]), such estimate is easily extene into analogous estimate about the x -erivatives of the resiual transverse fiels (the etails are left to the reaer): k ee R, k L (0,;L 2 (W)) + k eh R, k L (0,;L 2 (W)) apple C J 2,1,. (4.20) We can improve these estimates. Inee, if J 2W0 2 (W [0, ]), we can apply the estimate (4.) with (p, q) =(1, 0) an (p, q) =(1, 0) to obtain O() upper bouns for first orer erivatives in x an t of ee an eh. hen, the reaer will easily verify that substituting these inequalities into (4.17) an (4.1) an finally into (4.15) leas to k ee R, k L (0,;L 2 (W)) + k eh R, k L (0,;L 2 (W)) apple C 2 J 2,1, + J 1,2, ). (4.21)

15 14 ébastien Imperiale, Patrick Joly Finally, with one more egree of regularity on the source, namely J 2W0 (W [0, ]), this estimate can extene, as before, into estimate about the x -erivatives of the resiual transverse fiels: k ee R, k L (0,;L 2 (W)) + k eh R, k L (0,;L 2 (W)) apple C 2 J,1, + J 2,2, ). (4.22) tep 4 : proof of the error estimate (.14). We first write an equation in (v, i ) efine by v = V V an i = I I. Multiplying the first equation of (2.16) by rj s an the thir one by ry s an integrating over the section, we obtain (the etails are left to the reaer) >< C V L I +(! rot y s, rj s ) L2 () I 1 (! rot eh, rj s) L2 () = (! rot j s, ry s ) L2 () V + 1 (! rot ee, ry s) L2 () = eh R,, ee R,,! rot j s L 2 () + I, in R, t > 0,! rot y s L 2 (). in R, t > 0. (4.2) ome important simplifications now occur. First, one can use Green s formula (2.10) an the fact that rj s n = 0 along W, to obtain (! rot eh, rj s) L2 () =(eh, rot rj s) L2 () + hrj s n, eh i = 0. Next, again using Green s formula (2.10), where is replace by G, as well as the bounary conition (2.19), which gives ee = 0 along W we get, with an appropriate orientation of the normal vector n along G, (! rot ee, ry s) L2 () (! rot ee, ry s) L2 ( G ) which implies =(ee, rot ry s) L2 ( G ) + h [ry s n] G, ee i G + hry s n, E i = 0. (4.24) (! rot E, ry s) L2 () = 0, since [y s ] G = 1 inuces [ry s n] G = 0. Finally, we use (.), to en up with >< C V + I = eh R,!, rot j s L 2 () + I, in R, t > 0, L I + V = ee R,, (4.25)! rot y s L 2 (), in R, t > 0.

16 ubtracting (.7) to (4.25), we fin that (v, i ) are solution of >< C v + i = L i + v = eh R,, ee R,,! rot j s, in R, t > 0, L 2 () 15 (4.26)! rot y s L 2 (), in R, t > 0. o prove the convergence result of theorem.2, we will procee using energy techniques. We efine the energy E (t) =C v (x, t) 2 x + L i (x, t) 2 x, (4.27) R R where, thanks to the zero initial ata, we have E (0) =0. Next, we apply stanar energy analysis: we multiply the two equations of (4.26) respectively by v an i, we integrate along the x axis. After integration by parts (in the left han sies) an summation, we obtain 1 2 t E = R eh R,, i! rot js L 2 () x R ee R,, v! rot ys L 2 () x. (4.2) hen, we integrate by parts the right han sie to get 1 2 t E = R ee R,, i! rot ys L 2 () x R eh R,, v! rot js L 2 () x. (4.29) Using Cauchy-chwarz inequality, we obtain 1 2 t E applek v rj s k L2 (W) E R, L 2 (W) + k i ry s k L2 (W) eh R, L 2 (W). (4.0) On one han, thanks to the orthogonal ecomposition (4.11), we have which yiels k ee (, t)k 2 # = k V (, t) rj s k 2 # + k ee R, (, t)k 2 #, k V (, t) rj s k L2 (W) apple C k ee (, t)k L2 (W) apple C J 1,1,, (4.1) where we have use the stability estimate (2.22). On the other han, using separation of variables an the a priori estimate.1, we also have k V (, t) rj s k L2 (W) apple C J 1,1,. (4.2)

17 16 ébastien Imperiale, Patrick Joly Finally, by (4.1, 4.2) an the triangular inequality, we get In the same way, one has k v (, t) rj s k L2 (W) apple C J 1,1,. (4.) k i (, t) ry s k L2 (W) apple C J 1,1,. (4.4) hen, substituting (4.), (4.4) an (4.19) into (4.0), we get which yiels t E apple C apple C J 2 1,1,, kv k L (0,;L 2 (R)) + ki k L (0,;L 2 (R)) apple C () 1 2 J 1,1,. (4.5) Finally, the error estimate (.14) is obtaine by regrouping (4.14), (4.19) an (4.5). tep 6 : proof of the error estimate (.15). o improve the estimate (4.5) when more regularity is assume for the source term, we restart from (4.2) but o not integrate by part the right han sie. Instea, we apply irectly Cauchy-chwartz inequality to obtain 1 2 t E apple C (E ) 1 2 k ee R, which gives, after using Gronwall s lemma: k L2 (W) + k eh R, k L2 (W), E (t) 1 2 apple C k ee R, k L1 (0,;L 2 (W)) + k eh R, k L1 (0,;L 2 (W)). (4.6) hen, if J 2 W0 2 (W [0, ]), we can use the inequality (4.20) to get E (t) 1 2 apple C J 2,1, ), an if J 2 W0 (W [0, ]) we can use (4.22) to obtain E (t) 1 2 apple C 2 J,1, + J 2,2, ).

18 17 5 Appenix In this section, we prove the proposition 4.1. he proposition 4.2 may be proven in a very similar way. We recall the proposition: Proposition. here exist C > 0 epening only on (, #) such that, u 2 W(#), kuk L2 () apple C krot uk L2 () + kiv # uk L2 () + u, rj s #. he proof will be one in a classical way using contraiction arguments an the compactness properties proven in ( [1], [7]): Property 5.1. W(#) ( as efine by (4.6) ) is compactly embee in L 2 () 2. Proof Assuming the proposition 4.1 is not true, we can construct a sequence {u n } such that krot u n k L2 () + kiv # u n k L2 () + u n, rj s # apple 1 n, an ku n k L2 () = 1, u n n = 0. From the compactness property of W(#), we know that there exist u 2 W(#) such that u n! u in L 2 () 2 with rot u = iv # u = 0, # u, rj s = 0, (5.1) L 2 () an the limit u also satisfy kuk L2 () = 1, u n = 0. From the equation (5.1) an the efinition of U(#) given by (4.6), we know that u 2 U(#)? which means, from the ecomposition property (4.5) : u = a u rj s, with a u 2 R. An so u, rj s # = 0 ) a u = 0 ) u = 0, which contraicts the fact that we expecte kuk L2 () = 1. References [1] Caorsi, P Fernanes, an M Raffetto. On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. IAM Journal on Numerical Analysis, (2): (electronic), [2] R Dautray an J L Lions. Mathematical analysis an numerical methos for science an technology. Vol.. pringer-verlag, [] D A Hill an J R Wait. Propagation Along a Coaxial-Cable with a Helical hiel. Ieee ransactions on Microwave heory an echniques, 2(2):4 9, 190.

19 1 ébastien Imperiale, Patrick Joly [4] Imperiale an P Joly. Mathematical moeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section. (submitte), pages [5] Imperiale an P Joly. Mathematical an numerical moelling of piezoelectric sensors. EAIM: Mathematical Moelling an Numerical Analysis, (46):75 909, July [6] P Joly. Variational Methos for ime Depenant Wave Propagation. In M Ainsworth, P Davies, D Duncan, P Martin, an B Rynne, eitors, opics in computational wave propagation : Direct an Inverse Problems. Computational Methos in Wave Propagation, pages pringer Verlag, October 200. [7] P Monk. Finite element methos for Maxwell s equations. Oxfor science publications, 200. [] C Rose an M J Gans. A Dielectric-Free uperconucting Coaxial-Cable. Ieee ransactions on Microwave heory an echniques, (2): , [9] L W chmerr Jr an J ong. Ultrasonic Nonestructive Evaluation ystems. pringer, BONU Property 6.1. Let I be an interval of R, for all u 2 L 2 ( I) such that u/ 2 L 2 ( I) we have sup ku(x, x )k L2 () applek u k x 2I L2 ( I) + kuk L2 ( I). efine supp(j) =W J = I J, then from the step it is clear that k ee R, k L (0,;L 2 (W\W J )) + k eh R, k L (0,;L 2 (W\W J )) apple C J 1,1,. (6.1) Moreover, with one more egree of regularity on the source, namely if one assumes that J 2W0 2 (W [0, ]) we have k 2 ee R, 2 an from the step k ee R, Using the property?? we fin that an k L (0,;L2(W)) + k 2 eh R, 2 k L (0,;L 2 (W)) apple C J 2,1,. (6.2) k L (0,;L 2 (W\W J )) + k eh R, k L (0,;L 2 (W\W J )) apple C J 2,1,. (6.) sup k ee R, (x, x )k L2 () + sup k eh R, (x, x )k L2 () apple C ( J 1,1, + J 2,1, ) x 2I J x 2I J sup x 2R k ee R, (x, x )k L2 () + sup x 2R o conclue we use the following property k eh R, (x, x )k L2 () apple C ( J 1,1, + J 2,1, ).