SPECTRAL APPROXIMATION OF TIME-HARMONIC MAXWELL EQUATIONS IN THREE-DIMENSIONAL EXTERIOR DOMAINS. 1. Introduction

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1 PECTRAL APPROXIMATION OF TIME-HARMONIC MAXWELL EQUATION IN THREE-DIMENIONAL EXTERIOR DOMAIN LINA MA 1, JIE HEN 1 AND LI-LIAN WANG 2 Abstrct. We deveop in this pper n efficient nd robust spectr-gerkin method to sove the three-dimension time-hrmonic Mxwe equtions in exterior domins. We first reduce the probem to bounded domin by using the cpcity opertor which chrcterizes the nonrefecting boundry condition. Then, we dpt the trnsformed fied expnsion TFE pproch to reduce the probem to sequence of Mxwe equtions in seprbe spheric she. Finy, we deveop n efficient spectr gorithm by using Legendre pproximtion in the rdi direction nd vector spheric hrmonic expnsion in the tngenti directions. 1. Introduction We consider in this pper the pproximtion of the time-hrmonic Mxwe equtions in three-dimension exterior domin: iωµh + cur E = 0, iωεe cur H = 0, in R 3 \ D; E n D = g; im r r µ/ε H e r E = 0, 1.1 where D is three-dimension, simpy connected, bounded sctterer, i = 1 is the compex unit, g is resuted from given incident fied, µ is the mgnetic permebiity, ε is the eectric permittivity, ω is the frequency of the hrmonic wve, n is the unit outwrd norm of D nd e r = x/r with r = x. The boundry condition t infinity in 1.1 is known s the iver-müer rdition condition. The Mxwe equtions 1.1 py n importnt roe in mny scientific nd engineering ppictions, nd re so of fundment mthemtic interest see e.g., [13, 4, 11]. Despite its seemingy simpicity, the system 1.1 is notoriousy difficut to sove numericy. ome of the min chenges incude: i the indefiniteness when ω is not sm; ii highy oscitory soutions when ω is rge; iii the incompressibiity i.e., 1991 Mthemtics ubject Cssifiction. 65N35, 65N22, 65F05, 35J05. Key words nd phrses. Mthieu. 1 Deprtment of Mthemtics, Purdue University, West Lfyette, IN, 47907, UA. The work of this uthor is prtiy supported by NF grnt DM nd AFOR FA Division of Mthemtic ciences, choo of Physic nd Mthemtic ciences, Nnyng Technoogic University, , ingpore. The reserch of this uthor is prtiy supported by ingpore MOE AcRF Tier 1 Grnt RG 15/12, MOE AcRF Tier 2 Grnt MOE 2013-T , ARC 44/13 nd ingpore A TAR-ERC-PF Grnt 122-PF

2 2 L.N. MA, J. HEN & L.L. WANG divµh = divεe = 0, which is impicity impied by 1.1; nd iv the unboundedness of the domin. On the one hnd, one needs to construct pproximtion spces such tht the discrete probems re we posed nd ed to good pproximtions for wide rnge of wve number. On the other hnd, perhps more difficut probem is to deveop efficient gorithms for soving the indefinite iner system, prticury for rge wve numbers, from the given discretiztion. We refer to [11] nd the references therein, for vrious contributions with respect to numeric pproximtions of the time-hrmonic Mxwe equtions. Most notby, very popur nd effective method for deing with the unboundedness of the domin is to introduce perfecty mtched yer PML, initiy proposed in [3]. In this pper, we propose spectr pproximtion bsed on the tensor-product of vector spheric hrmonics VH, which forms compete orthogon bsis for L 2 -vector-vued functions on the spheric surfce, nd Legendre poynomis in the rdi direction. It is we-known tht the Mxwe equtions with constnt mgnetic permebiity nd eectric permittivity re seprbe if D is b, nd its soution cn be expicity expressed in terms of the VH nd the spheric Hnke functions [13]. Whie the expicit soution is very usefu for some theoretic considertions, it hs much ess vue in prctice, since most prctic probems woud hve one or more of the foowing situtions: non-spheric domins, non-constnt mgnetic permebiity nd eectric permittivity, non-homogeneous source etc., where n expicit soution woud not be vibe. In order to de with more gener sctterers D nd non-homogeneous source functions, we dpt the so-ced trnsformed fied expnsion TFE [15], which hs proven to be effective for vriety of situtions cf. [14, 5, 6, 9]. The TFE pproch consists of four steps: i reduce the probem in n unbounded domin to bounded domin with trnsprent boundry conditions; ii trnsform the reduced bounded domin to seprbe domin, consider the reduced domin s perturbtion of the seprbe domin, nd expnd the soution in term of the perturbtion prmeter ε; iii sove for ech expnsion coefficient in the seprbe domin; nd iv sum up the expnsion terms using robust Pdé pproximtion. The essenti step in the bove TFE pproch is the step iii, i.e., sove the Mxwe equtions in the seprbe domin which is spheric she in this cse with non-homogeneous source term nd non-oc boundry conditions t the outer spheric surfce. In this pper, we sh deveop n efficient nd robust spectr sover for the nonhomogeneous Mxwe equtions in spheric she. More precisey, we sh use VH to decoupe the probem into sequence of one-dimension probems tht cn be efficienty soved using direct spectr-gerkin method. Therefore, the entire TFE pproch does

3 MAXWELL EQUATION 3 not invove ny itertive sover, nd it is robust for ow to modertey high wve numbers nd to sctterers which hve sufficienty smooth boundries. The rest of the pper is orgnized s foows. In the next section, we introduce the VH nd present the formution of the cpcity opertor chrcterizing the exct nonrefecting boundry condition. In ection 3, we present the TFE gorithm, nd nd formue in Appendix B. In ection 4, we describe the Legendre spectr-gerkin method for the reduced one-dimension probems, nd give the numeric resuts in ection 5. In Appendix A, we provide some usefu formue for the VH, whie in Appendix B, we derive the Mxwe eqution in the trnsformed coordintes, nd the recursion formue in the TFE pproch. 2. Vector spheric hrmonics nd the cpcity opertor In this section, we rec some essenti properties of VH, nd derive the expicit formu for the cpcity opertor expressed in terms of VH, which chrcterizes the exct DtN boundry condition t the outer spheric surfce Vector spheric hrmonics. ever versions of VH with different nottion nd properties hve been used in prctice see e.g., [12, 10, 2, 13, 8, 7]. In wht foows, we dopt the fmiy of VH in [10, 13], nd remrk its retion with sever other fmiies documented in the bove iterture see Remrk 2.1 beow. Rec tht the spheric coordintes r, θ, φ re reted to the Crtesin coordintes x = x 1, x 2, x 3 by cf. [13]: x 1 = r sin θ cos φ, x 2 = r sin θ sin φ, x 3 = r cos θ, 2.1 with the moving right-hnded orthonorm coordinte bsis {e r, e θ, e φ } : e r = x/r, e θ = cos θ cos φ, cos θ sin φ, sin θ, e φ = sin φ, cos φ, For ny v = v 1, v 2, v 3, we denote by v r, v θ nd v φ the projections of v onto e r, e θ nd e φ, respectivey, tht is, v = v r e r + v θ e θ + v φ e φ with v r = v e r, v θ = v e θ, v φ = v e φ. 2.3 Herefter, et be the unit spheric surfce, nd denote by nd the Lpce- Betrmi nd tngent grdient opertors on. Rec tht u = 1 sin θ u u sin θ θ θ sin 2 θ φ 2, u = u θ e θ + 1 u sin θ φ e φ. 2.4 The spheric hrmonics {Y m } s normized in [13] re eigenfunctions of, nmey, Y m = + 1Y m, 0, m ; 2.5

4 4 L.N. MA, J. HEN & L.L. WANG nd form n orthonorm bsis for L 2 : Y m Y m d = δ δ mm. 2.6 The fmiy of VH is defined by T m V m W m = Y m e r = 1 sin θ = + 1Y m e r Y m Y m φ e θ Y m θ e φ, for 1, 0 m, 2.7, for 0, 0 m, 2.8 = Y m e r + Y m, for 1, 0 m. 2.9 Notice tht V 0 0 = e r / 4π. With the understnding of T 0 0 = W 0 0 = 0, the indexes {, m} run over {, m : 0, 0 m }. We coect in Appendix A the properties of VH to be used throughout the pper. Remrk 2.1. The VH in Hi [10] were denoted by {V m, X m, W m }. In fct, we hve the retion V m V m =, X m = it m, + 1 W m W m = Nédéec [13] empoyed the nottion {I m Y m e r = W m + V m 2 + 1, T m, N m }, nd there hod N+1 m = V m, = T m, I 1 m = W m The pherepck [18] used the nottion { Y m e r, Y m, } cur Y m in Morse nd Feshbch [12] so see [13, Thm ]. Noting tht cur Y m = Y m e r cf. [13, ], we hve, cur Y m = T m T m, Y m = + 1W m V m In the numeric experiments in ection 5, we sh use the VH in the pherepck. Define the vector L 2 -spce nd its tngenti vector spce: L 2 = L 2 3, TL 2 = { u L 2 : u e r = 0 } The fmiy of VH, {T m, V m, W m }, forms compete, orthogon bsis of L 2, whie the fmiy {T m, Y m } forms n orthogon bsis for TL 2 cf. 2.4 nd A.1-A The cpcity opertor. As the probem 1.1 is set in n unbounded domin, we first truncte the unbounded domin t n rtifici spheric surfce r = b. ince the exct soution for the homogeneous Mxwe equtions 1.1 exterior to the b r b with µ nd ε being constnt cn be obtined by using the seprtion of vribes [13], we cn set up the exct DtN nonrefecting boundry condition: H n T b E = 0, t r = b, 2.14 where for simpicity, we ssume herefter µ = ε = 1 so the wve number k = ω εµ = ω, nd the cpcity opertor T b, cting on the tngenti component of E i.e., E = E

5 MAXWELL EQUATION 5 n n, cn be determined s in [13] see 2.22 beow. Here, for the reders reference, we sketch the derivtion. Given the tngenti component of E on the rtifici surfce note tht E TL 2, we write Then the exterior probem: E r=b = =1 m =0 cur E e = ikh e, cur H e = ike e, r > b; E e e r = E, t r = b; [ c m T m + d m Y m ] im r r H e e r E e = 0, 2.16 cn be soved nyticy by using seprtion of vribes. The soution {H e, E e } cn be expressed in VH series in terms of {c m, d m } see [13, Thm ]. Then, the cpcity opertor T b, which ssocites E to H e e r on the rtifici spheric surfce, is given by see [13, ] with kb in pce of k: where T b E = H e e r r=b = =1 m =0 [ c m ikb Θ kbt m + ikb ] dm Θ kb Y m, 2.17 d dr h1 r Θ kb = z kb + 1 with z r = r h 1 r, 2.18 nd h 1 r is the spheric Hnke function of the first kind cf. [1]. By imposing H e r = H e e r t r = b, we obtin the exct boundry condition t r = b with T b E given by , but it is expressed by the expnsion coefficients {c m, d m } of E. Thus, it is necessry to represent it in terms of the expnsion coefficients of the fied E with r b. For this purpose, we write [ ] Er, θ, φ = v m rv m θ, φ + t m rt m θ, φ + w m rw m θ, φ, 2.19 =0 m =0 where we rec tht T 0 0 = W 0 0 = 0. Using 2.19 nd the identities: V m e r = T m, T m e r = Y m, W m e r = T m, Y m e r = T m, 2.20 we find E r=b = E e r er r=b = =1 m =0 =1 m =0 [ t m bt m + w m v m Compring the coefficients in 2.17 nd 2.21 eds to [ T b E r=b t m = b ikb Θ kbt m + ikb w m v m b Θ kb where Θ is defined by ] b Y m ] Y m, 2.22

6 6 L.N. MA, J. HEN & L.L. WANG 3. Trnsformed fied expnsion nd dimension reduction Eiminting H nd using the cpcity opertor, we reduce the probem 1.1 with µ = ε = 1 to cur cur E k 2 E = 0, in Ω b \ D; E n D = g; cure e r ikt b E = 0, t r = b, where Ω b is the b of rdius b, nd T b is defined in We now ppy the TFE pproch to Chnge of vribes. Assume tht the sctterer is given by D = { r < + hθ, φ : θ [0, π, φ [0, 2π }, 3.1 for some > 0. Let us choose b such tht b > mx θ,φ { + hθ, φ}, nd then mp the domin: Ω b \ D = { + hθ, φ < r < b} to the spheric she: Ω = { < r < b} with the chnge of vribes: where d = b. r = dr bhθ, φ d hθ, φ, θ = θ, φ = φ, 3.2 Let E = E r, E, where E r nd E re the xi component nd tngenti component of E, respectivey. We first notice tht we cn rewrite the Mxwe eqution fter mutipying both sides by r 2, s cur cur E k 2 E = 0, E r + r re r 2 k 2 E r = 0, 3.3 E e r e r + r r E r r 2 r re r 2 k 2 E = Let us denote the trnsformed fied by F r, θ, φ := Er, θ, φ = E r + Ar, θ, φ, θ, φ := F r, F, 3.5 d with Ar, θ, φ = hθ, φ b r. After some tedious mniputions see Appendix B, we find tht the system is trnsformed into: F r + r r F r 2 k 2 F r = f r, F e r e r + r r F r r 2 r r F r 2 k 2 F = f tp, F e r + F r h = g, t r =, cur F e r ikt b F = J, t r = b, where f r, f tp, g nd J re given in B.5-B.8 in Appendix B. 3.6

7 MAXWELL EQUATION Recursion by boundry perturbtion. Now we ssume h = εq, nd expnd F r, θ, φ ; ε = F n r, θ, φ ε n. 3.7 n=0 Writing F n = Fr n, F n, pugging the bove expnsion into 3.6, nd coecting the terms in powers of ε, we rrive t the foowing recursion for n 0: F n r + r r F n r 2 k 2 F n r = f n r F n e r e r + r r F n r r 2 r r F n r 2 k 2 F n = f n tp F n e r = g n, t r = cur F n e r ikt b F n = J n, t r = b, 3.8 where f n r, f n tp, g n nd J n re given in B.9-B.12 in Appendix B. We note in prticur tht f n r, f n tp nd J n ony depend on the previous four expnsion terms, nmey {F n i, i = 1, 2, 3, 4}. We cn rewrite the bove system in the more compct form: cur cur F n k 2 F n = 1 r 2 f n, in Ω, 3.9 F n e r r= = g n ; cur F n e r ikt b F n = J n, t r = b, 3.10 where f n = f n r, f n tp. Hence, using the TFE pproch, it bois down to soving sequence of non-homogeneous Mxwe equtions in the spheric she Ω. We re therefore concerned with deveoping n efficient, robust sover for this prototype system Dimension reduction. We now consider the foowing probem: cur cur E k 2 E = F, in Ω, 3.11 E e r r= = g; cure e r ikt b E = h, t r = b, 3.12 which hs to be soved for ech expnsion order n with given F, g nd h. It foows from [13, Thm ] tht the probem dmits unique soution, provided tht F L 2 Ω with F = 0 nd g, h TL 2. We refer the interested reders to [4, 13] for deicte regurity nysis of the bove probem. We first expnd, in terms of VH, the unknown function E s in 2.19, nd the source function F s: F r, θ, φ = =0 m =0 [ ] f,mrv v m θ, φ + f,mrt t m θ, φ + f,mrw w m θ, φ. 3.13

8 8 L.N. MA, J. HEN & L.L. WANG Then, we expnd the given dt g nd h in terms of VH bsis of TL 2 : [ ] gθ, φ = ĝ m T m θ, φ + g m Y m θ, φ ; hθ, φ = =1 m =0 =1 m =0 [ĥm T m θ, φ + h ] m Y m θ, φ. For simpicity of presenttion, we define the hndy differentition opertors: 3.14 d + = d dr + r, d = d dr r Inserting the expnsion 2.19 nd into 3.11, we find from the property A.9 tht 3.11 reduces to sequence of one-dimension probems in {v m, t m, w m }. More precisey, we hve v0 0 = 0, nd for 1 nd m, [ d d 1 wm d + ] +2 vm k 2 v m = f,m, v r, b, [ d+ +1 d + +2 vm d ] 1 wm k 2 w m = f,m, t r, b, d r 2 r 2 dtm dr dr r 2 t m k 2 t m = f,m, w r, b imiry, Inserting the expnsion 2.19 nd 3.14 into 3.12, nd using 2.20, 2.22 nd A.9, the boundry conditions 3.12 become Note tht the modes t m nd w m. w m v m = ĝ m, t m = g m ; 3.19 [ d 1 wm d + +2 vm ] b + k 2 b wm v m b Θ kb = ĥm, 3.20 r t m b b 1 z kbt m b = h m coefficients of T m re competey decouped from the modes v m In summry, we ony hve to sove the foowing sequence 1 nd m of onedimension probems with unknowns: v = v m, w = w m, u = t m, nd with given dt f v = f v,m, f w = f w,m, ĝ = ĝm, ĥ = ĥm, g = g m, h = h m : nd β d [ d 1 w d+ +2 v] k 2 v = f v, r, b, β d + [ +1 d + +2 v d 1 w] k 2 w = f w, r, b, 3.23 [ w v = ĝ, d 1 w d+ +2 v] b + k 2 w vb b = ĥ; 3.24 Θ kb 1 d r 2 r 2 du dr dr r 2 u k 2 u = f t, r, b, 3.25 u = g, u b b 1 z kbub = h, 3.26 where β = /2 + 1 nd z is defined in 2.18.

9 MAXWELL EQUATION 9 Remrk 3.1. We derive immeditey from the sovbiity of the 3D probem tht there exists unique tripe {v, w, u} for ech, m tht soves Remrk 3.2. Observe tht the probem is excty the eqution reduced from the time-hrmonic Hemhotz eqution with exct DtN boundry condition in spheric she cf. [17, 3.6]. ince efficient gorithms nd wve number-expicit priori estimtes for this probem hve redy presented in [17], we sh concentrte beow on pectr-gerkin method for the one-dimension systems We now construct the spectr-gerkin method for the couped system First, we mke simpe vribe trnsform, e.g., w ĝ w, to homogenize the Dirichet boundry condition in t r =. Hence, it suffices to consider β d [d 1 w d+ +2 v] k2 v = f 1, r, b, β d + +1 [d+ +2 v d 1 w] k2 w = f 2, r, b, 4.2 w v = 0, [d 1 w d+ +2 v]b + w vb k2 b = h b. 4.3 Θ kb Define the compex vector-vued functions v = v, w t, f = f 1, f 2 t, φ = φ 1, φ 2 t, nd the differenti opertors: = d + +2, d 1, v = d + +2 v d 1w Wek formution nd we-posedness. Let I =, b, nd P N be the set of re gebric poynomis of degree t most N. Define the pproximtion spce X N = { φ = φ 1, φ 2 t P N + ip N 2 : φ 1 φ 2 = 0 }, 4.5 nd the weighted inner product by u, v ω = I ur vrωrdr with ωr = r2, where v is the compex conjugte of v. Then, the spectr-gerkin pproximtion of is to find v N = v N, w N t X N such tht Bv N, φ := v N, φ ω k 2 Lv N, φ ω + k2 b 3 = I N Lf, φ ω + b 2 h b φ 1 φ 2 b, φ X N, Θ kb v N w N bφ 1 φ 2 b 4.6 where L is 2-by-2 digon mtrix: L = dig 2 + 1/, 2 + 1/ + 1,

10 10 L.N. MA, J. HEN & L.L. WANG nd I N is the Legendre-Guss-Lobtto interpotion opertor. In the derivtion of 4.6, we used the identities obtined from integrtion by prts nd the buit-in boundry condition in X N : b b d d 1 w N d + +2 v N φ 1 r 2 dr = d + +1 d+ +2 v N d 1 w N φ 2 r 2 dr = b d 1 w N d + +2 v Nd + +2 φ 1r 2 dr + d 1 w N d + +2 v N φ 1 r 2 b ; b d + +2 v N d 1 w Nd 1 φ 2r 2 dr + d + +2 v N d 1 w N φ 2 r 2 b. Proposition 4.1. For ny fixed, k, b, N, the probem 4.6 dmits unique soution v N X N. Proof. Rec tht cf. [11, Lemm 9.20]: c 1 Θ kb c 2, 1, 4.7 where c 1, c 2 re positive constnts depending on kb. Thus, tking φ = v N in 4.6, nd tking the re prt of the resuted eqution, eds to Re Bv N, v N v N 2 ω k 2 Lv N, v N ω k2 b 3 c 1 v N w N b 2, 4.8 where u 2 ω = u, u ω. In view of v N w N = 0, we derive from [16, Thm. 3.33] tht v N w N b = b r v N w N dr b r v N w N cn 2 v N w N, 4.9 where c is positive constnt independent of N. Notice tht v N 2 ω = r v N w N v N 1 2 w N r r ω 2 r v N w N v N 2 + w N 2 ; Lv N, v N ω v N 2 + w N 2. We find from the bove nd tht for > 0, Re Bv N, v N 2 r v N w N 2 C v N 2 + w N 2, 4.10 where C is positive constnt depending on, b, k, N. ince X N is finite dimension nd v N w N = 0, it is esy to check tht v N := r v N w N is norm on X N. Hence, 4.10 is indeed Gårding type inequity which impies the unique sovbiity of the probem 4.6.

11 MAXWELL EQUATION 11 We remrk tht since Re 1/Θ kb < 0 which cn be derived from [13, ], the corresponding term cn not contribute to the energy norm. Consequenty, we hve to use the trce inequity 4.9 to derive the Gårding type inequity Note so tht the bove proof does not provide wve-number expicit priori estimte on the energy norm. Hence, it is not possibe to derive, from the bove resut, wvenumber expicit error estimte for 4.6, s ws done for the decouped eqution in [17]. In forthcoming pper, we sh consider different pproch, which is more suitbe for nysis but ess convenient for impementtion, nd derive wve-number expicit error estimtes Impementtion. We now describe n efficient impementtion of the scheme 4.6. The efficiency of the gorithm essentiy reies on the choice of bsis functions for X N defined in 4.5. Let L n r be the re-vued Legendre poynomis of degree n, trnsformed from [ 1, 1] to [, b] vi iner mpping, which stisfies L n = 1 n nd L n b = 1. Define φ 0 = 1 + i 2 x + 1; φ j = 1 + il j 1 L j+1, 1 j N 1; φ N = 1 + i x 1. 2 et φj 0 φn ψ j =, ψ 0 N+j =, 0 j N 1; ψ φ 2N = j One verifies rediy tht ψ j X N for 0 j 2N nd tht they re inery independent. ince dimx N = 2N + 1, we hve X N = spn { ψ 0, ψ 1,, ψ 2N } Hence, the pproximte soution v N cn be written s 2N N 1 j=0 v N = α j ψ j = α jφ j + α 2N φ N N 1 j=0 j=0 α = N+jφ j + α 2N φ N etting ψ j = ψ 1,j, ψ 2,j t, 0 j 2N; α = α 0, α 1,, α 2N t ; s ij = b vn d + +2 ψ 1,j d 1 ψ 2,jd + +2 ψ 1,i d 1 ψ 2,ir 2 dr, 0 i, j 2N; w N φ N b ij = k 2 ψ 1,j ψ 1,i ψ 2,jψ 2,i r 2 dr, 0 i, j 2N; b ij = k2 b 3 Θ kb ψ 1,j ψ 2,j bψ 1,i ψ 2,i b, 0 i, j 2N; b f i = I N f 1 ψ 1,i I Nf 2 ψ 2,i r 2 dr b 2 g b ψ 1,i ψ 2,i b, f = f 0, f 1,, f 2N t ; = s ij, A = ij, B = b ij,

12 12 L.N. MA, J. HEN & L.L. WANG nz = 1085 Figure 4.1. Nonzero entries of the system mtrix in we find tht the iner system 4.6 reduces to the mtrix form: + A + B α = f We note tht the coefficient mtrices, A nd B re sprse, see Figure 4.1, nd Hermitin, i.e., = t, nd ikewise for A nd B. To compute their non-zero entries, we ony need to compute b φ jrφ i rr2 dr, b φ jrφ i rr 2 dr, b φ j rφ i rr 2 dr, which cn be evuted excty by using the properties of Legendre poynomis.

13 MAXWELL EQUATION 13 It is worthwhie to point out tht the bsis functions in 4.11 re constructed to minimize the couping of v N nd w N. Indeed, they re couped through the singe bsis function ψ 0. Hence, the system 4.14 cn be soved efficienty by using bock Gussin eimintion process to sove for α 2N first, foowed by soving two decouped systems of size N ech for α 0,, α N 1 t nd α N,, α 2N 1 t. 5. Numeric resuts In this section, we provide some numeric resuts to show the ccurcy nd efficiency of the proposed method. We use the exct mutipe soutions of cf. [13] s the reference soution. In the first exmpe, we tke the exct soution of to be M 0 { E = =1 m h 1 krt m θ, φ + cur h 1 krt m θ, φ }, 5.1 which is iner combintion of the trnsverse eectric nd mgnetic mutipoe soutions. By using 2.12, we find cur h 1 krt m θ, φ = k d dz h1 k d dz h1 Hence, the exct soution {v, w, u} of is v := v m = k d dz h1 w := w m = + 1 k d dz h1 u := t m = h 1 kr. We ook for the pproximte fied: E M0 N M0 r, θ, φ = =1 m =0 kr r h1 kr V m kr r kr r h1 kr kr r h 1 kr W m., h 1 kr, [ ] v,n m rv m θ, φ + t m,n rt m θ, φ + w,n m rw m θ, φ, 5.4 where {v m,n, wm,n } re computed from 4.6, i.e., spectr-gerkin pproximtion of {vm, w m }, nd {t m,n } re the spectr-gerkin pproximtion of {tm }. Using the orthogonity A.1, we hve the expression: E E M 0 2 M 0 N L 2 Ω = t m t m,n 2 L 2 I + 1 =1 m + vm v m,n 2 L 2 I wm w,n m 2 L 2 I In the computtion, we tke = 2, b = 4 nd M 0 = 10. In Figure 5.1, we pot the retive discrete L 2 -error: E E M0 N / E, 2 Ω 2 Ω

14 14 L.N. MA, J. HEN & L.L. WANG ginst vrious N for k = 10 eft nd k = 40 right. We observe tht the error decys exponentiy, s soon s N enters the symptotic rnge, which is for this cse roughy N > k k=40 k=60 k=100 discrete retive L 2 error Figure 5.1. Retive discrete 2 -errors ginst N for k = 10, 40, 100. In the second exmpe, we consider n exct soution generted by the boundry dt g t the sctterer s surfce. More precisey, the exct eectric fied E is given by where E = M 1 =1 m =0 { g1,h m 1 krt m θ, φ + g2, m 1 cur h krt m θ, φ }, g m 1, = g m 2, = 1 + 1h 1 k + 1h 1 kz k + 1 g Y m dσ; g cur Y m dσ. For given g, we cn compute g1, m nd gm 2, using pherepck [18]. Consider the incident wve: e ikx so tht g = e ikx. We tke = 2, b = 4 nd M 1 = 20, nd pot in Figure 5.2, the discrete retive L 2 -errors ginst N for k = 10, 20, 40 We observe tht the error behves very simiry s in the first exmpe. 6. Concuding remrks We deveoped in this pper n efficient nd robust spectr-gerkin method to sove the three-dimension time-hrmonic Mxwe equtions in exterior domins. The method is bsed on the trnsformed fied expnsion TFE pproch which reduces the origin probem in gener exterior domin to sequence of Mxwe equtions in seprbe spheric she. By using proper set of vector spheric hrmonic functions, we re

15 MAXWELL EQUATION k=10 k=20 k=30 discrete retive L 2 error N Figure 5.2. Retive discrete 2 -errors ginst N for k = 10, 20, 40. be to reduce the Mxwe equtions in seprbe spheric she to sequence of onedimension probems in the xi direction. Then, we proposed n efficient Legendre- Gerkin gorithm to sove the one-dimension probems. This method does not invove ny itertive gorithm for soving iner systems. Hence, it is robust to wve numbers s ong s the soution is we resoved by the spectr discretiztion. Aso, the method enjoys spectr ccurcy, i.e., the convergence rte increses s the smoothness of dt increses. To the best of the uthors knowedge, this is the first fu spectr method for soving the three-dimension time-hrmonic Mxwe equtions in exterior domins. Whie we hve restricted our ttention to probems with constnt mgnetic permebiity nd eectric permittivity, it is cer tht our method cn be esiy extended to yered mteris which wi ed to one-dimension probems with piecewise-constnt coefficients tht cn be soved efficienty with spectr-eement method. Appendix A. Properties of the vector spheric hrmonics The VH re mutuy orthogon in L 2 = L 2 3 : T m V m d = T m W m d = V m W m d = 0, V m V m d = δ δ mm, W m W m d = 2 + 1δ δ mm, which, together with 2.12, impies T m Y m d = 0, T m T m d = + 1δ δ mm, A.1 Y m Y m d = + 1δ δ mm. A.2

16 16 L.N. MA, J. HEN & L.L. WANG Let f nd v be differentibe scer nd vector functions, respectivey. Rec tht in spheric coordintes cf. [1]: grd f = f = f r e r + 1 f r θ e θ + 1 f r sin θ φ e φ, A.3 div v = v = 1 r 2 r 2 v r r + 1 sin θv θ + 1 v φ r sin θ θ r sin θ φ, A.4 cur v = v = 1 sin θvφ v θ e r v r r sin θ θ φ r sin θ φ rvφ e θ r + 1 rvθ rv r e φ. r r θ A.5 Let d + nd d be the differentition opertors defined in 3.15 nd we further define L = d2 dr d + 1 r dr r 2. A.6 In view of 2.10, the foowing properties cn be derived from [10]: The scr grdient: 2 + 1grd fy m = d f V m +1 + d + +1 f W m 1. A.7 The vector divergence: div fv m = + 1 d + +2 f Y m, div ft m = 0, div fw m = d 1 f Y m. A.8 The vector cur: cur fv m 2 + 1cur ft m = d + +2 f T m, cur fw m = d 1 f T m, = + 1 d + +1 f W m d f V m. A.9 The vector Lpce: fv m = L+1 fv m, ft m = L ft m, fw m = L 1 fw m. A.10 Appendix B. Formue reted to the trnsformed fied expnsion Rec tht we set F r, θ, φ = Er, θ, φ with the trnsform 3.2, nd we need to compute cur cure k 2 E in the new coordintes.

17 MAXWELL EQUATION 17 For ny scr function E nd vector function E, we hve the foowing formue under the spheric coordintes: E = r Ee r + 1 r θee θ + 1 r sin θ φee φ, E = 1 r 2 rr 2 E r + 1 r sin θ θsin θe θ + 1 r sin θ φe φ, E = θ Ee θ + 1 sin θ φee φ, E = 1 sin θ θsin θe θ + 1 sin θ φe φ, cur E = 1 θ sin θe φ φ E θ er r sin θ r sin θ φe r r re φ e θ + 1 r r re θ θ E r eφ, B.1 where E = E r, E = E r, E θ, E φ. We cn rewrite the st identity s cur E = E e r e r + E r e r 1 r rre e r = E e r e r + 1 r E r e r 1 r rre e r. Notice the st two terms ony invove the component of θ, φ. Consequenty, we cn derive cur cur E = E r e r e r 1 r rre e r e r e r + E e r e r 1 r rr E r e r e r r re e r e r = 1 r E r 1 r rre e r e r + E e r e r 1 r r E r r re e r e r [ 1 1 ] = r E r r rre e r e r + 1 r E e r e r + 1 r r E r + 1 r 2 r re e r e r [ = 1 r 2 E r + 1 ] r 2 r re + 1 r 2 E e r e r + 1 r r E r 1 r 2 r re. e r e r e r

18 18 L.N. MA, J. HEN & L.L. WANG The chnge of vribes eds to r F = d hθ, φ r E, d θ F = θ hθ, φ b r d φ F = φ hθ, φ b r d r E + θ E = θ hθ, φ b r d hθ, φ r F + θ E, r E + φ E = φ hθ, φ b r d hθ, φ r F + φ E. B.2 B.3 B.4 With the bove preprtion nd ddition ccutions, we cn derive the foowing formue for f r, f tp nd J in 3.6: d 2 f r = 2dh + h 2 F r d h A r F r d h r F r A r F r h A + A r r F r A 2dh + h 2 r r F hd h r r F d h r r F + dd h A r 2 r F h r AF 4 d h r AF + A r 2 AF + H i hk 2 F r, i=1 B.5 d 2 f tp = 2dh + h 2 F e r + d h r F e r A + d h A r F e r + A r F e r h r A r F e r A 2dh + h 2 r r F r hd hr r F r Ad h r F r + dr + A r r F r A + 2dh + h 2 r r 2 r F + 2dh h 2 + Adr r 2 r F 4 + r d + A r 2 AF + H i hk 2 F, i=1 B.6 nd dj = h b F e r hikt b F, g = r =b [1 + θh φh 2 ] sin 2 g θ e r + h b r F r e r + 1 =b b r AF e r r =b B.7 B.8 where F = F r, F, nd H 1 := 2dAr 2dhr 2, H 2 := h 2 r 2 4hAr + A 2, H 3 := 2 d h2 Ar 2 d ha2, H 4 := 1 d 2 h2 A 2.

19 MAXWELL EQUATION 19 imiry, we cn determine the foowing formue for f n r, f n tp nd J n in 3.8: d 2 f n r =2dq F n 1 r + q2 F n 2 r d A q r F n 1 r + q A q r F n 2 r d r F n 1 r A q + q r F n 2 r A q r F n 2 r q A q + A q r r Fr n 2 A q 2dq r r F n 1 q 2 r r F n 2 d q r r F n 1 q r A q F n 2 + A q 2 r A qf n 2 + dq r r F n 1 + q 2 r r F n 2 + d 2 A q r 2 r F n 1 dq A q r 2 r F n 2 i=1 d r A q F n 1 + q r A q F n 2 4 Hq i k 2 F n i r, B.9 d 2 f n tp = 2dq F n 1 e r q 2 F n 2 e r + d r F n 1 e r A q q r F n 2 e r A q + d A q r F n 1 e r q A q r F n 2 e r + A q r F n 2 e r q r A q r F n 2 e r A q 2dqr r F n 1 r + q 2 r r F n 2 r + dr r r F n 1 r + q 2 r 2 r r F n 2 q2 r r F n 2 dqr r F n 1 r r da q r F n 1 + qa q r F n 2 r r A q + A q r r F n 1 A q + 2dqr r 2 r F n 1 + 2dqr r 2 r F n 1 q 2 r r 2 r F n 2 r + A q dr r 2 r F n 1 + dr r 2 A qf n 1 + A q r 2 A qf n Hq i k 2 F n i, i=1 B.10 nd g n = [1 + θh φh 2 ] sin 2 θ dj n = q b F n 1 e r r =b + 1 b r A qf n 1 e r δ n0 g F r n 1 q e r + q b F n 1 r r =b qikt b F n 1, e r r =b B.11 B.12 where F m = F m r, F m for ny m, nd H 1 := 2dA q r 2dqr 2, H2 := q 2 r 2 4qA q r + A 2 q, H 3 := 2 d q2 A q r 2 d qa2 q, H4 := 1 d 2 q2 A 2 q.

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