On Spectral Approximations in Elliptical Geometries Using Mathieu Functions

MATHEMATICS OF COMPUTATION Volume 78, Number 266, April 2009, Pages 815–844 S 0025-5718(08)02197-2 Article electronically published on November 20, 2008

ON SPECTRAL APPROXIMATIONS IN ELLIPTICAL GEOMETRIES USING MATHIEU FUNCTIONS

JIE SHEN AND LI-LIAN WANG

Abstract. We consider in this paper approximation properties and applications of Mathieu functions. A ﬁrst set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the Mathieu-Legendre approximation to the modiﬁed Helmholtz equation and Helmholtz equation. Illustrative numerical results consistent with the theoretical analysis are also presented.

1. Introduction The Mathieu functions were first introduced by Mathieu in the nineteenth cen- tury, when he determined the vibrational modes of a stretched membrane with an elliptic boundary [27]. Since then, these functions have been extensively used in many areas of physics and engineering (cf., for instance, [25, 23, 19, 4, 21, 5, 32, 8, 17]), and many efforts have also been devoted to analytic, numerical and various other aspects of Mathieu functions (cf., for instance, [26, 6, 3, 36]). A considerable amount of mathematical results of Mathieu functions are contained in the books [28, 29, 1]. However, most of these results are concerned with classical properties such as identities, recursions and asymptotics. To the best of our knowledge, there are essentially no results on their approximation properties (in Sobolev spaces) which are necessary for the analysis of spectral methods using Mathieu functions. A main objective of this paper is to derive a first set of optimal approximation results for the Math- ieu functions. These approximation results will be the basic ingredients for the numerical analysis of Mathieu approximations to partial differential equations. Given a PDE in an elliptic or elliptic cylindrical geometry, a spectral method can be developed using two different approaches. In the first approach, we express the equation in the elliptic coordinates and then apply a Fourier approximation in the periodic direction combined with a polynomial approximation in nonperiodic direction(s), see, e.g., [9, 30]. However, unlike in the polar and spherical geometries, all Fourier components are usually coupled together by the nonconstant coefficients in the transformed equation (only the Poisson equation in elliptical geometries can be decoupled (see [24]) where a mixed Fourier and finite difference solver was

Received by the editor March 4, 2008. 2000 Mathematics Subject Classiﬁcation. Primary 65N35, 65N22, 65F05, 35J05. Key words and phrases. Mathieu functions, elliptic coordinates, approximation in Sobolev spaces, Helmholtz equations. The work of the ﬁrst author was partially supported by NSF Grant DMS-0610646. The work of the second author was partially supported by a Start-Up grant from NTU, Sin- gapore MOE Grant T207B2202, and Singapore NRF2007IDM-IDM002-010.

c 2008 American Mathematical Society Reverts to public domain 28 years from publication 815

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proposed for this very particular case). The second approach is to use a Mathieu expansion in the periodic direction which leads to a dimension reduction, as in the polar and spherical geometries. In this paper, we will explore the second approach, derive optimal approximation results for the Mathieu expansions and apply them to study the Mathieu spectral method for the modified Helmholtz equation and the Helmholtz equation. The rest of the paper is organized as follows. We first review some properties of the Mathieu functions in Section 2. The approximations by Mathieu functions in Sobolev spaces are studied in Sections 3 and 4. The applications of Mathieu approximations in spectral methods for two model Helmholtz-type equations are presented in Sections 5 and 6. Section 7 is devoted to some numerical results and discussions. We now introduce some notation to be used throughout this paper. Let Ω be a bounded, open domain in Rd,d =1, 2, 3, and let be a generic real weight 2 function defined in Ω. We denote by L(Ω) a Hilbert space of real- or complex- valued functions with inner product and norm 1/2 (u, v) = u vd¯ Ω, u =(u, u) , Ω s wherev ¯ is the complex conjugate of v. The weighted Sobolev spaces H(Ω) (s = 0, 1, 2, ···) can be defined as usual with inner products, norms and semi-norms · · · |·| s denoted by ( , )s,, s, and s,, respectively. For real s>0,H(Ω) is defined by space interpolation as in [2]. The subscript will be omitted from the 2 0 notation in cases of =1. In particular, we have L(Ω) = H(Ω). We will use 2 ·or · to denote the usual L -norm, and use · or · 2 to denote 0 0, L (Ω) 2 k dk the -weighted L -norm. We will use ∂x to denote the ordinary derivative dxk , whenever no confusion may arise. We will also use Sobolev spaces involving periodic functions. Namely, for m>0, m m we denote by Hp (0, 2π) the subspace of H (0, 2π) consisting of functions whose derivatives of order up to m − 1are2π−periodic. For any nonnegative integer N, let PN be the set of all algebraic polynomials of degree ≤ N. We denote by C a generic positive constant independent of any function, domain size and discretization parameters. We use the expression A B to mean that there exists a generic positive constant C such that A ≤ CB.

2. Mathieu functions We recall that under the elliptic transform: (2.1) x = c cosh µ cos θ, y = c sinh µ sin θ (where 2c is the focal distance), the two-dimensional Helmholtz equation in Carte- sian coordinates (2.2) ∆U + k2U =0

becomes 1 ∂2V ∂2V (2.3) + + k2V =0, c2 − ∂µ2 ∂θ2 2 cosh(2µ) cos(2θ) where V (µ, θ)=U(x, y). The Mathieu functions arise from applying the separation of variables approach in solving (2.3). More precisely, setting V (µ, θ)=R(µ)Φ(θ),

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we find that Φ(θ) satisfies the (angular) Mathieu equation d2Φ (2.4) +(a − 2q cos 2θ)Φ = 0, dθ2 and R(µ) satisfies the radial (or modified) Mathieu equation d2R (2.5) − (a − 2q cosh 2µ)R =0, dµ2 where a is the separation constant, and the parameter q = c2k2/4. We note that the radial Mathieu equation (2.5) can be transformed to√ the Mathieu equation (2.4), and vice versa, via the mapping θ = ±iµ (where i = −1). The Mathieu equation (2.4) supplemented with a periodic boundary condition admits two families of linearly independent periodic solutions (eigenfunctions), namely the even and the odd Mathieu functions of order m:

(2.6) Φm(θ; q)=cem(θ; q)orsem+1(θ; q),m=0, 1, ··· , with the corresponding eigenvalues classified into two categories: even and odd, denoted by am(q)andbm(q)forcem and sem, respectively. The notation ce and se, an abbreviation of “cosine-elliptic” and “sine-elliptic”, were first introduced in [7]. Notice that the Mathieu equation (2.4) becomes a harmonic equation when q =0, so the Mathieu functions reduce to the trigonometric functions, namely 2 (2.7) cem(θ;0)=cos(mθ), sem(θ;0)=sin(mθ),am(0) = bm(0) = m . { }∞ The set of Mathieu functions cem, sem+1 m=0 forms a complete orthogonal system in L2(0, 2π), and they are normalized so that 2π 2π π, if m = n, (2.8) cem(θ; q)cen(θ; q)dθ = sem(θ; q)sen(θ; q)dθ = 0 0 0, if m = n and 2π (2.9) cem(θ; q)sen(θ; q)dθ =0. 0 The parity, periodicity and normalization of the Mathieu functions are exactly the same as their trigonometric counterparts cos and sin, namely, cem is even and sem is odd, and they have period π when m is even, or period 2π when m is odd. Thanks to the periodicity and parity, the Mathieu functions can be expanded in the Fourier series ∞ ∞ (n) (n) (2.10) cen(θ; q)= Aj cos jθ, sen(θ; q)= Bj sin jθ, j=0 j=1 { (n) (n)} where the coefficients Aj ,Bj satisfy some five-term recursive relations (see Appendix A). This motivates most of the algorithms for computing Mathieu functions and their eigenvalues; see, e.g., [28, 1, 26, 36, 5, 3]. A package of Fortran and Matlab programs for manipulating Mathieu functions is also available (cf. [33]). For the Helmholtz equation (2.2), the wave number k>0sotheparameter q = c2k2/4 in the Mathieu equation is positive. However, for imaginary k in (2.2), q in (2.4)–(2.5) becomes negative and equation (2.2) becomes elliptic. Hence, the Mathieu functions with q>0 are mostly used in applications involving wave equations, while the Mathieu functions with q<0 are useful in solving elliptic and parabolic equations [32, 8, 18]. We see that with a change of variable θ → π/2 − θ,

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the Mathieu equation (2.4) is transformed into an equation of the same type with q<0. Thus, the Mathieu functions with q<0 can be deﬁned as follows (see, e.g., [28, 29, 1]): m ce2m(θ; −q)=(−1) ce2m(π/2 − θ; q),a2m(−q)=a2m(q), m ce2m+1(θ; −q)=(−1) se2m+1(π/2 − θ; q),a2m+1(−q)=b2m+1(q), (2.11) m se2m+1(θ; −q)=(−1) ce2m+1(π/2 − θ; q),b2m+1(−q)=a2m+1(q), m se2m+2(θ; −q)=(−1) se2m+2(π/2 − θ; q),b2m+2(−q)=b2m+2(q). While there have been quite a few papers/books devoted to analytic, numerical and various other aspects of Mathieu functions (cf., for instance, [26, 6, 3, 36]), there are essentially no results on the approximation of periodic functions using Mathieu functions (in Sobolev spaces). Such results are necessary for spectral methods for elliptic geometries.

3. Mathieu approximations with q>0 In this section, we consider approximation of periodic functions by Mathieu functions, and establish basic approximation results for Mathieu expansions. For clarity, we ﬁrst consider the Mathieu functions with q>0, and the case with q<0 will be presented in the proceeding section. To simplify the presentation, we shall use a uniform notation introduced in [29] to denote the odd and even Mathieu functions:

(3.1) me2m(θ; q)=cem(θ; q), me2m+1(θ; q)=sem+1(θ; q),m=0, 1, ··· and

(3.2) λ2m(q)=am(q)+2q, λ2m+1(q)=bm(q)+2q, m =0, 1, ··· .

By (2.4), the eigenpairs {(λm, mem)} satisfy the Sturm-Liouville equation:

(3.3) Aq mem = λm mem,m=0, 1, ··· , where the operator A − 2 2 (3.4) qv =( ∂θ +4q cos θ)v.

Let I := (0, 2π). One veriﬁes that the Sturm-Liouville operator Aq is compact, symmetric, strictly positive and self-adjoint in the sense that A A A ∀ ∈ 2 ( qu, v)=(v, qu)=(u, qv), u, v Hp (I), (3.5) A 2 2 ∀ ∈ 2 ( qv, v)= ∂θv +4q v 0,ω > 0, v Hp (I)withv =0, where ω =cos2 θ. Hence, we infer from the standard Sturm-Liouville theory (see, e.g., [11]) that: • The eigenvalues are all real, positive and distinct, i.e.,

(3.6) 0 <λ0(q) <λ1(q) < ···<λm(q) < ··· . • { }∞ The eigenfunctions, mem m=0, form a complete orthogonal system in L2(I), and they are normalized so that 1 2π (3.7) mem(θ; q)men(θ; q)dθ = δmn, π 0

where δmn is the Kronecker delta.

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• The Mathieu function mem has exactly m zeros, which are real, distinct and located in [0, 2π]. The following estimate for the eigenvalues is useful in the sequel. Lemma 3.1. For any q>0, we have that

2 2 (3.8) m <λm(q)

Multiplying the above equation by mem and integrating the resulting equation over (0, 2π), we deduce from the self-adjoint property of the operator Aq − λm that 2π ∂me 2π 0= (A − λ ) m me dθ − (λ (q) − 4cos2 θ)me2 dθ q m ∂q m m m 0 0 2π 2π A − ∂mem − − 2 2 = ( q λm)mem dθ λm(q) 4cos θ mem dθ. 0 ∂q 0 Therefore, by (3.3), the ﬁrst term vanishes, and thanks to the orthogonality (3.7), 2π 4 2 2 (3.9) λm(q)= cos θ mem(θ; q) dθ. π 0 Since 0 ≤ cos2 θ ≤ 1, the above identity implies that ⇒ − 0 <λm(q) < 4 0 <λm(q) λm(0) < 4q.

2 Since λm(0) = m , the desired result (3.8) follows.

3.1. Inverse inequalities. We ﬁrst point out that the Mathieu functions are or- 1 thogonal in Hp (I) equipped with the inner product

2 (3.10) aq(u, v)=(∂θu, ∂θv)+4q(u, v)ω,ω=cos θ. Indeed, by integration by parts, (3.3) and (3.7), (3.11) aq(mem, men)= Aqmem, men = λm(mem, men)=πλmδmn. For any integer M ≥ 0, we deﬁne the (M + 1)-dimensional space

q ≤ ≤ (3.12) XM =span mem :0 m M . Thanks to the orthogonality (3.11), we are able to prove the following inequalities. ∈ q Lemma 3.2. For any φ, ψ XM , we have 2 (3.13) |aq(φ, ψ)|≤λM φψ≤(M +4q)φψ, and the following inverse inequality holds l ≤ l ≥ (3.14) ∂θφ (M + ck) φ ,l 1, √ where ck = 4q.

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∈ q Proof. For any φ, ψ XM , we write M M ˆ ˆ φ = φm mem,ψ= ψn men. m=0 n=0 By the orthogonality (3.7), we have M M 2 ˆ 2 2 ˆ 2 φ = π |φm| , ψ = π |ψn| . m=0 n=0 Using the orthogonality (3.11), the Cauchy-Schwarz inequality and Lemma 3.1 yields

M M M ˆ ˆ ˆ ˆ aq(φ, ψ) = φmψnaq(mem, men) = πλmφmψm m=0 n=0 m=0 M 1/2 M 1/2 ˆ 2 ˆ 2 ≤ max λm π |φm| π |ψm| 0≤m≤M m=0 m=0 2 ≤ λM φψ≤(M +4q)φψ. Thus, by the deﬁnition (3.10), 2 2 ≤ 2 2 2 2 aq(φ, φ)= ∂θφ +4q φ 0,ω (M + c k ) φ , which implies (3.14) with l =1. The desired result with l>1canbederivedby induction.

Remark 3.1. Compared with the Fourier basis, the factor O(M)intheinverse inequality is optimal. 3.2. Error estimations of the L2-projection. For any v ∈ L2(I), we write ∞ (3.15) v(θ)= vˆmmem(θ; q), m=0 where 1 2π (3.16)v ˆm =ˆvm(q)= v(θ)mem(θ; q) dθ, m ≥ 0. π 0 Consider the orthogonal projection πq : L2(I) → Xq such that M M q − ∀ ∈ q (3.17) πM v v, vM =0, vM XM , or equivalently, M q (3.18) πM v (θ)= vˆmmem(θ; q). m=0 To measure the projection error, we first introduce a Hilbert space associated with the Sturm-Liouville operator defined in (3.4). Since Aq is a compact, sym- 1/2 metric and (strictly) positive self-adjoint operator, the fractional power Aq is well defined, and there holds (see, e.g., Chapter II of [35]): A1/2 2 (3.19) q v = aq(v, v),

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· · 1 where aq( , ) is the inner product of Hp (I) given in (3.10). Hence, for any integer s ≥ 0, we define the Hilbert space s ∈ s 2 As/2 2 As/2 As/2 ∞ (3.20) Hqp (I):= v Hp (I): v s := q v = q v, q v < . Hqp (I) s For real s>0, the space Hqp (I) and its norm are defined by space interpolation as in [2]. Formally, one verifies by using (3.3), (3.7) and (3.11) that for any integer n ≥ 0, (3.21) ∞ ∞ An 2 2n| |2 An+1/2 2 An An 2n+1| |2 q v = π λm vˆm , q v = aq( q v, q v)=π λm vˆm . m=0 m=0 s Therefore, as with the Fourier expansion, the norm of the space Hqp (I)withreal s ≥ 0 can be characterized in the frequency space ∞ 1/2 As/2 s | |2 (3.22) v s = q v = π λm vˆm . Hqp (I) m=0 We have the following fundamental approximation result. ∈ s ≥ Theorem 3.1. For any v Hqp (I) with s 0, q − ≤ −s/2 (3.23) π v v λ v s . M M+1 Hqp (I) Proof. We only need to prove this result with integer s ≥ 0. The general case can then be derived by space interpolation. We first assume that s =2n. By (3.16) and (3.3), 1 2π 1 2π vˆ = v(θ)me (θ; q)dθ = v(θ) An me dθ m π m πλn q m (3.24) 0 m 0 1 2π 1 = Anv (θ)me (θ; q)dθ = Anv , n q m n q m πλm 0 λm where Anv are the coefficients of the Mathieu expansion of Anv. Therefore, q m q ∞ ∞ − 2 πq v − v2 = π |vˆ |2 = π λ 2n Anv M m m q m (3.25) m=M+1 m=M+1

≤ −2n An 2 (3.6) −s 2 max λm q v = λM+1 v s , m>M Hqp (I) which implies (3.23) with s =2n. We now prove (3.23) with s =2n +1. By (3.21) and (3.6), ∞ ∞ q − 2 | |2 ≤ −2n−1 2n+1| |2 πM v v = π vˆm max λm πλm vˆm m>M (3.26) m=M+1 m=M+1 ≤ −2n−1 An An ≤ −s 2 λM+1 aq( q u, q u) λM+1 v s . Hqp (I) This completes the proof. The norm in the upper bound of the estimate (3.23) is expressed in the frequency space, and implicitly depends on the parameter q. To extract more explicit

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information from (3.23), we shall explore the explicit dependence of the approximation errors on the parameter q = c2k2/4, and express the norm in terms of the derivatives of v. It is worthwhile to point out that setting q = 0 in Theorem 3.1, we recover the classical Fourier approximation results. So, without loss of generality, we assume that the wave number ≥ ⇒ ≥ 2 2 (3.27) k k0 > 0 q q0 = c k0/4 > 0. It can be shown by induction that for any nonnegative integer n, n n−1 An − n 2n 2n−2j 2n−2j−1 (3.28) q v =( 1) ∂θ v + pj(θ; q)∂θ v + p˜j(θ; q)∂θ v, j=1 j=1

where pj(θ; q)and˜pj(θ; q)arepolynomialsof4q of degree j (without constant term, i.e., pj(θ;0)=p ˜j(θ; 0) = 0) with coefficients being polynomials of sin 2θ or cos 2θ. An 2n 2 We conclude that q is a continuous mapping from Hp (I)toL (I), since n−1 An ≤ 2n j 2n−2j 2n−2j−1 n (3.29) q v ∂θ v + C q ∂θ v + ∂θ v + q v . j=1 Similarly, using (3.19) and (3.28)–(3.29) leads to 1/2 √ An+1/2 An An ≤ An An q v = aq( q v, q v) ∂θ( q v) +2 q q v 0,ω (3.30) n−1 ≤ 2n+1 j+1/2 2n−2j 2n−2j+1 n+1/2 ∂θ v + C q ∂θ v + ∂θ v + q v , j=1 where we recall from (3.10) that ω =cos2 θ. As a consequence, we have the em- s ⊂ s bedding relation Hp (I) Hqp (I), and s−2 As/2 ≤| | (s−j)/2| | s/2 ≥ v Hs (I) = q v v s + C q v j (1 + q) v s,s 2, (3.31) qp √ j=0 ≤| | v s v s +2 q v 0,ω,s=1, 2. Hqp (I) Notice that the powers of q for derivatives of different order are different; such an explicit estimate is necessary when the wave number k 1(seeRemark3.2). The following estimate is a direct consequence of Theorem 3.1 and (3.31). ∈ s ≥ Corollary 3.1. For any v Hp (I) with s 0, s−2 q − ≤ −s | | (s−j)/2| | 1/2 (3.32) πM v v M v s + C q v j + q v 0,ω . j=0 Remark 3.2. As an example, let v(x)=eikx with k 1, and q = c2k2/4. A direct calculation leads to s−2 (s−j)/2 1/2 s/2 |v|s + C q |v|j + q v0,ω = O(q ); j=0

therefore, we have ks πq v − v = O , M M s

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k s/2 which will converge if M < 1. Notice that if we use the rough upper bound (1+q) k2 in (3.31), the condition for convergence will be M < 1. A more general approximation result is stated below. ∈ s ≤ ≤ Theorem 3.2. For any v Hqp (I), we have that for 0 r s, q − ≤ (r−s)/2 q − ≤ (r−s)/2 (3.33) π v v r λ π v v s λ v s . M Hqp (I) M+1 M Hqp (I) M+1 Hqp (I) Proof. By (3.22), we have ∞ ∞ q − 2 r | |2 ≤ r−s s | |2 πM v v r = π λm vˆm max λm πλm vˆm Hqp (I) m>M (3.34) m=M+1 m=M+1 ∞ r−s q − 2 ≤ r−s s | |2 r−s 2 = λM+1 πM v v s λM+1 πλm vˆm = λM+1 v s . Hqp (I) Hqp (I) m=0 This ends the proof.

As an important consequence of the above theorem, we have the following esti- r ≥ mates in the usual Sobolev space Hp (I)withr 1. ∈ s Theorem 3.3. For any v Hqp (I), √ | q − | ≤ (1−s)/2 −1/2 ≥ (3.35) π v v 1 λ 1+2 qλ v s ,s 1 M M+1 M+1 Hqp (I) and √ | q − | ≤ 1−s/2 −1 ≥ (3.36) π v v 2 λ 1+2 qλ v s ,s 2. M M+1 M+1 Hqp (I) In general, if k

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Hence, using Theorems 3.1 and 3.2 leads to √ | q − | ≤ q − q − π v v 1 π v v 1 +2 q π v v M M Hqp (I) M √ ≤ (1−s)/2 −1/2 ≥ λ 1+2 qλ v s ,s 1 M+1 M+1 Hqp (I) and √ | q − | ≤ 1−s/2 −1 ≥ π v v 2 λ 1+2 qλ v s ,s 2, M M+1 M+1 Hqp (I) which yield (3.35) and (3.36). We now prove (3.37) by induction. Assuming that (3.37) holds for j

4. Mathieu approximations with q<0 In this section, we extend the analysis of Mathieu approximations to the case with parameter q<0. Such results play an essential role in the analysis of spectral methods for elliptic and parabolic PDEs in elliptic geometries. For clarity, let q = −ρ with ρ>0, and let {cem, sem} be the Mathieu functions deﬁned in Section 2. In view of the symmetry (2.11), we deﬁne [ m ] Me2m(θ; ρ)=(−1) 2 cem(π/2 − θ; ρ), (4.1) [ m ] Me2m+1(θ; ρ)=(−1) 2 sem+1(π/2 − θ; ρ),

for m =0, 1, ··· , where [a] denotes the largest integer ≤ a. Let {λm(·)} be the eigenvalues defined in (3.2). One verifies that the pair {(λm, Mem)} satisfies the Sturm-Liouville problem

(4.2) AρMem = λmMem, where the Sturm-Liouville operator is deﬁned as A − 2 2 (4.3) ρv = ∂θ +4ρ sin θ v.

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As the operator Ap deﬁned in (3.4), Aρ is also compact, strictly positive, symmetric and self-adjoint. The following properties can be derived from the Sturm-Liouville theory or the symmetric property (2.11):

• The set of Mathieu functions {Mem(θ; ρ)} forms a complete orthogonal system in L2(I), and 1 2π (4.4) Mem(θ; ρ)Men(θ; ρ)dθ = δmn. π 0 1 Moreover, they are orthogonal in Hp (I) with the inner product 2 (4.5)a ˆρ(u, v)=(∂θu, ∂θv)+4ρ(u, v)ωˆ , withω ˆ =sin θ, namely, 2π 2π 2 aˆρ Mem, Men = ∂θMem∂θMendθ +4ρ MemMen cos θdθ (4.6) 0 0 = AρMem, Men = πλmδmn.

• The eigenvalues {λm} are real, distinct and arranged in ascending order (cf. (3.6))

(4.7) 0 <λ0(ρ) <λ1(ρ) < ···<λm(ρ) < ··· , and by (3.8),

2 2 (4.8) m <λm(ρ)

ρ ≤ ≤ (4.9) XM := span Mem :0 m M . The following inequalities can be proved in the same fashion as for Lemma 3.2. Lemma 4.1. We have | |≤ ≤ 2 ∀ ∈ ρ (4.10) aˆρ(φ, ψ) λM φ ψ (M +4ρ) φ ψ , φ, ψ XM , and the following inverse inequality holds: l ≤ l/2 ∀ ∈ ρ ≥ (4.11) ∂θφ λM φ , φ XM ,l 1. 2 ρ 2 → ρ We now consider the L -orthogonal projectionπ ˆM : L (I) XM deﬁned by ρ − ∀ ∈ ρ (4.12) (ˆπM v v, vM )=0, vM XM . s We introduce the Hilbert space Hρp (I) with the norm v s , which is deﬁned Hqp (I) by replacing Aq and aq(·, ·) in (3.19)–(3.20) by Aρ anda ˆρ(·, ·), respectively. Since the Sturm-Liouville operator Aρ is compact, strictly positive, symmetric and self- s adjoint, similar to (3.22), the norm of the space Hρp (I)canbeexpressedinthe frequency space 1/2 s 2π s | |2 1 (4.13) v Hs (I) = π λm vˆm withv ˆm = v(θ)Mem(θ; ρ)dθ. ρp π m=0 0

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In most applications, we may assume that the parameter ρ is a ﬁnite constant independent of M. One veriﬁes from a direct calculation (refer to (3.28)–(3.30)) s ⊂ s that the embedding relation Hρp (I) Hp (I)and ≤ ≥ (4.14) v s C v s,s 0, Hρp (I) where C is a positive constant independent of v (but depending on ρ). In the case of ρ 1, a more precise estimate with an explicit dependence of ρ can be carried out in the same fashion as in (3.31). Following the same argument as in the proofs of Theorems 3.1–3.4, we can derive the following approximation results. ∈ s Theorem 4.1. For any v Hρp (I), we have that ρ − ≤ (r−s)/2 ≤ ≤ (4.15) πˆ v v r λ v s , 0 r s. M Hρp (I) M+1 Hρp (I) ∈ s In particular, for any v Hp (I), | ρ − | r−s ≤ ≤ (4.16) πˆM v v r M v s, 0 r s and ρ − − ∞ 1/2 s ≥ (4.17) πˆM v v L M v s,s 1. In the preceding part, we presented Mathieu approximation results in Sobolev spaces, which are the main ingredients for the analysis of spectral approximation to PDEs in elliptical geometries, as illustrated in the succeeding two sections.

5. Application to the modified Helmholtz equation We ﬁrst consider the modiﬁed Helmholtz equation in an ellipse: (5.1) − ∆U + βU = F, in Ω; U(x, y)=0, on ∂Ω, where β ≥ 0,F is a given function and x2 y2 (5.2) Ω = (x, y): + < 1 c2 cosh2 1 c2 sinh2 1 with c being the semi-focal distance. Using the elliptic transform (2.1) and setting u(µ, θ)=U(x, y), we have that 1 ∂u 1 ∂u 1 (5.3) ∇U(x, y)= e + e := ∇˜ u h ∂µ µ h ∂θ θ h and 1 ∂2u ∂2u 1 (5.4) ∆U(x, y)= + := ∆˜ u, h2 ∂µ2 ∂θ2 h2 where 1/2 c 1/2 (5.5) h = h(µ, θ)=c cosh2 µ − cos2 θ = √ cosh 2µ − cos 2θ , 2 and the Jacobian from (x, y)to(µ, θ)ish2, i.e., dxdy = h2dµdθ. Hence, under the elliptic coordinates, equation (5.1) takes the form 1 − ∆˜ u + βu = f, (µ, θ) ∈Q:= (0, 1) × [0, 2π), (5.6) h2 u(1,θ)=0,θ∈ [0, 2π); u is 2π-periodic in θ,

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where f(µ, θ)=F (x, y). We note that curves of constant µ are all ellipses and the curves of constant θ are hyperbolas with foci x = ±c along the x-axis. The transform (2.1) is singular at µ = 0 when the ellipse is reduced to the line segment y =0and|x|≤c.1 Therefore, additional condition(s) should be imposed for the solution of (5.6) to be consistent with that of (5.1). We now derive the necessary condition(s) through weak formulations. The weak formulation of (5.1) is to find U ∈ H1(Ω) such that 0 ∇ ∇ ∀ ∈ 1 (5.7) U Vdxdy+ β UV dxdy = F V dxdy, V H0 (Ω). Ω Ω Ω By the Lax-Milgram Lemma, if β ≥ 0, this problem admits a unique solution ∈ 1 ∈ 2 U H0 (Ω), provided that F L (Ω). Let us define ∈ 2 Q ∇˜ ∈ 2 Q (5.8) X := u Lh2 ( ): u L ( ),u(1,θ)=0,u(µ, θ +2π)=u(µ, θ) . Using (5.3), we find that, under the elliptic coordinates, the weak formulation (5.7) ∈ becomes: find u X such that (5.9) B(u, v):= ∇˜ u∇˜ vdµdθ + β uvh2dµdθ = fvh2dµdθ, ∀v ∈ X. Q Q Q On the other hand, in order for (5.6) to be consistent with (5.9), the condition ∂u (5.10) (0,θ)=0, ∀θ ∈ [0, 2π) ∂µ should be supplemented to (5.6). 5.1. Dimension reduction. We now expand the solution in a series of Mathieu functions in θ-direction: ∞ 2 ˆ (5.11) u, h f = uˆm, fm Mem(θ; ρ) m=0 2 with ρ = c β/4. Since the Mathieu function Mem satisfies the Mathieu equation (4.2), substituting the expansion (5.11) into (5.6) and (5.10) leads to a sequence of one-dimensional problems in the µ-direction: 2 ˆ − uˆ +(λm +4ρ cosh µ)ûm = fm, ∀µ ∈ Λ=(0, 1), (5.12) m ··· uˆm(0) =u ˆm(1) = 0,m=0, 1, . Its weak formulation is: 1 1 Findu ˆm ∈ 0H (Λ) = φ : φ ∈ H (Λ),φ(1) = 0 such that (5.13) 1 Bm(ûm, vˆm)=(fˆm, vˆm), ∀vˆm ∈ 0H (Λ),m=0, 1, ··· , where B (5.14) m(ûm, vˆm)=(ûm, vˆm)+λm(ûm, vˆm)+4ρ(ûm, vˆm)χ, with the weight function χ =cosh2 µ. For each m ≥ 0, the problem is uniquely solvable thanks to the fact λm,ρ>0 and the Lax-Milgram lemma.

1The singularity here is diﬀerent from that of the polar coordinates, because µ = 0 corresponds to the whole interval [−c, c] rather than a single point.

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Moreover, we have the following aprioriestimate whose proof is given in Ap- pendix C.

2 Lemma 5.1. For each m ≥ 0, if fˆm ∈ L (Λ), then we have 2 2 2 2 ˆ 2 (5.15) uˆm 2 + λm uˆm 1 + λm uˆm fm . 5.2. Mixed Mathieu-Legendre spectral-Galerkin methods. We now describe our spectral approximation to (5.6) with (5.10). Given a cutoﬀ integer M>0, we deﬁne the Mathieu spectral approximation to the solution of (5.9) by M N (5.16) uMN (µ, θ)= uˆm(µ)Mem(θ; ρ), m=0 { N }M where uˆm m=0 are the solutions of the following Legendre spectral-Galerkin approximation to (5.13):

N ∈ ∈ Findu ˆm 0PN := φ PN : φ(1) = 0 such that (5.17) B N N ˆ N ∀ N ∈ ··· m(ˆum, vˆm)=(fm, vˆm), vˆm 0PN ,m=0, 1, ,M,

where the bilinear form Bm(·, ·) is deﬁned in (5.14).

5.3. Error estimations. We ﬁrst recall some relevant approximation results by Legendre polynomials (with a change of variable from (−1, 1) to Λ := (0, 1)). Con- 1 1 sider the orthogonal projection 0Π : 0H (Λ) → 0PN such that N 1 − 1 − ∀ ∈ (5.18) ∂µ(0ΠN v v),∂µvN + 0ΠN v v, vN =0, vN 0PN . As in [16], we introduce the weighted space to describe the approximation errors: k−1 r ∈ 2 − 2 2 k ∈ 2 ≤ ≤ (5.19) B (Λ) := v L (Λ) : (µ µ ) ∂µv L (Λ), 1 k r with the norm and semi-norm r 1 k−1 2 2 r−1 2 − 2 2 k | | − 2 2 r ≥ v Br (Λ) = v + (µ µ ) ∂µv , v Br (Λ) = (µ µ ) ∂µv ,r 1. k=1 In particular, we have B0(Λ) = L2(Λ). The following lemma can be derived from the existing Legendre polynomial approximation; see, e.g., [13, 16].

1 r Lemma 5.2. For any v ∈ 0H (Λ) ∩ B (Λ), 1 − σ−r − 2 (r−1)/2 r ≤ ≤ ≤ (5.20) 0ΠN v v σ N (µ µ ) ∂µv , 0 σ 1 r.

N We first estimate the error betweenu ˆm (the solution of (5.13)) andu ˆm (the − N N solution of (5.17)). For notational convenience, denotee ˆm =ûm uˆm ande ˆm = 1 − N 1 − N − N 0ΠN uˆm uˆm. Clearly, we have that 0ΠN uˆm uˆm =êm eˆm :=e ˜m. 1 r Lemma 5.3. For each m ≥ 0, if uˆm ∈ 0H (Λ) ∩ B (Λ) with r ≥ 1, then we have B − N − N 1/2 −r| | (5.21) m uˆm uˆm, uˆm uˆm (λm + N)N uˆm Br (Λ) and − N 2 −(2+r)| | (5.22) uˆm uˆm (λm + N )N uˆm Br (Λ).

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N ∈ Proof. By (5.13) and (5.17), we have that for anyv ˆm 0PN , B N ⇒B N N B N N (5.23) m(ˆem, vˆm)=0 m(ˆem, vˆm)= m(˜em, vˆm).

N N Takingv ˆm =êm in the above identity, we derive from (5.14), (5.18) and the Cauchy- Schwarz inequality that B N N − N N N N m(êm,eˆm)=(λm 1) e˜m, eˆm +4ρ e˜m, eˆm χ ≤ N N N N N N λm e˜m eˆm + e˜m eˆm +4ρ e˜m 0,χ eˆm 0,χ λ λ 1 ≤ m eˆN 2 + m eÑ 2 + |eˆN |2 +2eÑ 2 +2ρeˆN 2 +2ρeÑ 2 2 m 2 m 2 m 1 m m 0,χ m 0,χ 1 λ = B (êN , eˆN )+ m eÑ 2 +2ρeÑ 2 +2eÑ 2, 2 m m m 2 m m 0,χ m where we used the embedding inequality (B.1) for the following derivation: 1 (5.24) eˆN ≤2|eˆN | ⇒eÑ eˆN ≤2eÑ |eˆN | ≤ |eˆN |2 +2eÑ 2. m m 1 m m m m 1 2 m 1 m Therefore, we derive from Lemma 5.2 that for r ≥ 1, B N N ≤ N 2 N 2 N 2 m(êm, eˆm) λm e˜m +4 e˜m +4ρ e˜m 0,χ (5.25) −2r − 2 (r−1)/2 r 2 (λm +1)N (µ µ ) ∂µuˆm . Since by the triangle inequality, B ≤B N N B N N (5.26) m(êm, eˆm) m(êm, eˆm)+ m(˜em, e˜m), we obtain the desired result by using (5.25) and Lemma 5.2. We now estimate (5.22) by a duality argument. Given g ∈ L2(Λ), we consider 1 the dual problem of (5.13): Find w ∈ 0H (Λ) such that 1 (5.27) Bm(w, v)=(g, v), ∀v ∈ 0H (Λ). By an argument similar to that for Lemma 5.1, this problem admits a unique solution with the regularity 2 2 2 2 2 (5.28) w 2 + λm w 1 + λm w g .

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We now estimate the global errors between u (the solution of (5.9)) and uMN (the numerical solution deﬁned in (5.16)–(5.17)). For this purpose, we introduce the Sobolev space r,s Q 2 r ∩ s−1 1 ∩ s 2 ≥ (5.31) Hp ( ):=L (I; B (Λ)) Hp (I; 0H (Λ)) Hp (I; L (Λ)),r,s 1, with the norm 1 ∞ 2 2 s−1 2 s 2 (5.32) r,s | | u Hp (Q) = π uˆm Br (Λ) + λm uˆm H1(Λ) + λm uˆm L2(Λ) , m=0

where {uˆm} are the coeﬃcients of the Mathieu expansion of u. With the above preparations, we now estimate the global error.

Theorem 5.1. Let u be the solution of (5.9) and uMN its approximation given by ∈ r,s Q ≥ (5.16)–(5.17).Then,foru Hp ( ) with r, s 1, we have 1−s −r ∇˜ − 2 r,s (5.33) (u uMN ) L (Q) M +(M + N)N u Hp (Q) and −s 2 2 −2−r − 2 r,s (5.34) u uMN L (Q) M +(M + N )N u Hp (Q). − N Proof. Recall thate ˆm =ûm uˆm. Hence, by (5.11) and (5.16), M ∞ − (5.35) u uMN = eˆmMem + uˆmMem. m=0 m=M+1 Using (5.9), (4.2)–(4.6) and a direct calculation leads to B − − ∇˜ − 2 − 2 (u uMN ,u uMN )= (u uMN ) L2(Q) + β u uMN L2 (Q) h2 M 2 2 2 = πeˆ 2 +4ρπeˆm 2 + eˆm 2 aˆρ(Mem, Mem) m L (Λ) Lχ(Λ) L (Λ) m=0 ∞ (5.36) 2 2 2 + πuˆ 2 +4ρπuˆm 2 + uˆm 2 aˆρ(Mem, Mem) m L (Λ) Lχ(Λ) L (Λ) m=M+1 M ∞ = π Bm(êm, eˆm)+π Bm(ûm, uˆm), m=0 m=M+1 where h2 is defined in (5.5), and the weight function χ =cosh2 µ. By Lemma 5.3 and (5.14), M M 1/2 2 −2r 2 2 −2r 2 B (ê , eˆ ) (λ + N) N |uˆ | r ≤ (M + N) N u r,s . m m m M m B (Λ) Hp (Q) m=0 m=0 On the other hand, we have from (4.8) and (5.14) that ∞ ∞ B ≤ 1−s s−1B m(ûm, uˆm) λM+1 λm m(ûm, uˆm) m=M+1 m=M+1 (5.37) ∞ 2−2s s−1 2 s 2 M λm uˆm H1(Λ) + λm uˆm L2(Λ) . m=M+1

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Therefore, we deduce from the above estimates and (5.31)–(5.32) that 1−s −r ∇˜ − 2 − 2 r,s (u uMN ) L (Q) + β u uMN L (Q) (M +(M + N)N ) u Hp (Q). h2 This yields (5.33). We now prove (5.34). By (5.35), the orthogonality (4.4) and (5.22), M ∞ − 2 2 2 u uMN L2(Q) = π eˆm L2(Λ) + π uˆm L2(Λ) m=0 m=M+1 M ∞ (5.38) 2 2 2 −2(2+r) | |2 −2s 2s 2 (M + N ) N uˆm Br (Λ) + λm λm uˆm L2(Λ) m=0 m=M+1 −2s 2 2 2 −2(2+r) 2 M +(M + N ) N u r,s . Hp (Q) This ends the proof. We now present some numerical results. As a ﬁrst example, we consider (5.9) with β = c = 1 and the exact solution U(x, y)=cos(xy)=v(µ, θ) = cos(sinh(2µ)sin(2θ)/4). For this analytical solution, Theorem 5.1 holds for all r, s ≥ 0 hence an exponential convergence is expected. We plot the L2-errors against various M = N on the left side of Figure 5.1. An exponential convergence rate of order O(e−cN ) is clearly observed. For the second example, we consider (5.9) with the exact solution v(µ, θ)= γ ≥ 3 ∈ µ sin(2θ). It can be easily checked that for γ 2 but noninteger, we have v 2γ−ε,s Q 2 Hp ( ) for any ε, s > 0. Hence, Theorem 5.1 indicates an L convergence rate of N −2γ+ε with M = N for any ε>0. We plot the L2-errors vs. various M = N in log-log scale for γ =2.5, 2.8, 3.5 on the right side of Figure 5.1. We observe algebraic decays with slopes of roughly 2−2γ. We note that the diﬀerence between the observed rates and the rates predicted in Theorem 5.1 can be attributed to the fact that the forcing function f is replaced by its Mathieu-Legendre interpolation in the computation and that this interpolation error is not accounted for in Theorem 5.1.

Figure 5.1. Left: L2-errors in semi-log scale for the ﬁrst example. Right: L2-errors in log-log scale for the second example.

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6. Application to the Helmholtz equation Given r>0, we denote x2 y2 (6.1) Dr = (x, y): + < 1 , c2 cosh2 r c2 sinh2 r

where c>0 is the semi-focal distance. For b>a>0, we denote Ωa,b = Db\D¯a, i.e., the open domain between the two co-focal ellipses. We consider the following Helmholtz equation

2 (6.2) − ∆V − k V = G, in Ωa,b; V =0, ∂Da

together with a Robin boundary condition at ∂Db expressed in elliptic coordinates (see [14]): ∂v 1 (6.3) − ick sinh µ − tanh µ v =0, on ∂D , ∂µ 2 b where v(µ, θ)=V (x, y). We note that this boundary condition is a ﬁrst-order approximation to the exact Dirichlet-to-Neumann boundary condition (cf. [14]). Without loss of generality, we consider only the homogeneous boundary conditions since nonhomogeneous boundary conditions can be easily handled by boundary lifting. Under the elliptic coordinates (2.1), the equation (6.2)–(6.3) becomes 1 ∂2v ∂2v − + − k2v = g, ∀(µ, θ) ∈ (a, b) × [0, 2π), h2 ∂µ2 ∂θ2 (6.4) ∂v 1 v(a, θ)=0, − ick sinh µ − tanh µ v (b, θ)=0, ∀θ ∈ [0, 2π), ∂µ 2 v is 2π-periodic in θ,

where g(µ, θ)=G(x, y).

6.1. Dimension reduction. We expand the solution and given data in a series of Mathieu functions in the θ-direction ∞ 2 (6.5) v, h g = vˆm, gˆm mem(θ; q) m=0 with q = c2k2/4. For simplicity, hereafter, we denote kˆ = ck. The equation (6.4) is reduced to ˆ2 2 − vˆ +(λm − k cosh µ)ˆvm =ˆgm, ∀µ ∈ Λ=(a, b), (6.6) m −T ··· vˆm(a)=0, vˆm(b) k,bˆ vˆm(b)=0,m=0, 1, , where 1 (6.7) T := ikˆ sinh b − tanh b. k,bˆ 2

We note that the functions (v, g)and(ˆvm, gˆm) are complex-valued. With a slight abuse of notation, we shall still use Hs and L2, etc. to denote spaces of complex- valued functions.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MATHIEU APPROXIMATIONS 833 ∈ 0 1 Let (u, v):= Λ uvdµ. Then, a weak formulation of (6.6) is: Findv ˆm H (Λ) := {v ∈ H1(Λ) : v(a)=0} such that B − ˆ2 m(ˆvm, wˆm):= vˆm, wˆm + λm(ˆvm, wˆm) k vˆm, wˆm (6.8) χ −T ∀ ∈ 0 1 k,bˆ vˆm(b)wˆm(b)= gˆm, wˆm , wˆm H (Λ), where χ =cosh2 µ.

2 Theorem 6.1. Given m ≥ 0 and gˆm ∈ L (Λ), problem (6.8) admits a unique 2 0 1 solution vˆm ∈ H (Λ) ∩ H (Λ). T ˆ Proof. Observe that Im k,bˆ = k sinh b>0, so we deduce from the Fredholm 2 alternative theorem (see, for instance, [22, 31, 34]) that ifg ˆm ∈ L (Λ), the problem 0 1 (6.8) has a unique solutionv ˆm ∈ H (Λ). Then by a standard regularity argument, we have ˆ vˆm2 c(a, b, k)gˆm, where c(a, b, kˆ) is a positive constant depending on a, b and k.ˆ

We now derive a sharp apriorestimate with explicit dependence on k.ˆ ≥ ∈ 2 ∩ 2 Lemma 6.1. Given m 0 and gˆm L (Λ) Lχˆ−1 (Λ), we have

kˆ (6.9) vˆ + vˆ ≤ C gˆ −1 +(b − a +1)gˆ , m 2 m 0,χˆ a,b,kˆ m 0,χˆ m where χˆ(µ)=(µ − a + 1) sinh(2µ), (6.10) √ cosh(2b) tanh2 b 1 C ˆ = 2(b − a +1) + + . a,b,k sinh b 4kˆ2 sinh b kˆ Proof. In order to derive the desired aprioriestimate, we take three diﬀerent test − functions in (6.8), namely,v ˆm, (µ a)ˆvm andv ˆm. The following weight functions will be used in the sequel: χ(µ)=cosh2 µ, χ˜(µ)=(µ − a) sinh(2µ), χ¯(µ)=sinh(2µ), χˆ(µ)=(µ − a +1)sinh(2µ).

Step 1. Takingw ˆ m =ˆvm in (6.8) leads to b 2 2 − ˆ2 2 −T | |2 (6.11) vˆm + λm vˆm k vˆm 0,χ k,bˆ vˆm(b) = gˆmvˆmdµ, a whose real and imaginary parts read 1 (6.12) vˆ 2 + λ vˆ 2 + tanh b|vˆ (b)|2 = kˆ2vˆ 2 +Re(ˆg , vˆ ) m m m 2 m m 0,χ m m and ˆ 2 (6.13) −ik sinh b |vˆm(b)| =Im(ˆgm, vˆm).

Theorem 6.1 asserts that the solutionv ˆm of (6.8) also solves the original problem 1 (6.6). Therefore, we have that for anyw ˆm ∈ H (Λ), b b − − ˆ2 2 (6.14) vˆm +(λm k cosh µ)ˆvm wˆmdµ = gˆmwˆmdµ. a a

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ˆ2 2 − Step 2. In order to bound the term k vˆm 0,χ in (6.12), we takew ˆm =2(µ a)ˆvm in (6.14) and integrate by parts. Using the identities b b − − − − | |2 2Re vˆm(µ a)vˆmdµ = Re (µ a)d vˆm a a 2 − − | |2 = vˆm (b a) vˆm(b) , b b − − | |2 2Re (µ a)ˆvmvˆmdµ =Re (µ a)d vˆm (6.15) a a 2 2 =(b − a)|vˆm(b)| −vˆm , b b − − 2 − − 2 | |2 2Re (µ a)ˆvmvˆm cosh µdµ= Re (µ a)cosh µdvˆm a a − − 2 | |2 2 2 = (b a)cosh b vˆm(b) + vˆm 0,χ + vˆm 0,χ˜, and collecting all the real parts, we ﬁnd 2 ˆ2 2 ˆ2 2 − | |2 vˆm + k vˆm 0,χ + k vˆm 0,χ˜ + λm(b a) vˆm(b) − | |2 2 ˆ2 − 2 | |2 (6.16) =(b a) vˆm(b) + λm vˆm + k (b a)cosh b vˆm(b) − +2Re(ˆgm, (µ a)ˆvm). Step 3. We now takew ˆ m =2ˆvm in (6.14). Using the identities b b − | |2 −| |2 | |2 2Re vˆmvˆmdµ = vˆm(a) vˆm(b) , 2Re vˆmvˆmdµ = vˆm(b) , (6.17) a a b − 2 − 2 | |2 2 2Re (cosh µ)ˆvmvˆmdµ = cosh b vˆm(b) + vˆm 0,χ¯, a and collecting all the real parts, we get (6.18) ˆ2 2 | |2 | |2 | |2 ˆ2 2 | |2 k vˆm 0,χ¯ + λm vˆm(b) + vˆm(a) = vˆm(b) + k cosh b vˆm(b) +2Re(ˆgm, vˆm). A combination of (6.11), (6.16) and (6.18) leads to (6.19) 1 2vˆ 2 + kˆ2vˆ 2 + |vˆ (a)|2 + λ b − a +1 |vˆ (b)|2 + tanh b|vˆ (b)|2 m m 0,χˆ m m m 2 m ≤ − | |2 ˆ2 2 − | |2 (b a +1)vˆm(b) + k cosh b(b a +1)vˆm(b) | | | − | + Re(ˆgm, vˆm) +2Re(ˆgm, (µ a +1)ˆvm) , whereχ ˆ =(µ − a +1)sinh(2µ). Using the boundary condition at r = b in (6.6) yields 1 |vˆ (b)|2 = |T |2|vˆ (b)|2 = kˆ2 sinh2 b + tanh2 b |vˆ (b)|2. m k,bˆ m 4 m By (6.13) and the Cauchy-Schwarz inequality, cosh(2b) kˆ2 cosh(2b)(b − a +1)|vˆ (b)|2 = kˆ (b − a +1)|Im(ˆg , vˆ )| m sinh b m m (6.20) ˆ2 2 k 2 2 cosh (2b) 2 ≤ vˆm +2(b − a +1) gˆm −1 8 0,χˆ sinh2 b 0,χˆ

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and 1 tanh2 b tanh2 b(b − a +1)|vˆ (b)|2 = (b − a +1)|Im(ˆg , vˆ )| 4 m ˆ m m (6.21) 4k sinh b ˆ2 4 ≤ k 2 − 2 tanh b 2 vˆm 0,χˆ +(b a +1) gˆm −1 . 8 8kˆ4 sinh2 b 0,χˆ On the other hand, using the Cauchy-Schwarz inequality yields

ˆ2 | |≤k 2 1 2 (6.22) Re(ˆgm, vˆm) vˆm 0,χˆ + gˆm −1 4 kˆ2 0,χˆ and

| − |≤ 2 − 2 2 (6.23) 2 Re(ˆgm, (µ a +1)ˆvm) vˆm +(b a +1) gˆm .

Notice that cosh2 b +sinh2 b =cosh(2b), so the desired result follows from (6.19)– (6.23).

6.2. Spectral-Galerkin approximations. Given a cut-oﬀ integer M>0, we deﬁne the Mathieu-Legendre approximation to the solution v of (6.4) by

M N (6.24) vMN (µ, θ)= vm(µ)mem(θ; q), m=0 { N }M N ∈ 0 { ∈ where vm m=0 are the solutions of the problem: Find vm PN = φ PN : φ(a)=0} such that B N N N N N N m(vm,wm)= ∂µvm ,∂µwm + λm(vm,wm) (6.25) − ˆ2 N N −T N N N ∀ N ∈ 0 k vm ,wm χ k,bˆ vm (b)wm(b)= gˆm,wm , wm PN . As for (6.8), it is clear from the Fredholm Alternative Theorem that (6.25) admits a unique solution. The following result is needed for the error analysis.

∈ 2 Lemma 6.2. If gˆm Lχ˜−1 (Λ), we have ˆ N k N (6.26) ∂ v + v C gˆ −1 , µ m 2 m 0,χ˜ a,b,kˆ m 0,χ˜ where χ˜ =(µ − a) sinh(2µ) and √ 2 cosh(2b) tanh b 1 (6.27) C ˆ = 2(b − a) + + (b − a) sinh(2b) + . a,b,k sinh b 4kˆ2 sinh b kˆ N N − N ∈ 0 Proof. Since the first two test functions wm = vm, (µ a)∂µvm PN , but for N ∈ 0 the third one ∂µvm PN , the proof of Lemma 6.1 needs to be slightly modified to derive (6.26). For completeness, we sketch the derivation below. N N Note that (6.12)–(6.13) hold with vm in place ofv ˆm. We now take wm = − N 2(µ a)∂µvm in (6.25), and find that N − N − | N |2 N 2 (6.28) 2Re ∂µvm,∂µ((µ a)∂µvm ) =(b a) ∂µvm(b) + ∂µvm .

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N We also see that the second and third identities in (6.15) hold with vm in place of N − N vˆm. Thus, the real part of (6.25) with wm =2(µ a)∂µvm becomes N 2 ˆ2 N 2 ˆ2 N 2 ∂µvm + k vm 0,χ + k vm 0,χ˜ − | N |2 − | N |2 +(b a) ∂µvm (b) + λm(b a) vm(b) (6.29) N 2 2 2 N 2 N = λmv + kˆ (b − a)cosh b|v (b)| +2Re(ˆgm, (µ − a)∂µv ) m m m − T N N +2(b a)Re k,bˆ vm (b)∂µvm (b) . N Hence, a combination of (6.12) (replacingv ˆm by vm ) and (6.29) leads to 1 2∂ vN 2 + kˆ2vN 2 + tanh b|vN (b)|2 µ m m 0,χ˜ 2 m +(b − a)|∂ vN (b)|2 + λ (b − a)|vN (b)|2 (6.30) µ m m m ˆ2 2 N 2 N N = k (b − a)cosh b|v (b)| +2Re(ˆgm, (µ − a)∂µv )+Re(ˆgm,v ) m m m − T N N +2(b a)Re k,bˆ vm(b)∂µvm(b) . Using the Cauchy-Schwarz inequality yields − T N N 2(b a) Re k,bˆ vm(b)∂µvm (b) − ≤ b a| N |2 − |T |2| N |2 (6.31) ∂µvm(b) +(b a) k,bˆ vm(b) 4 b − a 1 ≤ |∂ vN (b)|2 +(b − a) kˆ2 sinh2 b + tanh2 b |vN (b)|2. 4 µ m 4 m Similarly, by (6.13) and the Cauchy-Schwarz inequality, cosh(2b) kˆ2 cosh(2b)(b − a)|vN (b)|2 = kˆ (b − a)|Im(ˆg ,vN )| m sinh b m m (6.32) ˆ2 2 k N 2 2 cosh (2b) 2 ≤ v +2(b − a) gˆm −1 8 m 0,χ˜ sinh2 b 0,χ˜ and 1 tanh2 b tanh2 b(b − a)|vN (b)|2 = (b − a)|Im(ˆg ,vN )| 4 m ˆ m m (6.33) 4k sinh b ˆ2 4 ≤ k 2 − 2 tanh b 2 vˆm 0,χ˜ +(b a) gˆm −1 . 8 8kˆ4 sinh2 b 0,χ˜ On the other hand, using the Cauchy-Schwarz inequality again yields ˆ2 | N |≤k N 2 1 2 (6.34) Re(ˆgm,vm) vm 0,χ˜ + gˆm −1 4 kˆ2 0,χ˜ and b | − N |≤ N 2 − 2| |2 2 Re(ˆgm, (µ a)∂µvm ) ∂µvm + (µ a) gˆm dµ a b | |2 (6.35) ≤ N 2 − 3 gˆm ∂µvm + (µ a) sinh(2µ) − dµ a (µ a) sinh(2µ) ≤ N 2 − 3 2 ∂µvm +(b a) sinh(2b) gˆm 0,χ˜−1 . Finally, a combination of (6.30)–(6.35) leads to the desired result.

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6.3. Error estimations. As in the previous section, we deﬁne the space Br(Λ) : k−1 r ∈ 2 − − 2 k ∈ 2 ≤ ≤ (6.36) B (Λ) := v L (Λ) : (µ a)(b µ) ∂µv L (Λ), 1 k r ,

with the norm and semi-norm r − 1 k 1 2 2 2 − − 2 k v Br (Λ) = v + (µ a)(b µ) ∂µv , k=1 r−1 | | − − 2 r v Br (Λ) = (µ a)(b µ) ∂µv . Let (µ)=(µ − a)(b − µ). The following approximation result will be used in the error analysis (see Appendix D for the proof). 1 1 → Lemma 6.3. There exists a mapping πN : H (Λ) PN satisfying 1 1 (6.37) πN w(a)=w(a),πN w(b)=w(b) and 1 − ∀ ∈ (6.38) ∂µ(πN w w),∂µwN =0, wN PN .

Moreover, for any w ∈ Br(Λ) with r ≥ 1, 1 1 1−r − − − | | (6.39) (πN w w) + N πN w w 0, 1 N w Br (Λ). With the aid of the aprioriestimates and the above approximation result, we derive the following error estimates.

N Theorem 6.2. Let vˆm and vm be the solutions of (6.8) and (6.25), respectively. 0 1 r If vˆm ∈ H (Λ) ∩ B (Λ) with r ≥ 1, we have − N ˆ − N 1−r (6.40) ∂µ(ˆvm vm) + k vˆm vm 0,χ˜ Dm,N,kˆN vˆm Br (Λ), where χ˜ =(µ − a) sinh(2µ), and − 3/2 1/2 ˆ −1 Dm,N,kˆ := 1+(b a) sinh (2b)kN (6.41) cosh2 b + C (b − a)1/2 λ + kˆ2 N −1 , a,b,kˆ m (sinh(2b))1/2 with Ca,b,kˆ being given by (6.27). N N − 1 N − 1 Proof. Let em = vm πN vˆm ande ˜m =ˆvm πN vˆm. By (6.6) and (6.25), we have B − N N N ∈ 0 N that m(ˆvm vm,wm)=0, for all wm PN . Hence, using the facte ˜m(b)=0and (6.38) leads to

B N N B − 1 N N N − ˆ2 N 2 N (6.42) m(em,wm)= m(ˆvm πN vˆm,wm)=λm(˜em,wm) k (˜em cosh µ, wm).

N − ˆ2 2 N Note that (6.42) can be viewed as (6.25) withg ˆm = λme˜m k (cosh µ)˜em. Thus, by (6.26), ˆ N k N N 2 N (6.43) ∂ e + e ≤ C λ e˜ −1 + kˆ e˜ ∗ µ m 2 m 0,χ˜ a,b,kˆ m m 0,χ˜ m 0,χ

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∗ cosh4 µ 2 −1 where we denote χ = (µ−a) sinh(2µ) =(cosh µ)˜χ . Hence, by Lemma 6.3, − 1/2 N (b ν1) 1 e˜ −1 = π vˆ − vˆ −1 m 0,χ˜ 1/2 N m m 0, (sinh(2ν1)) − 1/2 −r| | (b a) N vˆm Br (Λ), − 1/2 2 N (b ν2) cosh ν2 1 e˜ ∗ = π vˆ − vˆ −1 m 0,χ 1/2 N m m 0, (6.44) (sinh(2ν2)) − 1/2 2 (b a) cosh b −r N |vˆ | , (sinh(2b))1/2 m Br (Λ) N 1/2 1/2 1 − − − − e˜m 0,χ˜ =(ν3 a)(b ν3) sinh (2ν3) πN vˆm vˆm 0, 1 − 3/2 1/2 −r| | (b a) sin (2b)N vˆm Br (Λ), { }3 where νi i=1 are three constants in (a, b). Next, by the triangle inequality, (6.39) and (6.43)–(6.44), ∂ (ˆv − vN ) + kˆvˆ − vN ≤∂ eN + kêN + ∂ eÑ + kêÑ µ m m m m 0,χ˜ µ m m 0,χ˜ µ m m 0,χ˜ 1+(b − a)3/2 sinh1/2(2b)kNˆ −1

2 1/2 cosh b 2 −1 1−r + C (b − a) λ + kˆ N N |vˆ | . a,b,kˆ m (sinh(2b))1/2 m Br (Λ) This ends the proof.

− N Remark 6.1. To illustrate how the errorv ˆm vm behaves with respect to N, k and b ikµ with ﬁxed a>0andm ≥ 0, we consider a typical oscillatory functionv ˆm(µ)=e to be the solution of (6.8). Then, for any r>0, b r−1 |vˆ |2 = |∂r vˆ |2 (µ − a)(b − µ) dµ m Br (Λ) µ m (6.45) a b − r−1 b − a 2r 1 ≤ k2r (µ − a)(b − µ) dµ k k . a 2 − 3/2 2 −1 Notice that Dm,N,kˆ 1+(b a) cosh bk N . Therefore, Theorem 6.2 indicates that for this typical solution, we have that for any r ≥ 1, b − a k(b − a) r−1 ∂ (ˆv −vN )+kˆvˆ −vN 1+(b−a)3/2 cosh bk2N −1 k . µ m m m m 2 2N k(b−a) Hence, the error will decay exponentially as soon as 2N < 1. Therefore, we can signiﬁcantly reduce the computational cost by choosing b closer to a.

We now present some numerical results. We solved (6.25) with the functiong ˆm such that the exact solution isv ˆm =exp(ikµ). We refer to [12] for more details on the actual implementation of (6.25). In Figure 6.1, we plot numerical results which depict the convergence behaviors of our numerical scheme with (b − a)/2=1.On the left, we ﬁx k = 100 and plot the L2-errors with respect to N.Weobservethat k(b−a) the error decays exponentially fast as soon as N>k. Next, we denote α = 2N and plot on the right of Figure 6.1 the L2-errors with three diﬀerent α.Weobserve that in all three cases, the error converges exponentially.

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u(x)=sin(kx)*exp(ikx) u(x)=sin(kx)*exp(ikx) 2 −2 10 10 k=100 α=0.9 α=0.85 0 α=0.8 10 −4 10

−2 10

−6 10 −4 10

−6 −8 10 10 error error

−8 10 −10 10

−10 10

−12 10 −12 10

−14 −14 10 10 60 70 80 90 100 110 120 130 140 150 100 150 200 250 300 350 400 450 500 550 N k

Figure 6.1. Left: L2-error with respect to N for k = 100. Right: 2 k(b−a) L -errors with three diﬀerent α where α = 2N .

We now estimate the error of the Legendre-Mathieu spectral approximation. Let Q =Λ× I =(a, b) × (0, 2π), and introduce the space r,s 2 r ∩ s−1 0 1 ∩ s 2 ≥ (6.46) Hp (Q):=L (I; B (Λ)) Hp (I; H (Λ)) Hp (I; L (Λ)),r,s 1, with the norm 1 ∞ 2 2 s−1 2 s 2 (6.47) vr,s = |vˆm| + λ ∂µvˆm 2 + λ vˆm 2 , Hp (Q) Br (Λ) m L (Λ) m L (Λ) m=0

where {vˆm} are the coeﬃcients of the Mathieu expansion of v.

Theorem 6.3. Let v be the solution of (6.4),andletvMN be the approximate ∈ r,s ≥ solution given by (6.24)–(6.25).Then,forv Hp (Q) with r, s 1, we have −1 1−r −s/2 − 2 ˆ (6.48) v vMN L (Q) DM,N,kˆk N + λM+1 v Hr,s (Q), χ˜ p 1−r (1−s)/2 (6.49) ∂µ(v − v ) 2 D ˆN + λ vr,s , MN L (Q) M,N,k M+1 Hp (Q) and 1/2 1−r (1−s)/2 (6.50) ∂θ(v − v ) 2 (b − a)D ˆλ N + λ vr,s , MN L (Q) M,N,k M M+1 Hp (Q) − where χ˜ =(µ a) sinh(2µ),DM,N,kˆ is given in (6.41),andλm is the eigenvalue of the Mathieu equation (cf. (3.3)). Proof. We derive from the orthogonality of the Mathieu functions and (6.40) that M ∞ − 2 − N 2 2 v vMN 2 = π vˆm v 2 + π vˆm 2 Lχ˜ (Q) m Lχ˜(Λ) Lχ˜(Λ) m=0 m=M+1 M ∞ N 2 −s s 2 (6.51) ≤ πvˆm − v 2 + λ πλ vˆm 2 m Lχ˜(Λ) M+1 m Lχ˜(Λ) m=0 m=M+1 M ∞ ˆ−2 2 2−2r | |2 −s s 2 k D N vˆm r + λ λ vˆm 2 . M,N,kˆ B (Λ) M+1 m Lχ˜(Λ) m=0 m=M+1

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Similarly, using (6.40) leads to M 2 2 2−2r 2 ∂ (v − v ) 2 D N |vˆ | µ MN L (Q) M,N,kˆ m Br (Λ) m=0 ∞ 1−s s−1 2 + λM+1 λm ∂µvˆm L2(Λ). m=M+1 We now prove (6.50). One veriﬁes, by using the Hardy inequality (B.3), that for any w ∈ 0H1(Λ), wL2(Λ) ≤ 2(b − a)w L2(Λ). Hence, by (6.40), − N − − N − 1−r| | (6.52) vˆm vm L2(Λ) (b a) ∂µ(ˆvm vm ) L2(Λ) (b a)Dm,N,kˆN vˆm Br (Λ). By (3.10)–(3.11), (6.5) and (6.24), we have from (6.52) that − 2 ˆ2 − 2 ∂θ(v vMN ) + k (v vMN )cosθ M ∞ − N 2 2 = πλm vˆm vm L2(Λ) + πλm vˆm L2(Λ) m=0 m=M+1 M ∞ 2 2 2−2s 2 1−s s 2 (b − a) λ D N |vˆ | + λ λ |vˆ | 2 . M M,N,kˆ m Br (Λ) M+1 m m L (Λ) m=0 m=M+1 This ends the proof.

7. Concluding remarks The Mathieu functions are classical special functions which are important tools for solving certain partial differential equations in elliptic domains. In particular, they enable us to reduce the Helmholtz or the modified Helmholtz equation in a two- dimensional elliptic domain to a sequence of one-dimensional equations that are easy to solve numerically. However, to the best of our knowledge, there was no rigorous approximation result available for Mathieu expansions. This paper established a first set of such approximation results which will serve as basic ingredients for the numerical analysis of Mathieu approximations to PDEs. More precisely, we developed optimal approximation results for the angular Mathieu functions. These approximation results are very similar to those for the Fourier expansions. Namely, the orthogonal projection based on Mathieu expansions of a periodic function converges in the same way as that based on Fourier expansions. As examples of applications, we applied the approximation results developed in this paper to establish optimal error estimates for the Mathieu-Legendre approximation to the modified Helmholtz equation and the Helmholtz equation. We also presented illustrative numerical results which are consistent with our theoretical analysis.

Appendix A. Recurrence relations { (n) (n)} The coeﬃcients Aj ,Bj in the expansion (2.10) satisfy the following recursive relations:

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• Even functions of period π (i.e., ce2m and n =2m): − 2 (n) − (n) (n) ≥ (A.1) (an j )Aj q Aj−2 + Aj+2 =0,j 3, and (n) − (n) − (n) − (n) (n) (A.2) anA0 qA2 =0, (an 4)A2 q A0 + A4 =0.

• Even functions of period 2π (i.e., ce2m+1 and n =2m +1): − (n) − (n) (n) (A.3) (an 1)A1 q A1 + A3 =0, along with (A.1) for j ≥ 3. • Odd functions of period π (i.e., se2m and n =2m): − 2 (n) − (n) (n) ≥ (A.4) (bn j )Bj q Bj−2 + Bj+2 =0,j 3, and − (n) − (n) (A.5) (bn 4)B2 qB4 =0.

• Odd functions of period 2π (i.e., se2m+1 and n =2m +1): − (n) (n) − (n) (A.6) (bn 1)B1 + q B1 B3 =0, along with (A.1) for j ≥ 3.

Appendix B. An embedding inequality We have the following inequality: (B.1) w≤2w, ∀w ∈ H1(0, 1) with w(1) = 0.

Proof. We recall Hardy’s inequality ([20]) b b 2 b − d−2 ≤ 4 2 − d (B.2) φ(ν)dν (b µ) dµ − φ (µ)(b µ) dµ, d < 1 a µ 1 d a and b µ 2 b − d−2 ≤ 4 2 − d (B.3) φ(ν)dν (µ a) dµ − φ (µ)(µ a) dµ, d < 1. a a 1 d a Thanks to w(1) = 0, taking φ = w,a=0,b=1andd = 0 in (B.2) leads to 1 w2 ≤ |w|2(1 − µ)−2dµ ≤ 4w2. 0 This ends the proof.

Appendix C. Proof of Lemma 5.1

Taking vm =ˆum in (5.13)–(5.14), we use the Cauchy-Schwarz inequality and the embedding inequality (B.1) that 2 2 2 ≤| ˆ | uˆm + λm uˆm +4ρ uˆm 0,χ (fm, uˆm) (C.1) 1 ≤ 2fˆ uˆ ≤ uˆ 2 +2fˆ 2. m m 2 m m

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We verify by integration by parts and the factu ˆm(0) =u ˆm(1) = 0 that 1 1 1 − 2 | |2 2 uˆmuˆm cosh µdµ = uˆm cosh µdµ + uˆmuˆm sinh(2µ)dµ 0 0 0 1 (C.2) 2 1| |2 − | |2 = uˆm 0,χ + uˆm(1) sinh(2) uˆm cosh(2µ)dµ 2 0 1 = uˆ 2 + |uˆ (1)|2 sinh(2) + uˆ 2 − 2uˆ 2 . m 0,χ 2 m m m 0,χ − Multiplying the ﬁrst equation in (5.12) by uˆm and integrating the resulting equation over (0, 1), we derive from a direct calculation that 2 2 2 | |2 2 uˆm + λm uˆm +4ρ uˆm 0,χ +2ρ uˆm(1) sinh(2) + 4ρ uˆm (C.3) 1 1 ≤ 8ρuˆ 2 + |(fˆ , uˆ )|≤8ρuˆ 2 + fˆ 2 + uˆ 2. m 0,χ m m m 0,χ 2 m 2 m 2 ≤ ˆ 2 Since by (C.1), 8ρ uˆm 0,χ 4 fm , a combination of (C.1) and (C.3) leads to 2 2 ˆ 2 (C.4) uˆm 2 + λm uˆm 1 fm . Next, by the ﬁrst equation of (6.6) and (C.4), ≤ 2 ˆ ˆ (C.5) λm uˆm uˆm +4ρ cosh µuˆm + fm fm , which completes the proof of Lemma 5.1.

Appendix D. Proof of Lemma 6.3 We first recall a generalized Jacobi approximation (with a change variable from (−1, 1) to Λ = (a, b) and an extension to complex-valued functions) with parameters α = β = −1 (cf. Theorem 3.1 of [15]): − − −1 Let (µ)=(µ a)(b µ), and πN be the orthogonal projection from 2 0 1 ∩ L−1 (Λ) onto PN = H0 (Λ) PN such that b −1 − −1 ∀ ∈ 0 πN v v vN (µ)dµ =0, vN PN . a ∈ 2 ∩ r ≥ Then for any v L−1 (Λ) B (Λ) with r 1, we have −1 −1 1−r − − − | | (D.1) ∂µ(πN v v) + N πN v v 0, 1 N v Br (Λ). Next, using the Hardy inequalities (B.2)–(B.3) and an argument similar to that in 1 ⊆ 2 Appendix B, we have H0 (Λ) L−1 (Λ). Moreover, we find from [15] (see (2.5)) −1 1 that πN is also the H0 (Λ)-orthogonal projection, namely, b −1 − ∀ ∈ 0 (D.2) πN v v vN dµ =0, vN PN . a For any w ∈ H1(Λ), let w∗(µ) be the linear interpolation of w associated with the points µ = a, b, i.e., µ − b µ − a (D.3) w∗(µ)= w(a)+ w(b) ∈ P . a − b b − a 1 − ∗ ∈ 1 ⊆ 2 It is clear that w w H0 (Λ) L−1 (Λ). Now, we define 1 −1 − ∗ ∗ ∈ ∀ ∈ 1 (D.4) πN w = πN (w w )+w PN , w H (Λ).

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1 ∗ Obviously, πN w satisﬁes (6.37). Next we verify (6.38). For this purpose, let wN be the linear interpolation function of wN as deﬁned in (D.3). We derive from (D.2) and integration by parts that for any wN ∈ PN , b 1 − ∂µ πN w w ∂µwN dµ a b −1 − ∗ − − ∗ − ∗ ∗ = ∂µ πN (w w ) (w w ) ∂µ (wN wN )+wN dµ a b −1 − ∗ − − ∗ ∗ = ∂µ πN (w w ) (w w ) ∂µwN dµ a w∗ (b) − w∗ (a) b N N −1 − ∗ − − ∗ = πN (w w ) (w w ) =0. b − a a Hence, (6.38) holds. Next, by (D.1), 1 1 1−r ∗ − − − | − | (D.5) ∂µ(πN w w) + N πN w w 0, 1 N w w Br (Λ). r ∗ ≥ Since ∂µw =0forr 2, and for r =1, b ∗ −1/2 1 ∂µw =(b − a) |w(b) − w(a)|≤√ |∂µw|dµ ≤∂µw, b − a a | − ∗| ≤ | | we have w w Br (Λ) 2 w Br (Λ), which, together with (D.5), leads to the desired result. Acknowledgement The authors would like to thank Mr. Qirong Fang for performing the numerical computations presented in Section 6. References

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 E-mail address: [email protected] Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637616, Singapore E-mail address: [email protected]

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