Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

Abstract

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. We prove in this paper that both methods require a small increase (in the order of $k^\epsilon$ for any $\epsilon > 0$) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber k. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validating $\mathcal {O}(k^\epsilon )$ algorithms are presented.

Appendices

Appendix

A Asymptotic expansion of the total field

The ansatz representing the asymptotic expansion of the total field $\eta$ for the Neumann boundary value problem used in the present paper is given by Melrose and Taylor in [45]. However, the authors did not present all the mathematical steps needed in its derivation. In this section, we provide the missing details of this analysis.

Let $K \subset {\mathbb {R}}^{n+1}$ be a compact strictly convex obstacle such that $B= \partial K \subset {\mathbb {R}}^{n+1}$ is a smooth hyper-surface, and consider the Neumann-to-Dirichlet operator

$$\begin{aligned} N^{-1} : {\mathcal {E}}'({{\mathbb {R}}\times B}) \ni f(t,x)\mapsto v(t,x)|_{{\mathbb {R}}\times B} \in {\mathcal {D}}'({{\mathbb {R}}\times B}) \end{aligned}$$

(57)

where ${\mathcal {D}}'(B \times {\mathbb {R}})$ and ${\mathcal {E}}'({{\mathbb {R}}\times B})$ are the spaces of distributions and compactly supported distributions respectively, and v is the solution to the wave problem

$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _{tt} -\varDelta ) v(t,x)=0 &{} \text {in } {\mathbb {R}}\times \varOmega , \\ \partial _{\nu }v(t,x)|_{{\mathbb {R}}\times B} = f(t,x) &{} \text {on } {\mathbb {R}} \times B, \end{array} \right. \end{aligned}$$

(58)

wherein $\varOmega = {\mathbb {R}}^{n+1} \setminus K$ is the exterior domain, and $\nu$ is the outward unit normal.

In what follows, for an incident field $v^{i}(t,x) = \delta (t - \alpha \cdot x)$ ($\delta$ is the Dirac function) with direction $\alpha \in {\mathcal {S}}^n$, we denote the solution of the wave problem (58) associated with $f(t,x) = (\alpha \, \cdot \, \nu (x)) \partial _t v^{i}(t,x)$ by v. In this case, the total field $v^t := v + v^{i}$ can be expressed on the boundary as [45, p.296]

$$\begin{aligned} v^t(t,x)|_{{\mathbb {R}} \times B} = (I+N^{-1}(\alpha \cdot \nu (x)) \partial _t) \delta (t - \alpha \cdot x) \end{aligned}$$

(59)

where I is the identity operator. Using the same notation and procedure in [45], we define the Kirchhoff operator (see [45, Equation 8.26])

$$\begin{aligned} Q_N : \mathcal {E'}({\mathcal {S}}^n \times {\mathbb {R}}) \ni v(t,x) \mapsto (I+N^{-1}(\alpha \cdot \nu (x) ) \partial _t )Fv(t,x) \in \mathcal {D'}(B \times {\mathbb {R}}) \end{aligned}$$

where F is the Fourier integral operator [41, Equation 9]

$$\begin{aligned} Fv(t,x) := \int _{{\mathbb {R}}\times {\mathcal {S}}^n} \kappa _F(t-s,w,x) v(s,\alpha ) ds d \alpha \end{aligned}$$

(60)

with kernel $\kappa _{F}(t,\alpha ,x):=\delta (t-\alpha \cdot x)$.

As shown in [45], the asymptotic behavior of the total field $\eta$ is determined by the kernel $\kappa _{Q_N}$ of the Kirchhoff operator $Q_N$.

Lemma 6

[45, Lemma 9.1] The asymptotic behavior as $k \rightarrow \pm \infty$ of the total field $\eta (\alpha ,k,x)$ obtained by inverse Fourier transformation of the kernel $\kappa _{Q_N}(\alpha ,t,x)$

$$\begin{aligned} \eta (\alpha ,x,k)=\int e^{it k} \kappa _{Q_N}(\alpha ,t,x) dt \end{aligned}$$

(61)

is determined by the singularities of the kernel $\kappa _{Q_N}$ modulo rapidly decreasing terms.

The main goal is therefore the study of the kernel $\kappa _{Q_N}$. To this end, one first utilizes the general theory of Fourier integral operators with folding canonical relations [45, §5 and §6] to decompose the operator $Q_N$. In what follows, we use the notation in [45] except that $OPS^\theta _{\xi ,\zeta }$ denotes the collection of pseudodifferential operators with symbols in the Hörmander class $S^\theta _{\xi ,\zeta }$, and we write $OPS^\theta$ for $OPS^\theta _{1,0}$.

Theorem 6

[45, Theorem 8.30] The Kirchhoff operator $Q_N$ can be expressed as

$$\begin{aligned} Q_N = J_1D{\mathcal {A}}^{-1} J_2 \end{aligned}$$

(62)

where $J_1$ and $J_2$ are elliptic Fourier integral operators of order zero, $D\in OPS^{-\frac{n}{2}-\frac{1}{6}}_{\frac{1}{3},0}$ has an asymptotic expansion

$$\begin{aligned} D \sim \sum _{r \in {\mathbb {Z}}_+, \, \ell \in -{\mathbb {N}}} A_{r,\ell } \varPhi ^{r,\ell } \end{aligned}$$

(63)

with $A_{r,\ell } \in S^{-\frac{n}{2}-\frac{\ell }{3}-\frac{r}{3}+(\ell +1)_-}_{cl}$ and $\varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1)\in S^{(\frac{\ell }{3}-\frac{2r}{3})_{+}}_{\frac{1}{3},0} ({\mathbb {R}})$ (see [44, p.11-12]) so that

$$\begin{aligned} Dv(t,x) \sim \sum _{r \in {\mathbb {Z}}_+, \, \ell \in -{\mathbb {N}}} \int e^{i(x-y) \cdot \xi + i(t-t')k} a_{r,\ell }(t,x,k,\xi ) \varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _{1}) v(t',y) dt' dy dk d\xi \nonumber \\ \end{aligned}$$

(64)

where $(k,\xi )$ are variables dual to (t, x), and $a_{r,\ell } \in S^{-\frac{n}{2}-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_-}_{1,0}$ admits an asymptotic expansion

$$\begin{aligned} a_{r,\ell }(t,x,k,\xi ) \sim \sum _{q \in {\mathbb {Z}}_+} k^{-\frac{n}{2}-q-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_{-}} \, a_{q,r,\ell }(t,x,\xi ) \end{aligned}$$

(65)

wherein $a_{q,r,\ell }$ are $C^{\infty }$ functions uniformly bounded together with all their derivatives (cf. [42, Definition 2.5.6]), and ${\mathcal {A}}^{-1}$ is the convolution operator defined by Fourier transformation as [45, Equation 1.36]

$$\begin{aligned} \widehat{{\mathcal {A}}^{-1} v(t,x)} (\xi ) = \dfrac{{\hat{v}}(\xi )}{A_+(k^{-\frac{1}{3}}\xi _1)} \end{aligned}$$

where $A_+(z) := Ai(e^{\frac{2\pi i}{3}}z)$ and Ai is the Airy function [43].

The Fourier integral operator $J_2$ is given by

$$\begin{aligned} J_2 :&{\mathcal {E}}'({\mathbb {R}} \times {\mathcal {S}}^n) \rightarrow {\mathcal {D}}'({\mathbb {R}} \times {\mathbb {R}}^n) \\ :&v(s,\alpha ) \mapsto (J_2v)(t',y) = \int e^{i(y-\alpha ) \cdot \xi -i(t'-s)k} a_{J_2}(s,\alpha ,t',y) v(s,\alpha ) ds d\alpha dk d\xi \end{aligned}$$

where $a_{J_2} \in S^0_{1,0}$ ($a_{J_2}$ does not depend on $\xi$ and k because it is a symbol of order 0). Applying the Dirac function at the base point $(0,{\bar{\alpha }})$ (see [41, 45]) yields

$$\begin{aligned} (J_2\delta _{(0,{\bar{\alpha }})})(t',y) \nonumber&= \int e^{i(y-\alpha ) \cdot \xi -i(t'-s)k} a_{J_2}(s,\alpha ,t',y) \delta _{(0,{\bar{\alpha }})}(s,\alpha ) ds d\alpha dk d\xi \\&= \int e^{i(y-{\bar{\alpha }}) \cdot \xi -i t'k} a_{J_2}({\bar{\alpha }},t',y) dk d\xi \nonumber \\&= \int e^{i(y-{\bar{\alpha }}) \cdot \xi -i t'k} a_{J_2}({\bar{\alpha }},t',y)\widehat{\delta _{0}}(k,\xi )dk d\xi = (P\delta _{0})(t',y) \end{aligned}$$

(66)

where the operator $P\in OPS^{-\frac{n}{2}+\frac{1}{6}}$ is specified by

$$\begin{aligned} (Pv)(t',y) := \int e^{i(y-{\bar{\alpha }}) \cdot \xi -i t'k}a_{J_2}({\bar{\alpha }},y,t'){\widehat{v}}(k,\xi )dk d\xi . \end{aligned}$$

Accordingly, use of (66) in (62) implies

$$\begin{aligned} Q_N(\delta _{(0,{\bar{\alpha }})}(t,x)) = J_1D{\mathcal {A}}^{-1}J_2(\delta _{(0,{\bar{\alpha }})}(t,x)) = J_1D{\mathcal {A}}^{-1}P(\delta _{0}(t,x)). \end{aligned}$$

(67)

Finally, since $P\delta _0 = P^{\#}\delta _0 \mod OPS^{-\infty }$ and $P^{\#}$ commutes with ${\mathcal {A}}^{-1}$ [45, p.295], (67) can be rewritten as

$$\begin{aligned} Q_N(\delta _{(0,{\bar{\alpha }})}(t,x)) = J_1DP^{\#}{\mathcal {A}}^{-1}(\delta _{0}(t,x)) \end{aligned}$$

(68)

modulo rapidly decreasing terms.

In what follows, we briefly explain how representation (68) can be used to express the amplitude associated with the kernel $\kappa _N$ as an asymptotic series of oscillatory integrals each of which is amenable to an application of the stationary phase method [28].

To this end, we first use (63) to deduce for the composition $DP^{\#}$ of the pseudo-differential operators D and $P^{\#}$

$$\begin{aligned} DP^{\#}v(t,x) \sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i(x-y) \cdot \xi -i(t-t')k}a_{r,\ell }(t,x,k,\xi ) \varPhi ^{r,\ell }(k^{-\frac{1}{3}} \xi _1)P^{\#}v(t',y)dt'dydk d\xi \end{aligned}$$

(69)

with

$$\begin{aligned} P^{\#}v(t',y)&:= \int e^{i(y-{\bar{\alpha }}) \cdot \eta +it'\tau } p^{\#}({\bar{\alpha }},t',y) {\widehat{v}}(\tau ,\eta ) d\tau d\eta \nonumber \\&= \int e^{i(y-z-{\bar{\alpha }}) \cdot \eta +i(t'-t'')\tau }p^{\#}({\bar{\alpha }},t',y) v(t'',z) dt''dz d\tau d\eta \end{aligned}$$

(70)

where $p^{\#} \in S^0_{1,0}$, and $(\tau ,\eta )$ is the dual variable to $(t',y)$. Using (70) in (69), we therefore get

$$\begin{aligned} DP^{\#}v(t,x)&\sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i(x-z) \cdot \xi +(t-t'')k-i{\bar{\alpha }} \cdot \xi } b_{r,\ell }({\bar{\alpha }},t,x,t'',z,k,\xi ) \nonumber \\&\quad \varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1)v(t'',z)dt'' dz dk d\xi \end{aligned}$$

(71)

with

$$\begin{aligned} b_{r,\ell }({\bar{\alpha }},t,x,t'',z,k,\xi )&:= \int e^{i(y-z) \cdot (\eta -\xi )+i(t'-t'')(\tau -k)-i{\bar{\alpha }} \cdot (\eta -\xi )} \\&\quad a_{r,\ell }(x,t,\xi ,k)p^{\#}({\bar{\alpha }},t',y)dt'dy d\tau d\eta . \end{aligned}$$

As for the composition of the operator $DP^{\#}$ with the Fourier integral operator $J_1$ appearing in (62), let us first note that

$$\begin{aligned} J_1 :&\mathcal {D'}({\mathbb {R}} \times {\mathbb {R}}^n) \rightarrow {\mathcal {D}}'({\mathbb {R}}\times B) \nonumber \\ :&v(t',y) \mapsto J_1v(t,x) = \int e^{i\psi _1(x,\tau ,\eta )+i\tau (t-t')-iy \cdot \eta } a_{J_1}(x,t',y)v(t',y) dt'dy d\tau d\eta \end{aligned}$$

(72)

where $(\tau ,\eta )$ are variables dual to (t, x), $a_{J_1} \in S^0_{1,0}$, and the phase function $\psi _1$ is defined in a neighborhood of the base point as [45, Equations 7.11 and 7.13]

$$\begin{aligned} \psi _1(x,\tau ,\eta ) = -\frac{|\eta '|^2}{2\tau }-\frac{|x'|^2\tau }{2} - \left\{ \begin{array}{cl} \frac{3}{2} (-\eta _1 \tau ^{-\frac{1}{3}})^{\frac{3}{2}} {{\,\mathrm{sgn}\,}}(\alpha \cdot \nu (x)), &{} \text {if } \alpha \cdot \nu (x) \ne 0, \\ 0, &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

(73)

In (73), we have used the notation $x'=(x_2,\ldots ,x_n)$ for $x = (x_1,\ldots ,x_n) \in B$ and similarly for $\eta \in {\mathbb {R}}^n$. Combining (71) with (72), we obtain

$$\begin{aligned} J_1DP^{\#}v(t,x)&\sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i\psi _1(x,\tau ,\eta )+i\tau (t-t')-iy \cdot \eta } a_{J_1}(x,t',y) \Big [ \int e^{i(y-z)\xi +i(t'-t'')k -i{\bar{\alpha }} \cdot \xi } \\&\quad b_{r,\ell }({\bar{\alpha }},t',y,t'',z,k,\xi ) \varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1) v(t'',z) dt''dz dk d\xi \Big ] dt' dy d\tau d\eta , \end{aligned}$$

and we rewrite this as

$$\begin{aligned} J_1DP^{\#}v(t,x)&\sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i\psi _1(x,k,\xi )-i{\bar{\alpha }} \cdot \xi -itk -iz\xi -it'' k} \nonumber \\&\quad q_{N_{r,\ell }}({\bar{\alpha }},x,t'',z,k,\xi ) \varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1) v(t'',z)dt''dz dk d\xi \end{aligned}$$

(74)

where

$$\begin{aligned} q_{N_{r,\ell }}({\bar{\alpha }},x,t'',z,k,\xi )&:= a_{J_1}\# b_{r,\ell }({\bar{\alpha }},x,t'',z,k,\xi ) \\&= \int e^{i\psi _1(x,\tau ,\eta )-i\psi _1(x,k,\xi )+i(k-\tau )(t'-t) +i(\xi -\eta ) \cdot y} \\&\quad a_{J_1}(x,t',y) b_{r,\ell }({\bar{\alpha }},t',y,t'',z,k,\xi ) dt'dy d\tau d\eta . \end{aligned}$$

In light of (68), substituting [45, p.295]

$$\begin{aligned} {\mathcal {A}}^{-1}\delta _{0}(t'',z) = \int e^{i\xi \cdot z+ik t''} \frac{1}{A_+(k^{-\frac{1}{3}}\xi _1)} dk d\xi \end{aligned}$$

(75)

for v in (74), we get

$$\begin{aligned} Q_N\delta _{(0,{\bar{\alpha }})}(t,x)&\sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i\psi _1(x,k,\xi )-i{\bar{\alpha }} \cdot \xi -itk -iz \cdot \xi - it''k} q_{N_{r,\ell }}({\bar{\alpha }},x,t'',z,k,\xi ) \nonumber \\&\quad \varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1) {\mathcal {A}}^{-1}\delta _{0}(t'',z)dt''dz dk d\xi \nonumber \\&\sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i\psi _1(x,k,\xi )-i{\bar{\alpha }} \cdot \xi -itk} q_{N_{r,\ell }}({\bar{\alpha }},x,k,\xi ) \frac{\varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1)}{A_+(k^{-\frac{1}{3}}\xi _1)}dk d\xi \end{aligned}$$

(76)

where $q_{N_{r,\ell }}({\bar{\alpha }},x,k,\xi ):= a_J\# b_{r,\ell }({\bar{\alpha }},x,k,\xi )$.

For the kernel $\kappa _{Q_N}$, we accordingly have

$$\begin{aligned} \kappa _{Q_N}(\alpha ,t,x) \sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} \int e^{i\psi _1(x,k,\xi )-i{\bar{\alpha }} \cdot \xi -itk} q_{N_{r,\ell }}({\bar{\alpha }},x,k,\xi ) \frac{\varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1)}{A_+(k^{-\frac{1}{3}}\xi _1)} dk d\xi \end{aligned}$$

and, in virtue of Lemma 6, we deduce (see [45, p.298])

$$\begin{aligned} \eta (\alpha ,x,k) \sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in - {\mathbb {N}} \end{array}} \int e^{i\psi _1(x,\xi ,k)-i{\bar{\alpha }} \cdot \xi } q_{N_{r,\ell }}({\bar{\alpha }},x,k,\xi ) \frac{\varPhi ^{r,\ell }(k^{-\frac{1}{3}}\xi _1)}{A_+(k^{-\frac{1}{3}}\xi _1)} d\xi . \end{aligned}$$

(77)

Making the change of variable $\xi =k \zeta$ in (77), we therefore find [45, p.298]

$$\begin{aligned} \eta (\alpha ,x,k) \sim \sum _{\begin{array}{c} r \in {\mathbb {Z}}_+ \\ \ell \in - {\mathbb {N}} \end{array}} \int e^{ik \psi _2(\alpha , x,\zeta )} q_{N_{r,\ell }}({\bar{\alpha }},x,k,\zeta ) \frac{\varPhi ^{r,\ell }(k^{\frac{2}{3}}\zeta _1)}{A_+(k^{\frac{2}{3}}\zeta _1)} d\zeta \end{aligned}$$

(78)

where $\psi _2(\alpha , x,\zeta ) = \psi _1(x,1,\zeta )-{\bar{\alpha }} \cdot \zeta$, and with a slight abuse of notation we have written $q_{N_{r,\ell }}({\bar{\alpha }},x,k,\zeta )$ for $k^n q_{N_{r,\ell }}({\bar{\alpha }},x,k,\zeta )$. In this case, $q_{N_{r,\ell }}$ is a symbol of order $\frac{n}{2}+\frac{1}{6}-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_{-}$ so that

$$\begin{aligned} q_{N_{r,\ell }}({\bar{\alpha }},x,k,\zeta ) \sim \sum _{q \in {\mathbb {Z}}_+} k^{\frac{n}{2}+\frac{1}{6}-q-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_{-}} q_{N_{q,r,\ell }}({\bar{\alpha }},x,\zeta ). \end{aligned}$$

(79)

Using (79) in (78), we arrive at

$$\begin{aligned} \eta (\alpha ,x,k) \sim \sum _{\begin{array}{c} q,r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} k^{\frac{n}{2}+\frac{1}{6}-q-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_{-}} \int e^{ik\psi _2(\alpha , x,\zeta )} q_{N_{q,r,\ell }}({\bar{\alpha }},x,\zeta ) \frac{\varPhi ^{r,\ell }(k^{\frac{2}{3}}\zeta _1)}{A_+(k^{\frac{2}{3}}\zeta _1)} d\zeta . \end{aligned}$$

(80)

In order to further simplify (80), we introduce the function

$$\begin{aligned} \varPsi ^{r,\ell }(\tau ) := e^{-\frac{i\tau ^{3}}{3}} \int \frac{\varPhi ^{r,\ell }(s)}{A_+(s)}e^{-is\tau }ds. \end{aligned}$$

(81)

The symbolic behavior of $\varPsi ^{r,\ell }$ is as follows.

Lemma 7

[45, Lemma 9.34] The function $\varPsi ^{r,\ell }$ defined in (81) belongs to $S^{1+\ell -2r}({\mathbb {R}})$, admits an asymptotic expansion

$$\begin{aligned} \varPsi ^{r,\ell }(\tau ) \sim \sum _{j \in {\mathbb {Z}}_+} \alpha _{r,\ell ,j} \tau ^{1+\ell -2r-3j} \end{aligned}$$

(82)

as $\tau \rightarrow +\infty$, and is rapidly decreasing in the sense of Schwarz as $\tau \rightarrow -\infty$.

Rewriting (81) as

$$\begin{aligned} e^{\frac{i\tau ^{3}}{3}}\varPsi ^{r,\ell }(\tau ) =\int \frac{\varPhi ^{r,\ell }(s)}{A_+(s)}e^{-is\tau }ds =\widehat{\Big ( \frac{\varPhi ^{r,\ell }}{A_+} \Big )}(\tau ), \end{aligned}$$

and using ${\mathcal {F}}^{-1}$ to denote the inverse Fourier transform, we obtain

$$\begin{aligned} \frac{\varPhi ^{r,\ell }(k^{\frac{2}{3}}\zeta _1)}{A_+(k^{\frac{2}{3}}\zeta _1)}&= {\mathcal {F}}^{-1} \Big ( \widehat{\Big ( \frac{\varPhi ^{r,\ell }}{A_+} \Big )} (\tau ) \Big ) (k^{\frac{2}{3}}\zeta _1) = {\mathcal {F}}^{-1} \Big ( e^{\frac{i\tau ^{3}}{3}}\varPsi ^{r,\ell }(\tau ) \Big ) (k^{\frac{2}{3}}\zeta _1) \nonumber \\&= \int e^{ik^{\frac{2}{3}}\zeta _1 \tau } e^{\frac{i\tau ^{3}}{3}} \varPsi ^{r,\ell }(\tau ) d\tau = k^{\frac{1}{3}} \int e^{ik \zeta _1 t + ik\frac{t^3}{3}} \varPsi ^{r,\ell }(k^{\frac{1}{3}}t)dt . \end{aligned}$$

(83)

Using (83) in (80), we finally conclude

$$\begin{aligned} \eta (\alpha ,x,k) \sim k^{\frac{1}{3}} \sum _{\begin{array}{c} q,r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} k^{\frac{n}{2}+\frac{1}{6}-q-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_{-}} \, I_{q,r,\ell }(\alpha ,x,k) \end{aligned}$$

(84)

where

$$\begin{aligned} I_{q,r,\ell }(\alpha ,x,k) := \int e^{ik \psi _3(\zeta ,t)} q_{N_{q,r,\ell }}({\bar{\alpha }},x,\zeta ) \varPsi ^{r,\ell }(k^{\frac{1}{3}}t) d\zeta dt \end{aligned}$$

(85)

with the phase function given by

$$\begin{aligned} \psi _3(\zeta ,t) := \psi _2(\alpha , x,\zeta ) + t\zeta _1 + \frac{t^3}{3}. \end{aligned}$$

(86)

As shown in [45], the integrals $I_{q,r,\ell }(\alpha ,k,x)$ can be treated using the stationary phase method which results in

$$\begin{aligned} I_{q,r,\ell }(\alpha ,x,k) \sim k^{-\frac{1}{3}} \sum _{p \in {\mathbb {Z}}_+} k^{-\frac{n+1}{2}-\frac{2p}{3}} \, a_{p,q,r,\ell }(\alpha ,x) \, (\varPsi ^{r,\ell })^{(p)}(k^{\frac{1}{3}} Z(\alpha ,x)) \, e^{ik \alpha \cdot x}, \end{aligned}$$

(87)

and this leads into the following for the envelope $\eta ^{\mathrm{slow}}(\alpha ,k,x) = e^{-ik \alpha \cdot x} \eta (\alpha ,k,x)$.

Theorem 7

[45, Theorem 9.36] In a vicinity of the shadow boundary $\{ x \in B : \alpha \cdot \nu (x) = 0 \}$, $\eta ^{\mathrm{slow}}(\alpha ,k,x)$ belongs to the Hörmander class $S^{0}_{\frac{2}{3},\frac{1}{3}}$ and admits an asymptotic expansion

$$\begin{aligned} \eta ^{\mathrm{slow}}(\alpha ,x,k) \sim \sum _{\begin{array}{c} q,p,r \in {\mathbb {Z}}_+ \\ \ell \in -{\mathbb {N}} \end{array}} a_{p,q,r,\ell } (\alpha ,x,k) \end{aligned}$$

(88)

with

$$\begin{aligned} a_{p,q,r,\ell } (\alpha ,x,k) := k^{-\frac{1}{3}-\frac{2p}{3}-q-\frac{r}{3}-\frac{\ell }{3}+(\ell +1)_-} \, b_{p,q,r,\ell } (\alpha ,x) \, (\varPsi ^{r,\ell })^{(p)}(k^{\frac{1}{3}}Z(\alpha ,x)) \end{aligned}$$

where $b_{p,q,r,\ell }$ are complex-valued $C^{\infty }$ functions, Z is a real-valued $C^{\infty }$ function that is positive on the illuminated region $\{ x \in B : \alpha \cdot \nu (x) < 0 \}$, negative on the shadow region $\{ x \in B : \alpha \cdot \nu (x) >0 \}$, and that vanishes precisely to first order at the shadow boundary.

Under certain assumptions, Theorem 7 is in fact valid over the entire boundary B. This is given in the next theorem where we use the notation $B^{\epsilon }_{\gtrless } = \{ x \in B: \alpha \cdot \nu (x) \gtrless \epsilon \}$.

Theorem 8

Assume there exists $\epsilon \in (0,1)$ such that on $B^{\epsilon }_{<}$ the envelope $\eta ^{\mathrm{slow}}$ belongs to $S^0_{1,0}(B^{\epsilon }_{<} \times (0,\infty ))$ and admits an asymptotic expansion

$$\begin{aligned} \eta ^{\mathrm{slow}}(\alpha ,x,k) \sim \sum _{j \in {\mathbb {Z}}_+} k^{-j} \, a_j(\alpha ,x), \quad \text {as } k \rightarrow \infty , \end{aligned}$$

(89)

and it is rapidly decreasing in the sense of Schwarz on $B^{\epsilon }_{>}$ as $k \rightarrow -\infty$. Then $\eta ^{\mathrm{slow}} \in S^{0}_{\frac{2}{3},\frac{1}{3}}(B \times (0,\infty ))$ and the asymptotic expansion (88) is valid over the entire boundary B.

The proof of Theorem 8 follows the same lines as in the proof of [23, Corollary 5.3] (see also [27, Theorem 3.1 and Corollary 2.1]) and is based on the standard matching of asymptotic expansions technique (see e.g. [23] and the references therein). The expansion (89) related to Neumann problem is similar to the one given in [45, Equation 1.15] for the Dirichlet case. Furthermore, using the references provided in [23] (see the proof of Corollary 5.3) we can deduce that, for the two-dimensional Neumann boundary value problem, $\eta ^{\mathrm{slow}}$ decays exponentially in $B^{\epsilon }_{>}$ as $k \rightarrow -\infty$ which implies the assumption of its rapid decay in the sense of Schwarz in Theorem 8.

B Auxiliary results

Here we provide auxiliary results used in the proofs.

Lemma 8

[24, Lemma 14] Let $a(s,k) = k^{\theta } \, b(s) \, \varphi (k^{\omega }\varUpsilon (s))$ where $b,\varphi$ and $\varUpsilon$ are smooth functions, b and $\varUpsilon$ are periodic, and $\theta \in {\mathbb {R}} \backslash {\mathbb {N}}$ and $\omega \in {\mathbb {R}} \backslash {\mathbb {Z}}_{+}$. Then

$$\begin{aligned} |D_s^{n}D_{k}^{m} \, a(s,k)|&\lesssim k^{\theta -m} \sum _{j=0}^{n+m} k^{j \omega } |\varphi ^{(j)} (k^{\omega }\varUpsilon (s))| \end{aligned}$$

for all $n,m \in {\mathbb {Z}}_{+}$ and all $k >0$.

Theorem 9

[47, Corollary 3.12] Given a function $f \in C^{\infty }([a,b])$ and $n \in {\mathbb {Z}}_+$, there exists a constant $C_n > 0$ such that

$$\begin{aligned} \inf _{p \in {\mathbb {P}}_d} \Vert f-p \Vert _{L^2[(a,b )]} \le C_n \left[ \int _{a}^{b} \left| D^{n} f(s) \right| ^{2} \left( s-a \right) ^{n} \left( b-s \right) ^{n} ds \right] ^{\frac{1}{2}} \, d^{-n} \end{aligned}$$

for all $d \in {\mathbb {N}}$ with $d+1 \ge n$.

Lemma 9

[24, Lemma 14] Suppose that either $[\alpha ,\beta ] \subseteq [t_{1},t_{2}] \subseteq (c,d)$ or $[\alpha ,\beta ] \cap (t_{1},t_{2}) = \varnothing$ and $[c,d] \subseteq (t_1,t_2)$. Then, for any $a,b \in {\mathbb {R}}$, $n \in {\mathbb {N}} \cup \{ 0 \}$, $m \in {\mathbb {N}}$, there holds

$$\begin{aligned}&\int _{\alpha }^{\beta } \dfrac{(s-a)^{n} \, (b-s)^{n}}{(s-c)^{m} \, (d-s)^{m}} \, ds\\&\quad = \sum _{\begin{array}{c} 0 \le p,q \le n \\ 1 \le j \le m \end{array}} \left( {\begin{array}{c}2m-j-1\\ m-j\end{array}}\right) \left( {\begin{array}{c}n\\ p\end{array}}\right) \left( {\begin{array}{c}n\\ q\end{array}}\right) \dfrac{(-1)^{n} \, {\mathcal {F}}(\alpha ,\beta ;a,b;c,d;n,p,q;j)}{(d-c)^{2m-j}}. \end{aligned}$$

Here we have

$$\begin{aligned} {\mathcal {F}}(\alpha ,\beta ;a,b;c,d;n,p,q;j)= & {} \left( c-a \right) ^{p} \left( c-b \right) ^{q} \log \left( \dfrac{\beta -c}{\alpha -c} \right) \\&\quad + \left( a-d \right) ^{p} \left( b-d \right) ^{q} \log \left( \dfrac{d-\alpha }{d-\beta } \right) \end{aligned}$$

when $2n-(p+q+j) = -1$, and

$$\begin{aligned}&{\mathcal {F}}(\alpha ,\beta ;a,b;c,d;n,p,q;j) \\&\quad = \dfrac{\left( c-a \right) ^{p} \left( c-b \right) ^{q}}{2n-(p+q+j)+1} \left[ \left( \beta -c \right) ^{2n-(p+q+j)+1} - \left( \alpha -c \right) ^{2n-(p+q+j)+1} \right] \\&\quad + \dfrac{\left( a-d \right) ^{p} \left( b-d \right) ^{q}}{2n-(p+q+j)+1} \left[ \left( d-\alpha \right) ^{2n-(p+q+j)+1} - \left( d-\beta \right) ^{2n-(p+q+j)+1} \right] \end{aligned}$$

when $2n-(p+q+j) \ne -1$.