# A novel boundary integral formulation for the biharmonic wave scattering
problem

Heping Dong  School of Mathematics, Jilin University, Changchun, Jilin 130012,
China<EMAIL_ADDRESS>and  Peijun Li  Department of Mathematics, Purdue
University, West Lafayette, Indiana 47907, USA<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the cavity scattering problem in an infinite thin
plate, where the out-of-plane displacement is governed by the two-dimensional
biharmonic wave equation. Based on an operator splitting, the scattering
problem is recast into a coupled boundary value problem for the Helmholtz and
modified Helmholtz equations. A novel boundary integral formulation is
proposed for the coupled problem. By introducing an appropriate regularizer,
the well-posedness is established for the system of boundary integral
equations. Moreover, the convergence analysis is carried out for the semi- and
full-discrete schemes of the boundary integral system by using the collocation
method. Numerical results show that the proposed method is highly accurate for
both smooth and nonsmooth examples.

###### Key words and phrases:

Biharmonic wave equation, scattering problem, boundary integral equations,
collocation method, error estimates

###### 2010 Mathematics Subject Classification:

65N38, 45L05

The first author was supported in part by the NSFC grant 12171201. The second
author was supported partially by the NSF grant DMS-2208256.

##  1\. Introduction

Scattering problems for biharmonic waves have significant applications in thin
plate elasticity. For instance, the ultra-broadband elastic cloaking has been
experimentally realized at acoustic frequencies and could serve as a model for
seismic cloaks to protect infrastructure from earthquakes [5, 24]. Similar to
photonic and phononic crystals, platonic crystals (PlaCs) are fabricated by
using regularly spaced arrays of holes, called cavities, cut into thin elastic
plates [9, 18]. The design of PlaCs can be utilized to guide or harness
destructive wave energy for constructive purposes [4, 22, 25]. As an emerging
passive vibration control technique, the acoustic black hole (ABH) has
recently been developed for vibration and noise reduction of a system [20].
The basic idea of ABH is to arrange some inhomogeneous medium in a thin beam
or plate so that the wave can be trapped from propagating. In the wave motion
of these structures, the out-of-plane displacement of the thin plate satisfies
the biharmonic wave equation. Although the scattering theory has been well
developed for acoustic, elastic, and electromagnetic waves, many scattering
problems remain unsolved for biharmonic waves. This paper concerns the
biharmonic wave scattering of a time-harmonic plane wave by a cavity in an
infinite thin plate. A novel boundary integral formulation is proposed for the
boundary value problem and a spectrally accurate collocation method is
developed for the boundary integral system.

The method of boundary integral equations has several advantages: the solution
is represented by surface distributions which leads to fewer unknowns; the
radiation condition is exactly enforced that avoids the truncation error of
artificial boundary conditions; the surface geometry can be represented to
high precision using appropriate boundary elements. However, in a conventional
boundary integral method, these advantages may be offset by two factors: the
high cost of evaluating interactions among all the boundary elements and the
need to evaluate singular integrals. The former can be resolved by fast
evaluation methods such as the panel clustering algorithm [8] or the fast
multipole method [7], and the latter has to be treated using special
quadrature formulas or analytical expansions. Despite the difficulties, the
boundary integral method provides an efficient approach for solving many
problems, especially for those imposed in unbounded domains [1, 2].

We consider the cavity scattering problem in an infinite thin elastic plate,
where the wave motion is governed by the two-dimensional biharmonic wave
equation. As an exterior boundary value problem, it is analogous to the
obstacle scattering problem for the acoustic, elastic, or electromagnetic wave
equation. Compared with the obstacle scattering problem, the cavity scattering
problem is much less investigated. A numerical study was given in [22] on the
boundary integral method for solving the biharmonic wave scattering by smooth
cavities embedded in thin plates. In [23], boundary integral equations were
deduced for cavities with different boundary conditions, and the full
discretization of the boundary integral system was discussed by expanding the
unknown boundary values in terms of Fourier series. However, the mathematical
and numerical analysis are not available on the method of boundary integral
equations for solving the biharmonic wave scattering problem.

Motivated by [3], we present the first theoretical analysis on the biharmonic
wave scattering problem. Based on a splitting of the biharmonic wave operator,
two auxiliary functions are introduced to represent the Helmholtz and modified
Helmholtz wave components, respectively. The scattering problem is recast into
a coupled boundary value problem for the Helmholtz and modified Helmholtz
equations. The uniquneness is established for the scattering problem by
utilizing Rellich’s lemma and the exponentially decaying property of the
modified Helmholtz wave component. By the double- and single-layer potentials
representing the Helmholtz and modified Helmoltz wave components,
respectively, a novel boundary integral formulation is proposed for the
coupled boundary value problem. Following [3, 17], we introduce a regularizer
to the boundary integral system and split the singular operator into an
isomorphic operator plus a compact one. The Riesz–Fredholm theory is then
employed to show the well-posedness of the coupled boundary integral system.
Moreover, the collocation method is empolyed to examine the semi- and full-
discretization of the boundary integral equations. Special quadratures are
adopted for the hypersingular and logarithmic singular integrals to carry out
the convergence analysis for both schemes. In [16, 13], the quadratures were
investigated on the similar hypersingular and logarithmic singular integrals
for solving the acoustic obstacle scattering problem. In [14], a collocation
method was developed and the convergence was analyzed by using the
trigonometric interpolation to discretize the principal part of the
hypersingular operator. We refer to [11, 19, 21] for studies on general
singular integral equations. In addition to the convergence analysis,
numerical experiments are provided to demonstrate the high accuracy of the
proposed method for both smooth and nonsmooth examples.

The rest of this paper is structured as follows. In section 2, the scattering
problem is introduced and the uniqueness is proved for the boundary value
problem. Section 3 presents the boundary integral equations and discusses the
well-posedness of the coupled system. The convergence analysis is carried out
in section 4 for the semi- and full-discrete schemes of the system. Section 5
presents numerical experiments to verify the theoretical findings and to
demonstrate the superior performance of the proposed method. Section 6
concludes the paper with remarks and future work.

##  2\. Problem formulation

We consider the biharmonic wave scattering problem in a two-dimensional thin
plate with a cavity, which is represented by a bounded domain
$D\subset\mathbb{R}^{2}$ with an analytic boundary $\Gamma$. The plate is
assumed to be made of a homogeneous and isotropic elastic medium in
$\mathbb{R}^{2}\setminus\overline{D}$. Let $\tau=(\tau_{1},\tau_{2})^{\top}$
and $\nu=(\nu_{1},\nu_{2})^{\top}$ be the tangential and normal vectors on
$\Gamma$, respectively. It is clear to note $\tau_{1}=-\nu_{2}$ and
$\tau_{2}=\nu_{1}$. The plate is excited by a time-harmonic plane wave

$u^{\rm inc}(x)=e^{\mathrm{i}\kappa x\cdot d},\quad x\in\mathbb{R}^{2},$ (2.1)

where $\kappa>0$ is the wavenumber and $d=(\cos\theta,\sin\theta)^{\top}$ is
the incident direction with $\theta\in[0,2\pi)$ being the incident angle.

The out-of-plane displacement of the plate $u$ satisfies the two-dimensional
biharmonic wave equation

$\displaystyle\Delta^{2}u-\kappa^{4}u=0\quad{\rm
in}\,\mathbb{R}^{2}\setminus\overline{D}.$ (2.2)

Assuming that the cavity satisfies the clamped boundary condition, we have

$\displaystyle u=\partial_{\nu}u=0\quad{\rm on}\,\Gamma.$ (2.3)

The total displacement $u$ is composed of the incident wave field $u^{\rm
inc}$ and the scattered field $v$, i.e.,

$u=u^{\rm inc}+v.$

It can be verified from (2.1)–(2.2) that the scattered field $v$ satisfies

$\displaystyle\Delta^{2}v-\kappa^{4}v=0\quad{\rm
in}\,\mathbb{R}^{2}\setminus\overline{D}.$ (2.4)

By (2.3), $v$ satisfies the following boundary conditions on $\Gamma$:

$\displaystyle v=-u^{\rm inc},\quad\partial_{\nu}v=-\partial_{\nu}u^{\rm
inc}.$ (2.5)

In addition, $v$ and $\Delta v$ are required to satisfy the Sommerfeld
radiation condition

$\displaystyle\lim_{r\to\infty}r^{\frac{1}{2}}(\partial_{r}v-\mathrm{i}\kappa
v)=0,\quad\lim_{r\to\infty}r^{\frac{1}{2}}(\partial_{r}\Delta
v-\mathrm{i}\kappa\Delta v)=0,\quad r=|x|.$ (2.6)

Consider two auxiliary functions

$\displaystyle v_{\rm H}=-\frac{1}{2\kappa^{2}}(\Delta v-\kappa^{2}v),\quad
v_{\rm M}=\frac{1}{2\kappa^{2}}(\Delta v+\kappa^{2}v).$ (2.7)

It is clear to note $v=v_{\rm H}+v_{\rm M}$ and $\Delta v=\kappa^{2}(v_{\rm
M}-v_{\rm H})$. From (2.4), we have

$(\Delta-\kappa^{2})(\Delta+\kappa^{2})v=0\quad{\rm
in}\,\mathbb{R}^{2}\setminus\overline{D},$

which implies that $v_{\rm H}$ and $v_{\rm M}$ satisfy the Helmholtz equation
and the modified Helmholtz equation in $\mathbb{R}^{2}\setminus\overline{D}$,
respectively, i.e.,

$\displaystyle\Delta v_{\rm H}+\kappa^{2}v_{\rm H}=0,\quad\Delta v_{\rm
M}-\kappa^{2}v_{\rm M}=0.$ (2.8)

By (2.5), $v_{\rm H}$ and $v_{\rm M}$ satisfy the coupled boundary conditions
on $\Gamma$:

$\displaystyle v_{\rm H}+v_{\rm M}=f_{1},\quad\partial_{\nu}v_{\rm
H}+\partial_{\nu}v_{\rm M}=f_{2},$ (2.9)

where $f_{1}=-u^{\rm inc}$ and $f_{2}=-\partial_{\nu}u^{\rm inc}$. Combining
(2.6) and (2.7) yields that $v_{\rm H}$ and $v_{\rm M}$ satisfy the Sommerfeld
radiation condition

$\displaystyle\lim_{r\to\infty}r^{\frac{1}{2}}(\partial_{r}v_{\rm
H}-\mathrm{i}\kappa v_{\rm
H})=0,\quad\lim_{r\to\infty}r^{\frac{1}{2}}(\partial_{r}v_{\rm
M}-\mathrm{i}\kappa v_{\rm M})=0,\quad r=|x|.$ (2.10)

Clearly, the scattering problems (2.4)–(2.6) and (2.8)–(2.10) are equivalent.
Hence, it suffices to consider the scattering problem (2.8)–(2.10), where
$v_{\rm H}$ and $v_{\rm M}$ are called the Helmholtz and modified Helmholtz
wave components, respectively.

######  Remark 2.1.

Let $v_{\rm M}$ be a solution to the scattering problem (2.8)–(2.10). Denote
$D\subset B_{a}=\\{x\in\mathbb{R}^{2}:|x|<a\\}$. Then $v_{\rm M}$ has the
Fourier series expansion

$v_{\rm M}(r,\theta)=\sum_{n=-\infty}^{\infty}\frac{K_{n}(\kappa
r)}{K_{n}(\kappa a)}\hat{v}^{(n)}_{\rm M}(a)e^{\mathrm{i}n\theta},\quad
r=|x|>a,$

where $\hat{v}^{(n)}_{\rm M}(a)$ are the Fourier coefficients of $v_{\rm M}$
on $\partial B_{a}=\\{x\in\mathbb{R}^{2}:|x|=a\\}$ given by

$\hat{v}^{(n)}_{\rm M}(a)=\frac{1}{2\pi}\int_{0}^{2\pi}v_{\rm
M}(a,\theta)e^{-\mathrm{i}n\theta}\mathrm{d}\theta$

and $K_{n}(z)$ denotes the modified Bessel function of the second kind with
order $n$. By [26, p. 511], $K_{n}(z)$ has no zeros for $|\arg z|\leq\pi/2$,
which implies that $K_{n}(\kappa a)\neq 0$. Moreover, we have from [26, §3.7
and §7.23] that $K_{n}(t)=K_{-n}(t)$ and it admits the asymptotic expansion

$K_{n}(t)=\sqrt{\frac{\pi}{2t}}e^{-t}\Big{\\{}1+O\big{(}\frac{1}{t}\big{)}\Big{\\}},\quad
t\to+\infty,\quad t>0,$

which shows that $v_{\rm M}$ and $\partial_{r}v_{\rm M}$ decay exponentially
as $r\to\infty$ for a fixed $\kappa$.

######  Theorem 2.2.

The boundary value problem (2.8)–(2.10) has at most one solution.

###### Proof.

It is required to show that $v_{\rm H}=v_{\rm M}=0$ in
$\mathbb{R}^{2}\setminus\overline{D}$ when $f_{1}=f_{2}=0$. Applying Green’s
theorem in $D_{a}:=B_{a}\setminus\overline{D}$ and the boundary condition
(2.9), we obtain

$\displaystyle\int_{\partial B_{a}}v_{\rm H}\partial_{\nu}\overline{v_{\rm
H}}\mathrm{d}s$ $\displaystyle=\int_{D_{a}}\big{(}v_{\rm
H}\Delta\overline{v_{\rm H}}+\nabla v_{\rm H}\cdot\nabla\overline{v_{\rm
H}}\big{)}\mathrm{d}x+\int_{\Gamma}v_{\rm H}\partial_{\nu}\overline{v_{\rm
H}}\mathrm{d}s$ $\displaystyle=\int_{D_{a}}\big{(}-\kappa^{2}|v_{\rm
H}|^{2}+|\nabla v_{\rm H}|^{2}\big{)}\mathrm{d}x+\int_{\Gamma}v_{\rm
M}\partial_{\nu}\overline{v_{\rm M}}\mathrm{d}s,$ $\displaystyle\int_{\partial
B_{a}}v_{\rm M}\partial_{\nu}\overline{v_{\rm M}}\mathrm{d}s$
$\displaystyle=\int_{D_{a}}\big{(}\kappa^{2}|v_{\rm M}|^{2}+|\nabla v_{\rm
M}|^{2}\big{)}\mathrm{d}x+\int_{\Gamma}v_{\rm M}\partial_{\nu}\overline{v_{\rm
M}}\mathrm{d}s,$

which give

$\Im\int_{\partial B_{a}}v_{\rm H}\partial_{\nu}\overline{v_{\rm
H}}\mathrm{d}s=\Im\int_{\Gamma}v_{\rm M}\partial_{\nu}\overline{v_{\rm
M}}\mathrm{d}s=\Im\int_{\partial B_{a}}v_{\rm M}\partial_{\nu}\overline{v_{\rm
M}}\mathrm{d}s.$

It follows from the above equation and the Sommerfeld radiation condition
(2.6) that

$\displaystyle\lim_{a\to\infty}\int_{\partial B_{a}}|\partial_{\nu}v_{\rm
H}-\mathrm{i}\kappa v_{\rm H}|^{2}\mathrm{d}s$
$\displaystyle=\lim_{a\to\infty}\int_{\partial
B_{a}}\big{\\{}|\partial_{\nu}v_{\rm H}|^{2}+\kappa^{2}|v_{\rm
H}|^{2}+2\kappa\Im(v_{\rm H}\partial_{\nu}\overline{v_{\rm
H}})\big{\\}}\mathrm{d}s$ $\displaystyle=\lim_{a\to\infty}\int_{\partial
B_{a}}\big{\\{}|\partial_{\nu}v_{\rm H}|^{2}+\kappa^{2}|v_{\rm
H}|^{2}+2\kappa\Im(v_{\rm M}\partial_{\nu}\overline{v_{\rm
M}})\big{\\}}\mathrm{d}s=0.$ (2.11)

By Remark 2.1 and (2), we have

$\lim_{a\to\infty}\int_{\partial B_{a}}\big{\\{}|\partial_{\nu}v_{\rm
H}|^{2}+\kappa^{2}|v_{\rm H}|^{2}\big{\\}}\mathrm{d}s=0.$

Using Rellich’s lemma yields $v_{\rm H}=0$ in
$\mathbb{R}^{2}\setminus\overline{D}$. It follows from the continuity of
$v_{\rm H}$ and $\partial_{\nu}v_{\rm H}$ and (2.9) that $v_{\rm
H}=\partial_{\nu}v_{\rm H}=0$ and $v_{\rm M}=\partial_{\nu}v_{\rm M}=0$ on
$\Gamma$, which shows $v_{\rm M}=0$ in $\mathbb{R}^{2}\setminus\overline{D}$
by Green’s representation theorem (cf. [2, Theorem 2.5]). ∎

##  3\. Boundary integral formulation

In this section, we propose a novel formulation of boundary integral equations
for the scattering problem (2.8)–(2.10). Based on the Riesz–Fredholm theory, a
regularizer is constructed to split the operator equation into the form of an
isomorphic operator plus a compact one.

###  3.1. Coupled integral equations

Denote by $G_{\rm H}$ and $G_{\rm M}$ the fundamental solutions to the two-
dimensional Helmholtz and modified Helmholtz equations, respectively.
Explicitly, we have

$G_{\rm H}(x,y)=\frac{\mathrm{i}}{4}H_{0}^{(1)}(\kappa|x-y|),\quad G_{\rm
M}(x,y)=\frac{\mathrm{i}}{4}H_{0}^{(1)}(\mathrm{i}\kappa|x-y|),$

where $H_{0}^{(1)}$ is the Hankel function of the first kind with order zero.
Using the fundamental solutions, we represent the solutions of (2.8) as the
double- and single-layer potentials with densities $g_{1}$ and $g_{2}$,
respectively, i.e.,

$\displaystyle v_{\rm H}(x)=\int_{\Gamma}\frac{\partial G_{\rm
H}(x,y)}{\partial\nu(y)}g_{1}(y)\mathrm{d}s(y),\quad v_{\rm
M}(x)=\int_{\Gamma}G_{\rm M}(x,y)g_{2}(y)\mathrm{d}s(y),\quad
x\in\mathbb{R}^{2}\setminus\Gamma_{D}.$ (3.1)

It follows from the solution representation (3.1), the boundary conditions
(2.5), and the jump relations of the single- and double-layer potentials, we
may deduce the boundary integral equations on $\Gamma$:

$\displaystyle\begin{split}\frac{1}{2}g_{1}(x)+\int_{\Gamma}\frac{\partial
G_{\rm H}(x,y)}{\partial\nu(y)}g_{1}(y)\,\mathrm{d}s(y)+\int_{\Gamma}G_{\rm
M}(x,y)g_{2}(y)\,\mathrm{d}s(y)&=-u^{\rm inc}(x),\\\
\int_{\Gamma}\frac{\partial^{2}G_{\rm
H}(x,y)}{\partial\nu(x)\partial\nu(y)}g_{1}(y)\,\mathrm{d}s(y)-\frac{1}{2}g_{2}(x)+\int_{\Gamma}\frac{\partial
G_{\rm M}(x,y)}{\partial\nu(x)}g_{2}(y)\,\mathrm{d}s(y)&=-\frac{\partial
u^{\rm inc}(x)}{\partial\nu(x)}.\end{split}$ (3.2)

For $x\in\Gamma$, introduce the following integral operators:

$\displaystyle(\mathcal{L}g)(x)=2\int_{\Gamma}\frac{\partial G_{\rm
H}(x,y)}{\partial\nu(y)}g(y)\mathrm{d}s(y),$
$\displaystyle(\mathcal{T}g)(x)=2\int_{\Gamma_{D}}\frac{\partial^{2}G_{\rm
H}(x,y)}{\partial\nu(x)\partial\nu(y)}g(y)\mathrm{d}s(y),$
$\displaystyle(\mathcal{K}g)(x)=2\int_{\Gamma_{D}}\frac{\partial G_{\rm
M}(x,y)}{\partial\nu(x)}g(y)\mathrm{d}s(y),$
$\displaystyle(\mathcal{S}g)(x)=2\int_{\Gamma}G_{\rm
M}(x,y)g(y)\mathrm{d}s(y).$

Since the fundamental solution $G_{\rm M}$ decays exponentially, the modified
Helmholtz wave component $v_{\rm M}$ given in (3.1) does not propagate. Hence
we may obtain from (3.1) that the far-field pattern of the scattered wave $v$
is

$\displaystyle v_{\infty}(\hat{x})=\varrho\int_{\Gamma}\frac{\partial
e^{-\mathrm{i}\kappa\hat{x}\cdot y}}{\partial\nu(y)}g_{1}(y)\,\mathrm{d}s(y),$
(3.3)

where $\varrho=e^{\mathrm{i}\pi/4}/{\sqrt{8\kappa\pi}}$ and $\hat{x}=x/r$ is
the observation direction.

With the help of the integral operators, it is convenient to rewrite (3.2)
into the operator form

$\begin{cases}g_{1}+\mathcal{L}g_{1}+\mathcal{S}g_{2}=2f_{1},\\\
\mathcal{T}g_{1}-g_{2}+\mathcal{K}g_{2}=2f_{2}.\end{cases}$ (3.4)

######  Theorem 3.1.

The coupled boundary integral equations (3.4) has at most one solution
provided that the wavenumber $\kappa$ is not the Dirichlet eigenvalue of the
Helmholtz equation in $D$.

###### Proof.

It is required to show that $g_{1}=g_{2}=0$ if $f_{1}=f_{2}=0$. Let

$\displaystyle v_{\rm H}(x)=\begin{cases}v_{\rm H}^{i},&x\in D,\\\ v_{\rm
H}^{e},&x\in\mathbb{R}^{2}\setminus\overline{D},\end{cases}\qquad v_{\rm
M}(x)=\begin{cases}v_{\rm M}^{i},&x\in D,\\\ v_{\rm
M}^{e},&x\in\mathbb{R}^{2}\setminus\overline{D}.\end{cases}$

Since $v_{\rm H}^{e}$ and $v_{\rm M}^{e}$ satisfy the boundary value problem
(2.8)–(2.10), it follows from Theorem 2.2 that they are identically zero in
$\mathbb{R}^{2}\setminus\overline{D}$. Using the jump relations of the single-
and double-layer potentials, we have on $\Gamma$ that

$\displaystyle v_{\rm H}^{e}-v_{\rm H}^{i}=g_{1},\quad v_{\rm M}^{e}-v_{\rm
M}^{i}=0,$ $\displaystyle\partial_{\nu}v_{\rm H}^{e}-\partial_{\nu}v_{\rm
H}^{i}=0,\quad\partial_{\nu}v_{\rm M}^{e}-\partial_{\nu}v_{\rm M}^{i}=-g_{2}.$

Since neither $\kappa$ nor $\mathrm{i}\kappa$ is the Dirichlet eigenvalue of
the Helmholtz equation in $D$, we deduce $v_{\rm H}^{i}=v_{\rm M}^{i}=0$ in
$D$, which implies $g_{1}=g_{2}=0$. ∎

By Theorem 3.1, the uniqueness holds for the coupled system (3.4) if the
wavenumber $\kappa$ is not the Dirichlet eigenvalue of the Helmholtz equation
in $D$. In general, this assumption can be removed by using combined double-
and single-layer potentials for the boundary integral formulation [1]. In this
work, we assume that this assumption is satisfied and the coupled system (3.4)
has at most one solution.

###  3.2. Parameterization

Let the analytic boundary $\Gamma$ have a parametrization of the form

$\Gamma=\\{\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))\in\mathbb{R}^{2}:0\leq
t<2\pi\\},$

where $\gamma_{1}$ and $\gamma_{2}$ are $2\pi$-periodic functions satisfying
$|\gamma^{\prime}(t)|>0$ for $t\in[0,2\pi)$. Plugging the parameterization of
$\Gamma$ into $\mathcal{L},\mathcal{S},\mathcal{K}$, and $\mathcal{T}$, we
define the parameterized integral operators

$\displaystyle(L\phi)(t)=\int_{0}^{2\pi}l(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,$
$\displaystyle(K\phi)(t)=\int_{0}^{2\pi}k(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,$
$\displaystyle(S\phi)(t)=\int_{0}^{2\pi}s(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,$
$\displaystyle(R\phi)(t)=\int_{0}^{2\pi}r(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,$
$\displaystyle(H\phi)=\int_{0}^{2\pi}h(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,$

where the integral kernels are given by

$\displaystyle l(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}\kappa}{2}n(\zeta)\cdot[\gamma(t)-\gamma(\zeta)]\frac{H_{1}^{(1)}(\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|},\quad
s(t,\zeta)=\frac{\mathrm{i}}{2}H_{0}^{(1)}(\mathrm{i}\kappa|\gamma(t)-\gamma(\zeta)|),$
$\displaystyle k(t,\zeta)$
$\displaystyle=\frac{\kappa}{2}n(t)\cdot[\gamma(t)-\gamma(\zeta)]\frac{H_{1}^{(1)}(\mathrm{i}\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|},$
$\displaystyle r(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}\kappa^{2}}{2}H_{0}^{(1)}(\kappa|\gamma(t)-\gamma(\zeta)|)n(t)\cdot
n(\zeta),$ $\displaystyle h(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}}{2}\tilde{h}(t,\zeta)\Big{\\{}\kappa^{2}H_{0}^{(1)}(\kappa|\gamma(t)-\gamma(\zeta)|)-\frac{2\kappa
H_{1}^{(1)}(\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|}\Big{\\}}$
$\displaystyle\qquad+\frac{\mathrm{i}\kappa\gamma^{\prime}(t)\cdot\gamma^{\prime}(\zeta)}{2|\gamma(t)-\gamma(\zeta)|}H_{1}^{(1)}(\kappa|\gamma(t)-\gamma(\zeta)|)+\frac{1}{4\pi}\frac{1}{\sin^{2}\frac{1}{2}(t-\zeta)}.$

Here
$n(t):=\big{(}\gamma^{\prime}_{2}(t),-\gamma^{\prime}_{1}(t)\big{)}^{\top}$,
$n^{\perp}(t):=\gamma^{\prime}(t)=\big{(}\gamma^{\prime}_{1}(t),\gamma^{\prime}_{2}(t)\big{)}^{\top}$,
and

$\tilde{h}(t,\zeta)=\frac{\big{[}\gamma^{\prime}(t)\cdot(\gamma(t)-\gamma(\zeta))\big{]}\big{[}\gamma^{\prime}(\zeta)\cdot(\gamma(t)-\gamma(\zeta))\big{]}}{|\gamma(t)-\gamma(\zeta)|^{2}}.$

The kernels $l(t,\zeta)$, $s(t,\zeta)$, $k(t,\zeta)$, $r(t,\zeta)$, and
$h(t,\zeta)$ can be decomposed into the following forms:

$\displaystyle\begin{split}\chi(t,\zeta)&=\chi_{1}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}+\chi_{2}(t,\zeta),\\\
\chi_{2}(t,\zeta)&=\chi(t,\zeta)-\chi_{1}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)},\end{split}$
(3.5)

where $\chi=l,s,k,r,h$, and $\chi_{1},\chi_{2}$ are analytic functions.
Explicitly, we have

$\displaystyle l_{1}(t,\zeta)$
$\displaystyle=\frac{\kappa}{2\pi}n(\zeta)\cdot\big{[}\gamma(\zeta)-\gamma(t)\big{]}\frac{J_{1}(\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|},~{}~{}s_{1}(t,\zeta)=-\frac{1}{2\pi}J_{0}(\mathrm{i}\kappa|\gamma(t)-\gamma(\zeta)|),$
$\displaystyle k_{1}(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}\kappa}{2\pi}n(t)\cdot\big{[}\gamma(t)-\gamma(\zeta)\big{]}\frac{J_{1}(\mathrm{i}\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|},$
$\displaystyle r_{1}(t,\zeta)$
$\displaystyle=-\frac{\kappa^{2}}{2\pi}J_{0}(\kappa|\gamma(t)-\gamma(\zeta)|)n(t)\cdot
n(\zeta),$ $\displaystyle h_{1}(t,\zeta)$
$\displaystyle=-\frac{1}{2\pi}\tilde{h}(t,\zeta)\Bigg{(}\kappa^{2}J_{0}(\kappa|\gamma(t)-\gamma(\zeta)|)-\frac{2\kappa
J_{1}(\kappa|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|}\Bigg{)}$
$\displaystyle\qquad-\frac{\kappa\gamma^{\prime}(t)\cdot\gamma^{\prime}(\zeta)}{2\pi|\gamma(t)-\gamma(\zeta)|}J_{1}(\kappa|\gamma(t)-\gamma(\zeta)|),$

where $J_{0}$ and $J_{1}$ are the Bessel functions of the first kind with
order zero and one, respectively. The diagonal entries are given by

$\displaystyle l_{1}(t,t)$ $\displaystyle=0,\quad
l_{2}(t,t)=\frac{1}{2\pi}\frac{{n}(t)\cdot\gamma^{\prime\prime}(t)}{|\gamma^{\prime}(t)|^{2}},$
$\displaystyle s_{1}(t,t)$ $\displaystyle=-\frac{1}{2\pi},\quad
s_{2}(t,t)=\frac{\mathrm{i}}{2}-\frac{C}{\pi}-\frac{1}{\pi}\ln\big{(}\frac{\mathrm{i}\kappa}{2}|\gamma^{\prime}(t)|\big{)},$
$\displaystyle r_{1}(t,t)$
$\displaystyle=-\frac{\kappa^{2}}{2\pi}|\gamma^{\prime}(t)|^{2},\quad
r_{2}(t,t)=\kappa^{2}|\gamma^{\prime}(t)|^{2}\Big{\\{}\frac{\mathrm{i}}{2}-\frac{C}{\pi}-\frac{1}{\pi}\ln\big{(}\frac{\kappa}{2}|\gamma^{\prime}(t)|\big{)}\Big{\\}},$
$\displaystyle k_{1}(t,t)$ $\displaystyle=0,\quad
k_{2}(t,t)=\frac{1}{2\pi}\frac{{n}(t)\cdot\gamma^{\prime\prime}(t)}{|\gamma^{\prime}(t)|^{2}},\quad
h_{1}(t,t)=-\frac{\kappa^{2}|\gamma^{\prime}(t)|^{2}}{4\pi},$ $\displaystyle
h_{2}(t,t)$
$\displaystyle=\Big{(}\pi\mathrm{i}-1-2C-2\ln\frac{\kappa|\gamma^{\prime}(t)|}{2}\Big{)}\frac{\kappa^{2}|\gamma^{\prime}(t)|^{2}}{4\pi}+\frac{1}{12\pi}$
$\displaystyle\qquad+\frac{[\gamma^{\prime}(t)\cdot\gamma^{\prime\prime}(t)]^{2}}{2\pi|\gamma^{\prime}(t)|^{4}}-\frac{|\gamma^{\prime\prime}(t)|^{2}}{4\pi|\gamma^{\prime}(t)|^{2}}-\frac{\gamma^{\prime}(t)\cdot\gamma^{\prime\prime\prime}(t)}{6\pi|\gamma^{\prime}(t)|^{2}},$

where $C$ denotes Euler’s constant. We refer to [2, 13] for the details of the
decomposition.

By the identities (cf. [13, $(2.4)$ and $(2.6)$])

$(\mathcal{T}g)(x)=2\int_{\Gamma}\frac{\partial\Phi(x,y)}{\partial\tau(x)}\frac{\partial
g(y)}{\partial\tau}\mathrm{d}s(y)+2\kappa^{2}\int_{\Gamma}\Phi(x,y)\nu(x)\cdot\nu(y)g(y)\mathrm{d}s(y)$

and

$\displaystyle
2\int_{\Gamma}\frac{\partial\Phi(x(t),y)}{\partial\tau(x(t))}\frac{\partial
g(y)}{\partial\tau}\mathrm{d}s(y)$
$\displaystyle=\frac{1}{|\gamma^{\prime}(t)|}\int_{0}^{2\pi}\left(\frac{1}{2\pi}\cot\big{(}\frac{\zeta-t}{2}\big{)}\frac{\partial
g(y(\zeta))}{\partial\tau}-h(t,\zeta)g(y(\zeta))\right)d\zeta,$

the parameterized integral operator $T$ can be represented as

$|\gamma^{\prime}|T\phi=(T_{0}\phi-M_{0}\phi)-H\phi+R\phi,$

where the operator $T_{0}:H^{p}[0,2\pi]\rightarrow H^{p-1}[0,2\pi]$ and
$M_{0}$ are defined by

$(T_{0}\phi)(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\cot\big{(}\frac{\zeta-t}{2}\big{)}\phi^{\prime}(\zeta)\mathrm{d}\zeta+\frac{\mathrm{i}}{2\pi}\int_{0}^{2\pi}\phi(\zeta)\,\mathrm{d}\zeta\overset{\text{def}}{=}(\widetilde{T}_{0}\phi)(t)+(M_{0}\phi)(t).$

Substituting Maue’s formula into (3.4) and multiplying $|\gamma^{\prime}|$ on
both sides of the second equation of (3.4), we obtain

$\displaystyle\Bigg{(}\left[\begin{array}[]{cc}I&0\\\
T_{0}&-I\end{array}\right]+\left[\begin{array}[]{cc}L&S\\\ R-H-
M_{0}&K\end{array}\right]\Bigg{)}\left[\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right]=\left[\begin{array}[]{c}\eta_{1}\\\
\eta_{2}\end{array}\right],$ (3.14)

where $\eta_{1}=2(f_{1}\circ\gamma)$,
$\eta_{2}=2(f_{2}\circ\gamma)|\gamma^{\prime}|$,
$\psi_{1}=(g_{1}\circ\gamma)$, $\psi_{2}=(g_{2}\circ\gamma)|\gamma^{\prime}|$,
and $I$ is the identity operator.

###  3.3. Operator equations

For $p\geq 0$, we denote the space of $2\pi$-periodic functions by
$H^{p}[0,2\pi]$, which is equipped with the norm

$\|u\|_{p}^{2}:=\sum_{m=-\infty}^{\infty}(1+m^{2})^{p}|\hat{u}_{m}|^{2}<\infty,$

where the Fourier coefficients $\hat{u}_{m}$ are given by

$\hat{u}_{m}=\frac{1}{2\pi}\int_{0}^{2\pi}u(t)e^{-\mathrm{i}mt}\,\mathrm{d}t,\quad
m=0,\pm 1,\pm 2,\cdots$

Define the Sobolev space

$\displaystyle H^{p}[0,2\pi]^{2}$
$\displaystyle=\Big{\\{}\eta=(\eta_{1},\eta_{2})^{\top}:\eta_{j}\in
H^{p}[0,2\pi],j=1,2\Big{\\}},$

which has the standard norm $\|\eta\|_{p}=\|\eta_{1}\|_{p}+\|\eta_{2}\|_{p}$.

Define the integral operators $S_{0}:H^{p}[0,2\pi]\rightarrow
H^{p+1}[0,2\pi]$, $S_{1}:H^{p}[0,2\pi]\rightarrow H^{p+2}[0,2\pi]$, and
$S_{2}:H^{p}[0,2\pi]\rightarrow H^{p+3}[0,2\pi]$ by

$\displaystyle(S_{0}\phi)(t)=$
$\displaystyle-\frac{1}{2\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\phi(\zeta)\,\mathrm{d}\zeta+\frac{\mathrm{i}}{2\pi}\int_{0}^{2\pi}\phi(\zeta)\mathrm{d}\zeta$
$\displaystyle\overset{\text{def}}{=}$
$\displaystyle(\widetilde{S}_{0}\phi)(t)+(M_{0}\phi)(t),$
$\displaystyle(S_{1}\phi)(t)=$
$\displaystyle\frac{1}{4\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\sin(t-\zeta)\phi(\zeta)\mathrm{d}\zeta,$
$\displaystyle(S_{2}\phi)(t)=$
$\displaystyle-\frac{1}{8\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\sin^{2}(t-\zeta)\phi(\zeta)\mathrm{d}\zeta,$

which are bounded for any $p\geq 0$. We split the operators $S$, $L$, $K$,
$R$, and $H$ as

$\displaystyle S=\widetilde{S}_{0}+E_{1}S_{2}+\widetilde{S},\quad
L=E_{2}S_{2}+\widetilde{L},\quad K=-E_{2}S_{2}+\widetilde{K},$ $\displaystyle
R=E_{1}\widetilde{S}_{0}+2E_{3}S_{1}+E_{4}S_{2}+\widetilde{R},\quad
H=\frac{1}{2}E_{1}\widetilde{S}_{0}+E_{3}S_{1}+E_{5}S_{2}+\widetilde{H},$

where

$\displaystyle E_{1}\phi=\kappa^{2}|\gamma^{\prime}|^{2}\phi,\quad
E_{2}\phi=\kappa^{2}n\cdot\gamma^{\prime\prime}\phi,\quad
E_{3}\phi=\kappa^{2}\gamma^{\prime}\cdot\gamma^{\prime\prime}\phi,$
$\displaystyle
E_{4}\phi=\kappa^{4}|\gamma^{\prime}|^{4}\phi+2\kappa^{2}\gamma^{\prime}\cdot\gamma^{\prime\prime\prime}\phi,\quad
E_{5}\phi=-\frac{3}{4}\kappa^{4}|\gamma^{\prime}|^{4}\phi+\kappa^{2}\gamma^{\prime}\cdot\gamma^{\prime\prime\prime}\phi$

are bounded operators on $H^{p}[0,2\pi]$, and $E_{1}$ is isomorphic on
$H^{p}[0,2\pi]$ since $|\gamma^{\prime}(t)|>0$ is analytic (cf. [3, Theorem
3.1]). Here,
$\widetilde{S},\widetilde{L},\widetilde{K},\widetilde{R},\widetilde{H}:H^{p}[0,2\pi]\rightarrow
H^{p+4}[0,2\pi]$ are bounded, and they are integral operators of the form

$\displaystyle(Q\phi)(t)$
$\displaystyle=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}q_{1}(t,\zeta)\phi(\zeta)\mathrm{d}\zeta+\int_{0}^{2\pi}q_{2}(t,\zeta)\phi(\zeta)\mathrm{d}\zeta$
$\displaystyle\overset{\text{def}}{=}(Q_{1}\phi)(t)+(Q_{2}\phi)(t),$

where $q_{1}$ satisfies
$q_{1}(t,t)=\partial_{t}q_{1}(t,t)=\partial^{2}_{tt}q_{1}(t,t)=0$.

Hence the integral equations (3.14) can be reformulated into the operator form

$\displaystyle\mathcal{A}\psi:=(\mathcal{N}+\mathcal{D})\psi=\eta,$ (3.15)

where $\psi=(\psi_{1},\psi_{2})^{\top}$, $\eta=(\eta_{1},\eta_{2})^{\top}$,
and

$\displaystyle\mathcal{N}$ $\displaystyle=\left[\begin{array}[]{cc}I&0\\\
T_{0}&-I\end{array}\right]+\left[\begin{array}[]{cc}E_{2}S_{2}&S_{0}+E_{1}S_{2}\\\
\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2}&-E_{2}S_{2}\end{array}\right]\overset{\text{def}}{=}\mathcal{N}_{1}+\mathcal{N}_{2},$
$\displaystyle\mathcal{D}$
$\displaystyle=\left[\begin{array}[]{cc}\widetilde{L}&\widetilde{S}-M_{0}\\\
\widetilde{R}-\widetilde{H}-M_{0}-\frac{1}{2}E_{1}M_{0}&\widetilde{K}\end{array}\right]$

with $E_{6}\phi:=E_{4}\phi-E_{5}\phi$. It can be seen that
$\mathcal{N}:H^{p}[0,2\pi]^{2}\rightarrow H^{p-1}[0,2\pi]^{2}$ and
$\mathcal{D}:H^{p}[0,2\pi]^{2}\rightarrow H^{p+4}[0,2\pi]^{2}$ are bounded
operators.

######  Theorem 3.2.

For all $p\geq 0$, the operator $\mathcal{N}^{2}:H^{p}[0,2\pi]^{2}\rightarrow
H^{p+2}[0,2\pi]^{2}$ satisfies

$\mathcal{N}^{2}\psi=(\mathcal{U}+\mathcal{J})\psi\quad\forall\psi\in
H^{p}[0,2\pi]^{2},$

where $\mathcal{J}:H^{p}[0,2\pi]^{2}\rightarrow H^{p+2}[0,2\pi]^{2}$ is a
compact operator and $\mathcal{U}:H^{p}[0,2\pi]^{2}\rightarrow
H^{p+2}[0,2\pi]^{2}$ is an isomorphism given by

$\displaystyle\mathcal{U}=\left[\begin{array}[]{cc}I+S_{0}T_{0}+E_{1}S_{0}S_{0}&0\\\
0&I+T_{0}S_{0}+E_{1}S_{0}S_{0}\end{array}\right].$

###### Proof.

Let $f_{m}(t):=\mathrm{e}^{\mathrm{i}mt},m\in\mathbb{Z}$ be the trigonometric
basis functions. A simple calculation yields

$\displaystyle T_{0}f_{m}$
$\displaystyle=\lambda_{m}f_{m},\quad\lambda_{m}=\begin{cases}-|m|,&m=\pm
1,\pm 2,\cdots,\\\ \mathrm{i},&m=0,\end{cases}$ $\displaystyle S_{0}f_{m}$
$\displaystyle=\xi_{m}f_{m},\quad\xi_{m}=\begin{cases}\frac{1}{|m|},&m=\pm
1,\pm 2,\cdots,\\\ \mathrm{i},&m=0.\end{cases}$

It is clear to note that $S_{0}:H^{p}[0,2\pi]\rightarrow H^{p+1}[0,2\pi]$ and
$T_{0}:H^{p}[0,2\pi]\rightarrow H^{p-1}[0,2\pi]$ are isomorphic for any $p\geq
0$. Since $E_{1}$ is isomorphic from $H^{p}[0,2\pi]$ into $H^{p}[0,2\pi]$ and
$(I+T_{0}S_{0})f_{m}=(I+S_{0}T_{0})f_{m}=0$, we have

$\displaystyle\big{(}E_{1}^{-1}(I+T_{0}S_{0})+S_{0}S_{0}\big{)}f_{m}=\big{(}E_{1}^{-1}(I+S_{0}T_{0})+S_{0}S_{0}\big{)}f_{m}=\xi_{m}^{2}f_{m},$
(3.16)

which implies that $\mathcal{U}$ is isomorphic from $H^{p}[0,2\pi]^{2}$ to
$H^{p+2}[0,2\pi]^{2}$.

A straightforward calculation yields

$\displaystyle\mathcal{N}^{2}$
$\displaystyle=\left[\begin{array}[]{cc}\frac{1}{2}S_{0}E_{1}S_{0}+E_{1}S_{2}T_{0}&0\\\
T_{0}E_{2}S_{2}-E_{2}S_{2}T_{0}&\frac{1}{2}E_{1}S_{0}S_{0}+T_{0}E_{1}S_{2}\end{array}\right]+\mathcal{J}_{1}$
$\displaystyle\quad+\left[\begin{array}[]{cc}I+S_{0}T_{0}&0\\\
0&I+T_{0}S_{0}\end{array}\right]$

where

$\displaystyle\mathcal{J}_{1}$
$\displaystyle=\left[\begin{array}[]{cc}F_{1}&F_{2}\\\
F_{3}&F_{4}\end{array}\right]$

with the entries being defined by

$\displaystyle\begin{split}F_{1}&=2E_{2}S_{2}+E_{2}S_{2}E_{2}S_{2}+S_{0}(E_{3}S_{1}+E_{6}S_{2})+E_{1}S_{2}\big{(}\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2}\big{)},\\\
F_{2}&=E_{2}S_{2}(S_{0}+E_{1}S_{2})-(S_{0}+E_{1}S_{2})E_{2}S_{2},\\\
F_{3}&=\big{(}\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2}\big{)}E_{2}S_{2}-E_{2}S_{2}\big{(}\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2}\big{)},\\\
F_{4}&=2E_{2}S_{2}+E_{2}S_{2}E_{2}S_{2}+(E_{3}S_{1}+E_{6}S_{2})S_{0}+\big{(}\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2}\big{)}E_{1}S_{2}.\end{split}$
(3.17)

Since $F_{j},j=1,2,3,4$ are bounded from $H^{p}[0,2\pi]$ into
$H^{p+3}[0,2\pi]$, $\mathcal{J}_{1}$ is bounded from $H^{p}[0,2\pi]$ into
$H^{p+3}[0,2\pi]$, which shows that $\mathcal{J}_{1}$ is compact from
$H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$.

Using the identities

$\displaystyle(T_{0}E_{1}S_{2}\varphi)(t)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\cot\frac{\zeta-t}{2}(E_{1}S_{2}\varphi)^{\prime}(\zeta)\mathrm{d}\zeta+\frac{\mathrm{i}}{2\pi}\int_{0}^{2\pi}(E_{1}S_{2}\varphi)(\zeta)\mathrm{d}\zeta$
$\displaystyle=-\frac{1}{2\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-t}{2}\Big{)}(E_{1}S_{2}\varphi)^{\prime\prime}(\zeta)\mathrm{d}\zeta+(M_{0}E_{1}S_{2}\varphi)(t)$
$\displaystyle=\big{(}\widetilde{S}_{0}\\{(E_{1}S_{2}\varphi)^{\prime\prime}\\}\big{)}(t)+(M_{0}E_{1}S_{2}\varphi)(t)$

and

$\displaystyle(E_{1}S_{2}\varphi)^{\prime\prime}(\zeta)=$
$\displaystyle-\frac{\kappa^{2}}{8\pi}\int_{0}^{2\pi}\frac{\partial^{2}}{\partial{\zeta^{2}}}\biggl{\\{}|\gamma^{\prime}(\zeta)|^{2}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\sin^{2}(\zeta-s)\biggr{\\}}\varphi(s)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\frac{1}{2}(E_{1}\widetilde{S}_{0}\varphi)(\zeta)+(E_{7}S_{2}\varphi)(\zeta)+4(E_{3}B_{1}\varphi)(\zeta)+(E_{1}B_{2}\varphi)(\zeta),$

where
$E_{7}\phi=2\kappa^{2}(|\gamma^{\prime\prime}|^{2}+\gamma^{\prime}\cdot\gamma^{\prime\prime\prime})\phi$,
and

$\displaystyle(B_{1}\varphi)(\zeta)$
$\displaystyle=-\frac{1}{8\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\sin
2(\zeta-s)\varphi(s)\,\mathrm{d}s-\frac{1}{8\pi}\int_{0}^{2\pi}m_{1}(\zeta,s)\varphi(s)\mathrm{d}s,$
$\displaystyle(B_{2}\varphi)(\zeta)$
$\displaystyle=-\frac{1}{4\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\big{[}\cos
2(\zeta-s)-1\big{]}\varphi(s)\,\mathrm{d}s$
$\displaystyle\quad-\frac{1}{4\pi}\int_{0}^{2\pi}m_{2}(\zeta,s)\varphi(s)\mathrm{d}s$

with

$\displaystyle m_{1}(\zeta,s)$
$\displaystyle=\cot\frac{\zeta-s}{2}\sin^{2}(\zeta-s),$ $\displaystyle
m_{2}(\zeta,s)$ $\displaystyle=\cot\frac{\zeta-s}{2}\sin
2(\zeta-s)-\cos^{2}\frac{\zeta-s}{2}$

being analytic functions, we obtain from
$S_{0}S_{0}=\widetilde{S}_{0}\widetilde{S}_{0}+M_{0}S_{0}$ that

$\displaystyle\frac{1}{2}E_{1}S_{0}S_{0}+T_{0}E_{1}S_{2}$
$\displaystyle=\frac{1}{2}E_{1}S_{0}S_{0}+\Big{(}\frac{1}{2}\widetilde{S}_{0}E_{1}\widetilde{S}_{0}+\widetilde{S}_{0}E_{7}S_{2}+4\widetilde{S}_{0}E_{3}B_{1}+\widetilde{S}_{0}E_{1}B_{2}+M_{0}E_{1}S_{2}\Big{)}$
$\displaystyle=\frac{1}{2}E_{1}S_{0}S_{0}+\frac{1}{2}E_{1}\widetilde{S}_{0}\widetilde{S}_{0}+\frac{1}{2}\big{(}\widetilde{S}_{0}E_{1}-E_{1}\widetilde{S}_{0}\big{)}\widetilde{S}_{0}+\widetilde{S}_{0}E_{7}S_{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad~{}~{}+4\widetilde{S}_{0}E_{3}B_{1}+\widetilde{S}_{0}E_{1}B_{2}+M_{0}E_{1}S_{2}$
$\displaystyle=E_{1}S_{0}S_{0}+J_{1},$

where

$\displaystyle
J_{1}:=\frac{1}{2}\widetilde{B}_{3}\widetilde{S}_{0}+\widetilde{S}_{0}E_{7}S_{2}+4\widetilde{S}_{0}E_{3}B_{1}+\widetilde{S}_{0}E_{1}B_{2}-\frac{1}{2}E_{1}M_{0}S_{0}+M_{0}E_{1}S_{2}$
(3.18)

is compact from $H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$ due to the fact that
$B_{2}$ is bounded from $H^{p}[0,2\pi]$ into $H^{p+3}[0,2\pi]$, $B_{1}$ and

$\displaystyle(\widetilde{B}_{3}\varphi)(t):$
$\displaystyle=(\widetilde{S}_{0}E_{1}\varphi)(t)-(E_{1}\widetilde{S}_{0}\varphi)(t)$
$\displaystyle=-\frac{\kappa^{2}}{2\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\Big{(}|\gamma^{\prime}(\zeta)|^{2}-|\gamma^{\prime}(t)|^{2}\Big{)}\varphi(\zeta)\,\mathrm{d}\zeta$

are bounded from $H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$.

In addition, using the integration by parts yields

$\displaystyle(S_{2}\widetilde{T}_{0}\varphi)(t)$
$\displaystyle=-\frac{1}{8\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\sin^{2}(t-\zeta)(\widetilde{T}_{0}\varphi)(\zeta)\,\mathrm{d}\zeta$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-t}{2}\Big{)}\sin^{2}(t-\zeta)\frac{\partial}{\partial\zeta}\Biggl{\\{}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\varphi^{\prime}(s)\,\mathrm{d}s\Biggr{\\}}\,\mathrm{d}\zeta$
$\displaystyle=-\frac{1}{16\pi^{2}}\int_{0}^{2\pi}\Big{(}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\varphi^{\prime}(s)\,\mathrm{d}s\Big{)}\frac{\partial}{\partial\zeta}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Bigl{\\{}\ln\Big{(}4\sin^{2}\frac{\zeta-t}{2}\Big{)}\sin^{2}(t-\zeta)\Bigr{\\}}\,\mathrm{d}\zeta$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{2\pi}\Big{(}\int_{0}^{2\pi}\cot\frac{s-\zeta}{2}\varphi(s)\,\mathrm{d}s\Big{)}\Bigl{\\{}\cot\frac{\zeta-t}{2}\sin^{2}(\zeta-t)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\ln\Big{(}4\sin^{2}\frac{\zeta-t}{2}\Big{)}\sin
2(\zeta-t)\Bigr{\\}}\,\mathrm{d}\zeta$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{2\pi}\Bigl{\\{}2m_{2}(\zeta,t)+2\ln\Big{(}4\sin^{2}\frac{\zeta-t}{2}\Big{)}\cos
2(\zeta-t)\Bigr{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Big{(}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{\zeta-s}{2}\Big{)}\varphi(s)\,\mathrm{d}s\Big{)}\,\mathrm{d}\zeta$
$\displaystyle=\frac{1}{2}\widetilde{S}_{0}\widetilde{S}_{0}\varphi+B_{2}\widetilde{S}_{0}\varphi.$

Combining $E_{1}S_{2}T_{0}=E_{1}S_{2}\widetilde{T}_{0}+E_{1}S_{2}M_{0}$ and
$S_{0}E_{1}S_{0}=E_{1}S_{0}S_{0}+B_{3}S_{0}$, where

$\displaystyle(B_{3}\varphi)(t):$
$\displaystyle=-\frac{\kappa^{2}}{2\pi}\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\Big{(}|\gamma^{\prime}(\zeta)|^{2}-|\gamma^{\prime}(t)|^{2}\Big{)}\varphi(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\quad+\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{0}^{2\pi}\Big{(}|\gamma^{\prime}(\zeta)|^{2}-|\gamma^{\prime}(t)|^{2}\Big{)}\varphi(\zeta)\,\mathrm{d}\zeta,$

we obtain

$\displaystyle\frac{1}{2}S_{0}E_{1}S_{0}+E_{1}S_{2}T_{0}=$
$\displaystyle\Big{(}\frac{1}{2}E_{1}S_{0}S_{0}+\frac{1}{2}B_{3}S_{0}\Big{)}+\frac{1}{2}E_{1}\widetilde{S}_{0}\widetilde{S}_{0}+E_{1}B_{2}\widetilde{S}_{0}+E_{1}S_{2}M_{0}$
$\displaystyle=$ $\displaystyle E_{1}S_{0}S_{0}+J_{2},$

where
$J_{2}:=\frac{1}{2}B_{3}S_{0}+E_{1}B_{2}\widetilde{S}_{0}-\frac{1}{2}E_{1}M_{0}S_{0}+E_{1}S_{2}M_{0}$
is compact from $H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$, since
$B_{2}:H^{p}[0,2\pi]\rightarrow H^{p+3}[0,2\pi]$ and
$B_{3}:H^{p}[0,2\pi]\rightarrow H^{p+2}[0,2\pi]$ are bounded.

Analogously, we observe that

$\displaystyle T_{0}E_{2}S_{2}-E_{2}S_{2}T_{0}$
$\displaystyle=\Big{(}\frac{1}{2}\widetilde{S}_{0}E_{2}\widetilde{S}_{0}+\widetilde{S}_{0}E_{8}S_{2}+2\widetilde{S}_{0}E_{9}B_{1}+\widetilde{S}_{0}E_{2}B_{2}+M_{0}E_{2}S_{2}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad~{}~{}-\frac{1}{2}E_{2}\widetilde{S}_{0}\widetilde{S}_{0}-E_{2}B_{2}\widetilde{S}_{0}-E_{2}S_{2}M_{0}$
$\displaystyle=\frac{1}{2}(\widetilde{S}_{0}E_{2}-E_{2}\widetilde{S}_{0})\widetilde{S}_{0}+\widetilde{S}_{0}E_{8}S_{2}+2\widetilde{S}_{0}E_{9}B_{1}+\widetilde{S}_{0}E_{2}B_{2}-E_{2}B_{2}\widetilde{S}_{0}:=J_{3}$

is compact from $H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$, where
$E_{8}\phi=\kappa^{2}n\cdot\gamma^{(4)}\phi$ and
$E_{9}\phi=\kappa^{2}n\cdot\gamma^{\prime\prime\prime}\phi$ are bounded from
$H^{p}[0,2\pi]$ into $H^{p}[0,2\pi]$, and
$\widetilde{B}_{4}:=\widetilde{S}_{0}E_{2}-E_{2}\widetilde{S}_{0}$ is bounded
from $H^{p}[0,2\pi]$ into $H^{p+2}[0,2\pi]$.

The proof is completed by defining the operator

$\displaystyle\mathcal{J}=\mathcal{J}_{1}+\mathcal{J}_{2},$ (3.19)

where

$\displaystyle\mathcal{J}_{1}=\left[\begin{array}[]{cc}F_{1}&F_{2}\\\
F_{3}&F_{4}\end{array}\right],\quad\mathcal{J}_{2}=\left[\begin{array}[]{cc}J_{2}&0\\\
J_{3}&J_{1}\end{array}\right].$

∎

The well-posedness of the boundary integral equation (3.15) is stated as
follows.

######  Theorem 3.3.

For any $p\geq 0$, the operator equation (3.15) admits a unique solution in
$H^{p}[0,2\pi]^{2}$.

###### Proof.

It follows from Theorem 3.1 that the operator
$\mathcal{A}=\mathcal{N}+\mathcal{D}$ in (3.15) is injective. Hence
$\mathcal{A}^{2}$ is injective. Consider the operator equation

$\displaystyle\mathcal{A}^{2}\psi=(\mathcal{N}+\mathcal{D})^{2}\psi=\mathcal{A}\eta.$
(3.20)

Noting the boundedness of $\mathcal{N}:H^{p}[0,2\pi]^{2}\rightarrow
H^{p-1}[0,2\pi]^{2}$ and $\mathcal{D}:H^{p}[0,2\pi]^{2}\rightarrow
H^{p+4}[0,2\pi]^{2}$, we have from Theorem 3.2 that

$\displaystyle(\mathcal{N}+\mathcal{D})^{2}=\mathcal{U}+\mathcal{J}+\mathcal{N}\mathcal{D}+\mathcal{D}\mathcal{N}+\mathcal{D}^{2},$
(3.21)

where $\mathcal{U}$ is isomorphic and
$\mathcal{J}+\mathcal{N}\mathcal{D}+\mathcal{D}\mathcal{N}+\mathcal{D}^{2}$ is
compact from $H^{p}[0,2\pi]^{2}$ into $H^{p+2}[0,2\pi]^{2}$ for $p\geq 0$. It
follows from the Fredholm alternative that the operator equation (3.20) has a
unique solution $\psi\in H^{p}[0,2\pi]^{2}$, which shows that $\psi\in
H^{p}[0,2\pi]^{2}$ is also the solution to $\mathcal{A}\psi=\eta$ due to
$\mathcal{A}(\mathcal{A}\psi-\eta)=0$ and the injectivity of $\mathcal{A}$. ∎

##  4\. Collocation method

Combining (3.20) and (3.21) yields

$\mathcal{U}\psi+(\mathcal{J}+\mathcal{V})\psi=\mathcal{A}\eta,$

where
$\mathcal{V}=\mathcal{N}\mathcal{D}+\mathcal{D}\mathcal{N}+\mathcal{D}^{2}$ is
compact from $H^{p}[0,2\pi]^{2}$ into $H^{p+2}[0,2\pi]^{2}$. Equivalently, we
consider the operator equation

$\displaystyle\mathcal{E}^{-1}\mathcal{U}\psi+\mathcal{E}^{-1}(\mathcal{J}+\mathcal{V})\psi=\mathcal{E}^{-1}\mathcal{A}\eta,$
(4.1)

where

$\displaystyle\mathcal{E}=\left[\begin{array}[]{cc}E_{1}&0\\\
0&E_{1}\end{array}\right],\quad\mathcal{E}^{-1}\mathcal{U}=\left[\begin{array}[]{cc}E_{1}^{-1}(I+S_{0}T_{0})+S_{0}S_{0}&0\\\
0&E_{1}^{-1}(I+T_{0}S_{0})+S_{0}S_{0}\end{array}\right].$

It is clear to note that $\mathcal{E}^{-1}\mathcal{U}$ is isomorphic and
$\mathcal{E}^{-1}(\mathcal{J}+\mathcal{V})$ is compact from
$H^{p}[0,2\pi]^{2}$ into $H^{p+2}[0,2\pi]^{2}$ for $p\geq 0$.

In this section, we adopt the collocation method to study the convergence of
the semi- and full-discretization of the boundary integral equation (4.1). A
related work can be found in [3] on the convergence analysis of the
collocation method for solving the elastic obstacle scattering problem.

###  4.1. Semi-discrete scheme

Denote by $X_{n}$ the space of $n$-th order trigonometric polynomials. For any
$\varphi\in X_{n}$, it has the form

$\varphi(t)=\sum_{m=0}^{n}a_{m}\cos mt+\sum_{m=1}^{n-1}b_{m}\sin mt.$

Let $P_{n}:H^{p}[0,2\pi]\rightarrow X_{n}$ be the interpolation operator,
i.e., $(P_{n}g)(\zeta_{j}^{(n)})=g(\zeta_{j}^{(n)})$, where the interpolation
points $\zeta_{j}^{(n)}:=\pi j/n$, $j=0,\cdots,2n-1$. Clearly, $P_{n}$ is
bounded.

Let $X_{n}^{2}=\\{\psi=(\psi_{1},\psi_{2})^{\top}:\psi_{1}\in
X_{n},\psi_{2}\in X_{n}\\}$ and define the interpolation operator
$\mathcal{P}_{n}:H^{p}[0,2\pi]^{2}\rightarrow X_{n}^{2}$ by

$\mathcal{P}_{n}g=(P_{n}g_{1},P_{n}g_{2})^{\top}\quad\forall\,g=(g_{1},g_{2})\in
H^{p}[0,2\pi]^{2}.$

It is clear to note that $X_{n}^{2}$ is unisolvent with respect to the points
$\\{\zeta_{j}^{(n)}\\}_{j=0}^{2n-1}$. By [15, Theorem 11.8], we have

$\displaystyle\|\mathcal{P}_{n}g-g\|_{q}\leq\frac{C}{n^{p-q}}\|g\|_{p}\quad\forall\,g\in
H^{p}[0,2\pi]^{2},$ (4.2)

where $0\leq q\leq p,\frac{1}{2}<p$, and $C$ is a positive constant depending
on $p$ and $q$.

Hence $\psi=(\psi_{1},\psi_{2})^{\top}$ can be approximated by
$\psi^{n}=(\psi^{n}_{1},\psi^{n}_{2})^{\top}\in X_{n}^{2}$, which satisfies
the approximate operator equation

$\displaystyle\mathcal{E}^{-1}\mathcal{U}\psi^{n}+\mathcal{P}_{n}[\mathcal{E}^{-1}(\mathcal{J}+\mathcal{V})]\psi^{n}=\mathcal{P}_{n}(\mathcal{E}^{-1}\mathcal{A})\eta,$
(4.3)

where $\psi^{n}$ satisfies
$\mathcal{P}_{n}(\mathcal{E}^{-1}\mathcal{U})\psi^{n}=\mathcal{E}^{-1}\mathcal{U}\psi^{n}$
by noting (3.16).

The unique solvability of (4.3) and the error estimate of the solution are
given in the following theorem. Based on the fact that
$\mathcal{J},\mathcal{V}:H^{p}[0,2\pi]^{2}\rightarrow H^{p+3}[0,2\pi]^{2}$ are
bounded for any $p\geq 0$, the proof is similar to that of [3, Theorem 4.1]
and is omitted for brevity.

######  Theorem 4.1.

Let $\psi$ be the unique solution to (3.15). For sufficiently large $n$, the
approximate operator equation (4.3) has a unique solution $\psi^{n}$, which
satisfies

$\|\psi^{n}-\psi\|_{p}\leq
C\|\mathcal{P}_{n}(\mathcal{E}^{-1}\mathcal{U})\psi-\mathcal{E}^{-1}\mathcal{U}\psi\|_{p+2},$

where $C$ is a positive constant depending on $\mathcal{E}^{-1}\mathcal{J}$,
$\mathcal{E}^{-1}\mathcal{V}$ and $\mathcal{E}^{-1}\mathcal{U}$.

###  4.2. Full-discrete scheme

Based on the Lagrange basis (cf. [15, eqn. $(11.12)$])

$\mathfrak{L}_{j}(t)=\frac{1}{2n}\Big{\\{}1+2\sum_{k=1}^{n-1}\cos
k(t-\zeta_{j}^{(n)})+\cos n(t-\zeta_{j}^{(n)})\Big{\\}},\quad
j=0,1,\cdots,2n-1,$

an approximate solution $\widetilde{\psi}^{n}\in X_{n}^{2}$ of (4.3) can be
written as

$\widetilde{\psi}^{n}(t)=\big{(}\widetilde{\psi}^{n}_{1}(t),\widetilde{\psi}^{n}_{2}(t)\big{)}^{\top}=\bigg{(}\sum_{j=0}^{2n-1}\widetilde{\psi}^{n}_{1}(\zeta_{j}^{(n)})\mathfrak{L}_{j}(t),\sum_{j=0}^{2n-1}\widetilde{\psi}^{n}_{2}(\zeta_{j}^{(n)})\mathfrak{L}_{j}(t)\bigg{)}^{\top}.$

Moreover, it satisfies

$\displaystyle\mathcal{E}_{n}^{-1}\mathcal{U}_{n}\widetilde{\psi}^{n}+\mathcal{P}_{n}[\mathcal{E}_{n}^{-1}(\mathcal{J}_{n}+\mathcal{V}_{n})]\widetilde{\psi}^{n}=\mathcal{P}_{n}(\mathcal{E}_{n}^{-1}\mathcal{A}_{n})\eta,$
(4.4)

where
$\mathcal{A}_{n}=\mathcal{N}_{n}+\mathcal{D}_{n}=\mathcal{N}_{1,n}+\mathcal{N}_{2,n}+\mathcal{D}_{n}$,
$\mathcal{J}_{n}=\mathcal{J}_{1,n}+\mathcal{J}_{2,n}$,
$\mathcal{V}_{n}=\mathcal{N}_{n}\mathcal{D}_{n}+\mathcal{D}_{n}\mathcal{N}_{n}+\mathcal{D}_{n}\mathcal{D}_{n}$.
The involved quadrature operators are given by

$\displaystyle T_{0,n}=T_{0}P_{n},\quad S_{j,n}=S_{j}P_{n},\quad
E_{i,n}=E_{i}$ (4.5)

for $j=0,1,2$, $i=1,\cdots,9$, and the quadrature operators corresponding the
integral operators of the form

$\displaystyle(B\phi)(t)$
$\displaystyle=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}b_{1}(t,\zeta)\phi(\zeta)\,\mathrm{d}\zeta+\int_{0}^{2\pi}b_{2}(t,\zeta)\phi(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\overset{\text{def}}{=}(B_{1}\phi)(t)+(B_{2}\phi)(t)$

are defined by

$\displaystyle(B_{n}\phi)(t)$
$\displaystyle=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}P_{n}[b_{1}(t,\zeta)\phi(\zeta)]\,\mathrm{d}\zeta+\int_{0}^{2\pi}P_{n}[b_{2}(t,\zeta)\phi(\zeta)]\,\mathrm{d}\zeta$
$\displaystyle\overset{\text{def}}{=}(B_{1,n}\phi)(t)+(B_{2,n}\phi)(t).$

Then we have $\mathcal{N}_{1,n}=\mathcal{N}_{1}\mathcal{P}_{n}$,
$\mathcal{N}_{2,n}=\mathcal{N}_{2}\mathcal{P}_{n}$, and

$\displaystyle\mathcal{E}_{n}$
$\displaystyle=\left[\begin{array}[]{cc}E_{1,n}&0\\\
0&E_{1,n}\end{array}\right],\quad\mathcal{J}_{1,n}=\left[\begin{array}[]{cc}F_{1,n}&F_{2,n}\\\
F_{3,n}&F_{4,n}\end{array}\right],\quad\mathcal{J}_{2,n}=\left[\begin{array}[]{cc}J_{2,n}&0\\\
J_{3,n}&J_{1,n}\end{array}\right],$

where $F_{j,n}$ and $J_{i,n}$ are quadrature operators corresponding to the
integral operators $F_{j}$ and $J_{i}$, respectively. It is clear to note that
$\mathcal{E}_{n}^{-1}\mathcal{U}_{n}\widetilde{\psi}^{n}=\mathcal{E}^{-1}\mathcal{U}\widetilde{\psi}^{n}$
and $\mathcal{N}_{n}\widetilde{\psi}^{n}=\mathcal{N}\widetilde{\psi}^{n}$ for
any $\widetilde{\psi}^{n}\in X_{n}^{2}$.

Our goal is to examine the convergence of the fully discretized equation
(4.4). The function $\mathcal{D}\psi$ can be split into the following two
parts:

$(\mathcal{D}\psi)(t)=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\alpha(t,\zeta)\psi(\zeta)\,\mathrm{d}\zeta+\int_{0}^{2\pi}\beta(t,\zeta)\psi(\zeta)\,\mathrm{d}\zeta,$

where

$\alpha(t,\zeta)=\left[\begin{array}[]{cc}\alpha_{1}(t,\zeta)&\alpha_{2}(t,\zeta)\\\
\alpha_{3}(t,\zeta)&\alpha_{4}(t,\zeta)\end{array}\right],\quad\beta(t,\zeta)=\left[\begin{array}[]{cc}\beta_{1}(t,\zeta)&\beta_{2}(t,\zeta)\\\
\beta_{3}(t,\zeta)&\beta_{4}(t,\zeta)\end{array}\right]$

with
$\alpha_{j}(t,t)=\partial_{t}\alpha_{j}(t,t)=\partial^{2}_{tt}\alpha_{j}(t,t)=0,j=1,2,3,4$
and $\alpha_{j},\beta_{j}$ being analytic.

######  Theorem 4.2.

Let $0\leq q\leq p$ and $p>1/2$. Then for any $\psi\in X_{n}^{2}$ and $\chi\in
H^{p}[0,2\pi]^{2}$, it holds that

$\displaystyle\|\mathcal{D}_{n}\psi-\mathcal{D}\psi\|_{q+3}\leq
C\frac{1}{n^{p+1-q}}\|\psi\|_{p},\quad\|\mathcal{D}_{n}\chi-\mathcal{D}\chi\|_{q+3}\leq\widetilde{C}\frac{1}{n^{p-q}}\|\chi\|_{p},$
(4.6)

where $C$ and $\widetilde{C}$ are positive constants depending on $p$ and $q$.

###### Proof.

We write the derivative $\frac{\rm d^{2}}{{\rm d}t^{2}}(\mathcal{D}\psi)$ in
form of

$\displaystyle(\mathcal{D}^{\prime\prime}\psi)(t):=\frac{\rm d^{2}}{{\rm
d}t^{2}}(\mathcal{D}\psi)(t)=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\widetilde{\alpha}(t,\zeta)\psi(\zeta)\,\mathrm{d}\zeta+\int_{0}^{2\pi}\widetilde{\beta}(t,\zeta)\psi(\zeta)\,\mathrm{d}\zeta,$

where

$\displaystyle\widetilde{\alpha}(t,\zeta)=\partial^{2}_{tt}\alpha(t,\zeta),\quad\widetilde{\beta}(t,\zeta)=2\cot\frac{t-\zeta}{2}\partial_{t}\alpha(t,\zeta)-\alpha(t,\zeta)\frac{1}{2\sin^{2}\frac{t-\zeta}{2}}+\partial^{2}_{tt}\beta(t,\zeta).$

By the interpolatory quadrature, the full discretization of
$\mathcal{D}^{\prime\prime}$ can be written as

$\displaystyle(\mathcal{D}^{\prime\prime}_{n}\psi)(t)=\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\mathcal{P}_{n}\Big{\\{}\widetilde{\alpha}(t,\cdot)\psi\Big{\\}}(\zeta)\,\mathrm{d}\zeta+\int_{0}^{2\pi}\mathcal{P}_{n}\Big{\\{}\widetilde{\beta}(t,\cdot)\psi\Big{\\}}(\zeta)\,\mathrm{d}\zeta.$

Noting $p>1/2,0\leq q\leq p,\widetilde{\alpha}(t,t)=0$, and the analyticity of
the elements in $\widetilde{\alpha}(t,\zeta)$ and
$\widetilde{\beta}(t,\zeta)$, we have from [15, Lemma 13.21 and Theorem 12.18]
that

$\|\mathcal{D}^{\prime\prime}_{n}\psi-\mathcal{D}^{\prime\prime}\psi\|_{q+1}\leq
C_{1}\frac{1}{n^{p+1-q}}\|\psi\|_{p},\quad\|\mathcal{D}^{\prime\prime}_{n}\chi-\mathcal{D}^{\prime\prime}\chi\|_{q+1}\leq\widetilde{C}_{1}\frac{1}{n^{p-q}}\|\chi\|_{p}$

for any $\psi\in X_{n}^{2}$ and $\chi\in H^{p}[0,2\pi]^{2}$, where the
positive constants $C_{1}$ and $\widetilde{C}_{1}$ depend on $p$ and $q$.
Since $\mathcal{D}^{\prime\prime}_{n}\psi=\frac{\rm d^{2}}{{\rm
d}t^{2}}(\mathcal{D}_{n}\psi)$, the above inequalities reduce to

$\displaystyle\|\mathcal{D}_{n}\psi-\mathcal{D}\psi\|_{q+3}\leq
C_{2}\frac{1}{n^{p+1-q}}\|\psi\|_{p},\quad\|\mathcal{D}_{n}\chi-\mathcal{D}\chi\|_{q+3}\leq\widetilde{C}_{2}\frac{1}{n^{p-q}}\|\chi\|_{p},$

where $C_{2}$ and $\widetilde{C}_{2}$ are positive constants depending on $p$
and $q$. By [15, Theorem 8.13], the proof is completed since the above
inequalities hold for any $q$ satisfying $0\leq q\leq p$ and $p>1/2$. ∎

######  Remark 4.3.

For any $\varphi\in X_{n}$ and $\tilde{\chi}\in H^{p}[0,2\pi]$, we may follow
the proofs of [3, Theorem 4.2] and [15, Lemma 13.21] to show that
$S_{2},B_{2}$ satisfy the following type of estimates:

$\|S_{2,n}\varphi-S_{2}\varphi\|_{q+2}\leq
C_{1}\frac{1}{n^{p+1-q}}\|\varphi\|_{p},\quad\|S_{2,n}\tilde{\chi}-S_{2}\tilde{\chi}\|_{q+2}\leq\widetilde{C}_{1}\frac{1}{n^{p-q}}\|\tilde{\chi}\|_{p},$

and $S_{1},B_{1},B_{3},\widetilde{B}_{3},\widetilde{B}_{4}$ satisfy the
following type of estimates:

$\|S_{1,n}\varphi-S_{1}\varphi\|_{q+1}\leq
C_{2}\frac{1}{n^{p+1-q}}\|\varphi\|_{p},\quad\|S_{1,n}\tilde{\chi}-S_{1}\tilde{\chi}\|_{q+1}\leq\widetilde{C}_{2}\frac{1}{n^{p-q}}\|\tilde{\chi}\|_{p},$

where $0\leq q\leq p$, $p>\frac{1}{2}$, and the positive constants
$C_{j},\widetilde{C}_{j},j=1,2$ depend on $p$ and $q$.

Hereafter, the notation $a\preceq b$ stands for $a\leq Cb$, where $C>0$ is a
constant depending on $p$ and may change step by step in the proofs.

######  Theorem 4.4.

Let $p>\frac{3}{2}$. Then the following estimate holds:

$\|\mathcal{P}_{n}[\mathcal{E}_{n}^{-1}\mathcal{V}_{n}-\mathcal{E}^{-1}\mathcal{V}]\psi\|_{p+2}\preceq\frac{1}{n}\|\psi\|_{p}\quad\forall\,\psi\in
X_{n}^{2}.$

###### Proof.

The proof is motivated by [3, Theorem 4.3]. Recalling the boundedness of
$\mathcal{N}:H^{p}[0,2\pi]^{2}\rightarrow H^{p-1}[0,2\pi]^{2}$ and using
(4.2), we have for any $\chi\in H^{p}[0,2\pi]^{2},0\leq q\leq p,p>1/2$ that

$\displaystyle\|(\mathcal{N}_{n}-\mathcal{N})\chi\|_{q-1}=\|\mathcal{N}(\mathcal{P}_{n}\chi-\chi)\|_{q-1}\leq
C_{1}\|(\mathcal{P}_{n}\chi-\chi)\|_{q}\leq\frac{C_{2}}{n^{p-q}}\|\chi\|_{p},$

where $C_{1}$ and $C_{2}$ are positive constants depending on $q$ and $p,q$,
respectively. For any $p>1/2$, it is clear to note that $\mathcal{N}_{n}$ and
$\mathcal{N}_{n}-\mathcal{N}$ are uniformly bounded from $H^{p}[0,2\pi]^{2}$
to $H^{p-1}[0,2\pi]^{2}$.

For any $\psi\in X_{n}^{2}$, it follows from Theorem 4.2 and
$\mathcal{N}_{n}\psi=\mathcal{N}\psi$ that

$\displaystyle\|\mathcal{N}_{n}\mathcal{D}_{n}\psi-\mathcal{N}\mathcal{D}\psi\|_{p+2}$
$\displaystyle\leq\|\mathcal{N}_{n}(\mathcal{D}_{n}-\mathcal{D})\psi\|_{p+2}+\|(\mathcal{N}_{n}-\mathcal{N})(\mathcal{D}\psi-\mathcal{P}_{n}\mathcal{D}\psi)\|_{p+2}+\|(\mathcal{N}_{n}-\mathcal{N})\mathcal{P}_{n}\mathcal{D}\psi\|_{p+2}$
$\displaystyle\preceq\|(\mathcal{D}_{n}-\mathcal{D})\psi\|_{p+3}+\|\mathcal{D}\psi-\mathcal{P}_{n}\mathcal{D}\psi\|_{p+3}$
$\displaystyle\preceq 1/n\|\psi\|_{p}+1/n\|\mathcal{D}\psi\|_{p+4}\preceq
1/n\|\psi\|_{p}.$

Moreover, by Theorem 4.2, we have for any $p>1/2$ that $\mathcal{D}_{n}$ and
$\mathcal{D}_{n}-\mathcal{D}$ are uniformly bounded from $H^{p}[0,2\pi]^{2}$
to $H^{p+3}[0,2\pi]^{2}$. Combining (4.2) and (4.6), and noting the
boundedness of $\mathcal{D}:H^{p}[0,2\pi]^{2}\rightarrow H^{p+4}[0,2\pi]^{2}$
and the uniform boundedness of $\mathcal{P}_{n}:H^{p+2}[0,2\pi]^{2}\rightarrow
H^{p+2}[0,2\pi]^{2}$, we obtain

$\displaystyle\|\mathcal{D}^{2}_{n}\psi-\mathcal{D}^{2}\psi\|_{p+2}\leq\|\mathcal{D}^{2}_{n}\psi-\mathcal{D}^{2}\psi\|_{p+6}$
$\displaystyle\leq\|\mathcal{D}_{n}(\mathcal{D}_{n}-\mathcal{D})\psi\|_{p+6}+\|(\mathcal{D}_{n}-\mathcal{D})(\mathcal{D}\psi-\mathcal{P}_{n}\mathcal{D}\psi)\|_{p+6}+\|(\mathcal{D}_{n}-\mathcal{D})\mathcal{P}_{n}\mathcal{D}\psi\|_{p+6}$
$\displaystyle\preceq\|(\mathcal{D}_{n}-\mathcal{D})\psi\|_{p+3}+\|(\mathcal{D}\psi-\mathcal{P}_{n}\mathcal{D}\psi)\|_{p+3}+1/n\|\mathcal{P}_{n}\mathcal{D}\psi\|_{p+3}$
$\displaystyle\preceq
1/n\|\psi\|_{p}+1/n\|\mathcal{D}\psi\|_{p+4}+1/n\|\mathcal{D}\psi\|_{p+3}\preceq
1/n\|\psi\|_{p}.$

With the help of $\mathcal{N}_{1}\psi\in X_{n}^{2}$, the boundedness of
$\mathcal{N}_{2}:H^{p}[0,2\pi]^{2}\rightarrow H^{p+1}[0,2\pi]^{2}$ and the
uniform boundedness of $\mathcal{D}_{n}-\mathcal{D}:H^{p-1}[0,2\pi]^{2}$ to
$H^{p+2}[0,2\pi]^{2}$ and $\mathcal{P}_{n}:H^{p-1}[0,2\pi]^{2}\rightarrow
H^{p-1}[0,2\pi]^{2}$ for $p>\frac{3}{2}$, we deduce

$\displaystyle\|\mathcal{D}_{n}\mathcal{N}_{n}\psi-\mathcal{D}\mathcal{N}\psi\|_{p+2}=\|(\mathcal{D}_{n}-\mathcal{D})\mathcal{N}\psi\|_{p+2}$
$\displaystyle\leq\|(\mathcal{D}_{n}-\mathcal{D})(\mathcal{N}\psi-\mathcal{P}_{n}\mathcal{N}\psi)\|_{p+2}+\|(\mathcal{D}_{n}-\mathcal{D})\mathcal{P}_{n}\mathcal{N}\psi\|_{p+2}$
$\displaystyle\preceq\|\mathcal{N}\psi-\mathcal{P}_{n}\mathcal{N}\psi\|_{p-1}+1/n\|\mathcal{P}_{n}\mathcal{N}\psi\|_{p-1}$
$\displaystyle\leq\|\mathcal{N}_{1}\psi-\mathcal{P}_{n}\mathcal{N}_{1}\psi\|_{p-1}+\|\mathcal{N}_{2}\psi-\mathcal{P}_{n}\mathcal{N}_{2}\psi\|_{p-1}+1/n\|\mathcal{P}_{n}\mathcal{N}\psi\|_{p-1}$
$\displaystyle\preceq
1/n^{2}\|\mathcal{N}_{2}\psi\|_{p+1}+1/n\|\mathcal{N}\psi\|_{p-1}\preceq
1/n\|\psi\|_{p}.$

Combining the above estimates yields

$\displaystyle\|\mathcal{V}_{n}\psi-\mathcal{V}\psi\|_{p+2}$
$\displaystyle\leq\|\mathcal{N}_{n}\mathcal{D}_{n}\psi-\mathcal{N}\mathcal{D}\psi\|_{p+2}+\|\mathcal{D}_{n}\mathcal{N}_{n}\psi-\mathcal{D}\mathcal{N}\psi\|_{p+2}$
$\displaystyle\quad+\|\mathcal{D}_{n}^{2}\psi-\mathcal{D}^{2}\psi\|_{p+2}$
$\displaystyle\preceq 1/n\|\psi\|_{p},$

which completes the proof by noting that the operators
$\mathcal{E}^{-1},\mathcal{P}_{n}:H^{p+2}[0,2\pi]^{2}\rightarrow
H^{p+2}[0,2\pi]^{2}$ are uniformly bounded. ∎

######  Theorem 4.5.

Let $p>1/2$. Then the following estimate holds:

$\|\mathcal{P}_{n}[\mathcal{E}_{n}^{-1}\mathcal{J}_{n}-\mathcal{E}^{-1}\mathcal{J}]\psi\|_{p+2}\preceq\frac{1}{n}\|\psi\|_{p}\quad\forall\,\psi\in
X_{n}^{2}.$

###### Proof.

We claim for any $\varphi\in X_{n}$ that

$\displaystyle\|F_{j,n}\varphi-
F_{j}\varphi\|_{p+2}\preceq\frac{1}{n}\|\varphi\|_{p},\quad\|J_{i,n}\varphi-
J_{i}\varphi\|_{p+2}\preceq\frac{1}{n}\|\varphi\|_{p}.$ (4.7)

for $j=1,2,3,4$, $i=1,2,3$. In the following, we only show the proof for
$j=1$, $i=1$ since the other cases can be proved similarly.

First, we have from (3.17) that

$\displaystyle\|F_{1,n}\varphi-F_{1}\varphi\|_{p+2}\leq
I_{1}+I_{2}+I_{3}+I_{4}\quad\forall\,\varphi\in X_{n},$ (4.8)

where

$\displaystyle I_{1}$
$\displaystyle=\|2E_{2,n}S_{2,n}\varphi-2E_{2}S_{2}\varphi\|_{p+2},$
$\displaystyle I_{2}$ $\displaystyle=\|E_{2,n}S_{2,n}E_{2,n}S_{2,n}\varphi-
E_{2}S_{2}E_{2}S_{2}\varphi\|_{p+2},$ $\displaystyle I_{3}$
$\displaystyle=\|S_{0,n}(E_{3,n}S_{1,n}+E_{6,n}S_{2,n})\varphi-
S_{0}(E_{3}S_{1}+E_{6}S_{2})\varphi\|_{p+2},$ $\displaystyle I_{4}$
$\displaystyle=\|E_{1,n}S_{2,n}(\frac{1}{2}E_{1,n}S_{0,n}+E_{3,n}S_{1,n}+E_{6,n}S_{2,n})\varphi$
$\displaystyle\quad-
E_{1}S_{2}(\frac{1}{2}E_{1}S_{0}+E_{3}S_{1}+E_{6}S_{2})\varphi\|_{p+2}.$

It is clear to note from (4.5) that $I_{1}=0$. Using (4.2) and Remark 4.3, and
following the same techniques as those in [3, Theorem 4.4], we have

$\displaystyle I_{2}$ $\displaystyle\preceq\|S_{2,n}E_{2,n}S_{2,n}\varphi-
S_{2}E_{2}S_{2}\varphi\|_{p+2}=\|(S_{2,n}-S_{2})(E_{2}S_{2}\varphi-
P_{n}E_{2}S_{2}\varphi)\|_{p+2}$ $\displaystyle\preceq\|E_{2}S_{2}\varphi-
P_{n}E_{2}S_{2}\varphi\|_{p}\preceq\frac{1}{n^{3}}\|E_{2}S_{2}\varphi\|_{p+3}\preceq\frac{1}{n}\|\varphi\|_{p}.$

By the uniform boundedness of $S_{0,n}-S_{0}:H^{p}[0,2\pi]^{2}\rightarrow
H^{p+1}[0,2\pi]^{2}$, we get

$\displaystyle I_{3}$ $\displaystyle\preceq\|S_{0,n}E_{3,n}S_{1,n}\varphi-
S_{0}E_{3}S_{1}\varphi\|_{p+2}+\|S_{0,n}E_{6,n}S_{2,n}\varphi-
S_{0}E_{6}S_{2}\varphi\|_{p+2}$
$\displaystyle=\|(S_{0,n}-S_{0})(E_{3}S_{1}\varphi-
P_{n}E_{3}S_{1}\varphi)\|_{p+2}+\|(S_{0,n}-S_{0})(E_{6}S_{2}\varphi-
P_{n}E_{6}S_{2}\varphi)\|_{p+2}$ $\displaystyle\preceq\|E_{3}S_{1}\varphi-
P_{n}E_{3}S_{1}\varphi\|_{p+1}+\|E_{6}S_{2}\varphi-
P_{n}E_{6}S_{2}\varphi\|_{p+1}$
$\displaystyle\preceq\frac{1}{n}\|E_{3}S_{1}\varphi\|_{p+2}+\frac{1}{n^{2}}\|E_{6}S_{2}\varphi\|_{p+3}\preceq\frac{1}{n}\|\varphi\|_{p}.$

Analogously, we may show $I_{4}\preceq\frac{1}{n}\|\varphi\|_{p}$. Combining
$I_{j}\preceq\frac{1}{n}\|\varphi\|_{p},~{}j=1,2,3,4$ and (4.8) leads to the
first inequality in (4.7).

Next, it follows from (3.18) that

$\displaystyle\|J_{1,n}\varphi-J_{1}\varphi\|_{p+2}\leq
Q_{1}+Q_{2}+Q_{3}+Q_{4}+Q_{5}+Q_{6}\quad\forall\,\varphi\in X_{n},$

where

$\displaystyle Q_{1}$
$\displaystyle=\|\frac{1}{2}\widetilde{B}_{3,n}\widetilde{S}_{0,n}\varphi-\frac{1}{2}\widetilde{B}_{3}\widetilde{S}_{0}\varphi\|_{p+2},\qquad~{}~{}~{}~{}~{}~{}Q_{2}=\|\widetilde{S}_{0,n}E_{7,n}S_{2,n}\varphi-\widetilde{S}_{0}E_{7}S_{2}\varphi\|_{p+2},$
$\displaystyle Q_{3}$
$\displaystyle=\|4\widetilde{S}_{0,n}E_{3,n}B_{1,n}\varphi-4\widetilde{S}_{0}E_{3}B_{1}\varphi\|_{p+2},\quad~{}Q_{4}=\|\widetilde{S}_{0,n}E_{1,n}B_{2,n}\varphi-\widetilde{S}_{0}E_{1}B_{2}\varphi\|_{p+2},$
$\displaystyle Q_{5}$
$\displaystyle=\|\frac{1}{2}E_{1,n}M_{0,n}S_{0,n}\varphi-\frac{1}{2}E_{1}M_{0}S_{0}\varphi\|_{p+2},~{}Q_{6}=\|M_{0,n}E_{1,n}S_{2,n}\varphi-
M_{0}E_{1}S_{2}\varphi\|_{p+2}.$

Again, we have from Remark 4.3 that

$\displaystyle
Q_{1}\preceq\|(\widetilde{B}_{3,n}-\widetilde{B}_{3})\widetilde{S}_{0}\varphi\|_{p+2}\preceq\frac{1}{n}\|\widetilde{S}_{0}\varphi\|_{p+1}\preceq\frac{1}{n}\|\varphi\|_{p}.$

The inequality $Q_{2}\preceq\frac{1}{n}\|\varphi\|_{p}$ can be obtained by the
similar estimate of the second item of $I_{3}$. Noting (4.2) and the uniform
boundedness of
$\widetilde{S}_{0},~{}\widetilde{S}_{0,n}-\widetilde{S}_{0}:H^{p}[0,2\pi]\rightarrow
H^{p+1}[0,2\pi]$, we deduce

$\displaystyle Q_{3}$
$\displaystyle\preceq\|\widetilde{S}_{0,n}(E_{3,n}B_{1,n}-E_{3}B_{1})\varphi\|_{p+2}+\|(\widetilde{S}_{0,n}-\widetilde{S}_{0})(E_{3}B_{1}\varphi-
P_{n}E_{3}B_{1}\varphi)\|_{p+2}$
$\displaystyle\preceq\|(E_{3,n}B_{1,n}-E_{3}B_{1})\varphi\|_{p+1}+\|E_{3}B_{1}\varphi-
P_{n}E_{3}B_{1}\varphi\|_{p+1}$
$\displaystyle\preceq\|(B_{1,n}-B_{1})\varphi\|_{p+1}+\frac{1}{n}\|E_{3}B_{1}\varphi\|_{p+2}$
$\displaystyle\preceq\frac{1}{n}\|\varphi\|_{p}+\frac{1}{n}\|\varphi\|_{p}\preceq\frac{1}{n}\|\varphi\|_{p}.$

Analogously, $Q_{4}$ can be estimate as

$\displaystyle Q_{4}$
$\displaystyle\preceq\|(B_{2,n}-B_{2})\varphi\|_{p+1}+\frac{1}{n}\|E_{3}B_{2}\varphi\|_{p+2}$
$\displaystyle\leq\|(B_{2,n}-B_{2})\varphi\|_{p+2}+\frac{1}{n}\|E_{3}B_{2}\varphi\|_{p+3}\preceq\frac{1}{n}\|\varphi\|_{p}.$

Clearly, $Q_{5}=0$ by noting $S_{0,n}\varphi=S_{0}\varphi\in X_{n}$. It
follows from [10, Theorem A.45] that $M_{0}$ is a bounded operator from
$H^{p}[0,2\pi]$ into $H^{r}[0,2\pi]$ for every $-r\leq p\leq r$. Hence, we
have

$\displaystyle Q_{6}\preceq\|P_{n}E_{1}S_{2}\varphi-
E_{1}S_{2}\varphi\|_{p+2}\preceq\frac{1}{n}\|E_{1}S_{2}\varphi\|_{p+3}\preceq\frac{1}{n}\|\varphi\|_{p},$

which implies the second inequality of (4.7).

Combining (4.5) and (3.19), we complete the proof by noting that the operators
$\mathcal{E}^{-1},\mathcal{P}_{n}:H^{p+2}[0,2\pi]^{2}\rightarrow
H^{p+2}[0,2\pi]^{2}$ are uniformly bounded. ∎

The following result concerns the convergence of the full-discrete scheme.
Based on the uniform boundedness of the operators
$F_{j,n}-F_{j},J_{i,n}-J_{i}:H^{p}[0,2\pi]\rightarrow H^{p+2}[0,2\pi]$ from
Remark 4.3 and (4.6), the proof is similar to that of [3, Theorem 4.5] and is
omitted here for brevity.

######  Theorem 4.6.

Let $\psi$ be the unique solution to (3.15). For $p>3/2$ and sufficiently
large $n$, the fully discrete equation (4.4) admits a unique solution
$\widetilde{\psi}^{n}$, which satisfies

$\displaystyle\|\widetilde{\psi}^{n}-\psi\|_{p}$
$\displaystyle\preceq\|\mathcal{P}_{n}(\mathcal{E}^{-1}\mathcal{U})\psi-\mathcal{E}^{-1}\mathcal{U}\psi\|_{p+2}+\|\mathcal{P}_{n}[\mathcal{E}_{n}^{-1}(\mathcal{J}_{n}+\mathcal{K}_{n})-\mathcal{E}^{-1}(\mathcal{J}+\mathcal{K})]\psi\|_{p+2}$
$\displaystyle\quad+\|\mathcal{P}_{n}[\mathcal{E}_{n}^{-1}\mathcal{A}_{n}-\mathcal{E}^{-1}\mathcal{A}]\eta\|_{p+2}.$

##  5\. Numerical experiments

In this section, we present numerical implementation and show some examples to
demonstrate the superior performance of the proposed method. In particular, we
introduce a simple alternative boundary integral formulation by using a
combination of the single- and single-layer potentials.

###  5.1. Double-single layer potential formulation

As discussed in section 3, the boundary integral formulation is based on a
combination of the double- and single-layer potentials. Here we introduce two
different approaches to solve the corresponding full-discrete boundary
integral equations.

####  5.1.1. Approach 1

The first approach is to solve directly the equivalent full-discrete equation
of (3.14), i.e.,

$\displaystyle\Bigg{(}\left[\begin{array}[]{cc}I&0\\\
\widetilde{T}_{0}&-I\end{array}\right]+\left[\begin{array}[]{cc}L&S\\\
R-H&K\end{array}\right]\Bigg{)}\left[\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right]=\left[\begin{array}[]{c}\eta_{1}\\\
\eta_{2}\end{array}\right],$

where $\widetilde{T}_{0}=T_{0}-M_{0}$. By the decomposition (3.5), we apply
the simple trapezoidal for the smooth integrals:

$\displaystyle\int_{0}^{2\pi}f(\zeta)\mathrm{d}\zeta\approx\frac{\pi}{n}\sum_{j=0}^{2n-1}f(\zeta_{j}^{(n)}),$

while we employ the quadrature rules via the trigonometric interpolation for
the weakly singular integrals (cf. [2, 13]]:

$\displaystyle\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}f(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\approx\sum_{j=0}^{2n-1}R_{j}^{(n)}(t)f(\zeta_{j}^{(n)}),$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\cot\frac{\zeta-t}{2}f^{\prime}(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\approx\sum_{j=0}^{2n-1}T_{j}^{(n)}(t)f(\zeta_{j}^{(n)}),$

where the quadrature weights are given by

$\displaystyle
R_{j}^{(n)}(t)=-\frac{2\pi}{n}\sum_{m=1}^{n-1}\frac{1}{m}\cos\Big{[}m(t-\zeta_{j}^{(n)})\Big{]}-\frac{\pi}{n^{2}}\cos\Big{[}n(t-\zeta_{j}^{(n)})\Big{]},$
$\displaystyle
T_{j}^{(n)}(t)=-\frac{1}{n}\sum_{m=1}^{n-1}m\cos\Big{[}m(t-\zeta_{j}^{(n)})\Big{]}-\frac{1}{2}\cos\Big{[}n(t-\zeta_{j}^{(n)})\Big{]}.$

####  5.1.2. Approach 2

The second approach is to solve the equivalent full-discrete equation of
(3.15), i.e.,

$\displaystyle\mathcal{P}_{n}\mathcal{N}_{n}\widetilde{\psi}^{n}+\mathcal{P}_{n}\mathcal{D}_{n}\widetilde{\psi}^{n}=\mathcal{P}_{n}\eta~{}\Leftrightarrow~{}A_{n}\widetilde{\psi}^{n}=\eta^{n},$
(5.1)

where $A_{n}$ is the coefficient matrix of the full-discrete equation. As
mentioned in [3, section 5], it is more convenient to handle the equivalent
full-discrete equation (5.1) due to the simple quadrature operators
$\mathcal{N}_{n}$ and $\mathcal{D}_{n}$.

To handle the singular integrals $S_{1}$ and $S_{2}$, we use the trigonometric
interpolation in the quadrature rules

$\displaystyle\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\sin(t-\zeta)f(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\approx\sum_{j=0}^{2n-1}V_{j}^{(n)}(t)f(\zeta_{j}^{(n)}),$
$\displaystyle\int_{0}^{2\pi}\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}\sin^{2}(t-\zeta)f(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\approx\sum_{j=0}^{2n-1}W_{j}^{(n)}(t)f(\zeta_{j}^{(n)}),$

where the integral weights are defined by

$\displaystyle V_{j}^{(n)}(t)$
$\displaystyle=-\frac{\pi}{2n}\sin(\zeta_{j}^{(n)}-t)+\frac{2\pi}{n}\sum_{m=2}^{n-1}\frac{\sin[m(\zeta_{j}^{(n)}-t)]}{m^{2}-1}+\frac{\pi\sin[n(\zeta_{j}^{(n)}-t)]}{n(n^{2}-1)},$
$\displaystyle W_{j}^{(n)}(t)$
$\displaystyle=\frac{\pi}{n}\Big{\\{}\frac{1}{4}+\frac{2}{3}\cos(t-\zeta_{j}^{(n)})+\frac{1}{8}\cos[2(t-\zeta_{j}^{(n)})]\Big{\\}}+\frac{1}{2}R_{j}^{(n)}(t)$
$\displaystyle\quad+\frac{\pi}{n}\sum_{m=3}^{n-1}\frac{m}{m^{2}-4}\cos[m(t-\zeta_{j}^{(n)})]+\frac{\pi}{2(n^{2}-4)}\cos[n(t-\zeta_{j}^{(n)})].$

######  Remark 5.1.

By a straightforward calculation, we get

$\displaystyle W_{j}^{(n)}(t)-R_{j}^{(n)}(t)\sin^{2}(t-\zeta_{j}^{(n)})$
$\displaystyle=\frac{\pi}{2n^{2}}\sin[n(t-\zeta_{j}^{(n)})]\sin[2(t-\zeta_{j}^{(n)})]+\frac{\pi}{n^{2}-n}\sin[n(t-\zeta_{j}^{(n)})]\sin(t-\zeta_{j}^{(n)})$
$\displaystyle\quad-\frac{\pi}{n(n-1)(n+1)}\cos[(n-1)(t-\zeta_{j}^{(n)})]-\frac{\pi}{n(n-2)(n+2)}\cos[n(t-\zeta_{j}^{(n)})].$

It is clear to note that
$W_{j}^{(n)}(\zeta_{i}^{(n)})-R_{j}^{(n)}(\zeta_{i}^{(n)})\sin^{2}(\zeta_{i}^{(n)}-\zeta_{j}^{(n)})\not=0$,
which is different from [3, Remark 5.1]. However, we have
$W_{j}^{(n)}(\zeta_{i}^{(n)})-R_{j}^{(n)}(\zeta_{i}^{(n)})\sin^{2}(\zeta_{i}^{(n)}-\zeta_{j}^{(n)})\to
0$ as $n\to\infty$.

Table 1. The parameterized boundary curves. Boundary type | Parameterization
---|---
Apple-shaped | $\gamma(t)=\displaystyle\frac{0.55(1+0.9\cos{t}+0.1\sin{2t})}{1+0.75\cos{t}}(\cos{t},\sin{t}),\quad t\in[0,2\pi]$
Peanut-shaped | $\gamma(t)=0.275\sqrt{3\cos^{2}{t}+1}(\cos{t},\sin{t}),\quad t\in[0,2\pi]$
Peach-shaped | $\gamma(t)=0.22(\cos^{2}{t}\sqrt{1-\sin{t}}+2)(\cos{t},\sin{t}),\quad t\in[0,2\pi]$
Drop-shaped | $\displaystyle\gamma(t)=(2\sin{\frac{t}{2}}-1,-\sin{t}),\quad t\in[0,2\pi]$
Heart-shaped | $\displaystyle\gamma(t)=(\frac{3}{2}\sin{\frac{3t}{2}},\sin{t}),\quad t\in[0,2\pi]$

###  5.2. Single-single layer potential formulation

As a comparison, we introduce a computationally more convenient boundary
integral formulation by using the single- and single-layer potentials. Using
the fundamental solutions to the Helmholtz and modified Helmholtz equations,
we represent the solutions of (2.8) in terms of two single-layer potentials

$\displaystyle v_{\rm H}(x)=\int_{\Gamma}G_{\rm
H}(x,y)g_{1}(y)\mathrm{d}s(y),\quad v_{\rm M}(x)=\int_{\Gamma}G_{\rm
M}(x,y)g_{2}(y)\mathrm{d}s(y),\quad x\in\mathbb{R}^{2}\setminus\Gamma_{D},$
(5.2)

where $g_{1}$ and $g_{2}$ are densities.

Using the boundary condition (2.5) and the jump relation of the single-layer
potentials (5.2), we deduce the coupled boundary integral equations

$\displaystyle\begin{split}-u^{\rm inc}(x)&=\int_{\Gamma}G_{\rm
H}(x,y)g_{1}(y)\mathrm{d}s(y)+\int_{\Gamma}G_{\rm
M}(x,y)g_{2}(y)\mathrm{d}s(y),\\\ -\frac{\partial u^{\rm
inc}(x)}{\partial\nu(x)}&=-\frac{1}{2}g_{1}(x)+\int_{\Gamma}\frac{\partial
G_{\rm H}(x,y)}{\partial\nu(x)}g_{1}(y)\mathrm{d}s(y)\\\
&\quad-\frac{1}{2}g_{2}(x)+\int_{\Gamma}\frac{\partial G_{\rm
M}(x,y)}{\partial\nu(x)}g_{2}(y)\mathrm{d}s(y).\end{split}$ (5.3)

Similarly, we may obtain the following parametric form of (5.3):

$\displaystyle\mathcal{A}\psi\overset{\text{def}}{=}\left[\begin{array}[]{cc}S^{\rm
H}&S^{\rm M}\\\ -I+K^{\rm H}&-I+K^{\rm
M}\end{array}\right]\left[\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right]=\left[\begin{array}[]{c}\eta_{1}\\\
\eta_{2}\end{array}\right],$

where $\eta_{1}=2(f_{1}\circ\gamma)$,
$\eta_{2}=2(f_{2}\circ\gamma)|\gamma^{\prime}|$,
$\psi_{j}=(g_{j}\circ\gamma)|\gamma^{\prime}|$, $j=1,2$, and

$\displaystyle(S^{\sigma}\phi)(t)$
$\displaystyle=\int_{0}^{2\pi}s^{\sigma}(t,\zeta)\phi(\zeta)\mathrm{d}\zeta,\quad(K^{\sigma}\phi)(t)=\int_{0}^{2\pi}k^{\sigma}(t,\zeta)\phi(\zeta)\mathrm{d}\zeta$

with the kernels being given by

$\displaystyle s^{\sigma}(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}}{2}H_{0}^{(1)}(\kappa_{\sigma}|\gamma(t)-\gamma(\zeta)|),$
$\displaystyle k^{\sigma}(t,\zeta)$
$\displaystyle=\frac{\mathrm{i}\kappa_{\sigma}}{2}n(t)\cdot[\gamma(\zeta)-\gamma(t)]\frac{H_{1}^{(1)}(\kappa_{\sigma}|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|}.$

Here $\sigma={\rm H}$ or ${\rm M}$, $\kappa_{\rm H}=\kappa$, and $\kappa_{\rm
M}={\rm i}\kappa$. The kernels $s^{\sigma}(t,\zeta)$ and $k^{\sigma}(t,\zeta)$
can be decomposed into

$\displaystyle s^{\sigma}(t,\zeta)$
$\displaystyle=s_{1}^{\sigma}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}+s_{2}^{\sigma}(t,\zeta),$
$\displaystyle k^{\sigma}(t,\zeta)$
$\displaystyle=k_{1}^{\sigma}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}+k_{2}^{\sigma}(t,\zeta),$

where the functions

$\displaystyle s_{1}^{\sigma}(t,\zeta)$
$\displaystyle=-\frac{1}{2\pi}J_{0}(\kappa_{\sigma}|\gamma(t)-\gamma(\zeta)|),$
$\displaystyle s_{2}^{\sigma}(t,\zeta)$
$\displaystyle=m^{\sigma}(t,\zeta)-m_{1}^{\sigma}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)},$
$\displaystyle k_{1}^{\sigma}(t,\zeta)$
$\displaystyle=\frac{\kappa_{\sigma}}{2\pi}n(t)\cdot\big{[}\gamma(t)-\gamma(\zeta)\big{]}\frac{J_{1}(\kappa_{\sigma}|\gamma(t)-\gamma(\zeta)|)}{|\gamma(t)-\gamma(\zeta)|},$
$\displaystyle k_{2}^{\sigma}(t,\zeta)$
$\displaystyle=k^{\sigma}(t,\zeta)-k_{1}^{\sigma}(t,\zeta)\ln\Big{(}4\sin^{2}\frac{t-\zeta}{2}\Big{)}$

are analytic and their diagonal entries are given by

$\displaystyle s_{1}^{\sigma}(t,t)=-\frac{1}{2\pi},\quad
s_{2}^{\sigma}(t,t)=\frac{\mathrm{i}}{2}-\frac{C}{\pi}-\frac{1}{\pi}\ln\big{(}\frac{\kappa_{\sigma}}{2}|\gamma^{\prime}(t)|\big{)},$

and

$\displaystyle k_{1}^{\sigma}(t,t)=0,\quad
k_{2}^{\sigma}(t,t)=\frac{1}{2\pi}\frac{n(t)\cdot\gamma^{\prime\prime}(t)}{|\gamma^{\prime}(t)|^{2}}.$

Compared with (3.2), the coupled system (5.3) is simpler computationally since
it does not involve any hypersingular integral kernel. Theoretically, it is
unclear whether the boundary integral equations (5.3) and the corresponding
parametric form have a unique solution. The analysis will be left for a future
work.

###  5.3. Numerical examples

In this section, we present some numerical examples of both smooth and
nonsmooth cavities to demonstrate the performance of the proposed methods. The
code is written in Matlab using double precision arithmetic and the
computations are run on a personal computer with a 64 GB RAM, 3.70 GHz Intel
core i9 processor.

Table 2. Numerical errors of the wave fields for the apple- and peanut-shaped cavities by using the first approach for the double-single layer potential formulation. | Apple-shaped | Peanut-shaped
---|---|---
$n$ | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time
8 | 9.1028e-03 | 4.9959e-03 | 0.008s | 6.4240e-03 | 3.3783e-03 | 0.006s
16 | 1.9452e-03 | 7.4528e-04 | 0.01s | 2.6370e-04 | 7.9816e-05 | 0.01s
32 | 8.0695e-08 | 4.8943e-08 | 0.03s | 5.1058e-08 | 1.5750e-08 | 0.02s
64 | 2.2710e-12 | 6.4901e-13 | 0.08s | 7.9013e-14 | 2.7135e-14 | 0.07s
128 | 1.0169e-12 | 4.4399e-13 | 0.16s | 1.1928e-12 | 3.8851e-13 | 0.15s
Table 3. Numerical errors of the far-field patterns for the apple- and peanut-shaped cavities by using the first approach for the double-single layer potential formulation. | Apple-shaped | Peanut-shaped
---|---|---
$n$ | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time
8 | 2.9366e-02 | 0.007s | 1.7153e-03 | 0.004s
16 | 8.0457e-04 | 0.009s | 5.8490e-07 | 0.007s
32 | 2.2996e-07 | 0.02s | 5.0560e-10 | 0.02s
64 | 4.4398e-10 | 0.08s | 5.0548e-10 | 0.07s
128 | 4.4586e-10 | 0.16s | 5.0677e-10 | 0.14s

(a) $\Re(v^{(n_{*})}_{\infty})$ and $\Re(v^{(n)}_{\infty})$ (b)
$\Im(v^{(n_{*})}_{\infty})$ and $\Im(v^{(n)}_{\infty})$

Figure 1. The reference and numerical solutions for the apple-shaped cavity: (a) the real part of the far-field pattern; (b) the imaginary part of the far-field pattern. Table 4. Numerical errors of the wave fields for the apple- and peanut-shaped cavities by using the second approach for the double-single layer potential formulation. | Apple-shaped | Peanut-shaped
---|---|---
$n$ | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time
8 | 3.6851e-02 | 2.0239e-02 | 0.006s | 1.1121e-02 | 6.5680e-03 | 0.007s
16 | 1.7791e-03 | 1.2935e-03 | 0.008s | 2.6716e-04 | 8.4432e-05 | 0.01s
32 | 3.4434e-07 | 9.9280e-08 | 0.02s | 5.1173e-08 | 1.5776e-08 | 0.02s
64 | 2.2717e-12 | 6.4932e-13 | 0.08s | 7.8660e-14 | 2.7224e-14 | 0.08s
128 | 1.0171e-12 | 4.4408e-13 | 0.17s | 1.1929e-12 | 3.8858e-13 | 0.16s
Table 5. Numerical errors of the wave fields for the apple- and peach-shaped cavities by using the single-single layer potential formulation. | Apple-shaped | Peach-shaped
---|---|---
$n$ | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time
8 | 1.5871e-02 | 7.9131e-03 | 0.008s | 2.2946e-03 | 1.1103e-03 | 0.008s
16 | 1.6863e-02 | 9.6869e-03 | 0.01s | 3.1410e-04 | 1.1890e-04 | 0.01s
32 | 5.9174e-10 | 4.5291e-10 | 0.02s | 3.9650e-05 | 1.4962e-05 | 0.02s
64 | 7.1981e-15 | 5.9586e-15 | 0.06s | 4.9690e-06 | 1.8742e-06 | 0.06s
128 | 1.1133e-15 | 3.0029e-16 | 0.13s | 6.2163e-07 | 2.3443e-07 | 0.12s
Table 6. Numerical errors of the far-field patterns for the apple- and peach-shaped cavities by using the single-single layer potential formulation. | Apple-shaped | Peach-shaped
---|---|---
$n$ | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time
8 | 3.3854e-03 | 0.003s | 5.4010e-04 | 0.003s
16 | 1.0286e-03 | 0.005s | 1.4150e-05 | 0.005s
32 | 4.7122e-10 | 0.01s | 1.6853e-06 | 0.01s
64 | 1.8565e-14 | 0.04s | 2.0524e-07 | 0.05s
128 | 1.4716e-14 | 0.11s | 2.5289e-08 | 0.11s

(a) $\Re(v^{(n_{*})}_{\infty})$ and $\Re(v^{(n)}_{\infty})$ (b)
$\Im(v^{(n_{*})}_{\infty})$ and $\Im(v^{(n)}_{\infty})$

Figure 2. The reference and numerical solutions for the peach-shaped cavity:
(a) the real part of the far-field pattern; (b) the imaginary part of the far-
field pattern.

####  5.3.1. Smooth cavities

First, we consider the scattering by smooth cavities: the apple-, peanut-, and
peach-shaped cavities. Table 1 shows the parameterized equations for the
boundary curves, where the appled- and peanut-shaped cavities have analytic
boundary curves while the peach-shaped cavity has a $\mathcal{C}^{2}$ boundary
curve. These examples have been commonly used for benchmark tests of various
scattering problems, e.g., the elastic obstacle scattering problem [3].

Let us begin with an example which has an analytic solution in order to test
the accuracy of the methods. We consider a point source located at
$\bar{x}=(0.1,0.2)^{\top}\in D$ and construct the corresponding exact
solutions to the Helmholtz and modified Helmholtz equations

$\displaystyle v_{\rm H}^{*}(x)=H_{0}^{(1)}(\kappa|x-\bar{x}|),\quad v_{\rm
M}^{*}(x)=H_{0}^{(1)}(\mathrm{i}\kappa|x-\bar{x}|),\quad
x\in\mathbb{R}^{2}\setminus\overline{D},$ (5.4)

which satisfy the Sommerfeld radiation condition (2.10). Then the numerical
solution is obtained by solving the boundary value problem (2.8)–(2.10) with
the following boundary conditions on $\Gamma$:

$\displaystyle\tilde{f}_{1}=v_{\rm H}^{*}+v_{\rm
M}^{*},\quad\tilde{f}_{2}=\partial_{\nu}v_{\rm H}^{*}+\partial_{\nu}v_{\rm
M}^{*}.$

Tables 2, 4, and 5 list the $L^{2}(\partial B_{1})$ errors of the wave fields
$v_{\rm H}$ and $v_{\rm M}$ at the wavenumber $\kappa=2$, where $\partial
B_{1}=\\{x\in\mathbb{R}^{2}:|x|=1\\}$. Specifically, Tables 2 and 4 show the
results by using the first and second approaches for the double-single layer
potential formulation, respectively; Table 5 gives the results by using the
single-single layer potential formulation.

Now, we consider the scattering problem that the cavity is illuminated by a
plane wave. Since there are no analytic solutions, we use the numerical
solution with the number of collocation points $n_{*}=2048$ as the reference
solution. We take the incident angle $\theta=\pi/6$, solve the boundary
integral equations, and use (3.3) to calculate the corresponding far-field
patterns $v_{\infty}^{(n_{*})}$ at 32 observation points which are uniformly
distributed on the unit circle $\Omega$. To examine the convergence, the
reference solutions are compared with the numerical solutions computed by
using different numbers of collocation points. Table 3 shows the errors of the
far-field patterns for the apple- and peanut-shaped cavities by using the
first approach for the double-single layer potential formulation. Table 6
shows the errors of the far-field patterns for the apple- and peach-shaped
cavities by using the single-single layer potential formulation. Figures 1 and
2 plot the real and imaginary parts of the reference solution
$v_{\infty}^{(n_{*})}$ and the numerical solution $v_{\infty}^{(n)}$ with the
number of collocation points $n=32$ for the apple- and peach-shaped cavities,
respectively. Since the results are similar for the second approach to the
double-single layer potential formulation, they are not shown here for
brevity.

Based on the above numerical experiments, it can be observed that the errors
decrease exponentially as the number of collocation points increases, which
indicates the exponential convergence of the method and confirms our
theoretical analysis. It can also be found that the convergence rate of the
apple-shaped cavity is faster than that of the peach-shaped cavity, which is
consistent with the fact that the apple-shaped cavity has an analytic boundary
curve while the peach-shaped cavity has only a $\mathcal{C}^{2}$ boundary
curve.

####  5.3.2. Non-smooth cavities

Next we report the scattering by nonsmooth cavities: the drop- and heart-
shaped cavities. The parameterizations of the boundary curves of these two
cavities are shown in Table 1. Clearly, the drop-shaped cavity is convex but
the heart-shaped cavity is concave; both of the boundary curves have a single
corner at $x_{0}\in\Gamma$ and are analytic on $\Gamma\setminus\\{x_{0}\\}$.
The angle at the corner is defined as $\vartheta$ which satisfies
$0<\vartheta<2\pi$. It can be seen from the parameterizations that the drop-
and heart-shaped cavities have interior angles of $\vartheta=\pi/2$ and
$\vartheta=3\pi/2$, respectively, and the corner point $x_{0}$ corresponds to
the parameter $t=0$.

Again, to test the accuracy, we consider the point source located at
$\bar{x}=(0.1,0.2)^{\top}$ and $\bar{x}=(-0.5,0.2)^{\top}$ for the drop- and
heart-shaped cavities, respectively, and construct the corresponding exact
solution (5.4). Moreover, we employ the graded mesh in order to efficiently
resolve the wave field near the corner, i.e., more collocation points are
adopted on the boundary curve near the corner while less points are allocated
at the part of the boundary curve that is far from the corner. Specifically,
we take the substitution $t=w(s)$ in the parametric curve of the drop- and
hear-shaped cavities [2, 12], where

$\displaystyle w(s)=2\pi\frac{[v(s)]^{p}}{[v(s)]^{p}+[v(2\pi-s)]^{p}},\qquad
0\leq s\leq 2\pi$

with

$\displaystyle
v(s)=\Big{(}\frac{1}{p}-\frac{1}{2}\Big{)}\Big{(}\frac{\pi-s}{\pi}\Big{)}^{3}+\frac{1}{p}\frac{s-\pi}{\pi}+\frac{1}{2},\quad
p=2.$

In the experiments, the collocation points are chosen as $s_{j}:=\pi
j/n+\pi/(2n)$. The graded mesh of the collocation points $w(s_{j})$,
$j=0,\cdots,2n-1$ can be found in [3, Figure 3] for both of the drop- and
heart-shaped boundary curves.

Table 7. Numerical errors of the wave fields for the drop- and heart-shaped cavities by using the single-single layer potential formulation. | Drop-shaped | Heart-shaped
---|---|---
$n$ | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time | $\|v_{\rm H}^{*}-v_{\rm H}^{(n)}\|$ | $\|v_{\rm M}^{*}-v_{\rm M}^{(n)}\|$ | time
8 | 3.9875e-03 | 7.3325e-04 | 0.003s | 2.0080e-01 | 3.9261e-02 | 0.003s
16 | 1.2025e-05 | 2.2451e-06 | 0.007s | 6.2198e-02 | 3.5015e-02 | 0.007s
32 | 1.5096e-06 | 1.8009e-07 | 0.02s | 1.5920e-03 | 1.0946e-03 | 0.02s
64 | 1.8909e-07 | 2.2602e-08 | 0.07s | 4.4875e-05 | 3.6727e-05 | 0.07s
128 | 2.3294e-08 | 2.7856e-09 | 0.13s | 3.2170e-09 | 2.5981e-09 | 0.13s
256 | 4.5217e-09 | 5.4078e-10 | 0.67s | 3.2103e-12 | 1.5088e-12 | 0.68s
512 | 2.1147e-10 | 2.5291e-11 | 5.95s | 6.0989e-13 | 3.3518e-13 | 5.89s
1024 | 1.3734e-11 | 1.6426e-12 | 53.88s | 4.9250e-13 | 3.0598e-13 | 53.90s
Table 8. Numerical errors of the far-field patterns for the drop- and heart-shaped cavities by using the single-single layer potential formulation. | Drop-shaped | Heart-shaped
---|---|---
$n$ | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time | $\|v_{\infty}^{(n_{*})}-v_{\infty}^{(n)}\|$ | time
8 | 8.5518e-02 | 0.003s | 1.7155 | 0.003s
16 | 2.4203e-03 | 0.008s | 1.6035e-01 | 0.007s
32 | 3.5585e-04 | 0.02s | 4.1046e-04 | 0.02s
64 | 6.1901e-05 | 0.07s | 4.1476e-06 | 0.07s
128 | 6.7751e-06 | 0.13s | 3.9402e-10 | 0.12s
256 | 1.1587e-05 | 0.66s | 9.9265e-13 | 0.69s
512 | 3.6241e-06 | 5.99s | 2.0886e-13 | 5.99s
1024 | 1.5637e-06 | 53.62s | 2.0840e-13 | 54.15s

(a) $\Re(v^{(n_{*})}_{\infty})$ and $\Re(v^{(n)}_{\infty})$ (b)
$\Im(v^{(n_{*})}_{\infty})$ and $\Im(v^{(n)}_{\infty})$

Figure 3. The reference and numerical solutions for the drop-shaped cavity:
(a) the real part of the far-field pattern; (b) the imaginary part of the far-
field pattern.

(a) $\Re(v^{(n_{*})}_{\infty})$ and $\Re(v^{(n)}_{\infty})$ (b)
$\Im(v^{(n_{*})}_{\infty})$ and $\Im(v^{(n)}_{\infty})$

Figure 4. The reference and numerical solutions for the heart-shaped cavity:
(a) the real part of the far-field pattern; (b) the imaginary part of the far-
field pattern.

Table 7 lists the $L^{2}(\partial B_{2})$ errors between the numerical
solution and the exact solution (5.4) for the drop- and heart-shaped cavities,
where $\partial B_{2}=\\{x\in\mathbb{R}^{2}:|x|=2\\}$. Clearly, the errors
decay rapidly for both cases by using the graded meshes. This is due to the
reason that the exact solution is analytic. On the other hand, the results are
more accurate for the heart-shaped cavity than those of the drop-shaped
cavity. We believe that this phenomenon is related to the geometry of the
cavity. The analysis will be left for a future work. Similar to the scattering
by smooth cavities, we consider the plane wave incidence with incident angle
$\theta=\pi/6$ and compute the far-field patterns $v_{\infty}^{(n_{*})}$ with
$n_{*}=2048$ as reference solutions. Table 8 shows the errors of the far-field
patterns by using the single-single layer potential formulation. The numerical
solution $v_{\infty}^{(n)}$ and the reference solution $v_{\infty}^{(n_{*})}$
are shown in Figures 3 and 4 for the drop- and heart-shaped obstacles,
respectively. It can be seen that the two solutions agree with each other very
well.

##  6\. Conclusion

In this paper, we have studied the biharmonic wave scattering problem of a
cavity embedded in an infinite thin plate. Based on the biharmonic wave
operator splitting, the scattering problem of the fourth-order biharmonic wave
equation is reduced into a coupled boundary value problem of the Helmholtz and
modified Helmholtz equations. With the help of potential theory, a novel
boundary integral formulation is proposed for the scattering problem. Based on
an appropriate regularizer, the operator equation is split into an isomorphic
operator plus a compact one. The well-posedness is established for the coupled
system. The convergence analysis is carried out for both the semi- and full-
discrete schemes by using the collocation method. To demonstrate the superior
performance of the proposed method, numerical experiments are presented for
both smooth and nonsmooth cavities. The numerical results show that the
proposed method is highly accurate even for nonsmooth examples.

In this work, we examined only the clamped boundary condition. At the edge of
cavities, many other boundary conditions can be considered to take into
account of different physical behavior [6]. Clearly, a different boundary
condition will lead to a different formulation of boundary integral equations.
We are investigating these problems and will report the progress elsewhere in
the future.

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