Noname manuscript No.
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A Collocation Method for Weakly Singular Integral
Equations with Super-Algebraic Convergence Rate
Tilo Arens · Thomas Rösch
the date of receipt and acceptance should be inserted later
Abstract We consider biperiodic weakly singular integral equations of the
second kind such as they arise in boundary integral equation methods. These
are solved numerically using a collocation scheme based on trigonometric polynomials. The singularity is removed by a local change to polar coordinates.
The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence
of the scheme achieving algebraic convergence of any order under appropriate
regularity assumptions. The method can be applied to typical boundary value
problems such as potential and scattering problems both for bounded obstacles
and for periodic surfaces. We present numerical results demonstrating that the
expected convergence rates can be observed in practice.
Keywords numerical methods for integral equations · collocation method ·
super-algebraic convergence rate · Laplace equation · Helmholtz equation
Mathematics Subject Classification (2000) 65R20
1 Introduction
The boundary integral equation method consists of reformulating a boundary
value problem for an elliptic partial differential equation such as Laplace’s
equation or the Helmholtz equation equivalently as an integral equation
Z
λϕ(x) −
K(x, y) ϕ(y) ds(y) = ψ(x) ,
x ∈ ∂D .
(1.1)
∂D
Tilo Arens · Thomas Rösch
Karlsruhe Institute of Technology, Karlsruhe, Germany, E-mail: [email protected],
[email protected]
2
Tilo Arens, Thomas Rösch
This approach is attractive, in particular for exterior problems. For Collocation
and Galerkin methods, also carrying out matrix compression by the Fast Multipole Method or H-Matrix calculus, the complexity is comparable to Finite
Element Methods applied to the original boundary value problem. Low algebraic convergence orders with (close to) linearly growing operation count and
memory requirement are state of the art. On the other hand, for 2D problems,
originating in work by Kussmaul [12] and Martensen [13], Nyström methods
with exponential convergence rates at an O(N 2 ) operation count exist. For
smooth boundaries, such methods are particularly easy to implement, requiring only the composite trapezoidal rule and a rule with the same quadrature
points and weights given by a simple formula for integrating the singularity.
In 3D, the singularity in the kernel is more complex, depending both on distance and direction. Using a single regular grid, it is hard to find a quadrature
rule that achieves high order convergence. For an equation of the 2nd kind,
i.e. (1.1) with λ = 1, as we will consider throughout the paper, an approach
for smooth surfaces globally parametrizable over a sphere was suggested by
Wienert [15]. Using rotations and spherical coordinates, he removes the singularity. A full convergence analysis was given later in [8, 9].
A different approach was suggested by Bruno and Kunyanski [5, 6]. Let us
assume for simplicity that ∂D can be globally parametrized by a map η : Q →
∂D with Q = (−π, π)2 . This gives rise to an integral equation
Z
ϕ(t) −
k(t, τ ) ϕ(τ ) dτ = ψ(t) ,
t ∈ Q.
(1.2)
Q
Using a cut-off function χ with χ(t) = 0 for |t| ≥ %, the singularity is isolated,
Z
Z
ϕ(t)− k(t, τ ) χ(τ −t) ϕ(τ ) dτ − k(t, τ ) (1−χ(τ −t)) ϕ(τ ) dτ = ψ(t) (1.3)
Q
Q
for t ∈ Q. A transformation of the first integral in polar coordinates centered
on t removes the singularity, making all integrals computable to high order by
the composite trapezoidal rule, and a Nyström method is applied. Although
the idea is simple, a number of technicalities need to be addressed in the implementation. The papers [5, 6] include no analysis relating convergence rates to
overall complexity. Over the past decade, there have been some contributions
to closing this gap: In [10], Heinemeyer related the approach to the method of
locally corrected weights as proposed by [7]. He was able to prove point-wise
convergence of the discrete operators with super-algebraic convergence rates,
but did not give a convergence rate of the overall scheme. The most complete
analysis is given in [4], but the authors limit themselves to scattering problems.
In [1] the scheme of [5, 6] was interpreted as a collocation method based on
trigonometric polynomials. Evaluations in the polar coordinate grid points are
exact. Stability and a super-algebraic convergence rate at an O(N 2 ) computational complexity were shown for a semi-discrete scheme. However, the complexity estimate for the fully discrete scheme is much less favorable . In the
present paper, we improve the scheme of [1], reducing the overall complexity.
A Collocation Method with Super-Algebraic Convergence Rate
3
To start, we introduce Sobolev spaces of biperiodic functions and consider interpolation by trigonometric polynomials in Section 2. In Section 3, we analyse
the mapping properties of the integral operators and their discrete approximations. As our main result, in Theorems 3.10 and 3.11, we establish stability
and convergence of the overall scheme. Assuming that the transformed kernel
is infinitely often differentiable, any algebraic convergence rate is achieved.
In Section 4, we consider three different applications for the solver in the
context of boundary integral equations: an interior boundary value problem for
Laplace’s equation, a scattering problem for a bounded object and a scattering
problem for a periodic surface. Numerical results are presented in Section 5.
In this paper, we focus on single integral equations and the discussion of stability and convergence rates for approximating the weakly singular integral operator. The method has to compete against simpler strategies [3] that achieve 4th
order convergence by neglecting the singularity in a clever way. Thus we limit
ourselves to relatively simple settings and geometries. More complex geometries lead to systems of integral equations. It did not seem feasible to include
the corresponding extensions here. We also do not consider the question of
a data-sparse approximation of the matrix representing the integral operator
with a smooth kernel. The question of how to achieve this while maintaining
the overall high convergence rate will be the subject of future research.
2 Periodic Sobolev Spaces and Interpolation
Let us begin with some notational conventions. Throughout the paper, for x ∈
Cn , n ∈ N, |x| will denote the Euclidean norm of x, |x|∞ the maximum norm.
An exception are multi-indices α ∈ N20 for which, as usual, |α| = |α1 | + |α2 |.
The distinction will always be clear from the context.
Set Q = (−π, π)2 . We call a function u : R2 → C Q-periodic if
u(t) = u(t1 + 2ν1 π, t2 + 2ν2 π),
t = (t1 , t2 )> ∈ R2 ,
ν = (ν1 , ν2 )> ∈ Z2 .
Particular examples of Q-periodic functions are the trigonometric monomials,
1
exp (i ν · t) ,
t ∈ R2 , ν ∈ Z2 .
T (ν) (t) =
2π
The trigonometric monomials span spaces of trigonometric polynomials TN =
span{T (ν) : ν ∈ Z2N } where N ∈ N and Z2N = {µ ∈ Z2 : −N < µj ≤
N, j = 1, 2}. They also form a complete orthonormal system in L2 (Q), i.e.
every u ∈ L2 (Q) can be expanded into a Fourier series
X
u=
uν T (ν) .
ν∈Z2
The Fourier coefficients uν are given by
Z
(ν)
uν = u, T
=
u(t) T (−ν) (t) dt ,
2
L (Q)
Q
ν ∈ Z2 .
4
Tilo Arens, Thomas Rösch
s
Definition 2.1 Let s ≥ 0. The Sobolev space HQ
is given by
X
s
(1 + |ν|2 )s |uν |2 < ∞
HQ
= u ∈ L2 (Q) :
ν∈Z2
with inner product (u, v)H s =
Q
P
ν∈Z2
1/2
(1+|ν|2 )s uν vν and norm kukHQs = (u, u)H s .
Q
s
The 1D analogue of HQ
has been studied in depth in [11, 14]. Some results on
s
multivariate functions are also contained in [14]. HQ
is a Hilbert space and
0
2
s
there holds HQ = L (Q). For s > 1, by Sobolev’s imbedding theorem, HQ
is
compactly imbedded in (Cper , k · k∞ ), the space of Q-periodic continuous functions with the maximum norm. Hence, it makes sense to consider interpolation
by trigonometric polynomials in the interpolation points
π
N
N >
tN
µ , µ ∈ Z2N .
µ = (tµ,1 , tµ,2 ) =
N
s
Lemma 2.2 (Lemma 5.1 in [1]) Let s > 1 and 0 ≤ σ ≤ s. Given u ∈ HQ
and N ∈ N, there is a unique interpolation polynomial PN u ∈ TN such that
N
u(tN
µ ) = PN u(tµ ) ,
µ ∈ Z2N .
σ
s
is bounded with
→ HQ
The linear operator PN : HQ
kPN u − ukHQσ ≤ C N σ−s kukHQs ,
where C > 0 is a constant depending on σ and s.
Alternatively, PN can be expressed using the Lagrange basis representation,
X
N
PN u =
u(tN
(2.1)
µ ) Lµ ,
µ∈Z2N
with the Lagrange basis functions given by
X
π
T (ν) (t − tN
LN
µ ),
µ (t) =
2
2N
2
t ∈ R2 .
(2.2)
ν∈ZN
For t ∈ Q \ {tN
µ }, there also holds the expression
"
#
2
N
Y
t
−
t
1
j
µ,j
sin N (tj − tN
i + cot
.
LN
µ (t) =
µ,j )
4 N 2 j=1
2
This follows from the corresponding one-dimensional result in [11, Section 11.3]
with some obvious modifications due to a differently defined space TN .
2
Lemma 2.3 The set {LN
µ : µ ∈ ZN } is an orthogonal basis of (TN , k · kL2 (Q) ),
N
LN
µ , Lα
L2 (S)
=
π2
δα,µ ,
N2
µ, α ∈ Z2N .
A Collocation Method with Super-Algebraic Convergence Rate
5
2
Proof PN (TN ) = TN and (2.1) yield TN = span{LN
µ : µ ∈ ZN }. The orthogonality follows easily from (2.2) and the definition and orthogonality of the
trigonometric monomials.
t
u
s
In some instances, products of functions from HQ
with smooth functions occur.
m
m
For m ∈ N0 and ψ ∈ Cper = Cper ∩ C (Q), we set
kψk∞;m := sup |ψ(t)| + max sup |∂ β ψ(t)|.
t∈Q
|β|=m t∈Q
s
σ
Lemma 2.4 Let s ≥ 0 and σ ∈ N≥s . Suppose ϕ ∈ HQ
and let ψ ∈ Cper
. Then
s
ψϕ ∈ HQ and
kψϕkHQs ≤ Ckψk∞;σ kϕkHQs ,
where the constant C > 0 is independent of ϕ and ψ.
Proof The assertion follows from the equivalence of k · kHQs with the SobolevSlobodeckiı̌ norm (see [11, Section 11.3] for a detailed exposition in 1D). t
u
In particular, we are interested in an estimate of this kind when the smooth
factor is a trigonometric monomial.
Lemma 2.5 Let σ ∈ N. Then kT (µ) k∞;σ ≤ 1/(2π) (1 + |µ|σ ) for all µ ∈ Z2 .
Proof Let β ∈ N20 denote a multi-index with |β| = σ. Then, for µ ∈ Z2 and
t ∈ Q, ∂ β T (µ) (t) = i|β| µβ1 1 µβ2 2 T (µ) (t) and hence
|β|
β (µ) |µ1 |β1 |µ2 |β2
|µ|∞
∂ T (t) ≤
≤
.
2π
2π
Since |µ|∞ ≤ |µ|, we obtain kT (µ) k∞;σ ≤ 1/(2π) (1 + |µ|σ ).
t
u
In the later analysis, functions which are Q-periodic with respect to several independent variables will occur. Such functions can be expanded into a Fourier
series with respect to one of these variables. The behaviour of the Fourier
coefficients in such expansions will be of importance.
Lemma 2.6 Let F ∈ C ∞ (R2 × R2 ) be Q-periodic with respect to both arguments and define
Z
(λ)
F (t) =
F (t, τ ) T (−λ) (τ ) dτ,
λ ∈ Z2 , t ∈ R2 .
Q
Then
F (t, τ ) =
X
F (λ) (t) T (λ) (τ ),
t, τ ∈ R2 ,
λ∈Z2
∞
holds pointwise, where (F (λ) )λ∈Z2 ⊆ Cper
= Cper ∩ C ∞ (Q). Moreover, for any
2
m ∈ N0 and any multi-index β ∈ N0 there exists a constant C > 0 such that
m
sup sup 1 + |λ|2 |∂ β F (λ) (t)| ≤ C kF k∞;|β|+2m .
λ∈Z2 t∈R2
6
Tilo Arens, Thomas Rösch
Proof We expand F into a Fourier series with respect to the second argument.
Pointwise convergence holds due to the smoothness of F (t, ·) for all t ∈ R2 .
In particular, the definition of F (λ) and well-known facts about parameter∞
dependent integrals yield F (λ) ∈ Cper
. Furthermore,
|∂ β F (λ) (t)| ≤ 2π kF k∞;|β| ,
For m ∈ N0 , we have |λ|2m =
m!
α1 !α2 !
α∈N20 :|α|=m
P
t ∈ R2 ,
β ∈ N20 .
1 2α2
λ2α
1 λ2 .
Let λ ∈ Z2 , t ∈ R2 , β ∈ N20 and α ∈ N20 with |α| = m. Then
2α1 2α2 β (λ)
m!
∂ F (t)
α1 !α2 ! λ1 λ2
Z
2|α| 2α1 2α2 (−λ)
β
= α1m!
λ
λ
T
(τ
)
∂
F
(t,
τ
)
−
i
dτ
t
1
2
!α2 !
|
{z
}
Q
=∂ (2α1 ,2α2 ) T (−λ) (τ )
Z
m! = α1 !α2 ! ∂tβ ∂τ(2α1 ,2α2 ) F (t, τ ) T (−λ) (τ ) dτ ≤ 2π
Q
m!
α1 !α2 ! kF k∞;|β|+2m
,
where we have used integration by parts in the third line. Hence,
X
m+1
m!
|λ|2m ∂ β F (λ) (t) ≤ 2π kF k∞;|β|+2m
π kF k∞;|β|+2m .
α1 !α2 ! = 2
|α|=m
m
≤ 2m 1 + |λ|2m , λ ∈ Z2 , we obtain
m 1 + |λ|2 ∂ β F (λ) (t) ≤ 2m+1 π kF k∞;|β| + 2m kF k∞;|β|+2m .
From 1 + |λ|2
Since λ ∈ Z2 and t ∈ R2 were chosen arbitrarily, the proof is completed by
|β|
|β|+2m
t
u
into Cper .
observing the boundedness of the imbedding of Cper
3 The Approach for a Single Biperiodic Integral Equation
We now return to considering the integral equation (1.2). We will impose the
following assumptions on the kernel function and the right hand side:
Assumption 1 Let k = k1 + k2 , where k1 ∈ C ∞ (Q × Q \ {(t, t) : t ∈ Q})
and k2 ∈ C ∞ (R2 × R2 ) are Q-periodic with respect to both variables. For every
multi-index α ∈ N20 , the estimate
|∂ α k1 (t, τ )| ≤
C
,
minν∈Z2 |t − τ − 2π ν|1+|α|
t, τ ∈ R2 ,
is satisfied. For some 0 < %0 < π, setting `(t, r, v) = |r| k1 (t, t + rv), t ∈ Q,
r ∈ [−%0 , %0 ], v ∈ S1 , we assume that ` ∈ C ∞ (Q × [−%0 , %0 ] × S1 ).
s
We also assume ψ ∈ HQ
.
A Collocation Method with Super-Algebraic Convergence Rate
7
Hence, k1 is assumed to have a particular type of weak singularity that can
be removed by a transformation to polar coordinates around the singularity.
Examples of such kernels will be discussed in Section 4.
To make use of Assumption 1 in the numerical method, we require appropriate
cut-off functions. For 0 < δ < ε < π define


|τ | ≤ δ ,

1 ,
ε−|τ |
δ < |τ | < ε ,
χδ,ε (τ ) = χ̃ ε−δ ,


0 ,
τ ∈ Q, |τ | ≥ ε ,
with
χ̃(s) =
e−1/s
,
e−1/s + e−1/(1−s)
s ∈ (0, 1) .
Note that χδ,ε is infinitely often differentiable. On all of R2 , χδ,ε is assumed
to be Q-periodic. Furthermore, an argument by induction shows that
m
X
ε − |τ |
pα
(`)
` (τ )
χ̃
,
α ∈ N20 , |α| = m ∈ N ,
∂ α χδ,ε (τ ) =
(δ − ε)` |τ |2m−`
ε−δ
`=1
where pα
` are either homogeneous polynomials of degree m or the zero function.
From this representation, we obtain the estimate
α
|∂ χδ,ε (t)| ≤ Cα
m
X
`=1
εm
,
δ 2m−` (ε − δ)`
t ∈ R2 .
(3.1)
0 < % < π/δ2 ,
(3.2)
Fixing numbers 0 < δ1 < δ2 , (3.1) implies
|∂ α χδ1 %,δ2 % (t)| ≤ Cα,δ1 ,δ2 %−m ,
with a constant Cα,δ1 ,δ2 independent of %.
With the help of these cut-off functions, we localize the singularity in the
integral operator from (1.2). Fixing numbers 0 < ε1 < ε2 < 1 and 0 < % < %0 ,
define
ksmooth (t, τ ) = k1 (t, τ ) (1 − χε1 %,ε2 % (τ − t)) + k2 (t, τ ) ,
(3.3)
and introduce the operators
Z
J1 ϕ(t) =
k1 (t, τ ) χε1 %,ε2 % (τ − t) χε2 %,% (τ − t) ϕ(τ ) dτ ,
Q
Z
J2 ϕ(t) =
ksmooth (t, τ ) ϕ(τ ) dτ ,
t ∈ Q.
Q
Then (1.2) can be rewritten as
ϕ − J1 ϕ − J2 ϕ = ψ
on Q .
The reason for introducing χε2 %,% will be explained below.
(3.4)
8
Tilo Arens, Thomas Rösch
Next, we rewrite J1 ϕ(t) using polar coordinates around t. We set
% cos ϑ
Π(p) = r
,
p = (r, ϑ)> ∈ Q ,
π sin ϑ
and
kpolar (t, p) =
|r| %2
k1 (t, t + Π(p)) χε1 %,ε2 % (Π(p)) ,
2 π2
t, p = (r, ϑ)> ∈ Q .
Substituting τ = t + Π(p) in the expression for the operator J1 gives
Z
J1 ϕ(t) =
kpolar (t, p) χε2 %,% (Π(p)) ϕ(t + Π(p)) dp ,
t ∈ Q.
(3.5)
Q
By Assumption 1 and kpolar (t, p) = 0 for |Π(p)| ≥ ε2 %, we have that kpolar ∈
C ∞ (Q × Q) can be extended Q-periodically with respect to both arguments to
C ∞ (R2 ×R2 ). The reason for introducing χε2 %,% becomes clear now: χε2 %,% (Π(·))
ϕ(t + Π(·)) can also be Q-periodically extended to C ∞ (R2 ).
We want to solve (3.4) numerically using a collocation method on the space
TN . Thus, the semidiscrete problem is to find ϕN ∈ TN such that
ϕN − PN J1 ϕN − PN J2 ϕN = PN ψ
on Q.
(3.6)
A fully discrete method is obtained in several steps. Firstly, both integrals are
replaced by composite trapezoidal rules which are highly efficient for periodic
s
functions. For M , N ∈ N, we set for ϕ ∈ HQ
Z
J1,M ϕ(t) =
PM [kpolar (t, ·) χε2 %,% (Π(·)) ϕ(t + Π(·))] (p) dp
Q
2
π X
M
M
kpolar (t, tM
ν ) χε2 %,% (Π(tν )) ϕ(t + Π(tν ))
M2
ν∈Z2M
Z
J2,N ϕ(t) =
PN [ksmooth (t, ·) ϕ] (τ ) dτ
=
(3.7)
Q
π2 X
N
= 2
ksmooth (t, tN
ν ) ϕ(tν ) .
N
2
(3.8)
ν∈ZN
While both operators are discrete in principle, only J2,N can be used directly.
The expression for J1,M involves the evaluation of ϕ(t + Π(tM
ν )). An exact
M
evaluation requires the knowledge of LN
(Π(t
))
for
all
µ
∈
Z2N , ν ∈ Z2M
µ
ν
2 2
which amounts to O(N M ) operations. In [4–6] the quadrature rule in radial
direction is slightly perturbed and the values of ϕ(t + Π(·)) in the quadrature
points are obtained to high accuracy by fixed degree polynomial interpolation.
However, this approach limits the asymptotic convergence rate.
The approach of [1] is a collocation method and uses the exact values of ϕ(t +
Π(·)) in the quadrature points. Here, we modify the scheme by reducing the
A Collocation Method with Super-Algebraic Convergence Rate
9
cost in the approximation of J1 . We require the orthogonal projection OM
from L2 (Q) onto TM ,
OM v =
X v, T (µ)
µ∈Z2M
L2 (Q)
T (µ) =
M2 X
M
(v, LM
µ )L2 (Q) Lµ
π2
2
(3.9)
µ∈ZM
for v ∈ L2 (Q), where the second representation is due to Lemma 2.3. Let
1 < ε3 denote a number such that ε3 % ≤ %0 . A scaled projection for functions
on Q% = (−ε3 %, ε3 %)2 is given by
h ε % i π 3
ÕM v = OM v
·
· ,
v ∈ L2 (Q% ) .
π
ε3 %
We define for M , M̃ ∈ N,
Z
J1,M,M̃ ϕ(t) =
PM kpolar (t, ·)
Q
hn
o
i
OM χε2 %,% ÕM̃ [χ%,ε3 % ϕ(t + ·)] ◦ Π (p) dp . (3.10)
The operator OM was already used in [1]. It makes the derivation of (3.17)
below possible which is central to the proof of Theorem 3.9. The projection
ÕM̃ reduces the complexity of the scheme when compared to the approach
in [1]. We analyse this in the beginning of Section 5.
For the remaining part of this section, we focus on the convergence analysis of
the approach introduced above. We will start with properties of the operators
J2 and J2,N which are simpler to analyse.
s+1
s
→ HQ
is a well-defined, bounded
Theorem 3.1 Let s ≥ 0. Then J2 : HQ
− max{5,s+3}
linear operator with kJ2 k ≤ C %
for all % ≤ %0 with C dependent
on k, ε1 and ε2 .
Proof We write ksmooth using its Fourier series representation from Lemma 2.6,
X
ksmooth (t, τ ) =
k (λ) (t) T (λ) (τ ),
t, τ ∈ R2 .
λ∈Z2
s
Let ϕ ∈ HQ
and denote by σ = bsc the largest integer smaller or equal to s.
By Lemma 2.4, there holds
(λ) k s+1 = 2π k (λ) T (0) H s+1 ≤ C k (λ) ∞;σ+2
HQ
Q
for all λ ∈ Z2 . Therefore,
X Z
X
T (λ) (τ ) ϕ(τ ) dτ k (λ) s+1 =
kJ2 ϕkH s+1 ≤
|ϕ−λ | k (λ) H s+1
H
Q
λ∈Z2
Q
Q
λ∈Z2
Q
10
Tilo Arens, Thomas Rösch
≤ C kϕkHQs
!1/2
2 −σ
X
(λ) 2
k ∞;σ+2
1 + |λ|
λ∈Z2
≤ C kϕkHQs sup
h
σ
2 max{1− 2 ,0}
1 + |λ|
λ∈Z2
i
(λ) k ∞;σ+2
X
2 −2
!1/2
1 + |λ|
λ∈Z2
≤ C kϕkHQs ksmooth ∞;σ+2+max{2−σ,0} = C kϕkHQs ksmooth ∞;max{4,σ+2} ,
where the last estimate is due to Lemma 2.6. Here, and throughout the paper,
we denote by C a generic constant that may be different in each occurence.
Define the set Ω% = {(t, τ ) ∈ R2 × R2 : |t − τ + 2πν| ≥ ε2 % for all ν ∈ Z2 }. We
proceed to bound for m ∈ N0 using Assumption 1 and (3.2)
ksmooth ≤ kk2 ∞;m + kk1 ∞
∞;m
X
k∂ α k1 (·, ·)k∞;Ω% k∂ β (1 − χε1 %,ε2 % )k∞;R2 ≤ C%−m−1
+C
|α|+|β|=m
for % ≤ %0 , which completes the proof.
t
u
t+1
s
Theorem 3.2 Let s > 1 and t ∈ [0, s]. Then J2,N : HQ
→ HQ
is a welldefined, bounded linear operator. Moreover,
k(J2 − J2,N ) ϕkH t+1 ≤ C %−2s−6 N t−s kϕkHQs
Q
for all ϕ ∈
s
,
HQ
% ≤ %0 and all N ∈ N2 , where C depends on k, ε1 and ε2 .
Proof Let σ ∈ N≥s . From Lemma 2.2, we conclude
Z
I − PN [T (λ) ϕ](τ ) dτ ≤ 2π I − PN [T (λ) ϕ]H t ≤ C N t−s T (λ) ϕH s .
Q
Q
Q
From Lemmas 2.4 and 2.5, we obtain
(λ) σ
σ
2 2
T ϕ s ≤ C T (λ) s ≤ C 1+|λ|
s ≤ C 1+|λ|
kϕk
kϕk
kϕkHQs
H
H
Q
Q
H
∞;σ
Q
so that
Z
Q
σ/2
I − PN [T (λ) ϕ](τ ) dτ ≤ C N t−s 1 + |λ|2
kϕkHQs .
Thus
k(J2 − J2,N ) ϕkH t+1 ≤
Q
X Z
λ∈Z2
Q
I − PN [T (λ) ϕ](τ ) dτ k (λ) H t+1
≤ C N t−s kϕkHQs
Q
X
1 + |λ|2
σ/2 (λ) k ∞;σ+1
λ∈Z2
and again Lemma 2.6 completes the proof as the remaining argument is very
similar to that at the end of Theorem 3.1.
t
u
A Collocation Method with Super-Algebraic Convergence Rate
11
The derivation of a similar result for the approximation J1,M,M̃ of J1 as introduced in (3.10) is more complicated. Although the singularity has been removed, the integral operator now takes on a non-standard form which makes
the analysis of its mapping properties much more involved.
To simplify the considerations, let us rewrite J1 in terms of expressions that
are easier to analyse. Writing kpolar as a Fourier series with respect to p,
X (λ)
kpolar (t) T (λ) (p),
kpolar (t, p) =
t, p ∈ Q ,
(3.11)
λ∈Z2
we formally have
X
J1 ϕ(t) =
(λ)
kpolar (t)
λ∈Z2
Z
T (λ) (p) χε2 %,% (Π(p)) ϕ(t + Π(p)) dp .
Q
The later analysis will show that interchanging integration and summation is
indeed justified.
Recalling Q% = (−ε3 %, ε3 %)2 , and defining the scaled trigonometric monomials
π
1
(ν)
exp i
τ ·ν ,
τ ∈ Q% ,
TQ% (τ ) =
2ε3 %
ε3 %
consider functions u of t ∈ Q, τ ∈ Q% . These can be expanded into Fourier
series with respect to both variables,
X
(ν)
u(t, τ ) =
uµ,ν T (µ) (t) TQ% (τ ) .
µ,ν∈Z2
For s ≥ 0, we introduce the vector space
n
o
s
2
HQ,Q
=
u
∈
L
(Q
×
Q
)
:
p
(u)
<
∞
for
all
σ
≥
0
,
%
s,σ
%
where
X
ps,σ (u) =
(1 + |µ|2 )s (1 + |µ −
2 σ
2
π
ε3 % ν| ) |uµ,ν |
,
σ ≥ 0.
µ,ν∈Z2
s
t
Remark 3.3 For all 0 ≤ t ≤ s, HQ,Q
is a subspace of HQ,Q
.
%
%
s
and σ ≥ 0 also set
For u ∈ HQ,Q
%
X
qs,σ (u) =
(1 + | επ3 % ν|2 )s (1 + |µ −
2 σ
2
π
ε3 % ν| ) |uµ,ν | .
µ,ν∈Z2
Between ps,σ and qs,σ , there holds a certain equivalence relation. For u ∈
s
HQ,Q
and σ ≥ 0, we estimate
%
X
σ
s
ps,σ (u) =
1 + |µ|2 1 + |µ − επ3 % ν|2 |uµ,ν |2
µ,ν∈Z2
12
Tilo Arens, Thomas Rösch
≤ 2s
X
1 + |µ −
2
π
ε3 % ν|
+ | επ3 % ν|2
s
1 + |µ −
2 σ
π
|uµ,ν |2
ε3 % ν|
µ,ν∈Z2
≤ 2s
X
1 + | επ3 % ν|2
s
1 + |µ −
2 σ+s
π
|uµ,ν |2
ε3 % ν|
= 2s qs,σ+s (u) , (3.12)
µ,ν∈Z2
and by similar arguments also qs,σ (u) ≤ 2s ps,σ+s (u).
Two technical lemmas are required to establish the mapping properties of J1 .
Lemma 3.4 Denote by χ
\
%,ε3 % the Fourier transform of the extension of χ%,ε3 % |Q
to R2 by 0. Then for any σ ∈ N0 and ε3 % ≤ %0 ,
−2σ+2
sup (1 + |x|2 )σ χ
\
,
%,ε3 % (x) ≤ C %
x∈R2
where the constant C depends only on σ and ε3 .
Proof We note
χ%,ε3 % (%t) = χ1,ε3 (t) ,
t ∈ Q.
2
Let Bδ = {t ∈ R : |t| < δ}. Then
Z
Z
−it·x
2
χ
(t)
e
dt
=
%
χ
\
(x)
=
%,ε3 %
%,ε3 %
χ1,ε3 (t) e−i% t·x dt .
B ε3
B ε3 %
Let R > 0 and consider x = |x| x̂ with |x| ≥ R. We rewrite the integral using
the divergence theorem as
Z
χ1,ε3 (t) e−i% t·x dt
Bε3
Z
=
B ε3
1
=
i%|x|
x̂ χ1,ε3 (t) e−i%|x| t·x̂
x̂ · ∇χ1,ε3 (t) e−i%|x| t·x̂
− ∇t ·
dt
i |x| %
i |x| %
Z
x̂ · ∇χ1,ε3 (t) e−i%|x| t·x̂ dt .
Bε3
We repeat this argument 2σ − 1 times to obtain
Z
%2
χ
\
(x)
=
h(t) e−i%|x| t·x̂ dt
%,ε3 %
(i%|x|)2σ Bε3
with some function h depending on ε3 and continuously on x̂. The assertion
follows by applying the triangular inequality for integrals and taking the maximum with respect to x̂.
For |x| ≤ R, we use |\
χ%,ε3 % (x)| ≤ C (ε3 %)2 and ε3 % ≤ %0 .
Lemma 3.5 Let s ≥ 0, ε3 % ≤ %0 .
t
u
A Collocation Method with Super-Algebraic Convergence Rate
13
s
(a) For ϕ ∈ HQ
define
Mϕ(t, τ ) := χ%,ε3 % (τ ) ϕ(t + τ ) ,
(t, τ ) ∈ Q × Q% .
s
Then Mϕ ∈ HQ,Q
, and for all σ ≥ 0,
%
ps,σ (Mϕ) ≤ C %−2σ−2 kϕk2HQs ,
where the constant C depends only on σ and ε3 .
s
(b) For λ ∈ Z2 and u ∈ HQ,Q
, set
%
Z
J (λ) u(t) :=
T (λ) (p) χε2 %,% (Π(p)) u(t, Π(p)) dp,
t ∈ Q.
Q
s+1
s
→ HQ
with
(J (λ) )λ∈Z2 is a family of linear operators mapping HQ,Q
%
q
s
and λ ∈ Z2 ,
for all u ∈ HQ,Q
kJ (λ) ukH s+1 ≤ C %−1 (1+|λ|2 ) ps,3 (u),
%
Q
where c > 0 is a constant only depending on ε2 , ε3 and s.
s
→
(c) (J (λ) ◦M)λ∈Z2 is a family of linear and bounded operators mapping HQ
s+1
HQ . In particular,
kJ (λ) M ϕkH s+1 ≤ C %−5 (1 + |λ|2 )kϕkHQs ,
Q
s
for all ϕ ∈ HQ
and all λ ∈ Z2 , the constant C > 0 only depending on ε2 , ε3 and s.
s
Proof (a) Let s ≥ 0 and ϕ ∈ HQ
. In a first step, we calculate the Fouriercoefficients uµ,ν of u = Mϕ. Therefore, let µ, ν ∈ Z2 . Then
Z Z
(−ν)
uµ,ν =
u(t, τ ) T (−µ) (t) TQ% (τ ) dτ dt
Q
Q%
Z
1
−iµ·(t+τ
−τ
)
χ%,ε3 % (τ ) e 3
ϕ(t + τ ) e
dt dτ
2π Q
Q%
Z
Z
π
0
1
1
=
χ%,ε3 % (τ ) e−i( ε3 % ν−µ)·τ
ϕ(t0 ) e−iµ·t dt0 dτ
2ε3 % R2
2π τ +Q
1
π
=
χ
\
%,ε3 % ( ε3 % ν − µ)ϕµ ,
2ε3 %
1
=
2ε3 %
Z
−i επ% ν·τ
where the last step holds due to the Q-periodicity of ϕ. Now, in a second step,
for σ ≥ 0, there holds
σ+2
X
2
1+| επ3 % ν−µ|2
1
2 s
π
2 |ϕµ |2 χ
ps,σ (u) =
1 + |µ|
\
%,ε3 % ( ε3 % ν − µ) .
π
2
2
(2ε3 %)
1+|
ν−µ|
2
ε3 %
µ,ν∈Z
From
Z
R2
1
dx ≥ h2
(1 + |x|2 )2
X
ν∈Z2
ν1 ,ν2 6=0
1
,
(1 + |hν|2 )2
h > 0,
14
Tilo Arens, Thomas Rösch
and similar estimates for the remaining terms in the sum, we see that the value
−2
P
is uniformly bounded in µ and % for
of the series ν∈Z2 1 + | επ3 % ν − µ|2
ε3 % ≤ %0 . Thus from Lemma 3.4, the assertion follows.
(b) Using the Fourier series expansion of u, there holds
Z
X
(ν)
uµ,ν
J (λ) u =
T (λ) (p) χε2 %,% (Π(p)) TQ% (Π(p)) dp T (µ) .
Q
µ,ν∈Z2
Suppose ν 6= 0. We write (ν1 , ν2 )> = qν (cos ϑν , sin ϑν )> for some qν > 0 and
some ϑν ∈ (−π, π], and obtain
(ν)
TQ% (Π(p)) =
1
exp (i qν (r/ε3 ) cos(ϑ − ϑν )) ,
2ε3 %
p = (r, ϑ) ∈ Q.
Hence, the substitution ϑ0 = ϑ − ϑν and the 2π-periodicity with respect to ϑ
yield
Z
(ν)
T (λ) (p) χε2 %,% (Π(p)) TQ% (Π(p)) dp
Q
Z
0
1 i λ2 ϑν
T (λ) (r, ϑ0 ) [χε2 %,% ◦ Π] (r, ϑ0 + ϑν )ei qν (r/ε3 ) cos ϑ d(r, ϑ0 ).
e
=
2ε3 %
Q
The behaviour of the integral in this expression with respect to λ and ν can
be estimated by the method of stationary phase. A detailed proof is given
in [1, Lemma 6.2]. We obtain
(λ)
T [χε2 %,% ◦ Π] (·, · + ϑν )
Z
h
i
∞;2
(ν)
(λ)
T (p) χε2 %,% TQ% ◦ Π(p) dp ≤ C
.
q
ν
Q
Similarly as in the proof of Lemma 3.4 we observe that
r cos(ϑ + ϑν )
[χε2 %,% ◦ Π] (r, ϑ + ϑν ) = χε2 ,1
π sin(ϑ + ϑν )
is independent of %. Hence
(λ) (λ) T T Z
h
i
√
∞;2
∞;2
(ν)
T (λ) (p) χε2 %,% TQ% ◦ Π(p) dp ≤ C
≤ 2C
.
2
q
(1 + |ν| )1/2
ν
Q
(3.13)
Note that the final estimate is also true for ν = 0. Using Lemma 2.5, gives
Z
2
T (λ) (p)χε2 %,% (Π(p))T (ν) (Π(p)) dp ≤ C 1 + |λ|
.
Q%
(1 + |ν|2 )1/2
Q
We proceed with
(λ) 2
J u
s+1
HQ
!2
≤C
2
X
µ∈Z2
2 s+1
(1 + |µ| )
X
ν∈Z2
1+|λ|2
|u |
(1+|ν|2 )1/2 µ,ν
A Collocation Method with Super-Algebraic Convergence Rate
= C 2 (1 + |λ|2 )2
X
X 1+|µ|2 1/2
(1 + |µ|2 )s
1+|ν|2
µ∈Z2
2
2 2
≤ C (1 + |λ| )
X
≤C %
−2
2 2
(1 + |λ| )
!2
|uµ,ν |
ν∈Z2
X (1+|
2 s
(1 + |µ| )
µ∈Z2
2
15
1/2
π
π
2
2
ε3 % ν| ) (1+|µ− ε3 % ν| )
(1+|ν|2 )
!2
|uµ,ν |
ν∈Z2
X
2 s
(1 + |µ| )
µ∈Z2
X
ν∈Z2
π
(1+|µ− ε % ν|2 )3/2
3
π
1+|µ− ε % ν|2
3
!2
|uµ,ν |
. (3.14)
−2
P
As in the proof of part (a), the series ν∈Z2 1 + |µ − επ3 % ν|2
is bounded
independently of µ and % < 1, so that we can apply the Hölder inequality for
`2 -series to obtain
X
(λ) 2
(1 + |µ|2 )s (1 + |µ − επ3 % ν|2 )3 |uµ,ν |2
J u s+1 ≤ C 2 %−2 (1 + |λ|2 )2
HQ
µ,ν∈Z2
2
=C %
−2
2 2
(1 + |λ| ) ps,3 (u) .
(c) The assertion follows directly by combining (a) and (b).
t
u
With these preliminary considerations, we are now able to investigate the
mapping properties of J1 .
s+1
s
→ HQ
defined in (3.5) is a bounded
Theorem 3.6 Let s ≥ 0. Then J1 : HQ
linear operator with
kJ1 ϕkH s+1 ≤ C %−5 kϕkHQs
Q
for ε3 % ≤ %0 with C depending only on k, s, ε1 , ε2 and ε3 .
Proof By definition, J1 is a linear integral operator. To show boundedness
s+1
s
to HQ
, we insert χ%,ε3 % (Π(·)) in the integrand and expand kpolar
from HQ
into its Fourier series (3.11). With J (λ) from Lemma 3.5 we obtain
X (λ)
J1 ϕ(t) =
kpolar (t) J (λ) Mϕ(t) .
λ∈Z2
This is justified by the estimates for any σ ∈ N≥s+1 using Lemma 3.5
X (λ)
kJ1 ϕkH s+1 ≤ C
kkpolar k∞;σ J (λ) M ϕH s+1
Q
Q
λ∈Z2
≤ C %−5 kϕkHQs
X
(λ)
1 + |λ|2 kkpolar k∞;σ
λ∈Z2
≤ C %−5 kϕkHQs
X
λ∈Z2
3 (λ)
1
1 + |λ|2 kkpolar k∞;σ .
2
2
(1 + |λ| )
16
Tilo Arens, Thomas Rösch
The series converges as the two last factors are bounded by Lemma 2.6 with
3 (λ)
sup 1 + |λ|2 kkpolar k∞;σ ≤ C kkpolar k∞,σ+6 .
λ∈Z2
With t, p = (r, ϑ)> ∈ Q and setting p̂ = (cos ϑ, sin ϑ)> , we write kpolar (t, p) as
r % %r ` t,
, p̂ χε1 ,ε2
p̂
kpolar (t, p) =
2π
π
π
with the function ` from Assumption 1. It follows that kkpolar k∞,σ+6 ≤ C
uniformly for ε3 % ≤ %0 with C depending only on σ, ε1 and ε2 . This completes
the proof.
t
u
We next derive an analogue of Theorem 3.2 for J1 , i.e. an estimate for the
difference J1 − J1,M,M̃ . To this end, we write
h
i h
i
J1 − J1,M,M̃ = J1 − J˜1,M̃ + J˜1,M̃ − J1,M,M̃ ,
where
J˜1,M̃ ϕ(t) =
Z
kpolar (t, p)
Q
hn
o
i
χε2 %,% ÕM̃ [χ%,ε3 % ϕ(t + ·)] ◦ Π (p) dp
for t ∈ Q. Note that, using the projection
X X
(ν)
OM̃ u(·, ··) =
uµ,ν T (µ) (·) TQ% (··) ,
µ∈Z2
s
u ∈ HQ,Q
,
%
(3.15)
(3.16)
ν∈Z2
M̃
P (λ)
we can write J˜1,M̃ as J˜1,M̃ ϕ =
kpolar J (λ) OM̃ M ϕ.
λ∈Z2
Lemma 3.7 Let s ≥ 0, ε3 % ≤ %0 , M ∈ N and recall the definitions of J (λ)
and M from Lemma 3.5.
s
and λ ∈ Z2 ,
(a) For all u ∈ HQ,Q
%
kJ (λ) ukH s+1 ≤ C %−1 (1 + |λ|2 )
Q
q
qs,s+3 (u) ,
where C depends only on s, ε2 and ε3 .
s
(b) For 0 ≤ t ≤ s, σ ≥ 0 and all u ∈ HQ,Q
,
%
qt,σ (I − OM )u ≤ M 2(t−s) qs,σ (u) .
Here I denotes the identity operator.
t+1
s
(c) Let 0 ≤ t ≤ s, λ ∈ Z2 . Then J (λ) (I − OM )M : HQ
→ HQ
is bounded
with
kJ (λ) (I − OM )M ϕkH t+1 ≤ C %−2s−5 (1 + |λ|2 ) M t−s kϕkHQs
Q
for all ϕ ∈
s
HQ
,
where the constant C > 0 only depends on s, t, ε2 and ε3 .
A Collocation Method with Super-Algebraic Convergence Rate
17
Proof (a) This follows from Lemma 3.5 (b) together with (3.12).
(b) Let 0 ≤ t ≤ s and σ ≥ 0. Then
X X
qt,σ (I − OM )u =
t
1 + |µ −
2 σ
π
|uµ,ν |2
ε3 % ν|
s
1 + |µ −
2 σ
π
|uµ,ν |2
ε3 % ν|
1 + | επ3 % ν|2
µ∈Z2 ν∈Z2 \Z2M
≤
X
X
1 + |ν|2
t−s
1 + | επ3 % ν|2
µ∈Z2 ν∈Z2 \Z2M
≤ M 2(t−s) qs,σ (u)
s
holds for all u ∈ HQ,Q
.
%
s
(c) Let ϕ ∈ HQ
and set u = Mϕ. Then, by Lemma 3.5 (a) and Remark
t
t
3.3, u ∈ HQ,Q% , and hence also (I − OM )u ∈ HQ,Q
. From part (a) and (b)
%
together with (3.12) and Lemma 3.5 (a), we obtain the estimate
q
kJ (λ) (I − OM ) ukH t+1 ≤ C %−1 (1 + |λ|2 ) qt,t+3 (I − OM )u
Q
q
q
≤ C %−1 (1 + |λ|2 ) M t−s qs,s+3 (u) ≤ C %−1 (1 + |λ|2 ) M t−s ps,2s+3 (u)
≤ C %−2s−5 (1 + |λ|2 ) M t−s kϕkHQs ,
t
u
which is the desired result.
t+1
s
→ HQ
Theorem 3.8 Let M̃ ∈ N, s ≥ 0 and t ∈ [0, s]. Then J˜1,M̃ : HQ
defined in (3.15) is a well-defined, linear and bounded operator with
k(J1 − J˜1,M̃ )ϕkH t+1 ≤ C %−2s−5 M̃ t−s kϕkHQs
Q
for all ϕ ∈
s
HQ
and ε3 % ≤ %0 , where C > 0 only depends on k, s, t, ε2 and ε3 .
s
and σ ∈ N≥s+1 . Proceeding analogously as in the proof of
Proof Let ϕ ∈ HQ
Theorem 3.6, from Lemma 3.7 (c) we obtain
k(J1 − J˜1,M̃ )ϕkH t+1 ≤ C
X
Q
(λ)
kkpolar k∞;σ J (λ) (I − OM̃ )M ϕH t+1
Q
λ∈Z2
≤ C %−2s−5 M̃ t−s kϕkHQs
X
(λ)
1 + |λ|2 kkpolar k∞;σ .
λ∈Z2
The remainder of the proof goes like the end of the proof of Theorem 3.6. t
u
s
→
Theorem 3.9 Let M, M̃ ∈ N, s ≥ 0 and t ∈ [0, s]. Then J1,M,M̃ : HQ
t+1
HQ defined in (3.10) is a well-defined, linear, bounded operator. Moreover,
k(J˜1,M̃ − J1,M,M̃ )ϕkH t+1 ≤ C %−4 M t−s kϕkHQs
Q
for all ϕ ∈
s
HQ
and ε3 % ≤ %0 , where C > 0 only depends on k, s, t, ε2 and ε3 .
18
Tilo Arens, Thomas Rösch
s
Proof We follow the proof of Theorem 6.5 in [1]. Let ϕ ∈ HQ
. We set
v(p) = χε2 %,% ÕM̃ [χ%,ε3 % ϕ(t + ·)] ◦ Π(p),
p ∈ Q,
and write the operators as
X
J˜1,M̃ ϕ =
Z
(λ)
kpolar
λ∈Z2
X
J1,M,M̃ ϕ =
Z
(λ)
T (λ) (p) v(p) dp ,
Q
kpolar
h
i
PM T (λ) OM v (p) dp .
Q
λ∈Z2
A central observation regarding this representation of J1,M,M̃ is
Z
Z
X
M
PM T (λ) OM v (p) dp =
T (λ) (tM
)
O
v(t
)
LM
M
ι
ι
ι (p) dp
Q
Q
ι∈Z2M
=
X
T (λ) (tM
ι )
ι∈Z2M
π2
OM v(tM
ι )
M2
Z
π2 X M 2
M
M M
L
(p)
v(p)
dp
L
(t
)
=
T
ν
ν
ι
M2
π2 Q
ι∈Z2M
ν∈Z2M
Z
Z
X
M
=
T (λ) (tM
)
L
(p)
v(p)
dp
=
v(p) PM T (λ) (p) dp ,
ι
ι
X
(λ)
(tM
ι )
Q
ι∈Z2M
Q
so that we obtain
(J˜1,M̃ − J1,M,M̃ )ϕ =
X
λ∈Z2
(λ)
kpolar
Z
h
i
v(p) T (λ) (p) − PM T (λ) (p) dp .
(3.17)
Q
τ
is
Let τ > 3 and ω ≥ s − t + τ . By Sobolev’s Imbedding Theorem, HQ
continuously imbedded in the space of twice continuously differentiable Qperiodic functions. Hence, by Lemma 2.2
(λ)
T
− PM T (λ) ∞;2 ≤ C T (λ) − PM T (λ) H τ
Q
ω/2
τ −ω (λ) ≤CM
T
≤ C M t−s 1 + |λ|2
.
Hω
Q
Let u = Mϕ with M from Lemma 3.5 and recall OM̃ from (3.16). We obtain
Z
Q
T (λ) (p) − PM T (λ) (p) χε2 %,% (Π(p)) OM̃ u(·, Π(p)) dp
Z
i
X X
h
(ν)
=
uµ,ν
T (λ) − PM T (λ) (p) χε2 %,% TQ% ◦ Π(p) dp T (µ) .
µ∈Z2 ν∈Z2
Q
M̃
Slightly modifying the estimate in (3.13), we proceed as in (3.14) to obtain
A Collocation Method with Super-Algebraic Convergence Rate
Z
Q
19
T (λ) (p) − PM T (λ) (p) χε2 %,% (Π(p)) OM̃ u(·, Π(p)) dp t+1
HQ
≤ C M t−s 1 + |λ|2
ω/2 q
pt,3 (u).
Now, by setting σ = btc from (3.17), we arrive at
X (λ)
J˜
kkpolar k∞;σ+2
1,M̃ − J1,M,M̃ ϕ H t+1 ≤ C
Q
Z
×
T
(λ)
(p) − PM T
λ∈Z2
(λ)
(p) χε2 %,% (Π(p))OM̃ u(·, Π(p)) dp
t+1
HQ
Q
q
X (λ)
ω/2
≤ C M t−s pt,3 (u)
kkpolar k∞;σ+2 1 + |λ|2
.
λ∈Z2
Using Lemma 3.5 (a), Lemma 2.6 and arguing as in the proof of Theorem 3.6,
we establish the bound
−4
J˜
kϕkHQt .
1,M̃ − J1,M,M̃ ϕ H t+1 ≤ C %
Q
s
t
The assertion follows from the continuous imbedding of HQ
in HQ
.
t
u
We now consider the approximation of the solution of the integral equation
(3.4) by the fully discrete version of (3.6) which is to find ϕN ∈ TN such that
ϕN − PN (J1,M,M̃ + J2,N )ϕN = PN ψ .
(3.18)
Based on our results so far, we now prove stability and convergence for (3.18).
We introduce the meshsize h = π/N and set M = M̃ = d%/he where dxe
denotes the smallest integer larger than or equal to x. Further set
A = J1 + J2 ,
Ah = PN (J1,M,M̃ + J2,N ) .
s
, s ≥ 0.
We will assume that I − A is boundedly invertible on any HQ
Theorem 3.10 Let t > 1 and assume that % = hα for some α ∈ (0, 1/(2t+6)).
t
t
has a bounded inverse
→ HQ
Then there exists h0 > 0 such that I − Ah : HQ
for any 0 < h ≤ h0 with norm bounded independently of h.
Proof We write
A − Ah = (J1 − J1,M,M̃ ) + (J2 − J2,N ) + (I − PN ) (J1,M,M̃ + J2,N ) .
From Theorems 3.2, 3.9 and 3.8, we have the estimates
k(J1 − J1,M,M̃ )ϕkHQt ≤ C h %−1 %−2t−5 + %−4 kϕkHQt ,
k(J1 − J1,M,M̃ )ϕkH t+1 ≤ C %−2t−5 + %−4 kϕkHQt ,
Q
k(J2 − J2,N )ϕkHQt ≤ C h %−2t−6 kϕkHQt ,
20
Tilo Arens, Thomas Rösch
k(J2 − J2,N )ϕkH t+1 ≤ C %−2t−6 kϕkHQt .
Q
t+1
t
By Lemma 2.2, the norm of I − PN : HQ
→ HQ
is bounded by Ch. Thus
k(A − Ah ) ϕkHQt ≤ C h %−2t−6 kϕkHQt −→ 0
(h → 0) .
The assertion follows from standard results for operator approximation.
t
u
Theorem 3.11 Let α ∈ (0, 1/3) and % = hα . Assume t ≥ 0 and s > max{1, t,
10α+3αt+t
}. Assume further that (3.18) is a stable approximation of (3.4) in
1−3α
s
HQ
, i.e. there exists c > 0 such that kϕh kHQs ≤ ckϕkHQs for sufficiently small
h. Then there exists h0 > 0 such that
kϕ − ϕh kHQt ≤ C h(s−t)(1−3α)/2 kϕkHQs
for all 0 < h ≤ h0 .
Proof From ϕh = PN ψ + Ah ϕh = PN (ϕ − Aϕ + J1,M,M̃ ϕh + J2,N ϕh ) we obtain
(I − A)(ϕ − ϕh )
= ϕ − Aϕ + J1,M,M̃ ϕh + J2,N ϕh − ϕh − (J1,M,M̃ + J2,N − A)ϕh
= (I − PN )(ϕ − Aϕ + J1,M,M̃ ϕh + J2,N ϕh ) − (J1,M,M̃ + J2,N − A)ϕh .
From Theorems 3.2, 3.8 and 3.9, we have
kJ1,M,M̃ ϕh + J2,N ϕh kHQs ≤ kAk + C h %−2s−6 kϕh kHQs .
Similarly, we have k(J1,M,M̃ + J2,N − A) ϕh kHQ1 ≤ C hs %−3s−5 kϕh kHQs .
Thus from the boundedness of (I −A)−1 in L2 (Q), Lemma 2.2 and the stability
estimate, we conclude
kϕ − ϕh kL2 (Q) ≤ C k(I − A)(ϕ − ϕh )kL2 (Q) ≤ C hs %−3s−5 kϕkHQs
for all h ≤ h0 such that also % ≤ %0 .
For the general result, we observe that for T ∈ TN , the estimate kT kHQt ≤
t
. Using
C h−t kT kL2 (Q) follows directly from the definition of the norm in HQ
the orthogonal projection ON , we have
kϕ − ϕh kHQt ≤ kϕ − ON ϕkHQt + kON ϕ − ϕh kHQt
≤ kϕ − ON ϕkHQt + C h−t kON ϕ − ϕh kL2 (Q)
≤ kϕ − ON ϕkHQt + C h−t kϕ − ϕh kL2 (Q) ,
where the last estimate follows from the Pythagorean theorem. For ϕ − ON ϕ,
bounds as for ϕ − PN ϕ have been shown in the proof of Lemma 2.2. Thus
kϕ − ϕh kHQt ≤ C hs−t 1 + %−3s−5 kϕkHQs .
From s ≥
10α+3αt+t
1−3α
follows (s − t)(1 − 3α)/2 ≥ α(5 + 3t). Thus
hs−t %−3s−5 = hs−t−α(3s+5) = h(s−t)(1−3α) h−α(5+3t) ≤ h(s−t)(1−3α)/2 .
This concludes the proof.
t
u
A Collocation Method with Super-Algebraic Convergence Rate
21
4 Applications
In this section we present applications for the method analysed in Section
3. There are two classes of examples: boundary value problems for bounded
domains that are globally parametrizable over Q and boundary value problems in periodic media. More general geometries lead to systems of Q-periodic
integral equations. We will discuss such cases in a forthcoming paper.
Example 4.1 Let D ⊆ R3 with ∂D the image of a C ∞ smooth regular Qperiodic parametrization η : R2 → R3 , i.e. D is torus-shaped. We orientate η
so that ∂1 η × ∂2 η points out of D. The Dirichlet boundary value problem
∆u = 0
u=f
in D ,
(4.1)
on ∂D ,
can be solved by a double layer potential ansatz,
Z
n(y) · (x − y)
ϕ̃(y) ds(y) ,
u(x) = DL ϕ̃(x) =
3
∂D 4π |x − y|
x ∈ D,
where n(y) denotes the outward drawn unit normal to ∂D in y. Through the
jump relations for the double layer potential we see that u is a solution to
(4.1) if ϕ̃ is a solution to the integral equation
Z
n(y) · (x − y)
ϕ̃(x) −
ϕ̃(y) ds(y) = −2 f (x) ,
x ∈ ∂D .
3
∂D 2π |x − y|
We insert η and set ϕ(t) = ϕ̃(η(t)) to obtain, for t ∈ Q,
Z
(∂1 η(τ ) × ∂2 η(τ )) · (η(t) − η(τ ))
ϕ(t) −
ϕ(τ ) dτ = −2 f (η(t)) .
2π |η(t) − η(τ )|3
Q
(4.2)
Lemma 4.2 The kernel function in (4.2) satisfies Assumption 1 with k2 = 0.
Proof We denote the kernel of (4.2) by k(·, ··). The bound on derivatives of k
is obvious from iterated applications of the quotient rule.
Representing τ = t + r v, t ∈ Q, r ∈ R, v ∈ S1 , we can apply various variants
of the residual representation in Taylor’s theorem to obtain
Z 1
∂
η(t) − η(τ ) = r
η(τ − s v)
dσ
0 ∂s
s=σr
Z 1
∂2
0
2
= −r η (τ ) v + r
(1 − σ) 2 η(τ − sv)
dσ .
(4.3)
∂s
0
s=σr
Note that ∂1 η(τ ) × ∂2 η(τ )) · [η 0 (τ ) v] = 0, as this is a scalar product of the
normal and a tangential vector to ∂D. It follows that both enumerator and
denominator in |r| k(t, t + rv) are C ∞ .
22
Tilo Arens, Thomas Rösch
It remains to show that |η(t + rv) − η(t)|/r is uniformly bounded away from
zero. Fixing any r0 > 0, this is clear for r0 ≤ r ≤ %0 = π/2. From the first
representation in (4.3), we see
|η(t + rv) − η(t)|2
=
r2
Z
0
1
Z
1
v > η 0 (t + σ1 r v)> η 0 (t + σ2 r v) v dσ2 dσ1 .
0
The matrix η 0 (t)> η 0 (t) is the first fundamental form of ∂D, which is uniformly
positive definite as η is regular. Hence we can choose r0 such that the integrand
above is bounded away from 0 uniformly in t ∈ Q, v ∈ S1 and r ≤ r0 . Thus `
from Assumption 1 is C ∞ smooth.
t
u
Example 4.3 Consider the scattering of a time-harmonic acoustic wave by a
smooth bounded obstacle. Such a problem can be modelled as an exterior
boundary value problem for the Helmholtz equation. For D ⊆ R3 we assume
again that ∂D is the image of a C ∞ smooth regular Q-periodic parametrization
η : R2 → R3 . For a wavenumber k > 0, we assume that the total field u satisfies
∆u + k 2 u = 0
u=0
in R3 \ D ,
(4.4)
on ∂D .
Given a solution ui to the Helmholtz equation in all of R3 called the incident
field, we additionally assume that the scattered field us = u − ui satisfies the
Sommerfeld radiation condition at infinity.
A solution to this problem is given by the combined potential layer ansatz,
Z ∂Φ(x, y)
us (x) = DL ϕ̃(x) − iλ SL ϕ̃(x) =
− iλ Φ(x, y) ϕ̃(y) ds(y)
∂n(y)
∂D
for x ∈ R3 \ D provided that ϕ̃ is a solution to the boundary integral equation
Z ∂Φ(x, y)
ϕ̃(x) + 2
− iλ Φ(x, y) ϕ̃(y) ds(y) = −2 ui (x) ,
x ∈ ∂D.
∂n(y)
∂D
Here, λ is some complex number satisfying Re(λ) 6= 0 and Φ denotes the
fundamental solution to the Helmholtz equation
Φ(x, y) =
eik |x−y|
,
4π |x − y|
x 6= y .
After inserting the parametrization and substituting ϕ(t) = ϕ̃(η(t)), we obtain
the Q-periodic integral equation
Z
ϕ(t) +
Q
eik |η(t)−η(τ )|
2π
h
(∂1 η(τ )×∂2 η(τ ))·(η(t)−η(τ ))
|η(t)−η(τ )|3
(1 − ik |η(t) − η(τ )|)
i
η(τ )×∂2 η(τ )|
− iλ |∂1|η(t)−η(τ
ϕ(τ ) dτ = −2ui (η(t)) ,
)|
t ∈ Q . (4.5)
A Collocation Method with Super-Algebraic Convergence Rate
23
Lemma 4.4 The kernel function in (4.5) satisfies Assumption 1.
Proof We set
D(y) n(y) · (x − y)
(cos(k |x − y|) + k |x − y| sin(k|x − y|))
|x − y|3
D(y)
− iλ
cos(k |x − y|) ,
|x − y|
1 D(y) n(y) · (x − y)
K2 (x, y) =
(i sin(k |x − y|) − ik |x − y| cos(k|x − y|))
2π
|x − y|3
D(y)
+λ
sin(k |x − y|) ,
|x − y|
K1 (x, y) =
1
2π
where D(y) = |∂1 η(τ ) × ∂2 η(τ )||y=η(τ ) is the determinant of the first fundamental form. We denote the kernel of (4.5) by
k(t, τ ) = k1 (t, τ ) + k2 (t, τ ) = K1 (η(t), η(τ )) + K2 (η(t), η(τ )) .
Note that
K2 (x, y) = D(y) n(y) · (x − y) p1 (k 2 |x − y|2 ) + λ D(y) p2 (k 2 |x − y|2 )
with analytic functions p1 , p2 . The smoothness of D(η(τ )) follows again from
the positive definiteness of the first fundamental form. Thus k2 is smooth.
Similarly, we have
K1 (x, y) =
D(y)
D(y) n(y) · (x − y)
p3 (k 2 |x − y|2 ) + λ
p4 (k 2 |x − y|2 )
3
|x − y|
|x − y|
with analytic functions p3 , p4 . The smoothness of ` now follows with similar
arguments as in the proof of Lemma 4.2.
t
u
Example 4.5 For the scattering of a plane acoustic wave by a biperiodic smooth
surface, the problem formulation is similar to that of Example 4.3. Again the
total field satisfies (4.4), but here D is unbounded and given by
D = {x = (t, x3 )> ∈ R3 : t ∈ R2 , x3 < f (t)} ,
where f ∈ C ∞ (R2 ) is Q-periodic, and the incident field is given by
ui (x) = ei k d·x ,
x ∈ R3 ,
d ∈ S2 , d3 < 0 .
An important feature of ui and correspondingly also the scattered field is that
it is kd-quasi-periodic, i.e. ui (t + 2π ν, x3 ) = exp(i 2πk (d1 , d2 )> · ν) ui (t, x3 ).
As a radiation condition we use the condition that us is a linear superposition
of upward propagating plane waves and evanescent waves, see [1] for details.
Then us can again be found as a combined double- and single-layer potential,
Z ∂G(x, y)
− iλ G(x, y) ϕ̃(y) ds(y)
us (x) = DL ϕ̃(x) − iλ SL ϕ̃(x) =
∂n(y)
∂D
24
Tilo Arens, Thomas Rösch
for x ∈ R3 \ D. Here G is the kd-quasi-periodic fundamental solution to the
Helmholtz equation and ϕ̃ is a kd-quasi-periodic density. Representations and
algorithms for the efficient evaluation of this function are discussed in [1, 2].
The unit normal n(y) to ∂D at y is assumed to point upward, i.e. n3 > 0.
Setting ϕ(t) = exp(−ik (d1 , d2 )> · t) ϕ̃(t, f (t)), we obtain a Q-periodic density
ϕ. us is a solution to the scattering problem, if ϕ solves the integral equation
Z
>
∂G(x, y)
ϕ(t) + 2
− iλ G(x, y)
ei k(d1 ,d2 ) ·(τ −t)
∂n(y)
Q
p
>
t ∈ Q . (4.6)
× 1 + |∇f (τ )|2 ϕ(τ ) dτ = −2 e−i k(d1 ,d2 ) ·t ui (t, f (t)) ,
For simplicity, we have used the abbreviations x = (t, f (t)), y = (τ, f (τ )) here.
From [1, Theorem 3.8] we have the representation
G(x, y) =
cos(k |x − y|)
+ P (k 2 |x − y|2 ) ,
4π |x − y|
|x − y| ≤
π
,
2
with an analytic function P . Hence, by a similar analysis as in Lemma 4.4, we
see that the kernel in (4.6) satisfies Assumption 1.
5 Implementation and Numerical Examples
The implementation of the operator J2,N is explicitly given in (3.8). The implementation of J1,M,M̃ as given by (3.10) is more complicated as it involves an
approximation of two orthogonal projections. To simplify notation, we define
X µ,ι (ι)
µ
wM̃
= ÕM̃ χ%,ε3 % ϕ(tN
wM̃ TQ% ,
µ + ·) =
ι∈Z2
M̃
µ
vM,
M̃
= OM
hn
µ
χε2 %,% wM̃
o
i
◦Π ,
so that
J1,M,M̃ ϕ(tN
µ)=
Z
Q
h
i
µ
PM kpolar (tN
µ , ·) vM,M̃ (p) dp
π2 X
µ
M
M
kpolar (tN
= 2
µ , tν ) vM,M̃ (tν ) ,
M
2
µ ∈ Z2N .
(5.1)
ν∈ZM
From (3.9), using the Lagrange basis function, we obtain for µ ∈ Z2N , ν ∈ Z2M ,
X X µ,ι Z
π2 µ
(ι)
M
M M
v
(t ) =
wM̃
χε2 %,% (Π(p)) TQ% (Π(p)) LM
λ (p) dp Lλ (tν )
M 2 M,M̃ ν
Q
2
2
λ∈ZM ι∈Z
M̃
X µ,ι Z
(ι)
(5.2)
=
wM̃
χε2 %,% (Π(p)) TQ% (Π(p)) LM
ν (p) dp .
ι∈Z2
M̃
Q
A Collocation Method with Super-Algebraic Convergence Rate
25
Preparation (carry out once)
– compute bµ,ν = kpolar (tN
, tM ), µ ∈ Z2N , ν ∈ Z2M
Z hn µ ν
i
o
(ι)
– compute cν,ι ≈
(p) dp, ν ∈ Z2M ,
χε2 %,% TQ% ◦ Π LM
ν
Q
ι ∈ Z2
M̃
Matrix-Vector-Multiplication (ϕ(tN
µ ))µ 7→ (ψµ )µ
– for
-
each µ ∈ Z2N
N
N
compute dµ,λ = χ%,ε3 % (tN
λ ∈ Z2
λ ) ϕ(tµ + tλ ),
M̃
perform (wµ,ι )ι ←− FFTM̃ (dµ,λ )λ
M̃
P
compute v µ
=
cν,ι wµ,ι , ν ∈ Z2M
M,M̃ ,ν
ι∈Z2
M̃
M̃
- compute ψµ =
bµ,ν v µ
M,M̃ ,ν
ν∈Z2
M
P
Fig. 5.1 Algorithm for Matrix-Vector-Multiplication ψ ← A1 ϕ
The evaluation of (5.2) requires accurate evaluation of the integrals. As χε2 %,%
(ι)
is radially symmetric, we can apply the Jacobi-Anger expansion for TQ% (Π(·))
to obtain a sum of products of two 1D integrals. Integration over p2 can be
carried out analytically, while for the integration over p1 we apply the composite trapezoidal rule. The computation of an approximation of the Fourier
µ,ι
coefficients wM̃
is done by interpolation instead of the orthogonal projection.
s
The convergence rate of both operations are identical for functions in HQ
.
For a description of the implementation, we assume again that N = π/h and
that % = hα for some fixed α ∈ (0, 1). Denote by N = 4N 2 the total number
of unknowns. With M̃ = M = d%/he, we have #Z2M = #Z2M̃ = O(N1−α ).
Below, we will use indices in Z2N for the coefficients of vectors in CN , the map
from Z2N → {1, . . . , N} being implicit, and likewise for matrices in CN×N . We
write the fully discrete version of (3.18) as the linear system
(I − A1 − A2 )ϕ = ψ ,
2
N
where ϕµ = ϕ(tN
µ ), ψµ = ψ(tµ ), µ ∈ ZN , I denotes the identity matrix in
N×N
N×N
C
and Aj = (aj,µ,ν ) ∈ C
denotes the discretization of Jj , j = 1, 2.
The linear system will be solved using GMRES. The computation of Aj ϕ,
j = 1, 2, is required in each step. The coefficients a2,µ,ν are given by (3.3)
and (3.8). The operation count and storage requirements for this are O(N2 ).
However, various approaches are known to reduce this considerably, such as
Fast Multipole Methods, H-Matrix calculus, etc. The focus of the present
paper does not lie in this aspect.
The matrix vector multiplication A1 ϕ can be carried out using (5.1)–(5.2).
The algorithm is described in detail in Figure 5.1. From this description, we
see that the preparation step requires O(N2−α + N2−2α ) memory locations for
storage and O(N2−α + SN2−2α ) operations, where S denotes the number of
26
Tilo Arens, Thomas Rösch
−3
10
−4
error
10
−5
10
−6
10
1
2
10
10
N
Fig. 5.2 Errors for Dirichlet Problem for Laplace’s equation. Green symbols are used for
ε1 = 0.1667 with α = 0.15 (×), α = 0.25 (), blue symbols for ε1 = 0.6667, with α = 0.15
(+), α = 0.25 (◦), α = 0.4 (∗), magenta symbols are used for ε1 = 0.6667 and α = 0.6 ().
operations for acurate evaluation of the integrals in (5.2). For the matrix vector
multiplication itself, only the operational count is of interest. It amounts to
O(N (N1−α + N1−α log(N1−α ))) = O(N2−α (1 + (1 − α) log(N)))
operations.
In order for the preparation step not to dominate the overall complexity, it
is necessary that S is at most proportional to Nα . This requires some careful
balancing. Note however that (5.2) only involves the parameters % and εj and
is independent of k. Hence, these calculations can be carried out once for a
given parameter set and the resulting values stored in a data base.
As a first example, we consider the Dirichlet problem for Laplace’s equation
from Example 4.1. As D we choose a torus given by the parametrization


(R1 + R2 cos(t2 )) cos(t1 )
t
t = 1 ∈ R2 ,
η(t) =  (R1 + R2 cos(t2 )) sin(t1 )  ,
(5.3)
t2
R2 sin(t2 )
with R1 = 1.0 and R2 = 0.25. We chose f for the exact solution u(x) =
−1
4π x − (2, 1, 1)> , x ∈ D. The integral equation (4.2) was solved numerically for various sets of parameters and the double layer potential with the
resulting density was evaluated at a number of points x(`) ∈ D, ` = 1, ..., P .
Tests were run with % = 0.15 (20/N )α for various α ∈ (0, 1) and for two
choices for ε1 . In all cases ε2 = 0.8333, ε3 = 2.0 as well as M̃ = d%/he and M
proportional to M̃ with a fixed constant. Figure 5.2 shows the maximum error
ErrN for evaluations of the double layer potential in the points x(`) . Table 5.1
lists estimated orders of convergence
EOC =
log(ErrN1 / ErrN2 )
log(N2 /N1 )
A Collocation Method with Super-Algebraic Convergence Rate
N
total # of
unknowns
24
32
48
64
96
128
192
2304
4096
9216
16384
36864
65536
147456
N
total # of
unknowns
24
32
48
64
96
128
192
2304
4096
9216
16384
36864
65536
147456
27
ε1 = 0.1667
α = 0.15
α = 0.25
M
EOC
M
EOC
3
4
5
6
9
11
16
-1.1848
2.2589
0.7081
3.5330
2.9765
5.2162
3
4
5
6
8
10
13
-1.1996
2.4786
0.6326
2.4481
3.2272
3.3908
α = 0.15
M
EOC
ε1 = 0.6667
α = 0.25
α = 0.4
M
EOC
M
EOC
3
4
5
6
9
11
16
3
4
5
6
8
10
13
-0.2149
1.5702
1.0533
3.1685
3.4198
5.2133
0.0386
1.2810
1.8965
2.3366
3.2304
3.8985
3
4
5
5
6
7
11
-0.1475
2.3161
-0.1309
1.1316
3.2355
3.9480
α = 0.6
M
EOC
3
3
3
4
4
5
6
1.2710
0.0382
0.5098
0.7219
2.7811
1.2084
Table 5.1 Estimated orders of convergence for the Dirichlet problem for Laplace’s equation.
for subsequent values N1 , N2 of N . The value of N is also listed for each
problem.
For the small choices of α we see that the increasing convergence rates to be
expected from Theorem 3.11 are indeed achieved. On the other hand, the result
for α = 0.6 does not exhibit an increasing convergence rate. The requirement
α < 1/3 from Theorem 3.11 may be pessimistic, but the tests clearly indicate
that a large value of α will prohibit super-algebraic convergence.
As the second example, we carried out computations
√ for a scattering
√ problem
from Example 4.3 for the wave numbers k1 = 2 2 and k2 = 4 2. As the
obstacle, the torus defined by (5.3) was used and ρ was chosen by the same
recipe as before. After solution of the boundary integral equation, the field
was evaluated on a 51 × 51 grid G in the horizontal plane x3 = −1 beneath
the obstacle. To assess the achievable convergence rates, we used the “incident
field” ui = Φ(·, z) with z = (0.9, 0.0, 0.15)> ∈ D. In this case us = −ui
in R3 \ D, so that we know the scattered field exactly. We denote the field
computed by our method for a given value of N by uN and evaluated
ErrN = max |usN (x) + ui (x)| .
x∈G
The values obtained have been plotted in Figure 5.3. In Table 5.2 we report
on estimated
√ convergence rates computed as√in the previous example for the
case k = 2 2. Corresponding rates for k = 4 2 are presented in Table 5.3.
A true scattering
√ problem is displayed in Figure 5.4. A plane wave with incident
direction d = ( 3/2, 0, −1/2)> is scattered by the torus given by (5.3). The
28
Tilo Arens, Thomas Rösch
−2
10
−3
error
10
−4
10
−5
10
1
2
10
10
N
√
Fig. 5.3 Errors for the scattering problem. Blue symbols used for√k = 2 2 with α = 0.15
(+), α = 0.25 (◦), α = 0.4 (∗), red symbols are used for k = 4 2 with α = 0.15 (×),
α = 0.25 (), α = 0.4 ().
N
total # of
unknowns
M
24
32
48
64
96
128
192
2304
4096
9216
16384
36864
65536
147456
3
4
5
6
9
11
16
α = 0.15
EOC
-0.5278
1.1799
1.6587
2.8249
3.8533
5.6597
M
3
4
5
6
8
10
13
α = 0.25
EOC
-0.1241
0.7187
2.8525
2.3226
3.4798
3.2667
M
3
4
5
5
6
7
11
α = 0.4
EOC
-0.4843
2.0920
-0.3633
1.7294
3.3664
3.9864
√
Table 5.2 Estimated orders of convergence for the scattering problem with k = 2 2.
N
total # of
unknowns
M
24
32
48
64
96
128
192
2304
4096
9216
16384
36864
65536
147456
3
4
5
6
9
11
16
α = 0.15
EOC
0.5676
1.1900
1.5574
2.9099
4.0797
4.7555
M
3
4
5
6
8
10
13
α = 0.25
EOC
0.8457
0.6782
3.0471
2.4527
3.6469
3.2173
M
3
4
5
5
6
7
11
α = 0.4
EOC
0.2843
2.0810
-0.5332
1.8940
2.7877
4.6971
√
Table 5.3 Estimated orders of convergence for the scattering problem with k = 4 2.
√
wavenumber is k = 12 2, the other parameters are chosen as N = 128,
M = 11 and % = 0.35671.
The torus is visibly enhanced by a superimposed grid that is in no way related
to the computational grid used. On the surface of the torus, the real part of
−ui is displayed, on the horizontal and vertical plane, the potential evaluated
for the density computed with our solver is presented.
A Collocation Method with Super-Algebraic Convergence Rate
29
√
Fig. 5.4 Scattering of a plane wave by a torus at k = 12 2.
Finally, we also carried out computations for scattering problems involving
periodic surfaces as described in Example 4.5. Specifically, we used the surface
given as the graph of the function
1
cos(t1 ) exp(sin(t1 )) ,
t ∈ R.
6
√
Computations were carried
out at k = 2 2 with the plane wave ui (x) =
√
exp(ik x · d) with d = ( 3/2, 0, −1/2)> used as the incident field. Figure 5.5
shows the real part of the scattered field in this situation. The parameters
used for this plot are N = 48, M = 5 and % = 0.41325. Two periods of the
surface are shown in both directions. The field on the biperiodic surface is
simply the Dirichlet boundary value, on the vertical planes the potential with
the computed density was evaluated.
f (t) =
In conclusion, these examples show comprehensively that the super-algebraic
convergence rates predicted by Theorem 3.11 are achievable in practice in a
variety of applications. Moreover, there is clear numerical evidence, that too
large values of α will destroy high order of convergence, although convergence
orders of approximately 4 were still achievable with α = 0.4.
Acknowledgements This research was supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant Super-Algebraisch konvergente numerische Verfahren für
biperiodische Integralgleichungen.
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