Numer. Math. https://doi.org/10.1007/s00211-018-0960-8
Numerische Mathematik
Collocation with trigonometric polynomials for integral equations to the mixed boundary value problem Ernst P. Stephan1 · Matthias T. Teltscher1
Received: 30 March 2017 / Revised: 27 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We consider the direct boundary integral equation formulation for the mixed Dirichlet–Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and trigonometric polynomials. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. Special care is taken for handling the hypersingular operator as in Hartmann and Stephan (in: Dick, Kuo, Wozniakowski (eds) Festschrift for the 80th birthday of Ian Sloan, Springer, Berlin, 2018). With the indirect method used in Elschner et al. (Numer Math 76(3):355–381, 1997) this was avoided. Using Mellin transformation techniques a stability and solvability analysis of the transformed integral equations can be performed, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space. Mathematics Subject Classification 65 R 20 · 65 N 35 · 65 N 38
1 Introduction We consider the mixed boundary value problem in a bounded, simply connected domain Ω ⊂ R2 with piecewise smooth boundary Γ = Γ N ∪ Γ D :
Dedicated to Prof. Dr. Wolfgang L. Wendland on the occasion of his 80th birthday.
B 1
Ernst P. Stephan
[email protected] Leibniz University, Hannover, Germany
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E. P. Stephan, M. T. Teltscher
For given g on Γ N , f on Γ D , find u on Ω with Δu = 0 in Ω, u = f on Γ D , ∂u ∂n = g on Γ N ,
(1)
where ∂u ∂n denotes the normal derivative of u with respect to the exterior normal n and 2 Δ = ∂x + ∂ y2 the Laplace operator. It is well known that the solution of the mixed boundary value problem for the Laplacian may not be smooth even for smooth Dirichlet and Neumann data. Let {P0 , P1 } be the interface points (i.e. Pi ∈ Γ N ∪ Γ D , i = 0, 1) and let us assume the polygon Γ forms an interior angle ωi at Pi . Then by [3] for P ∈ Ω u(P) = C(θ )r π/2ωi + smoother terms
(2)
where (r, θ ) are the polar coordinates centered at Pi . With the representation formula and jump relations one has for P ∈ Γ ∂ 1 ∂u log |P − Q| u(Q) d S Q = − log |P − Q| (Q) d S Q , (3) ∂n Q π ∂n Γ Γ ∂ ∂ 1 ∂u (P) − log |P − Q| u(Q) d S Q ∂n Q π ∂n P ∂n Q Γ ∂ 1 ∂u =− log |P − Q| (Q) d S Q , (4) π ∂n P ∂n
1 u(P) − π
Γ
where |P − Q| is the Euclidean distance between P and Q, and d S Q is the element of arc length. With (3) and (4) we transform the mixed boundary value problem (1) into a system of integral equations which is solved then approximately by a collocation method. We proceed as follows: We consider (3) for P ∈ Γ D and (4) for P ∈ Γ N , and decompose u and ∂u ∂n in known and unknown parts, i.e. v on Γ N u= f on Γ D
and
∂u g on Γ N = . ∂n ψ on Γ D
(5)
This gives
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Wnn Knd − Kdn Vdd
− Wnd v g I − Knn = , − Vdn I + Kdd ψ f
(6)
Collocation with trigonometric polynomials for integral...
with the identity I and Wnn ϕ(P) =
1 π
ΓN
∂ ∂ log |P − Q| ϕ(Q) d S Q , ∂n P ∂n Q
∂ ∂ log |P − Q| ϕ(Q) d S Q , ∂n P ∂n Q ΓD ∂ 1 Kdn ϕ(P) = − log |P − Q| ϕ(Q) d S Q , π ∂n Q ΓN ∂ 1 Kdd ϕ(P) = − log |P − Q| ϕ(Q) d S Q , π ∂n Q ΓD ∂ 1 Knn ϕ(P) = − log |P − Q| ϕ(Q) d S Q , π ∂n P ΓN ∂ 1 ϕ(P) = − log |P − Q| ϕ(Q) d S Q , Knd π ∂n P ΓD 1 Vdn ϕ(P) = − log |P − Q| ϕ(Q) d S Q , π ΓN 1 Vdd ϕ(P) = − log |P − Q| ϕ(Q) d S Q , π
1 Wnd ϕ(P) = π
P ∈ ΓN , P ∈ ΓN , P ∈ ΓD , P ∈ ΓD , P ∈ ΓN , P ∈ ΓN , P ∈ ΓD , P ∈ ΓD .
ΓD
Throughout the paper we assume: (Assumption 1) System (6) with f = g = 0 has the unique solution v = ψ = 0 in L p (Γ ) for any p > 1. The solutions of the integral equations have appropriate singularities which correspond to the behaviour (2). In detail there is the following situation: Let P0 and P1 be vertices on Γ with Γ N ∩ Γ D = {P0 , P1 }, and let ωi = (1 − χi )π , 0 < |χi | < 1 denote the interior angle at Pi , i = 0, 1. From [3,4] we have with H˜ s (Γ ) = {w ∈ H s (Γ ) : w = 0 on Γ \Γ }: Proposition 1 Let f ∈ H˜ s (Γ D ) and g ∈ H˜ s−1 (Γ N ) for s > −1/2. Then there exist vs ∈ H˜ s (Γ N ) , ψs ∈ H s−1 (Γ D ), such that the solution of (6) has the form with Pi i = 0, 1 from above 2 |P − Pi |ηi + |P − Pi |ηi log |P − Pi | v v (P) ∼ + s , P∈Γ ψs ψ |P − Pi |ηi −1 + |P − Pi |ηi −1 log|P − Pi | i=1
with ηi =
π 2ωi
=
1 2(1−χi ) ,
i = 0, 1.
These singularities yield only slow convergence for a numerical approximation scheme like the collocation method. This can be overcome by an appropriate mesh grading
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E. P. Stephan, M. T. Teltscher
transformation. This procedure was investigated in [7] for the integral equations resulting from a single layer potential ansatz, i.e. an indirect method (see also for other situations [5,8,9] and especially the initiating work [2] by Chandler and Graham). Here we investigate the mesh grading transformation together with suitable trigonometric polynomials for the collocation method applied to the direct method, where the hypersingular operator plays a crucial role. We show for the direct integral equation formulation that mesh grading yields a rapidly converging collocation solution. Let I1 denote the open interval (0, 1) und I2 the open interval (1, 2). Let α : [0, 2] → Γ be a continuous, piecewise smooth, parametrization such that |α | is bounded from above and below on each arc by a positive constant with α(I1 ) = Γ N ,
α(I2 ) = Γ D ,
(7)
and define for v, ψ from (5) gα (x) := g(α(x))|α (x)| , 0 ≤ x ≤ 1, vα (x) := v(α(x)), , 1 ≤ x ≤ 2. ψα (x) := ψ(α(x))|α (x)|, f α (x) = f (α(x))
(8)
Taking a mesh-grading transformation such that α(x) ∼ x q
(9)
near Pi = 0 = α(0) there holds with some constants C, C’ vα (x) = v(α(x)) = C x qηi log(|x|) + smoother terms,
ψα (x) = ψ(α(x))|α (x)| = C x qηi −1log(|x|) + smoother terms.
This means that ψα , vα can become arbitrarily smooth when q is chosen large enough; more precisely: Proposition 2 Let l ∈ N and let the given data f, g in Propsition 1 be chosen sufficiently smooth. Then the solution (vα , ψα ) of (12), (13) for q > maxi=0,1 (2l − 1)(1 − χi ) satisfies (vα , ψα ) ∈ H l (I1 ) × H l−1 (I2 ). Thus vα , ψα can be approximated by an evenly spaced high order spline or a trigonometric function. With a mesh grading transformation an analysis in the L 2 space is possible. Our procedure is as follows: the restriction ψ of ∂u ∂n to Γ D is approximated by trigonometric cosine functions, whereas the restriction v of u to Γ N is approximated by trigonometric polynomials. For a polygon Γ with more than two corners or transition points between Γ D and Γ N the mesh grading transformation would be carried out for each corner or transition point and the restriction ψ to each smooth arc expressed by different cosine series whereas the restriction v by different trigonometric polymonials, respectively. Our analysis (like the one for the indirect method in [7,10]) has an unusual feature. It is that to each smooth arc (after parametrization as above) we associate a separate periodic Sobolev space. The perodic setting is done differently for ψ and v. As in [7]
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we extend the unknown Neumann data ψ = ∂u ∂n on a given arc (after parametrization) to twice the natural range of the variable x by requiring the function to be even about each endpoint of Γ D . This approach has been useful for single open arcs and was introduced by Sloan and Yan in [14]. But the unkown Dirichlet data v is treated differently. v is extended from Γ N by zero to Γ D and afterwards extended with period 2 and approximated by trigonometric polynomials. The paper is organized as follows: in Sect. 2 we describe our collocation method. In Sect. 3 we introduce a suitable periodic function space setting. In Sect. 4 some preliminary results on weakly singular and hypersingular integral operator acting on trigonometric functions are given. Also collocation projections on trigonometric polynomials/cosine functions are introduced. In Sect. 5 we present the crucial mapping properties of the integral operators. Section 6 gives the error analysis for our collocation method, which is based on a finite section method. We can show that our collocation method converges with a rate as high as justified by the order of the mesh grading and the regularity of the given data. Following the spirit of the results in the pioneering work by Chandler and Graham [2] (for further references see[6]) we only prove stability if the approximate solution is cut off by zero on some number of intervals near each corner (transition point). This modification of the collocation method seems not to be necessary in practice. Our numerical example in Sect. 7 shows that the method with no cut off appears to be perfectly stable. An important role in deriving the error estimates is played by Mellin techniques as in [2,7]. In Sect. 7 we comment on the numerical aspects of our collocation method and present a discrete version of it by use of trapezoidal quadrature. In Sect. 8 we give some details on the Mellin techniques which are used in this paper. A short discussion of Mellin operators is given in the appendix. Finally we assume that the parts Γ N and Γ D of the curved polygon Γ are given by straight line pieces near the corners (and transition points between Γ N and Γ D ). This assumption is necessary to apply Mellin techniques. However this assumption seems not to be necessary for the numerical realization of the collocation method. Always c and C mean generic constants, and f (x) ∼ g(x) means there exist positive constants c1 , c2 such that c1 g(x) ≤ f (x) ≤ c2 g(x) for all x.
2 A numerical method The condition (9) on α is realised as follows: Let α˜ : [0, 2] → Γ be a continuous, piecewise smooth parametrization such that ˜ 1 ) = Γ N , α(I ˜ 2) = ΓD . |α˜ | satisfies the conditions for α above, and α(I Let now γ : [0, 1] → [0, 1] be a grid transformation with order q, i.e. let there exist a with 0 < < 1/2 such that γ (x) =
xq, 0≤x ≤ . q 1 − (1 − x) , 1 − ≤ x ≤ 1
(10)
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E. P. Stephan, M. T. Teltscher
Now we define a new mesh graded transformation α(x) :=
α(γ ˜ (x)), 0≤x ≤1 . α(2 ˜ − γ (2 − x)), 1 ≤ x ≤ 2
(11)
Note, α defined this way satisfies (9). Next we insert this parametrization in the system (6). Substituting P = α(x) and Q = α(y) and multiplying the first row of (6) with |α (x)| we obtain with gα (x) := g(α(x))|α (x)|, 0 ≤ x ≤ 1, and f α (x) = f (α(x)), 1 ≤ x ≤ 2, the new system 1
2 (w1 (x, y) + w2 (x, y))vα (y)dy +
0
1 = gα (x) −
k (x, y)gα (y)dy −
0
k (x, y)ψα (y)dy
1 2
(w1 (x, y) + w2 (x, y)) f α (y)dy, 0 ≤ x ≤ 1, 1
(12) 1 −
2 k(x, y)vα (y)dy +
0
v(x, y)ψα (y)dy 1
1 =−
2 v(x, y)gα (y)dy + f α (x) +
0
k(x, y) f α (y)dy, 1 ≤ x ≤ 2,
(13)
1
with 1 (n α(x) , n α(y) ) |α (x)||α (y)|, π |α(x) − α(y)|2 2 (α(x) − α(y), n α(x) )(α(x) − α(y), n α(y) ) |α (x)||α (y)|, w2 (x, y) := π |α(x) − α(y)|4 1 (α(x) − α(y), n α(y) ) |α (y)|, k(x, y) := − π |α(x) − α(y)|2 1 (α(y) − α(x), n α(x) ) k (x, y) := k(y, x) = − |α (x)|, π |α(x) − α(y)|2 1 v(x, y) := − log|α(x) − α(y)|, π w1 (x, y) := −
(14) (15) (16) (17) (18)
where (·, ·) denotes the Euclidean inner product. N imx and The numerical method is simply to approximate vα by m=−N +1 a1,m e N −1 3h ψα by l=0 a2,l cos(πlx) and then collocate (12) at the points lh + 4 for l = 0, . . . , N − 1 and to collocate (13) at the points lh + 3h 4 for N ≤ l ≤ 2N − 1. This choice of collocation points allows us to use the mid points for the trapezoidal rule for computing the arizing integrals by quadrature.
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Collocation with trigonometric polynomials for integral...
3 The periodic function space setting For simplicity we consider two arcs Γ N , Γ D . As mentioned in the introduction the first step in the analysis of the collocation method is to introduce a periodic function space setting. Appropriate Sobolev spaces are given in the next section. Here we rewrite the boundary integral equations (12) and (13) so that this system has an appropriate periodic structure. We want to consider Γ N and Γ D separately and follow [14] and [10]. The idea is to assign to Γ N and Γ D each an own periodic Sobolev space. We achieve this by the following construction. Recall that the parametrization function α has values on Γ N for 0 ≤ x ≤ 1, and values on Γ D for 1 ≤ x ≤ 2. Let us introduce the corresponding 2-periodic functions α(x), 0≤x ≤1 α1 (x) := , α(−x), −1 ≤ x ≤ 0 α(x), 1≤x ≤2 α2 (x) := , α(2 − x), 0 ≤ x ≤ 1
(19) (20)
together with α j (x) = α j (x + 2) ∀x ∈ R,
j = 1, 2.
(21)
In this way α| I1 and α| I2 each are extended to 2-periodic functions defined on R. Note that α1 is the transformation function corresponding to Γ N and α2 the one corresponding to Γ D . (We would have to define further functions α3 …, if Γ contained further arcs.) Next we must extend the solutions vα and ψα in a similar way. However we treat them differently: We extend vα by its 2-periodic zero extension and change ψα into a 2-periodic, even function. Before we do that we introduce the necessary function spaces and canonical operators. First we introduce the space V0 of complex-valued, measurable functions defined on I1 and the space V of complex-valued, measurable, 2-periodic functions. Let F1 be the subspace of 2-periodic functions which disappear on the interval [1, 2], F2 the subspace of 2-periodic functions vanishing on [0, 1] and E the subspace of 2-periodic functions which are even at zero. V0 := {ϕ : I1 → C ∪ {∞}} , F1 := {ϕ ∈ V : ϕ(x) = ϕ(2 + x), ϕ(x) ≡ 0 on [1, 2]} , F2 := {ϕ ∈ V : ϕ(x) = ϕ(2 + x), ϕ(x) ≡ 0 on [0, 1]} , E = {ϕ ∈ V : ϕ(x) = ϕ(2 + x), , ϕ(x) = ϕ(−x)} .
(22) (23) (24) (25)
We define the following canonical mappings: the zero extension ∼: V0 → F1 maps f onto its 2-periodic extension of
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E. P. Stephan, M. T. Teltscher
f˜(x) :=
f (x), x ∈ I1 0, else.
(26)
Let ρ : V → V0 be the restriction on I1 , i.e. ρ f (x) := f (x), x ∈ I1 ,
(27)
and ρ˜ : V → F1 be defined as ρ˜ f = ρf .
(28)
Further we define μ j : V → E, j = 1, 2 as the 2-periodic extension of μ j f (x) :=
f (x), x ∈ Ij . f (2 − x), else.
(29)
Now we can define our solution z := (z 1 , z 2 )t ∈ F1 × E of (12), (13) with u α , φα from (8) as follows z 1 := ρv ˜ α,
(30)
z 2 := μ2 ψα .
(31)
Further we define z 1 as 2-periodic extension of z 1 (x) := u(α(x)).
(32)
The reason for the different treatment of z 1 and z 2 are the different behaviours of v and ψ. Now we create our final system of equations by rewriting (12), (13) with the following operators: W11 K21 K12 V22
: : : :
V V V V
→ → → →
E, E, E, E
and
W12 K22 K11 V21
: : : :
V V V V
→ → → →
E, E, E, E
where W11 ϕ(x) :=
1 π
1 0
+2
123
(n α1 (x) , n α1 (y) ) |α1 (x) − α1 (y)|2
|α1 (x)||α1 (y)|
(α1 (x) − α1 (y), n α1 (x) )(α1 (x) − α1 (y), n α1 (y) ) |α1 (x) − α1 (y)|4
|α1 (x)||α1 (y)| ϕ(y)dy,
(33)
Collocation with trigonometric polynomials for integral...
W12 ϕ(x) :=
1 π
2
K21 ϕ(x) := −
K22 ϕ(x) := −
ϕ(x) := − K11
ϕ(x) := − K12
1 π 1 π 1 π 1 π
1 V22 ϕ(x) := − π V12 ϕ(x) := −
|α1 (x) − α2 (y)|2
1
+2
1 π
(n α1 (x) , n α2 (y) )
|α1 (x)||α2 (y)|
(α1 (x) − α2 (y), n α1 (x) )(α1 (x) − α2 (y), n α2 (y) ) |α1 (x) − α2 (y)|4 1
(α2 (x) − α1 (y), n α1 (y) ) |α2 (x) − α1 (y)|2
0
2
(α2 (x) − α2 (y), n α2 (y) ) |α2 (x) − α2 (y)|2
1
1
(α1 (y) − α1 (x), n α1 (x) ) |α1 (y) − α1 (x)|2
0
2
(α2 (y) − α1 (x), n α1 (x) ) |α2 (y) − α1 (x)|2
1
|α1 (x)||α2 (y)| ϕ(y)dy,
(34)
|α1 (y)|ϕ(y)dy,
(35)
|α2 (y)|ϕ(y)dy,
(36)
|α1 (x)|ϕ(y)dy,
(37)
|α2 (x)|ϕ(y)dy,
(38)
1 log|α2 (x) − α2 (y)|ϕ(y)dy,
(39)
log|α1 (x) − α2 (y)|ϕ(y)dy.
(40)
0
2 1
With
− W12 W11 K12 I − K11 B := , C := − K21 V22 − V21 I + K22
(41)
and f := (μ1 gα , μ2 f α )t ∈ E × E we thus have to solve instead of (6) or (12),(13) the new system Bz = Cf
(42)
for z = (z 1 , z 2 )t ∈ F1 × E. Note that the integrals (12), (13) extend only over half of the period. Thus the periodic extensions of each of our solution functions z 1 , z 2 have little effect on the appearance of the equations but allow us later on a simplified analysis which is only possible in periodic spaces. With the collocation points s = h + 3h/4, = 0, . . . , 2N − 1, N ∈ N the collocation method then reads Find zh = (z 1h , z 2h )t ∈ Th × Te,h , such that z h (s ) = g(s ) − K g(s ) − W W11 z 1h (s ) + K12 12 f (s ), = 0, . . . , N − 1, 2 11 h − K21 z 1 (s ) + V22 z 2h (s ) = − V21 g(s ) + f (s ) + K22 f (s ), = n, . . . , 2N − 1.
(43)
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4 Spaces and mapping properties 4.1 Key operators For s ∈ R let H s be the Sobolev space of 2-periodic functions with norm
f 2s =
max{1, |m|}2s | fˇ(m)|2 ,
(44)
m∈Z
where 1 fˇ(m) = 2
1
f (x)e−iπ mx d x
(45)
−1
denotes the Fourier coefficient of f , and f is given by f (x) =
fˇ(m)eiπ mx
(46)
m∈Z
Let H be the Hilbert transform on H s defined by 1 Hϕ(x) = − pv 2 =i
1 cot
−1
π 2
(x − y) ϕ(y)dy
iπ mx (sign m)ϕ(m)e ˇ ,
(47)
m∈Z
where sign 0 = 0. Due to (47) there holds H : H s → H s isometrically, i.e. Hϕ s =
ϕ s , and H2 = −I + T ,
(48)
1 where T ϕ = ϕ(0) ˇ = 21 −1 ϕ(x)d x. Let A be the single layer potential operator on a circle parameterised from −1 till 1 with the radius e−1/2 1 Aϕ(x) = − π
1
log|2e−1/2 sin
−1
π 2
(x − y) |ϕ(y)dy.
(49)
A has the representation Aϕ(x) =
1 ϕ(m) ˇ eiπ mx π max{1, |m|} m∈Z
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(50)
Collocation with trigonometric polynomials for integral...
From (50) we get, A−1 ϕ(x) = π
iπ mx max{1, |m|}ϕ(m)e ˇ
(51)
m∈Z
With A : H s → H s+1
(52)
A−1 : H s+1 → H s
(53)
DA = AD = H,
(54)
and
we see
where D denotes the derivative operator. Together with (48) and (50) this yields A−1 = −DH + T = −HD + T
(55)
and thus A−2 = (−DH + T )A−1 = −DH(−HD + T ) + T A−1 = DH2 D − DHT + T A−1 = −D 2 + DT D + T A−1 = −D 2 + T A−1 . (56) and for k ∈ N D k A = AD k .
(57)
D k A−1 = A−1 D k .
(58)
This gives immediately
Next we define some spaces which are related to H s and which we need in our analysis. First we consider spaces which are used for vα on Γ D . After that we consider the space Hes which was introduced by Yan, Sloan in [14] and turned out to be very useful for handling open arcs. Let H s (I1 ) be the usual Sobolev space of functions defined on I1 . Take f˜ as in (26). Further let s := { f ∈ H s (I1 ) : f˜ ∈ H s }, H H∼s := { f ∈ H s : f = 0 on [1, 2]} s }, = {ϕ˜ : ϕ ∈ H
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E. P. Stephan, M. T. Teltscher
s := { f ∈ V : f | I1 = h| I1 for a h ∈ H s }. H Since H∼s ⊂ H s we can set ϕ H∼s := ϕ s for ϕ ∈ H∼s Since we assume that the Dirichlet data are continuous on whole Γ we can pose with Proposition 2 and (32) even the bit sharper condition for z 1 in (32).
1 z1 ∈ H
(59)
u(P0 ) = z 1 (0) = 0, u(P1 ) = z 1 (1) = 0
(60)
Additionally we suppose
hence (59) is equivalent to z 1 ∈ H∼1 . The proof of Proposition 2 shows furthermore: If there holds (60), z 1 (m−1) (0) = z 1 (m−1) (1) = 0
m for m ∈ N, then z 1 ∈ H∼m . and z 1 ∈ H
(61)
Let Hes be the space of 2-periodic even functions and Hos the corresponding space of odd functions so that H s = Hes ⊕ Hos . That means, that each f ∈ H s can be split uniquely into f = f e + f o with f e ∈ Hes and f o ∈ Hos . Since Hes ⊂ H s , there holds · 2H s ∼ · 2s (hence on H∼s we can take e the H s -norm and write from now on always only · s ), where (45) becomes fˇe (m) =
1 f e (x) cos(π mx)d x
(62)
1
since f e (x) cos(π mx) is the even and f e (x)i sin(π mx) is the odd part of the integrand. Thus (46) gives f e (x) = 2
fˇe (m) cos(π mx),
(63)
m∈N
where the apostroph in the sum means that the m = 0 term has to be multiplied by 1/2. For z 2 defined by (31) we assume that z 2 = (z 2 )e ∈ Hem for some m ≥ 0, which is possible due to Proposition 2 (here (z 2 )e denotes the even part of z 2 ).
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(64)
Collocation with trigonometric polynomials for integral...
Next we consider the operators H and A on Hes : From (47) we see that H maps an even function onto an odd function and vice versa, i.e. H : Hes → Hos ,
H : Hos → Hes .
(65)
Now we get: 1 Hϕ(x) = 2 1 = 2
1 −1
1 −1
sin(π x) + sin(π y) ϕ(y)dy cos(π x) − cos(π y) 1 sin(π x) ϕ(y)dy + cos(π x) − cos(π y) 2
1 −1
sin(π y) ϕ(y)dy cos(π x) − cos(π y)
=: He ϕ(x) + Ho ϕ(x).
(66)
From the integral kernels of He and Ho one sees that for ϕ = ϕe + ϕo with ϕe ∈ Hes and ϕo ∈ Hos there holds: He ϕe = Hϕe , He ϕo = 0,
Ho ϕe = 0, Ho ϕo = Hϕo .
(67)
Also A can be split into even and odd parts. For ϕ ∈ Hes (49) gives: 1
π (x − y) log 2e−1/2 sin 2 −1
π + log 2e−1/2 sin (x + y) ϕe (y)dy 2 1 1 =− log|2e−1 (cos(π x) − cos(π y))|ϕe (y)dy π
1 Aϕe (x) = − 2π
0
=: Ae ϕe (x).
(68)
Thus (50) and (63) yield Ae ϕe (x) =
fˇe (m) 2 cos(π mx). π max{1, |m|}
(69)
m∈N
−1 : H s+1 → H s is given Hence Ae : Hes → Hes+1 isometrically and A−1 e = (Ae ) e e by
A−1 e ϕe (x) = 2π
max{1, |m|} fˇe (m) cos(π mx)
(70)
m∈N
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E. P. Stephan, M. T. Teltscher
where (68) implies −1 A−1 e ϕe (x) = A ϕe (x)
(71)
Finally (54), (55), (67) and (71) give: A−1 e = −DHe + T = −Ho D + T
(72)
and DAe = Ao D = He , DAo = Ae D = Ho .
(73)
4.2 The collocation projection Since we are looking for z 1 and z 2 in different spaces we must approximate them differently. We use trigonometric polynomials. In the following let N ∈ N and h = 1/N . We define Th := span{eiπ mx : −N + 1 ≤ m ≤ N }. Then with ξm (x) := eiπ mx the collocation projection Ph : H s → Th for s > 1/2 is given as:
Ph f :=
N
( f, ξm )h ξm
(74)
m=−N +1
with ( f, g)h :=
2N h ( f · g)(kh − h/4). 2
(75)
k=1
Ph is an interpolation operator due to the following result: Lemma 1 The operator Ph satisfies on H s , s > 1/2, the following properties: Let f ∈ H s with s > 1/2. Then there holds with a constant C > 0 independent of h and f: (P1) (P2) (P3) (P4)
(Ph f, ξ )h = ( f, ξ )h , ξ ∈ Th . Ph2 = Ph . Ph f (kh − h/4) = f (kh − h/4), k = 1, . . . , N , h = 1/N .
f − Ph f t ≤ Ch s−t f s for s > 1/2, s > t ≥ 0.
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Proof From [1] it is known that (P1) till (P4) hold for the collocation projection Q h with nodal points kh −h/2. Let us denote by ·, · the discrete scalar product belonging to Q h . Then there holds 2N h ( f · ξm )(kh − h/4) 2 k=1 2N −iπ mh/4 h =e ( f 1 · ξm )(kh − h/2) = e−iπ mh/4 f 1 , ξm 2
( f, ξm )h =
k=1
with f 1 (x) := f (x + h/4), and thus Ph f (x) =
N
( f, ξm )h ξm (x)
m=−N +1
=
N
e−iπ mh/4 f 1 , ξm ξm (x)
m=−N +1
=
N
f 1 , ξm ξm (x − h/4) = Q h f 1 (x − h/4),
m=−N +1
especially Ph f (kh − h/4) = Q h f 1 (kh − h/2). Hence (P1) till (P4) follow from properties of Q h . For z 2 we proceed a bit differently. First we take Te,h := span{cos(π mx) : 0 ≤ m ≤ N − 1} and set ζm (x) := cos(π mx). As collocation projector we take correspondingly to (62) and (63) the operator Pe,h : Hes → Te,h with Pe,h f := 2
N −1
( f, ζm )e,h ζm
(76)
m=0
with ( f, g)e,h := h
N
( f · g)(kh − h/4).
k=1
Lemma 2 The operator Pe,h satisfies on H s , s > 1/2, the following properties: Let f ∈ Hes with s > 1/2. Then there holds with a constant C > 0 independent of h and f: ˜ ( P1)) (Pe,h f, ζ )e,h = ( f, ζ )e,h ζ ∈ Th . 2 = P ˜ ( P2)) Pe,h e,h
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E. P. Stephan, M. T. Teltscher
˜ ( P3)) Pe,h f (kh − h/4) = f (kh − h/4), k = 1, . . . , N , h = 1/N . ˜ ( P4))
f − Pe,h f t ≤ Ch s−t f s for s > 1/2, s > t ≥ 0. ˜ till ( P4) ˜ hold for the collocation projection P˜h Proof From [7] it is known that( P1) with nodal points kh −h/2. Let us denote by [., .] the discrete scalar product belonging to P˜h . Then there holds ( f, ζm )e,h = h
N −1
( f · ζm )(kh + h/2) = cos(π mh/4)[ f 1 , ζm ]
k=0
with f 1 (x) := f (x + h/4), and thus Pe,h f (x) = 2
N −1
( f, ζm )e,h ζm (x)
m=0
=2
N −1
[ f 1 , ζm ]ζm (x − h/4)
m=0
˜ till ( P4) ˜ follow from especially Pe,h f (kh − h/4) = P˜h f 1 (kh − h/2). Hence ( P1) properties of P˜h . Altogether for Ph :=
Ph 0 0 Pe,h
and ϕ ∈ H s+1 × Hes we have
ϕ − Ph ϕ H t+1 ×Het ≤ Ch s−t ϕ H s+1 ×Hes for s > 1/2, s > t ≥ 0.
(77)
The collocation method reads as: Find z 1h ∈ Th , z 2h ∈ Te,h , such that
Ph W11 z 1h + K12 z 2h = Ph g − K11 g − W12 f ,
Pe,h − K21 z 1h + V22 z 2h = Pe,h (− V21 g + f + K22 f ) . This can be written in matrix form as: Find zh = (z 1h , z 2h )t ∈ Th × Te,h with Ph B zh = Ph C f.
(78)
5 Mapping properties of the integral operators and Mellin techniques We write Eq. (42) as: B z = (A + K) z = C f,
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(79)
Collocation with trigonometric polynomials for integral...
with −1 B K12 A 0 A := , K := 0 U K21 Ae
(80)
B := W11 −A−1 , U := V22 −Ae .
(81)
and
From (81) follows Lemma 3 The operator B is bounded from H∼1 → H 1 and compact from H∼1 → H 0 . Lemma 4 ([7]) The operators in (35), (38) and (81) 0 0 A−1 e U : He → He ,
K21 : H 1 → He1 , : He0 → He0 K12
are bounded. Remark 1 Note that the transpose double layer potential operator K is denoted in [7] by K. Next we define the matrix 0 A−1 A := −1 . −A−1 e K21 A Ae ∗
(82)
Lemma 5 A∗ is an isomorphism from H 0 × He1 to H −1 × He0 , with inverse (A∗ )−1 =
A 0 . K21 A2 Ae
(83)
Proof Due to Lemma 4 K21 : H 1 → He1 is bounded. Hence (83) is a bounded operator from H −1 × He0 to H 0 × He1 . Next we define the operators −2 0 A D := A A = 0 I ∗
and
E1 A−1 K12 , M := A K = E2 M ∗
(84)
where E1 := A−1 B, E2 :=
−A−1 e K21 AB,
(85) (86)
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E. P. Stephan, M. T. Teltscher −1 M := −A−1 e K21 A K12 +Ae U
=
−A−1 e K21 Ae
K12 +A−1 e U.
(87) (88)
Note that E1 : H∼1 → H 0 and E2 : H∼1 → He1 are bounded operators. This follows immediately from Lemmas 3 and 4. With Lemma 5 and (84) we can rewrite (79) as: (D + M) z = e, e := A∗ C f.
(89)
For the proof of the next lemma we use results from [7]. The difficulty in handling the integral operators, defining M: He0 → He0 , lies in the behaviour of the kernels for x, y near 0 or 1, thus in the corners of Γ . We describe shortly the method which is used in [7] to solve this problem by using Mellin techniques: at each vertex of Γ we assume that there exists a neighborhood where Γ N and Γ D each can be represented by half axis. Then for each operator one realises separately that the kernel in a neighborhood of 0 and 1 can be represented as a Mellin convolution. Cut-off functions are introduced which allow to split their kernels into Mellin convolutions (near the corners) and a smooth remainder. With this localization the operators split into Mellin convolution operators and compact remainder operators. Then one computes the symbols of these Mellin operators (see e.g. [4,5]) and observes that the symbols belong to the class −∞ −1,0 . With the standard theorem from the theory of Mellin operators [2] one obtains that the corresponding Mellin operators are bounded and hence also the original operators are bounded. In the following let χ be a smooth cut-off function on the interval [0, 1]. There is a 0 < < 1/2, with χ (x) = 1, x ∈ [0, ], supp(χ ) ⊂ [0, 1). Further we introduce by the translation operators Ri,1 2−ξ mapping I1 to Ii etc. as follows(always defined 2-periodically): R1,i 2−ξ ϕ(x) := ϕ(2 − x) j,1
R2−ξ ϕ(x) := ϕ(2 − x) R2,i ξ +1 ϕ(x) := ϕ(x + 1) j,2
Rξ −1 ϕ(x) := ϕ(x − 1)
This is needed in constructions of the form 1, j
i,2 1,2 Li j = Ri,1 2−ξ Lk R2−ξ + Rξ +1 Lk Rξ −1 + remainder, k ∈ {0, 1},
where Li j represents an integral operator, which integrates over I j and restricts on Ii . These translations are necessary in order to consider the integral operators Li j as Mellin convolution operators. As it is clear from the context we omit the upper indices
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Collocation with trigonometric polynomials for integral...
in R in the following. We also introduce certain Mellin convolution operators on the half axes (0, ∞) (see [7]). o u(x) = 1/π H
∞
u(y) e u(x) = 1/π Ho (x/y) dy, H y
0
0
∞
∞
ω u(x) = 1/π K
K ω (x/y)
u(y) ω u(x) = 1/π dy L y
0
where
∞
Ho (t) = K ω (t) = −
2 , 1−t 2
He (x/y)
u(y) dy y
L ω (x/y)
u(y) dy y
0
He (t) =
2t , 1−t 2
qt q − 1 sin ω qt q−1 (t − cos ω) (t) = − , L ω t 2q − 2t q cos ω + 1 t 2q − 2t q cos ω + 1
Next we consider the operator M: Lemma 6 For the operator M defined in (87) there holds: 1. M has the representation 0 χ R2−ξ + Rξ +1 χ M 1 χ Rξ −1 + E M = R2−ξ χ M 0 , M 1 and compact E, and the symbols with Mellin convolution operators M −∞ M j , j = 0, 1 are from −1,0 . j (y − i/2))}∞ 2. {arg(1 + M −∞ = 0 for all q ≥ 2. −1 Proof 1. Let G := −A−1 e K21 Ae K12 , then M = G + Ae U. The assertion for the −1 operator G is given in Lemma 12 and for A U in [7, Proof of Lemma 5.3]. Hence the assertion also holds true for M. e 2. Again with Lemma 12 and [7, Proof of Lemma 5.3], there holds with H Ho = −1:
)2 − H o (L −H = 1+ 0 − H e ) = (K o L 0 . G(z) 1 + M(z) ω
(90)
Due to [4, Pages 201–202] there holds (z)| ≤ c1 < 1, z = −1/2, for q ≥ 1. |K ω
(91)
From [7, Proof of Lemma 5.4] we take o L 0 )(y − i/2) = sinh(π y) sinh(2π y/q) + sin(π/q) . h(y) := −(H cosh(π y)(cosh(2π y/q) − cos(π/q)) A closer look into this expression for q > 0 shows that h(y) = h(−y) and h(y) → 1 for y → ±∞, furthermore h(y) is monotone on R+ . Hence h(y) has an absolute
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E. P. Stephan, M. T. Teltscher
maximum or absolute minimum at y = 0. For q ≥ 2 we have h(0) = We therefore have
sin(π/q) 1−cos(π/q)
o L 0 )(z) ≥ 1, z = −1/2, for q ≥ 1. −(H
≥ 1.
(92)
With (90), (91) and (92) we therefore have − i/2) ≥ (1 − c1 ) > 0, 1 + M(y
and thus 2. is proved. Corollary 1 Let q ≥ 2. Then there holds: 1. M : He0 → He0 is bounded. j )χ , j = 0, 1, are invertible on L 2 (0, 1/2). 2. The operators χ (I + M 3. I + M is Fredholm of index 0 on He0 .
Proof All three assertions follow immediately from the last Lemma and the results on Mellin convolution in “Appendix A”: The first assertion follows with Lemma 6, 1. and A(iv); The second assertion follows from Lemma 6, 2. and A(vii); and the third assertion is obtained as follows: by the last Lemma and A(vi) there holds 0 )χ R2−ξ + Rξ +1 χ (I + M 1 )χ Rξ −1 + E, I + M = R2−ξ χ (I + M j )χ , j = 0, 1, are invertible due to 2. and the Mellin convolution operators χ (I + M . E is compact, and hence I + M is Fredholm of index 0 (see [13]). Theorem 1 Let (Assumption 1) hold and let q ≥ 2. Then the operator B : H∼1 × He0 → He0 × He1 has a bounded inverse. Proof The operator D from (84) maps H 1 × He0 to H −1 × He0 isomorphically. Therefore we first consider instead of B the expression D−1 A∗ B = D−1 (D + M) = I + D−1 M =: I + M,
(93)
where I denotes the unitmatrix. Now I + M and I + E3 are both Fredholm of index 0, due to Lemma 6 and the compactness of E3 := A2 E1 . Hence with E2 compact I+M=
I + E3 A K12 : H∼1 × He0 → H 1 × He0 E2 I + M
is Fredholm of index 0 due to [13, Theorem 1.7]. Also B is Fredholm of index 0 due to Lemma 5 and (93). It remains to show that B is injective. Now let z ∈ H∼1 × He0 with Bz = 0. Analogously to [5, Proof of Lemma 4], Z (P) := |(α −1 ) (P)|z(α −1 (P)),
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P ∈ Γ,
Collocation with trigonometric polynomials for integral...
where α −1 : Γ → [0, 2] is the inverse transformation of (11). Z solves the homogenuous equations (3), (4), and because z ∈ H 0 × H 0 , there holds Z ∈ L p (Γ ) for 2q due to [5]. Therefore applying (Assumption 1) the proof is complete. 1 < p < 2q−1 Next we analyse the stability of the collocation method (78) and derive an error estimate in the H∼1 × He0 -norm. It is well known that the use of Mellin operators implies that the stability can only be shown for a slightly modified collocation method [2,6]. For this we define for 0 < r < 1/2 the cut-off operator Tr as follows: For ϕ ∈ V let Tr ϕ be the 2-periodic extension of Tr ϕ(x) =
ϕ(x), x ∈ (r, 1 − r ), 0, x ∈ (0, r ) ∪ (1 − r, 1).
(94)
With this we define an approximation of M by T E1 A−1 K12 r . M r := E2 MTr
(95)
Lemma 7 There exists a r0 > 0, such that
(D + M r )z H −1 ×He0 ≥ c z H∼1 ×He0 , z ∈ H∼1 × He0 , for all r ≤ r0 . Proof We use Lemma 6 to write M r = N r + F r , with T 0 E 0 A−1 K12 r , Fr = 1 , E2 E Tr 0 N Tr 0 χ R2−ξ + Rξ +1 χ M 1 χ Rξ −1 . N = R2−ξ χ M
Nr =
Therefore we have D + Mr = D + Nr + Fr =
−2 −1 A K12 Tr A + Fr . 0 I + N Tr
With Corollary 1, 2. I + N is invertible on He0 , and due to Theorem 1 and Lemma 5 also D + M : H 1 × He0 → H −1 × He0 is invertible. Since furthermore Tr for r → 0 converges strongly towards the identity and since the operators E, E1 and E2 (and hence F r ) are compact, due to [12, Chapter 17, Theorem 1.1] it is only left to show that the stability estimate holds for the operator D + N r . But for this it suffices to show that I + N Tr is stable on He0 . Due to [5, Proof of Theorem 6], [13] this is equivalent to the stability of Tr (I +N )Tr 0 )χ Tr on He0 , which follows from the stability of the finite section operators Tr χ (I +M and Tr χ (I + M1 )χ Tr on L 2 (0, 1/2) and L 2 (1/2, 1) respectively. Lemma 6, 2. and “Appendix A”(vii) imply therefore the assertion.
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E. P. Stephan, M. T. Teltscher
6 Error analysis We apply the cut-off operator Tr defined above to modify the collocation method. With r = i ∗ h in (94) we consider instead of (78) the new collocation equation. Ph (A + K i ∗ h ) zh = Ph C f, zh ∈ Th × Te,h ,
(96)
where Ki ∗ h =
T∗ B K12 i h . 0 U Ti ∗ h
(97)
Like in the transformation from (79) to (89) we want to reformulate the collocation equation (96). For this we define implicitely the projection operator Rh : H −1 × He0 → Th × Te,h via the condition Ph (A∗ )−1 Rh z = Ph (A∗ )−1 z, z ∈ H −1 × He0 .
(98)
As the next lemma shows, Rh is well defined. Lemma 8 The unique solution of (98) is given by Rh 0 z Rh z = Q h Sh
(99)
where Rh = A−1 Ph A, Sh = A−1 e Pe,h Ae , 2 −1 Q h = A−1 e Pe,h K21 A (I − A Ph A).
Furthermore there holds for z ∈ H m−1 × Hem , m ≥ 1, the error estimate
(I − Rh )z H −1 ×He0 ≤ ch m z H m−1 ×Hem .
(100)
Proof From (50) and (51) follows, that A and A−1 commute Ph on Th . Therefore A 0 0 A−1 z, Rh z = −1 P h K A2 Ae −A−1 21 e Pe,h K21 A Ae
which yields (99). We have:
(I − Rh )z H −1 ×He0 ≤ c { (I − Rh )z 1 −1 + Q h z 1 0 + (I − Sh )z 2 0 } . There holds
Q h z 1 0 ≤ c A2 (I − Rh )z 1 1 ≤ c (I − Rh )z 1 −1 ,
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Collocation with trigonometric polynomials for integral...
thus altogether we have with (77):
(I − Rh )z H −1 ×He0 ≤ c (I − Rh )z 1 −1 + c (I − Sh )z 2 0 ≤ c (I − Ph )Az 1 0 + c (I − Pe,h )Ae z 2 1 ≤ ch m Az 1 m + ch m Ae z 2 m+1 ≤ ch m z H m−1 ×Hem . With Lemma 8 and (98) one obtains that (96) is equivalent to (D + Rh M i ∗ h ) zh = Rh e, zh ∈ Th × Te,h ,
(101)
where M i ∗ h = A∗ K i ∗ h =
T∗ E1 A−1 K12 i h . E2 MTi ∗ h
(102)
M i ∗ h corresponds to M r in (95) with r = i ∗ h. Next we give a further lemma, before we can give the final convergence and stability result. But we need the following result from [7]: Lemma 9 Let the operator L be a Mellin convolution operator of the form (129) (in
∈ −∞ . Then for ϕ ∈ H 0 and 0 < r < 1 there the Appendix) on R+ with symbol L −1,1 holds: 1. DLTr ϕ 0 ≤ (c/r ) ϕ 0 . 2. DL(I − Tr )x m ϕ 0 ≤ (cr m−1 ) ϕ 0 . Proof 1. See [7, Proof of Lemma 6.2 (6.15)]. 2. See [7, Proof of Lemma 6.3].
Lemma 10 Let q ≥ 2 and suppose that i ∗ is sufficiently large. Then the estimate
(I − Rh )M i ∗ h z H −1 ×He0 ≤ ε z H∼1 ×He0 , z ∈ H∼1 × He0 , holds for all sufficently small h where ε is independent of z and h. Proof There holds
(I − Rh )M i ∗ h z H −1 ×He0 ∼ (I − Rh )(E1 z 1 + A−1 K12 Ti ∗ h z 2 ) −1 + (Q h (E1 z 1 + A−1 K12 Ti ∗ h z 2 ) 0 ≤ c (I − Rh )E1 z 1 −1 + c (I − Rh )A−1 K12 Ti ∗ h z 2 −1 + c (I − Sh )E2 z 1 0 + c (I − Sh )MTi ∗ h z 2 0
(103)
123
E. P. Stephan, M. T. Teltscher 2 −1 due to the definition of Q h and the boundedness of A−1 e Pe,h K21 A from H∼ to 0 0 He . Since now Rh and Sh converge strongly towards the identity on H and He0 respectively, and E1 is bounded from H∼1 to H 0 , there holds
(I − Rh )E1 z 1 −1 + (I − Sh )E2 z 1 0 < ε z 1 1 for h sufficently small. Hence it is only left to estimate the second and fourth term in (103). Since both I − Ph and I − Sh eliminate the zeroth Fourier coefficient, the Theorem 2 and (100) together with the Poincaré inequality give
(I − Rh )ϕ1 −1 ≤ c (I − Ph )Aϕ1 0 ≤ ch DAϕ1 0 (54)
= ch ADϕ1 0
≤ ch Dϕ1 −1 ,
(I − Sh )ϕ2 0 ≤ ch Dϕ2 0 ,
ϕ1 ∈ H 0 , ϕ2 ∈ H 1 ,
such that we have
(I − Rh )A−1 K12 Ti ∗ h z 2 −1 + (I − Sh )MTi ∗ h z 2 0 ≤ ch DA−1 K12 Ti ∗ h z 2 −1 + DMTi ∗ h z 2 0 (55) = ch DHD K12 Ti ∗ h z 2 −1 + DMTi ∗ h z 2 0 ≤ ch D K12 Ti ∗ h z 2 0 + DMTi ∗ h z 2 0 Ti ∗ h z 2 0 + D 2 He K21 Ae K12 Ti ∗ h z 2 0 ≤ ch D K12 2 + D He U Ti ∗ h z 2 0 Ti ∗ h z 2 0 + D 2 He K21 Ae K12 Ti ∗ h z 2 0 ≤ ch D K12 2 (104) + D U Ti ∗ h z 2 0 , DU and From Lemma 12 and [7, Remark 2] we know that the operators K12 K21 Ae K12 have a representation
D 2 He
1 χ R(2) + E 0 χ + R(1) χ L χL 0 , L 1 are Mellin convolution with appropriate translation operators R(1) , R(2) , where L −∞ + operators on R with symbols on −1,1 , and E is bounded from He0 to He1 . Thus we only have to apply Lemma 9 with r = i ∗ h, and then the foregoing estimate implies
(I − Rh )A−1 K12 Ti ∗ h z 2 −1 + (I − Sh )MTi ∗ h z 2 0 ≤ (c/i ∗ ) z 2 0 ,
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Collocation with trigonometric polynomials for integral...
if h is small enough. If we now choose i ∗ large enough (note that: i ∗ and h depend on ε, but not on each other), then we obtain the desired estimate. Finally we can present the convergence result of our collocation method: Theorem 2 Suppose (Assumptiion 1) holds and q ≥ 2, and i ∗ is suffiently large. Then for all f ∈ H∼s × Hes+1 , s > 1/2, there exists a unique solution zh ∈ Th × Te,h of (96). Suppose the exact solution z = (z 1 , z 2 )t of (42) satisfies z ∈ H∼m+1 × Hem , m z 2 = x(1 − x) w, w ∈ He0 ,
(105) (106)
then we have the estimate
z − zh H 1 ×He0 ≤ ch m ( z H∼m+1 ×H m + w 0 ). e
(107)
Remark 2 1. Indeed it can be shown that the solution z of (42) has the form (106) for sufficiently large m, if f and g are sufficiently smooth and the grading parameter q is large enough [7, Remark 3]. 2. Due to (61) the assumption (105) is equivalent to:
m+1 × Hem and z 1 (m) (0) = z 1 (m) (1) = 0. (z 1 , z 2 )t ∈ H Proof of Theorem 2 The stability of the method (101) follows from Lemmas 7 and 10, because there holds
(D + Rh M i ∗ h )zh H −1 ×He0 ≥ (D + M i ∗ h )zh H −1 ×He0 − (I − Rh )M i ∗ h zh H −1 ×He0 ≥ (c − ε) zh H∼1 ×He0 , zh ∈ Th × Te,h ,
(108)
and ε < c for i ∗ h sufficiently large. Since the right hand side of (96) is well defined for f ∈ H∼s × Hes+1 , s > 1/2, the first assertion is therefore shown. It remains to show the error estimate (107). Note due to (77) we have
z − zh H 1 ×He0 ≤ (I − Ph )zh H 1 ×He0 + zh − Ph z H 1 ×He0 ≤ ch m z H∼m+1 ×H m + zh − Ph z H 1 ×He0 e
The second term on the right hand side we estimate as follows: First we apply (108) with zh − Ph z instead of zh ; then with (101) and (89) we have Rh (D + M)z = Rh e = (D + Rh M i ∗ h )zh and Rh (D + M i ∗ h )Ph z = (D + Rh M i ∗ h )Ph z. Finally we apply (77) and (100) and obtain:
zh − Ph z H 1 ×He0 ≤ c (D + Rh M i ∗ h )(zh − Ph z) H −1 ×He0 = c Rh [(D + M)z − (D + M i ∗ h )Ph z] H −1 ×He0
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E. P. Stephan, M. T. Teltscher
= c ( Rh D − DPh )z + Rh (M − M i ∗ h Ph )z H −1 ×He0 =Rh −I+I
≤ c (Rh − I)Dz H −1 ×He0 + c D(I − Ph )z H −1 ×He0 + c Rh (M − M i ∗ h Ph )z H −1 ×He0 =Rh −I+I
≤ ch m Dz H m−1 ×Hem + c (I − Ph )z H 1 ×He0 + c (M − M i ∗ h Ph )z H −1 ×He0 + c (I − Rh )(M − M i ∗ h Ph )z H −1 ×He0 ≤ ch m z H m+1 ×Hem + c (M − M i ∗ h Ph )z H −1 ×He0 + ch (M − M i ∗ h Ph )z H 0 ×He1 .
(109)
The second term on the right hand side of (109) we can estimate with (77) and the assumption (106). Note that the sequence of operators M i ∗ h : H 1 × He0 → H −1 × He0 is uniformly bounded, since the operator M is bounded and the sequence of operators Ti ∗ h is uniformly bounded. Hence we have
((M − M i ∗ h Ph )z H −1 ×He0 ≤ M i ∗ h (I − Ph )z H −1 ×He0 + (M − M i ∗ h )z H −1 ×He0 ≤ c (I − Ph )z H 1 ×He0 + c (I − Ti ∗ h )z 2 0 ≤ ch m z H∼m+1 ×H m + ch m w 0 . e
Next we estimate the last expression of (109).
(M − M i ∗ h Ph )z H 0 ×He1 ≤ c E1 (I − Ph )z 1 0 + c E2 (I − Ph )z 1 1 + c A−1 K12 (I − Ti ∗ h Pe,h )z 2 0 + c M(I − Ti ∗ h Pe,h )z 2 1 m ≤ ch z 1 m+1 + c D K12 (I − Ti ∗ h Pe,h )z 2 0 + c DM(I − Ti ∗ h Pe,h )z 2 0 m ≤ ch z 1 m+1 + c D K12 (I − Ti ∗ h )z 2 0 + c D K12 Ti ∗ h (I − Pe,h )z 2 0 + c DM(I − Ti ∗ h )z 2 0 + c DMTi ∗ h (I − Pe,h )z 2 0 .
(110)
Due to Lemma 9, 1., see also the proof of Lemma 10 we know that, the third and the fifth term of the right hand side of (110) are bounded from above by c i ∗h
(I − Pe,h )z 2 0 ≤ ch m−1 z 2 m .
In the same way we can estimate the second and the fourth term with the help of Lemma 9, and obtain c(i ∗ h)m−1 w 0 as upper bound (i ∗ is independent of h). Thus the proof is complete.
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Collocation with trigonometric polynomials for integral...
Remark 3 In the above theorem we have derived an estimate for the collocation error z 1 −z h1 in the H 1 -norm. In our derivation we have taken into account the approximation error on the interval [1, 2] in our estimate, despite we know that z 1 ≡ 0 on [1, 2] (z 1 was defined as zero extension of u α : I1 → R). Hence it would be inappropriate if we would take in our numerical experiments the larger error z 1 − z h1 1 as approximation error. Instead we will take below the error
ρz 1 − ρz 1h H 1 (I1 ) = z 1 − ρz 1h 1 ≤ z 1 − z h1 1 (where ρ is the restriction defined in (27)).
7 Numerical results To approximate the arising integrals in (43) we use a trapezoidal quadrature with equidistant nodes σ j = j h + h/2, j = 0, . . . , 2n − 1, n ∈ N. With the integral kernels (14)–(18), the system (43) becomes the system n 2n h w1 (s , σ j ) + w2 (s , σ j ) z 1h (σ j ) + hk(σ j , s )z 2h (σ j ) j=1
= g(s ) −
j=n+1 n
hk(σ j , s )g(σ j ) +
j=1
2n
h w1 (s , σ j ) + w2 (s , σ j ) f (σ j ),
j=n+1
= 0, . . . , n − 1, n 2n h − hk(s , σ j )z 1 (σ j ) + hv(s , σ j )z 2h (σ j ) j=1
= f (s ) −
j=n+1 n
hv(s , σ j )g(σ j ) +
j=1
2n
hk(s , σ j ) f (σ j ),
j=n+1
= n, . . . , 2n − 1. Here we have already k (s, σ ) replaced by k(σ, s). We must regularise the integrals, if our quadrature should work well. By choosing z 1 (0) = z 1 (1) = 0 in (60) we have already done some regularisation, since in this way we weaken the poles of the integral kernels for σ → 0. The hypersingular operator we regularise with the help of A−1 from (83). This we have already done in our error analysis in Sect. 2: there we have shown in Lemma 3 that B = W11 −A−1 is compact from H∼1 into H 0 . We do this in our computation using the integral kernel a˜ of A−1 .
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From (55) follows: π
π 1 ∂ cot (s − σ ) + 2 ∂s 2 2 π 1 −π π = − − 1 . = + 2 2 1 − cos(π(s − σ )) 4 sin2 (π/2(s − σ ))
a(s, ˜ σ) =
For the single layer potential part V22 we proceed as follows: We regularise V22 : He0 → He1 with the operator Ae . From [7] we know that the kernel of U = V22 −Ae is bounded on each compact subset of the interval [1,2]. The integral kernel ae of Ae is known from (68) ae = −
1 log|2e−1 (cos(π x) − cos(π y))|. π
With these regularisations the system becomes: A−1 z 1h (s ) +
n h w1 (s , σ j ) + w2 (s , σ j ) − a(s ˜ , σ j ) z 1h (σ j ) j=1
+
2n
hk(σ j , s )z 2h (σ j )
j=n+1
= g(s ) −
n
hk(σ j , s )g(σ j ) +
j=1
2n
h w1 (s , σ j ) + w2 (s , σ j ) f (σ j ),
j=n+1
= 0, . . . , n − 1, n 2n − hk(s , σ j )z 1h (σ j ) + Ae z 2h (s ) + h(v(s , σ j ) − ae (s , σ j ))z 2h (σ j ) j=1
= f (s ) −
j=n+1 n
hv(s , σ j )g(σ j ) +
j=1
2n
hk(s , σ j ) f (σ j ),
j=n+1
= n, . . . , 2n − 1. Next we show the performance of our collocation method with an example Let ν ∈ C ∞ [0, 1], with ν(0) = 0, ν(1) = 1, ν (x) > 0, 0 ≤ x ≤ 1,
(111)
and take γ (x) :=
123
ν q (x) . ν q (x) + ν q (1 − x)
(112)
Collocation with trigonometric polynomials for integral...
We choose from [11] ν(x) =
1 1 1 1 − (2x − 1)3 + (2x − 1) + , 2 q q 2
with q ≥ 2. Let Γ be the boundary of a deformed circle with vertices P0 and P1 . As parametrisation we take: α(x) ˜ = cos π x +
π π (x − 1)2 − x 2 , sin π x 2 tan(ω0 /2) 2 tan(ω1 /2)
t ,
Γ is symmetric to the real axis; Ω is left of Γ , and Γ N is above and Γ D is below the x-axis. Take α(γ ˜ (x)), 0≤x ≤1 α(x) := . α(2 ˜ − γ (2 − x)), 1 ≤ x ≤ 2 With the polar coordinates (ri , θi ), i = 0, 1 (centered in Pi ) of a point P ∈ R2 with 0 ≤ θ0 ≤ 2π, −π ≤ θ1 ≤ π we take f 0 (P) := r0λ0 sin (λ0 (θ0 − π + ω0 /2)) , f 1 (P) := r1λ1 sin (λ1 (θ1 + ω1 /2)) , with λi =
π 1 = , i = 0, 1. ωi 1 − χi
The functions f 0 , f 1 are harmonic and their normal derivatives in P0 and in P1 respectively are singular. We take f (P) := f 0 (P) + f 1 (P), and set in (12), (13) gα (x) =
∂f (α(x))|α (x)|, x ∈ I1 , ∂n
f α (x) = f (α(x)), x ∈ I2 .
In our example we choose (see Fig. 1) 6 π, 5 7 ω1 = π, 5
ω0 =
i.e.
χ0 = −0.2,
i.e.
χ1 = −0.4.
We compute the approximation errors e1h := ρz 1 − ρz 1h H 1 (I1 ) and e2h := z 2 − z 2h 0 for n = 1, . . . , 1024 with these Sobolev norms defined by Fourier coefficients.
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E. P. Stephan, M. T. Teltscher
Fig. 1 Distribution of the collocation points for χ0 = −0.2, χ1 = −0.4, n = 64, q0 = 2, q1 = 4
Tabel 1 shows our numerical results with the empirically determined convergence rates (EOC). Those numbers demonstrate the improvement of the convergence order for increasing values of the grading exponent q as expected from Theorem 2. The error estimate (107) gives for q = 2, 3, 4, 5 the theoretical convergence rates 0.929, 1.64, 2.36 and 3.07 for e1h and e2h . One clearly observes the improvement of the convergence rates for higher grading parameters . While the (EOC)’s for e2h are in good agreement with the theoretical convergence rates the approximation for ρz 1h is mostly better. Note that we consider the error e1h := ρz 1 − ρz 1h H 1 (I1 ) and not the worse error z 1 − z 1h 1 (see Remark 3). The case q0 = q1 = 5 led to numerical problems, therefore we listed the results for q0 = 5, q1 = 4 in the last column with qi at vertex Pi .
8 Proofs with Mellin techniques In this section we use properties of Mellin transformation and Mellin convolutions which are collected in the appendix. ∞ Lemma 11 Let lϕ(x) := − πq 0 log(y)ϕ(y)dy, and let Vω , Kω , Kω and Wω be the and W in the case operators V21 − l, K21 , K21 21 α1 (x) = P + C x q , x ∈ [0, ∞), α2 (x) = P + Ceiω x q ,
123
(113)
Collocation with trigonometric polynomials for integral... Table 1 Error and convergence rates (χ0 = − 0.2, χ1 = − 0.4) n
q=2
q=3
e1h 16
EOC
4.499e−4
e2h 3.518e−2
0.918 32
2.127e−3
64
9.478e−4
128
3.950e−4
0.893
2.862e−3
512
7.088e−5
1024
3.244e−5
n
q=4
1.13
1.66 1.811e−4
2.06 3.231e−6
1.06
EOC
1.704e−4
e2h
1.66 5.738e−5
1.94 8.439e−7
32
1.198e−4
64
1.915e−5
128
3.281e−6
256
5.930e−7
512
1.113e−7
EOC
2.861e−3 2.53
1.69 1.777e−5
2.42
2.38
2.37
2.41
2.38
2.37
2.04
2.37 3.362e−6
2.37 1.078e−7
2.35 1.288e−7
2.38 1.737e−5
5.565e−7
6.549e−7
2.41 9.058e−5
2.888e−6
3.376e−6
2.73 4.808e−4
2.42
2.39
2.47
EOC
3.188e−3
1.507e−5
1.750e−5
e2h
2.91 8.055e−5
9.179e−5 2.54
EOC
1.542e−4
4.912e−4
2.711e−8
e1h
2.54
2.65
1024
2.16 1.344e−5
7.007e−4
1.66 5.704e−4
q0 = 5, q1 = 4
e1h 16
2.23
0.971 1.460e−3
1.68 1.805e−3
5.989e−5 0.930
1.21
1.72 5.782e−3
2.28
0.907
1.27
EOC
1.905e−2
2.804e−4
5.454e−3
e2h
2.34 1.362e−3
1.023e−2 1.26
EOC
1.557e−4
1.899e−2
1.642e−4
e1h
0.889
1.17
256
EOC
2.36 6.533e−7
2.02 2.658e−8
2.34 1.286e−7
with a complex-valued constant C. Then the above operators can be given as Mellin convolution operators with the Mellin kernels 1 log|1 − t q eiω |, π qt q sin ω 1 , K ω (t) = − · 2q π t − 2t q cos ω + 1 qt q−1 sin ω 1 ∂ 1 K ω (t) = − · 2q = Vω (t)qt q−1 , q π t − 2t cos ω + 1 x ∂ω Vω (t) =
(114) (115) (116)
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E. P. Stephan, M. T. Teltscher
Wω (t) =
1 ∂ K ω (t)qt q−1 , x ∂ω
(117)
where t = x/y, and with the symbols
cosh (π − ω) qλ ϕ (λ − i), λ λ q sinh π q
sinh (π − ω) qλ ϕ (λ), K ω (λ) ϕ (λ) = K ω ϕ(λ) = sinh π qλ
sinh (π − ω) λ+i q
K ϕ (λ), ϕ (λ) = ω ϕ(λ) = K ω (λ + i) sinh π λ+i q λ+i 2 ϕ(λ) = W (λ + i) ϕ (λ + i) = ϕ (λ + i) W Vω (λ + i) ω ω q
λ+i λ + i cosh (π − ω) q
ϕ (λ + i), =− q sinh π λ+i ϕ (λ − i) = − V ω ϕ(λ) = V ω (λ)
λ ∈ (−q, 0),
(118)
λ ∈ (−q, q),
(119)
λ ∈ (−q − 1, q − 1),
(120) (121)
λ ∈ (−q − 1, q − 1).
(122)
q
Proof The kernel (116) and the symbol (120) of Kω we can take from [7] (there K is written instead of K ). With Kω we can proceed as in [7]: due to (113) there holds: 1 Kω ϕ(x) = − π =−
1 π
∞ 0
(α2 (x) − α1 (y), n α1 (y) ) |α1 (y)|ϕ(y)dy |α2 (x) − α1 (y)|2
∞ x 2q 0
q x q y q sin ω ϕ(y) dy. · q q 2q − 2x y cos ω + y y
Therefore K ω (t) = −
qt q sin ω 1 · 2q = t K ω (t). π t − 2t q cos ω + 1
From the last equation and the convolution theorem (“Appendix A” (viii)) we obtain (119). Furthermore one gets: 1 1 qy q−1 , K ω (x, y) := K ω (x/y) = y π x q eiω − y q 1 1 q x q−1 eiω . K ω (x, y) := K ω (x/y) = y π x q eiω − y q
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Collocation with trigonometric polynomials for integral...
The representation (114) is immediate, if one inserts (113) into the definition of V21 − l, and one calculates K ω (x/y) =
1 ∂ Vω (x/y) q x ∂ω
Then the convolution theorem yields the desired symbol (118); whereas application of the convolution theorem to (117) yields (121). Next we complete the proof of Lemma 6. Lemma 12 Let G := −A−1 e K21 Ae K12 . Then
0 χ R2−ξ + Rξ +1 χ G 1 χ Rξ −1 + E G = R2−ξ χ G
(123)
= −W0 Kωi V0 Kω + E = with E compact and Mellin convolution operators G i i i := D H e Kωi Ae Kω has the Mellin symbol e Kωi Ae Kω + E where G DH i
i
2 −∞
= G . Kω i ∈ i −1,1
Proof We know from [7] K21 = R2−ξ χ Kω0 χ + Rξ +1 χ Kω1 χ R1−ξ + E, K12 = χ Kω 0 χ R2−ξ + R1−ξ χ Kω 1 χ Rξ −1 + E. Writing G = (−W22 − A−1 e + W22 ) K21 (V11 + Ae − V11 ) K12
one observes +E G = −W22 K21 V11 K12
with a compact operator E since the operators −A−1 e + W22 and Ae − V11 are smoothing for q = 2 (see[7]). Now the representation (123) follows with the Mellin convolution operators = −W0 Kωi V0 Kω + E G i i where W0 and V0 denote the Mellin convolution operators with kernels W0 and V0 , respectively. +E. Therefore the symbol Furthermore from (72) follows G = DHe K21 Ae K12 of Gi := D He Kωi Ae Kωi we calculate as follows. First ϕ (λ) = (−iλ + 1) ϕ (λ + i)
(124)
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E. P. Stephan, M. T. Teltscher
Let the Mellin operator L have the forn 1 Lϕ(x) = π
∞
L(x/y)y β
ϕ(y) dy. y
0
Then there holds 1 DLϕ(x) = π =
1 π
∞ 0
∞
∂ ϕ(y) L(x/y)y β dy ∂x y L (t)|t=x/y y β−1
ϕ(y) dy. y
(125)
0
Due to the convolution theorem we have Lu(λ) = L(λ) u (λ − iβ). Therefore there holds with (124) and (125):
(λ) DLϕ(λ) =L ϕ (λ − i(β − 1)) = (−iλ + 1) L(λ + i) ϕ (λ − i(β − 1)).
(126)
e be the Mellin operator belonging to He (i.e. the part of the operator which is Let H localized in a corner P). With [7], (73), (126) and “Appendix A”(v) we have e (λ + i) e ϕ(λ) = (−iλ + 1)H ϕ (λ + i) D H
(127)
e ϕ(λ) = DA e (λ) H H ϕ (λ) = Ae ϕ(λ + i) e ϕ(λ) = (−iλ + 1)
(128)
and
Therefore we have e (λ + i)(Kω Gϕ(λ) = (−iλ + 1)H Ae Kω ϕ)(λ + i) e (λ + i)K ω (λ + i)A = (−iλ + 1)H e Kω ϕ(λ + i) e (λ + i)K e (λ)K ω (λ + i) i H = (−iλ + 1)H ω ϕ(λ) λ+i e (λ + i)H e (λ) K ω (λ + i) 2 =H ϕ (λ), hence
e (λ + i)H e (λ) K =H ω (λ + i) 2 G(λ)
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Collocation with trigonometric polynomials for integral...
2 λ + i λ coth π Kω (λ + i) = coth π 2 2 2 2 sinh (π − ω)(λ + i)/q , = Kω (λ + i) = sinh2 π(λ + i)/q i.e. 2
. = K G ω ∈ Now from [7] we know K ω
−∞
−1,1 .
On the other hand we get
ϕ(λ) = W 0 (λ + i) K ω (λ + i)V 0 (λ + i) K ω (λ + i) G ϕ (λ). yielding 2
(λ) = ( K ω (λ + i))2 coth π λ + i G 2 0 and the proof is complete. when taking q = 2 in the Mellin symbols V 0 and W
A Appendix Here we list some results on Mellin convolution operators (see [5,7] and [3]). (i) The Mellin transform v of a function v : R+ → C is defined as:
∞
v (z) =
x i z−1 v(x)d x.
0
s (R+ ) into the completion of The operator v → v is an isomorphism from H ∞ C (z = s − 1/2) under the norm
1/2 (1 + |z| ) | v (z)| |dz| 2 s
z=s−1/2
2
.
The inverse to v → v is given by v(x) =
1 2π
z=−1/2
x −i z v (z)|dz|.
v is analytic on C. For v ∈ C0∞ (R+ ), (ii) Let K be a Mellin convolution operator, i.e. K has the form Kv(x) =
∞
K 0
x v(y) dy. y y
(129)
123
E. P. Stephan, M. T. Teltscher
with kernel K and symbol
= symbol(K) = K
(z). K(z)
v (z), and K is a continuous If x −1/2 K (x) ∈ L 1 (R+ ), then Kv(z) = K(z) operator on L 2 (R+ ) with norm bounded by
K 0 ≤
sup |K(z)|.
z=−1/2
(130)
(iii) If K and L are Mellin convolution operators with bounded symbols on z = =K
· L.
−1/2, then KL is a Mellin operator with bounded symbol KL
≥ c > 0, z = −1/2, then K continuously invertible on L 2 (R+ ), where If |K|
−1 . its inverse is again a Mellin operator with symbol K−1 = (K)
(iv) The symbol K(z) of a Mellin convolution operator K is said to be from the , class −∞ α,β α < −1/2 < β, if it is analytic in the strip α < z < β and the estimates
= O((1 + |z|)−k ), |z| → ∞, k ∈ N, K(z) hold in each substrip α < z < β , α < α < 1/2 < β < β . Then the kernel K of K satisfies the estimates sup |x k−ρ D k K (x)| < ∞, k ∈ Z+ , α < ρ < β.
x∈R+
Especially for k = 0 this implies x −1/2 K (x) < cx ρ−1/2 , such that x −1/2 K (x) ∈ L 1 (R+ ) (choose ρ > −1/2 near 0, ρ < −1/2 near ∞), and due to (ii) K is then a bounded operator on L 2 (R+ ) with bound (130). (v) The Cauchy singular operator defined on R+ 1 Hv(x) = p.v. π
∞ 0
v(y)dy y−x
is a Mellin operator with symbol H(z) = −i coth π z, which is analytic for −1 < z < 0, and there holds
= ∓i + O((1 + |z|)−k ), z → ±∞, k ∈ N, H(z) on each substrip α < z < β , 0 < α < 1/2 < β < 1. For the kernel H (x) = (1 − x)−1 then (130) holds.
∈ −∞ or K(z) = (vi) Let χ be a smooth functions with supp(χ ) ⊂ [0, 1]. If K α,β −i coth π z hold for Mellin convolution operator K, then the commutator χ K − / supp(χ ), then also χ K is Kχ I is compact on L 2 (R+ ). If furthermore 0 ∈ compact on L 2 (R+ ).
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Collocation with trigonometric polynomials for integral...
(vii) Let φ and φr , 0 < r < 1, be the characteristic functions on the intervals (0, 1) and (r, 1). Assume there holds 1. x −1/2 K (x) ∈ L 1 (R+ ),
2. 1 + K(−i/2 + y) = 0, y ∈ R, and
3. {arg(1 + K(−i/2 + y))}∞ −∞ = 0, ∞ where {arg ·}−∞ denotes the change of the arguments , when y runs from −∞ to ∞. Then the Mellin convolution operator φ(I + K)φ is continuously invertible on L 2 (0, 1), and the finite section operators φr (I + K)φr are stable, i.e. there exist a r0 > 0 and a c > 0, such that
φr (I + K)φr v 0 ≥ c φr v 0 , v ∈ L 2 (0, 1), for all r ≤ r0 . (viii) (Convolution theorem) For v ∈ C0∞ (0, ∞)
∞
Kv(x) = 0
v(y) x x α yβ dy, α, β ∈ C, K y y
there holds
− iα) Kv(z) = K(z v (z − i(α + β)).
References 1. Atkinson, K.E., Sloan, I.H.: The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math. Comput. 56, 119–139 (1991) 2. Chandler, G.A., Graham, I.G.: Product integration-collocation methods for noncompact integral operator equations. Math. Comput. 50, 125–138 (1988) 3. Costabel, M., Stephan, E.P.: The method of Mellin transformation for boundary integral equations on curves with corners. In: 5th IMACS Conference, Lehigh Univ. Computer methods for Cauchy singular integral equations (1984) 4. Costabel, M., Stephan, E.P.: Boundary Integral Equations for Mixed Boundary Value Problems in Polygonal Domains and Galerkin Approximation. Mathematical Models and Methods in Mechanics, vol. 15, pp. 175–251. Banach Center Publications, PWN-Polish-Scientific Publishers, Warsaw (1985) 5. Elschner, J., Graham, I.G.: An optimal order collocation method for first kind boundary integral equations on polygons. Numer. Math. 70, 1–31 (1995) 6. Elschner, J., Graham, I.G.: Numerical methods for integral equations of Mellin type. J. Comput. Appl. Math. 125, 423–437 (2000) 7. Elschner, J., Jeon, Y., Sloan, I.H., Stephan, E.P.: The collocation method for mixed boundary value problems on domains with curved polygonal boundaries. Numer. Math. 76(3), 355–381 (1997) 8. Elschner, J., Stephan, E.P.: A discrete collocation method for Symm’s integral equation on curves with corners. J. Comput. Appl. Math. 75, 131–146 (1996) 9. Hartmann, T., Stephan, E.P.: A discrete collocation method for a hypersingular integral equation on curves with cornes. In: Dick, J., Kuo, E.Y., Wozniakowski, H. (eds.) Festschrift for the 80th Birthday of Ian Sloan. Springer, Berlin (2018) 10. Jeon, Y., Sloan, I.H., Stephan, E.P., Elschner, J.: Discrete qualocation methods for logarithmic-kernel intregral equations on a piecewise smooth boundary. Adv. Comput. Math. 7, 547–571 (1997) 11. Kress, R.: A Nyström method for boundary integral equations on domains with corners. Numer. Math. 58, 145–161 (1990) 12. Mikhlin, S.G., Prößdorf, S.: Singular Integral Operators. Springer, Berlin (1986)
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E. P. Stephan, M. T. Teltscher 13. Prößdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Operator Theory, vol. 52. Birkhäuser, Basel-Stuttgart (1991) 14. Yan, Y., Sloan, I.H.: On integral equations of the first kind with logarithmic kernels. J. Integr. Equ. Appl. 1(4), 549–579 (1988)
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