Numerical Algorithms 22 (1999) 327–351
327
Full collocation methods for some boundary integral equations ∗ Ricardo Celorrio a and Francisco-Javier Sayas b,∗∗ a
Departamento de Matem´atica Aplicada – EUITIZ, Universidad de Zaragoza, Corona de Arag´on, E-50009 Zaragoza, Spain E-mail:
[email protected] b Departamento de Matem´atica Aplicada – CPS, Universidad de Zaragoza, Mar´ıa de Luna, 3, E-50015 Zaragoza, Spain E-mail:
[email protected]
Received 26 October 1998; revised 19 November 1999 Communicated by C. Brezinski
In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given. Keywords: logarithmic integral equations, collocation, full discretization, extrapolation AMS subject classification: 65R20, 45L10
1.
Introduction
In this paper we deal with the numerical solution of some Fredholm linear integral equations of the first kind with logarithmic kernel Z 1 2 A( · , t) log x( · ) − x(t) + B( · , t) g(t) dt = f , (1.1) 0
related to a smooth curve in the plane given by its parameterization x. These equations and more complicated models, including systems with some integral restrictions and possibly some additional scalar unknowns, arise in a vast number of situations as parameterized forms of boundary integral equations. Boundary integral representations have been widely used as practical means of obtaining a dimension reduction or of dealing with unbounded domains for some homogeneous boundary value problems ∗ ∗∗
The work on this paper was partially supported by the DGES project No. PB97-1013. Corresponding author.
J.C. Baltzer AG, Science Publishers
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related to elliptic partial differential operators for which a fundamental solution is known. For a large set of examples where these equations appear see [2,13]. The use of spline collocation methods for the numerical solution of (1.1), as well as many other equations fitting in the frame of pseudodifferential operators, has been of great importance both in theory and in practice. However, the analysis of these methods is recent (see [2,9,11]). The advantages of collocation methods in comparison to the classical Galerkin schemes are well known. Despite their poorer convergence properties, the idea of a collocation method is basically simpler (requiring no “variational formulation”) and in the evaluation of the matrix of the method, single integrals have to be computed instead of the double integrals of Galerkin methods. A generalization of the collocation methods, by means of adequate linear combinations of collocation matrices in several grids, has created the new family of the so-called qualocation methods (see [14] for a review). On the other hand, some new results of superconvergence in nodes plus the existence of an asymptotic expansion of the error have been proven in [3]. Such an expression for the error allows the use of Richardson extrapolation (i.e., several discretization levels are used to cancel terms in the error expansion) as a way of accelerating the convergence and estimating the error a posteriori. As is the case with Galerkin methods, the collocation and qualocation families have the drawback of not being fully discrete. Therefore, an adequate strategy has to be given for a not too costly but good enough computation of the integrals appearing in the linear system. Some work has been done in this direction: see [7,10], for instance. In our work we develop some ideas originally applied to the full discretization of the Galerkin equations, given in [5,6]. The basic idea is to split the kernel into a part containing most of the singularity, which is evaluated exactly, plus a smoother part. The numerical integration is adapted from a single basic formula taking a B-spline as weight function. Furthermore, some advantages are drawn from taking some particular points, even outside the integration interval, as nodes of the basic quadrature rule. Namely, we can greatly reduce the number of evaluations of the kernel functions, which, together with the solution of the linear system, is the costliest part of the algorithm in terms of CPU time. After introducing the problem, the functional setting and recalling the properties of the spline collocation method in section 2, we split the matrix related to the B-spline basis into two parts. This decomposition is used in section 4 to define a fully discrete method, which will be called a full collocation method. In section 5 we analyze the stability of the method and give some first convergence estimates, which will be improved for smoother solutions in section 6. With the help of some technical results gathered in section 7, an asymptotic expansion of the error of the full collocation method is derived in section 6. For reasons of keeping the problem as simple as possible we restrict our attention to the situation when the number of divisions of the interval is even. A parallel analysis can be easily given for an odd number of intervals, although the resulting asymptotic expansion can be seen to be different. The method of section 4 is defined so that the good approximation properties of the collocation
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method in weak norms are kept. We also show how the nodal superconvergence of the method for splines of even degree is maintained. Finally, in section 8 we face two additional questions: the use of a linear postprocessing of the numerical solution of the integral equation and numerical quadrature in the right-hand side. Postprocessing aims at solving a boundary value problem via an integral representation. As a simple application of a general theory developed in [13], we show how the asymptotic expansions of section 6 are transformed into new asymptotic expansions of the application of these postprocesses, i.e., to the calculation of potentials. On the other hand, the right-hand side of the integral equation often appears in integral form. We briefly discuss the effect of numerical quadrature at this level. Section 9 is devoted to illustrating the results of the preceding sections with some numerical examples. 2.
Statement of the problem
Let us consider a 1-periodic continuous function f : R → C. Let x : R → Γ ⊂ R2 , be a regular 1-periodic parameterization of a simple Jordan curve in the plane Γ. 6 0 for all s. Therefore, x(s) 6= x(t), for all s, t such that 0 < |s − t| < 1, and |x0 (s)| = We consider the logarithmic kernel 2 (2.1) V (s, t) := A(s, t) log x(s) − x(t) + B(s, t), where A, B : R2 → C are smooth functions, 1-periodic in both variables, and A satisfies A(s, s) 6= 0,
∀s ∈ R.
(2.2)
The problem whose solution we intend to approximate is the following weakly singular integral equation of the first kind: Z 1 V g := V ( · , t)g(t) dt = f. (2.3) 0
We will further assume that if V g = 0 and g is smooth, then g = 0. The functional setting to study this kind of equation is usually that of periodic Sobolev spaces. If u ∈ C ∞ := {u : R → C: u ∈ C ∞ (R), u(1 + · ) = u}, let Z 1 u(t)e−2kπit dt, k ∈ Z, u ˆ(k) := 0
be its Fourier coefficients. For r ∈ R, the space H r is the completion of C ∞ with the norm 2 1/2 2 X 2r kukr := u ˆ(k) ˆ(0) + |k| u . k6=0
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It is well known that H r is a Hilbert space which can be identified with the space of 1-periodic distributions u such that kukr < ∞, where Fourier coefficients are defined by duality. Furthermore, the elements of H 0 can be identified with the 1-periodization of functions in L2 (0, 1) and the inner products of both spaces coincide. From the structure of the kernel, hypothesis (2.2) and the emptiness of the kernel assumed above, it can be easily proven that V is an isomorphism from H r to H r+1 . For questions related to this and similar operators as well as of the Sobolev spaces, we refer to [17]. Among the most widely used numerical methods for equations of the form (2.3), spline collocation methods have been the object of greatest interest, both from the theoretical and the practical points of view. Let N be a positive integer and h := 1/N . Consider a uniform partition of R given by the points si := s0 + i h,
i ∈ Z\{0},
where s0 can be arbitrarily chosen. Let m be a non-negative integer and let Vh be the space of 1-periodic smoothest splines of degree m with knots on the grid {si }, i.e., Vh := uh ∈ H m : uh |(si ,si+1 ) ∈ Pm , ∀i , where Pm is the space of complex polynomials of degree not greater than m. From the Sobolev embedding theorem, it is simple to see that, for m > 1, uh ∈ Vh has m − 1 continuous derivatives. For m even let h zi := si + , i ∈ Z, 2 and for m odd, let zi := si ,
i ∈ Z.
The set of points {zi } will be referred to as the collocation nodes of the grid. The collocation method with splines of degree m for the approximate solution of (2.3) is the following numerical scheme: gh ∈ Vh
such that
V gh (zi ) = f (zi ),
i = 1, . . . , N.
(Ph )
To avoid an unnecessary amount of parameters, explicit reference to the degree m will be avoided. The following result gathers the classical theory on the collocation method for logarithmic equations [2,9,11]. Proposition 2.1. There exists h0 such that for all h 6 h0 the equations (Ph ) are uniquely solvable. Moreover, we have the convergence bounds: for all s ∈ [−1, m + 1/2), s 6 t 6 m + 1 and t > −1/2, there exists C independent of h such that for all g ∈ H t, kgh − gks 6 Cht−s kgkt .
(2.4)
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Moreover, if m is even we have for all g ∈ H m+2 kgh − gk−2 6 Chm+3 kgkm+2 . 3.
(2.5)
On the matrices of the collocation method
Once a basis of Vh is chosen, problem (Ph ) becomes an N × N linear system for the coordinates of gh . In this way, we define the functions µ0 := χ(−1/2,1/2) ,
µm := µm−1 ∗ µ0 ,
m > 1,
where χE denotes the characteristic function of the set E. Note that suppµm = [−(m+ 1)/2, (m+1)/2]. For i ∈ Z, let ψi be the 1-periodic extension of µm (( · −zi )/h)). For the sake of simplicity we will not make explicit the order of the splines. Obviously, we have ψi ≡ ψj , if i ≡ j (mod N ). It is easy to prove that for N > m + 1, Vh = span{ψ1 , . . . , ψN }. This basis of minimally supported functions of Vh is often referred to as the B-spline basis. Then, (Ph ) is equivalent to gh =
N X j=1
gj ψj ∈ Vh ,
such that
N X
V ψj (zi )gj = f (zi ),
i = 1, . . . , N.
j=1
In this section we give a decomposition of the matrix of the spline collocation which will be essential for the forthcoming numerical approximation. Let us denote the coefficient matrix A := (ai,j )N i,j=1 with elements ai,j := V ψj (zi ),
i, j = 1, . . . , N.
In the sequel we will make extensive use of the following notation: m + 2, m even, m := m + 1, m odd. We will also consider the banded open set Ω := (s, t) ∈ R2 : |s − t| < 1 , and the class of diagonally-periodic functions T := C : Ω → C: C(s + 1, t + 1) = C(s, t), ∀(s, t) ∈ Ω . Consider a function C ∈ T ∩ C ∞ (Ω), such that ∂k A ∂k C (s, s) = (s, s), ∂tk ∂tk
k = 0, . . . , m − 2.
(3.1)
(3.2)
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We emphasize the fact that C does not need to be periodic. Let now F ∈ T be given by ( V (s, t) − C(s, t) log(s − t)2 , if s 6= t, 2 F (s, t) := A(s, s) log x0 (s) + B(s, s), if s = t. It is clear that F ∈ C m−2 (Ω) and that we have the decomposition V (s, t) = F (s, t) + C(s, t) log(s − t)2 ,
∀(s, t) ∈ Ω.
For i, j ∈ Z such that |i − j| 6 N/2 we consider the coefficients Z (1) αi,j := h F (zi , zj + ht)µm (t) dt, ZR (2) αi,j := h C(zi , zj + ht) log h2 (j − i + t)2 µm (t) dt. R
(2) For such values of i, j we have V ψj (zi ) = α(1) i,j + αi,j . We remark that in the case (k) N = 2p, in general, α(k) i,i−p 6= αi,i+p for k = 1, 2, although V ψi−p (zi ) = V ψi+p (zi ). We now decompose A = A(1) + A(2) as follows: for k = 1, 2 (k) |i − j| < N/2, αi,j , α(k) , i − j > N/2, i,j+N (k) ai,j = j − i > N/2, α(k) i,j−N , α(k) + α(k) /2, |i − j| = N/2. i,j−N
i,j+N
Obviously, the case |i − j| = N/2 arises only when N is even. 4.
The full collocation method
To give a full discretization of the equations of the collocation method (i.e., to compute A approximately) we will make use of the decomposition given in section 3, approximating in fact the coefficients α(k) i,j . Let L be a basic quadrature formula Z l X ck u(tk ) ≈ u(t)µm (t) dt, Lu := R
k=−l
where l is a non-negative integer, the nodes are assumed to be taken in increasing order (t−l < t−l+1 < · · · < tl ) and we have tk = −t−k ,
ck = c−k ,
k = −l, . . . , l.
We admit the possibility that c0 = 0. In that case, the formula has an even number of nodes. We demand that L be exact for polynomials of degree not greater than m. By
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symmetry we will have an additional degree of exactness up to m + 1. Then for i, j such that |i − j| 6 N/2 we make the approximation (1) α(1) i,j ≈ βi,j := h LF (zi , zj + h · ).
Let now {yk : k = −m/2, . . . , m/2} be a set of symmetric points (yk = −y−k ) and consider the polynomials Πi,j ∈ Pm such that Πi,j (zj + hyk ) = C(zi , zj + hyk ),
|k| 6
m , 2
again for |i − j| 6 N/2. For these values of i, j we define Z (2) βi,j := h Πi,j (zj + ht) log h2 (j − i + t)2 µm (t) dt ≈ α(2) i,j . R
(k) βi+N ,j+N
(k) = βi,j . From these coefficients we define the matrices B(1) and Notice that B(2) following the rules for A(1) and A(2) . We finally set B := B(1) + B(2) .
Definition 4.1 (Full collocation method). The full collocation method for approximately solving the equation V g = f is the method defined by the equations gh∗ =
N X
gj∗ ψj ∈ Vh ,
such that
j=1
N X
bi,j gj∗ = f (zi ),
i = 1, . . . , N ,
(P∗h )
j=1
where bi,j are the elements of the matrix B defined above. We remark that the calculations for the coefficients B(2) can be carried out analytically, since writing m X t − zj n ρi,j,n Πi,j (t) = h n=0
we have (2) βi,j
=h
m X n=0
where
Z
ρi,j,n log h
t µm (t) dt + γj−i,n , n
2 R
Z γk,n :=
R
tn log(k + t)2 µm (t) dt = (−1)n γ−k,n .
These quantities are independent of the particular kernel or data of the problem, and even of N . Therefore, they can be computed beforehand. A very simple situation arises when we choose C(s, t) ≡ η(s) (see below). Then (2) Πi,j (t) ≡ η(zi ) are constant polynomials. In this case βi,j = hη(zi )(log h2 + γ|j−i|,0). If in a more general situation we take the points tk = k and yk = k as nodes for the numerical quadrature and interpolation process, respectively, we arrive at a method
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which only requires the evaluation of F (zk , zn ) for |k − n| 6 N/2 + l and of C(zk , zn ) for |k − n| 6 N/2 + m/2. We finally detail a particularly simple scheme arising from the collocation with piecewise constant functions (m = 0). We take C(s, t) = A(s, s) and Lf :=
11 1 1 f (−1) + f (0) + f (1). 24 12 24
(4.1)
In this case the only needed coefficients γk,0 can be given by γ0,0 = −2(1 + log 2),
γk,0 = 2 log k −
+∞ X n=1
5.
1 , (2n2 + n)(2k)2n
∀k > 1.
(4.2)
Stability of the method
In this section we show that the family of full collocation methods introduced in section 4 gives numerical schemes which are uniquely solvable and stable in the norm H 0 as the collocation method. Finally, we compare the solutions of both methods and deduce some initial convergence estimates. The main result of this section is the following theorem. Theorem 5.1. The equations of the full collocation method are uniquely solvable for h small enough. Furthermore,
∗
g 6 Ckgk0 . h 0 In the sequel, we denote Z E[f ] :=
∞
−∞
f (t)µm (t) dt − Lf ,
and recall that E[p] = 0 for all polynomials of degree not greater than m + 1 and for all odd functions. The analysis begins with a comparison of the matrices A(k) and B(k) in the norm X |Dx|∞ = max |di,j |, i=1,...,N |x|∞ N
|D|∞ :=
sup 06=x∈CN
with |x|∞ := maxi=1,...,N |xi |. Proposition 5.2. |A(1) − B(1) |∞ 6 Chm | log h|.
j=1
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Proof. Let p = N/2 if N is even and p = (N − 1)/2 if N is odd. Obviously it is enough to prove that uniformly for all i p X (1) (1) m α − β i,i+k i,i+k 6 Ch | log h|. k=−p
Let ϕ ∈ C ∞ (R) be an even cut-off function such that 1, for |x| 6 1/6, ϕ(x) = 0, for |x| > 1/3.
(5.1)
We denote aq (s) :=
1 ∂ q (A − C) (s, s) q! ∂tq
and define m X
a(s, t) := V (s, t) − log(s − t)2 C(s, t) + ϕ(s − t)
! aq (s)(t − s)q .
q=m−1
Obviously, we have F (s, t) = a(s, t) + ϕ(s − t) log(s − t)
2
m X
aq (s)(t − s)q .
q=m−1
If we write
we obtain α(1) i,j
−
(1) βi,j
Ek,q (h) := E ϕ(zk + h · )(zk + h · )q log(zk + h · )2 , ! m X = hE F (zi , zj + h · ) = h E a(zi , zj + h · ) + aq (zi )Ej−i,q (h) . q=m−1
Since a ∈ C m (Ω) we have |E[a(zi , zj + h · )]| 6 Chm , for all i, j such that |i − j| 6 p. From lemma 5.3, it can be easily proven that for q = m − 1, m and k ∈ Z Ek,q (h) 6 Chm−1 | log h| Hence, the result follows readily.
1 . 1 + |k|3
Lemma 5.3. Let cq (x) := ϕ(x) xq log x2 , with q a positive integer and ϕ ∈ C ∞ (R) satisfying (5.1). Then for all k ∈ Z hq m+2 E cq (zk + h · ) 6 C| log h| . (5.2) +h (max(1, |k|))m+2−q
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Proof. Since E[cq (z−k + h · )] = (−1)q E[cq (zk + h · )], we just have to study nonnegative values of k. Let η := max{(m + 1)/2, tl } with tl the maximum of the quadrature nodes in L. If k > N/3 + η, then E[cq (zk + h · )] = 0 by the compactness of supp ϕ. Let λ φλq (x; h) := ϕ(h x) log x2 xq with λ ∈ {0, 1}. Notice that E cq (zk + h · ) = hq E φ1q (k + · ; h) + log h2 E φ0q (k + · ; h) . Since φλq is continuous in R we have E cq (zk + h · ) 6 Chq | log h|,
0 6 k 6 η,
because we are dealing with a fixed number of cases. Then (5.2) holds in a much more trivial form for these cases. Let now k be such that η < k < η + N/3. Then we decompose (5.3) E cq (zk + h · = log(kh)2 E ϕ(zk + h · )(zk + h · )q + kq hq E[φk,h ], where φk,h
· q · · 2 1+ := ϕ kh 1 + log 1 + . k k k
Notice that we have | log(kh)2 | 6 C| log h| for these values of k. Hence, since xq ϕ(x) ∈ C ∞ (R) is compactly supported and the degree of L is m + 1 we obtain log(kh)2 E ϕ(zk + h · )(zk + h · )q 6 C| log h|hm+2 . For the second term on the right-hand side of (5.3), it is easy to prove that for all n>0 −n , max φ(n) k,h (t) 6 C|k| −η6t6η
uniformly for h > 0 and k > η. Then, considering the Taylor expansion of φk,h (t) of degree m + 1 at t = 0, we have q q k h E[φk,h ] 6 Chq |k|q−(m+2) , uniformly for h > 0 and k > η. The result follows by a simple grouping of terms. Proposition 5.4. |A(2) − B(2) |∞ 6 Chm+2 | log h|. Proof. Let Jj := [zj − h(m + 1)/2, zj + (m + 1)/2]. As a simple consequence of lemma 7.3 in section 7 (see (7.1)), since C ∈ T ∩ C ∞ (Ω), it follows that t − zj m+1 + O hm+2 Qm+1 C(zi , t) − Πi,j (t) = dm+1 (zi , zj )h h
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uniformly for all |i − j| 6 N/2 and t ∈ Jj , with dm+1 ∈ T ∩ C ∞ (Ω) and Qm+1 an odd polynomial of degree m + 1. Therefore, Z (2) (2) 2 α − β = C(zi , t) − Πi,j (t) log(zi − t) ψj (t) dt i,j i,j Jj Z t − zj m+1 2 log(zi − t) ψj (t) dt 6h Qm+1 dm+1 (zi , zj ) h Jj Z log(zi − t)2 dt + Chm+2 Z 6 Chm+2
Jj (m+1)/2
Qm+1 (v) log h2 (j − i + v)2 µm (v) dv
−(m+1)/2 Z (m+1)/2 m+3
+ Ch
−(m+1)/2
| log h2 (j − i + v)2 dv.
A simple analysis gives that for all |k| 6 N/2 Z (m+1)/2 log h2 (k + v)2 dv 6 C| log h|. −(m+1)/2
Since Qm+1 is odd and µm is even we have Z (m+1)/2 Qm+1 (v) log(hv)2 µm (v) dv = 0 −(m+1)/2
and for 0 6= k ∈ Z Z (m+1)/2 2 2 Qm+1 (v) log h (k + v) µm (v) dv −(m+1)/2 Z (m+1)/2 C v 2 . Qm+1 (v) log 1 + µm (v) dv 6 = k |k| −(m+1)/2 Hence, for |k| 6 N/2 (2) (2) m+2 α − β i,i+k i,i+k 6 Ch
1 + Chm+3 | log h| max(1, |k|) PN/2 and the result follows by applying the bound k=1 1/k 6 C| log h|.
Proof of theorem 5.1. As a straightforward consequence of [3, proposition 8 and lemma 9], it follows that for all x ∈ CN |x|∞ 6 |x|2 6 Ch−3/2 |Ax|∞ , where | · |2 denotes the Euclidean norm in CN and C is independent of N .
(5.4)
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On the one hand, by propositions 5.2 and 5.4 we have that |A−B|∞ 6 Ch3/2 ε(h), being ε(h) := hm−3/2 | log h|. Therefore I − BA−1 6 |A − B|∞ A−1 6 Cε(h), ∞ ∞ by (5.4). The Neumann series then proves that BA−1 is invertible for h such that Cε(h) < 1 and that 1 BA−1 −1 6 6 C 0. ∞ 1 − Cε(h) Obviously, B−1 = A−1 (BA−1)−1 and by (5.4), −1 B x 6 Ch−3/2 (BA−1 )−1 x 6 C 0 h−3/2 |x|∞ . 2 ∞
(5.5)
Let now g and g∗ be respectively the vectors of coefficients of gh and gh∗ in the B-spline basis of Vh . Then
gh − g∗ 6 Ch1/2 g − g∗ 6 Ch−1 |Bg − Ag|∞ h 0 2 6 C 0 ε(h)h1/2 |g|∞ 6 C 00 ε(h)kgh k0 ,
(5.6)
by (5.5), the fact that Ag = Bg∗ , well-known discrete inequalities for splines and the stability of the collocation method on H 0 . This finishes the proof of the theorem. Notice also that (5.6) gives a first convergence estimate of the full collocation method assuming no regularity of the solution. If g ∈ H 1 , it follows readily that
gh − g∗ 6 Chm−1 | log h| kgk1 . (5.7) h 0 In section 6 we will prove that these estimates can be improved for smooth solutions. 6.
Asymptotic expansion of the error
For the sake of simplicity, in this section (and in its auxiliary counterpart, section 7) we will restrict our attention to the case N = 2p, for a positive integer p. Since we are mainly interested in having an asymptotic expansion of the error to apply Richardson extrapolation and doubling the number of subintervals is one of the most used strategies, this case is the most important. Again for simplicity we will assume that g ∈ C ∞ . Nevertheless, such a smoothness requirement on the exact solution can be relaxed if one looks for limited expansions in the following results. Definition 6.1 (Error operators). For k = 1, 2 we define the operators Sh[k] : Vh → CN given by Sh[k] gh := A(k) − B(k) g, P where g = (g1 , . . . , gN )T , gh = N j=1 gj ψj .
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When necessary, if g is the vector of coefficients of gh in the B-spline basis, we will use the components gj with j ∈ Z, understanding that gj = gi if i ≡ j (mod N ). We will use two modified versions of the Landau O symbol. If ah = (a1 , . . . , aN ) ∈ CN , we write ai = O∗ (hn ) if |ah |∞ 6 Chn , with C independent of h = 1/N but possibly depending on all remaining parameters, unless explicitly stated. In that case we also write ah = O∞ (hn ). Finally, we consider the following symbol for a trapezoidal sum: n X j=k
∗
1 1 aj := ak + ak+1 + · · · + an−1 + an . 2 2
Definition 6.2 (Nodal evaluation and interpolation). Let ∆h : C → CN be the nodal evaluation map, i.e., if e ∈ C, ∆h e := (e(z1 ), . . . , e(zN ))T . We consider the interpolation operator Qh : C → Vh given by the conditions ∆h Qh g = ∆h g. ∞ Proposition 6.3. Let g ∈ C ∞ . Then there exists a sequence of functions {e[1] k }⊂C such that for all integer T
Sh[1] Qh g
T X
=
T +1 hk ∆h e[1] + O h . ∞ k
k=m+1
P Proof. Let p := N/2. Take Qh g = N j=1 aj ψj , with the usual N -periodization of the coefficients ai . By lemma 7.2 and the definition of the matrices A(1) and B(1) it follows that Sh[1] Qh g i
=h
i+p X
∗
j=i−p
=
T X
E F (zi , zj + h · ) aj "
hn E h
p X
# ∗
F (zi , zi + zj + h · )g(n) (zi + zj + h · )Rn
j=−p
n=0
T +1
+ O∗ h
.
(6.1)
For n > 0, let an (t; x) := F (x, x + t)g(n) (x + t). We introduce the decomposition F (s, t) = D(s, t) log(s − t)2 + H(s, t), where D, H ∈ C ∞ (Ω) ∩ T and ∂k D (s, s) ≡ 0, ∂sk
k = 0, . . . , m − 2.
(6.2)
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Let also bn (t; x) := D(x, x + t)g(n) (x + t). We are going to apply lemma 7.4 to study the displaced trapezoidal sums h
p X
∗
an (zj + hu, x),
j=−p
for x ∈ [0, 1] and u ∈ [−η, η], with η := max{(m + 1)/2, tl } (see the definitions of L and E). We first make explicit the terms of the forthcoming expansion: Z 1/2 [1] c0,n (x) := an (t; x) dt, −1/2 1 1 [1] (k−1) 1 (k−1) a − ; x , k > 1, ; x − an ck,n (x) := k! n 2 2 1 (k−1) c[2] (0; x), k > 1. k,n (x) := k! bn ∞ Since D, H ∈ T are smooth and g ∈ C ∞ it follows readily that c[l] k,n ∈ C . Hence, by lemma 7.4 we have for h small enough (so that all evaluations of F take place inside Ω),
h
p X
∗
an (zj + hu, x) =
j=−p
ν X
hk Tk (u)c[1] k,n (x) +
k=0
ν−1 X
ν+1 hk+1 Dk (u)c[2] , k+1,n (x) + O h
k=1
uniformly for all u ∈ [−η, η] and x ∈ [0, 1]. Therefore, taking ν = T − n we obtain # " p X ∗ an (zj + h · , zi )Rn hn E h j=−p
=
TX −n
hk+n E[Rn Tk ]c[1] k,n (zi ) +
k=0
TX −n
T +1 hk+n E[Rn Dk−1 ]c[2] . (6.3) k,n (zi ) + O∗ h
k=2
Let us first remark the following facts: (a) if n + k 6 m + 1, then E[Rn Tk ] = 0; (b) by (6.2), c[2] k,n ≡ 0 if k 6 m − 1; (c) by symmetry E[Rn Tk ] = E[Rn Dk ] = 0 whenever n + k is odd. Hence, the first sum in (6.3) contains only even powers of h, beginning at m + 2. The second sum includes only odd powers of h, beginning at m + n or m + n + 1, depending on the parity of n. By (6.1) and (6.3) the result holds.
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341
∞ Proposition 6.4. Let g ∈ C ∞ . Then there exists a sequence of functions {e[2] k }⊂C such that for all integer T T X
Sh[2] Qh g =
T +1 hk ∆h e[2] . k + O∞ h
k=m+2
P Proof. Let again p := N/2, Qh g = N j=1 aj ψj and Jj := [zj − h(m + 1)/2, zj + h(m + 1)/2]. A direct application of lemmas 7.2 and 7.3 (see (7.1)) yields for h small enough Sh[2] Qh g i
=
i+p X
∗
Z
j=i−p
=
C(zi , t) − Πi,j (t) log(zi − t)2 ψj (t) dt
aj Jj
k+n6T X
k+n
h
i+p Z X ∗
j=i−p
n>0,k>m+1
× ωk,n
dk (zi , t) log(zi − t)2 g(n) (t) Jj
! t − zj dt + O∗ hT +1 , h
where ωk,n (t) := Qk (t)Rn (t)µm (t). By the properties of the polynomials {Rn } and {Qk }, it can be easily seen that ωk,n (−t) = (−1)k+n ωk,n (t) for all t. With a change of variable from Jj to [−(m + 1)/2, (m + 1)/2] and a simple index reordering, the addendum related to the pair (k, n) is transformed into ! Z (m+1)/2 p X k+n ∗ h βk,n (zj + hu; zi ) ωk,n(u) du, (6.4) h −(m+1)/2
j=−p
with βk,n (t; x) := dk (x, x + t)g(n) (x + t) log t2 . By proceeding as in the proof of proposition 6.3, applying now lemma 7.5 instead of lemma 7.4, each term of the form (6.4) can be expanded in powers of h. It is also simple to see that the first power of h to appear is that of m + 2. Then, the result follows readily. Definition 6.5 (Collocation and full collocation operators). For g ∈ H r , r > −1/2, let Ch g be the solution of the collocation equations (Ph ) with right-hand side f = V g. Likewise, let Ch∗ g be the solution to the full collocation equations (P∗h ) with the same right-hand side. By (5.7), we can easily prove that for all g ∈ H 1
∗
C g 6 Ckgk1 , h ∞ where k · k∞ denotes the maximum norm.
(6.5)
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Theorem 6.6. Let g ∈ C ∞ and N even large enough. Then there exists a sequence of functions {gk } ⊂ C ∞ such that for all M
M
X
∗
hk Qh gk 6 ChM +1 . (6.6)
Ch g − Qh g −
k=m
Moreover,
∞
∗
C g − Ch g 6 Chm+1 . h ∞
(6.7)
Proof. Let g and g∗ be the respective vectors of coefficients of Ch g and Ch∗g in the B-spline basis. Obviously, B g∗ − g = (A − B)g = Sh[1] + Sh[2] Ch g. Theorem 3 of [3] states the existence of a sequence of functions {fk } ⊂ C ∞ such that for all M
M +2
X
hk Qh fk 6 ChM +3. (6.8)
Ch g − Qh g −
k=m
∞
Hence, by propositions 6.3 and 6.4 and the boundedness of the error operators Sh[k] , we obtain ! M +2 X [1] [2] [1] [2] k h Qh fk + O∞ hM +3 Qh g + Sh + Sh Ch g = Sh + Sh k=m M +2 X
=
hk ∆h uk + O∞ hM +3 ,
k=m+1
for a new sequence {uk } ⊂ C ∞ . Applying (5.5) it follows that g − g∗ −
M +2 X
hk B−1 ∆h uk = O∞ hM +(3/2) .
k=m+1
Notice that B−1 ∆h uk is the vector of coefficients of Ch∗ V −1 uk in the B-spline basis. Then by the fact that the | · |∞ norm of the vector of coefficients of a spline is uniformly equivalent to the k · k∞ norm of the spline and by (6.5), it follows that
M
X
∗
hk Ch∗ V −1 uk 6 ChM +1 . (6.9)
Ch g − Ch g −
k=m+1
∞
This proves (6.7). To prove (6.6), we consider expansions (6.9), (6.8) and follow the proof of [3, theorem 3] to inductively substitute the full collocation operators by interpolation operators.
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343
Notice that (6.7), when m is even, implies that for smooth enough g
g − C ∗ g 6 Chm+3 . h −2 Therefore, the weak norm superconvergence of the collocation method is conserved. Corollary 6.7. If m is even, kCh∗ g − Qh gk∞ 6 Chm+2 , i.e., we keep the nodal superconvergence of the collocation method (cf. [3]). The case N = 2p + 1 is somewhat simpler and arises by analyzing rectangular rules instead of trapezoidal rules. Let us emphasize that the asymptotic expansions of N even or odd are a priori different. However, a detailed analysis reveals that some terms in both expansions coincide. 7.
Auxiliary results
To shorten the statements of the following technical results we introduce the following concept: Definition 7.1 (Symmetry property (S)). We say that a sequence of functions {Yk : k > k0 } satisfies property (S) if Yk (−t) = (−1)k Yk (t),
∀t.
7.1. Local expansions related to interpolation The spline interpolation operator Qh (see definition 6.2) has the following property. Lemma 7.2. There exists a sequence of polynomials {Rk : k > 1}, with R0 ≡ 1, = k for all k and satisfying (S), such that: for all M and g ∈ C M +1 , if degRk P Qh g = N j=1 aj ψj , then aj =
M X
hk g(k) (s)Rk
k=0
s − zj h
+ O hM +1
uniformly for all j and for all s ∈ supp ψj . Proof.
It is a straightforward modification of [12, proposition 5.2].
Let us recall the interpolation process defined in section 4. Let {yk : |k| 6 m/2} be a fixed set of points in the conditions given there. Then, for c ∈ C(−1, 1) we associate the polynomials {πj c}j6N/2 of degree not greater than m satisfying πj c(zj + hyk ) = c(zj + hyk ),
k = −m/2, . . . , m/2.
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Lemma 7.3. There exists a sequence of polynomials {Qk : k > m + 1} satisfying (S) and degQk = k for all k, such that for all M and c ∈ C M +1 (−1, 1) M X s − zj k (k) M + Rj,h h c (s)Qk c(s). c(s) − πj c(s) = h k=m+1
The remainder can be bounded as follows: for all η > 0 there exist h0 > 0 and ξ ∈ [1/2, 1) such that for all h 6 h0 M R c(s) 6 ChM +1 max c(M +1) (s) . max max j,h −N/26j6N/2 s∈[zj −hη,zj +hη]
s∈[−ξ,ξ]
Proof. It is a straightforward application of Taylor expansions and of the affine equivalence of the interpolation processes for all j. Property (S) follows from the symmetry of the nodes yk . In the notation of section 4, if pi,j := πj C(zi , zi + · ) for |j| 6 N/2, then Πi,j = pi,j−i ( · − zi ) for |i − j| 6 N/2. Therefore, denoting dk := ∂ k C/∂tk , we can obtain an expansion of the form M X t − zj k + O hM +1 , (7.1) h dk (zi , t)Qk C(zi , t) − Πi,j (t) = h k=m+1
uniformly in i, j ∈ Z such that |i − j| 6 N/2 and for all t ∈ supp ψj . 7.2. Extended Euler–Maclaurin formulae Let {Bk } be the sequence of Bernoulli polynomials (cf. [15]). For k > 0 we define 1 1 Tk (t) := Bk (t) + (−1)k Bk (−t) = Bk (t) + Bk (1 + t) . 2 2 It is clear that T0 ≡ 1 and that Tk is a sequence of polynomials of increasing order satisfying property (S). Lemma 7.4. There exists a sequence of 1-periodic continuous functions Dk for k > 1, satisfying (S) and such that: for N = 2p, for all M and for all a, b ∈ C M +1 (−1, 1) with a(0) = 0 we have the asymptotic expansion of the displaced trapezoidal rule applied to f (x) := a(x) log x2 + b(x), Z 1/2 p M X X 1 hk ∗ (k−1) 1 (k−1) Tk (u) f −f − h f (zj + hu) = f (t) dt + k! 2 2 −1/2 j=−p
k=1
+
M −1 X k=1
hk+1 Dk (u)a(k) (0) + RM ,h f (u), k!
R. Celorrio, F.-J. Sayas / Full collocation for boundary integral equations
where for all η > 0, there exist h0 and ξ ∈ [1/2, 1) max max Rh,M f (u) 6 ChM +1 max u∈[−η,η]
Proof.
06j6M +1 t∈[−ξ,ξ]
345
such that for h 6 h0 (j) a (t) + max b(M +1) (t) . t∈[−ξ,ξ]
The result is proven in [4] for rectangular rules of type p−1 X
h
f (zj + hu).
(7.2)
j=−p
To obtain the trapezoidal rules, we average (7.2) for u and u + 1.
Lemma 7.5. Let ω ∈ C[−η, η] and N = 2p. There exists D0ω ∈ R, independent of h, such that for all f (x) := a(x) log x2 with a ∈ C T +1 (−1, 1) we have the expansion ! Z η p X ∗ h f (zj + hu) ω(u) du −η
j=−p
Z
=
1/2
M −1 X
hk+1 ω (k) D a (0) k! k −1/2 k=0 M X 1 hk ω (k−1) 1 ω − f (k−1) − + RM Tk f + ,h f , k! 2 2
T0ω
f (t) dt +
k=1
where Tkω
Z
η
Z
−η η
:=
Dkω :=
−η
Tk (u)ω(u) du,
k > 0,
Dk (u)ω(u) du,
k > 1.
If ξ and h0 are those given in lemma 7.4, then for all h 6 h0 ω R f 6 ChM +1 max max a(j) (s) . M ,h 06j6M +1 s∈[−ξ,ξ]
Moreover, if ω is an odd function, D0ω = 0. The proof of this result can be derived from [4]. Notice that by the symmetry property (S) satisfied by Dk and Tk we have: (a) if ω is an even function Tkω = Dkω = 0 for k odd; (b) if ω is an odd function, Tkω = Dkω = 0 for k even.
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8.
R. Celorrio, F.-J. Sayas / Full collocation for boundary integral equations
Further approximations
Again for simplicity of exposition, in this section we will assume that N is even, without explicitly stating this fact in the results. 8.1. Approximation of potentials Let Γ be the closed curve defined in section 2. Let F : (R2 \Γ) × R → C be in and 1-periodic in its “second” variable. Consider the topological vector space X := C ∞ (R2 \Γ) endowed with its usual topology of uniform convergence of all derivatives on compact sets (see [8]). Then we consider the operator T : H 0 → X, Z 1 F (y, t)g(t) dt. T g(y) := C ∞ ((R2 \Γ) × R)
0
We also P consider the following approximation by quadrature of T , Th : Vh → X. For gh = N j=1 gj ψj , we define Th gh (y) := h
N X
gj L F (y, zj + h · ) .
(8.1)
j=1
Theorem 8.1. If g ∈ C ∞ , there exist two sequences of functions in X, {vk } and {wk }, such that T Ch∗ g = T g +
M X
hk vk + O hM +1 ,
(8.2)
hk wk + O hM +1 ,
(8.3)
k=m+1
Th Ch∗ g = T g +
M X k=m+1
where the symbols O are understood in the sense defined by the topology of X. Proof. Expansion (8.2) can be derived from theorem 6.6, following a similar result in [3] for the collocation method. To prove (8.3) one can see that [13, theorem 11], applies to this situation or work directly on obtaining an expansion of Th Qh g − T g (with the same techniques as used in proposition 6.3) and apply theorem 6.6. 8.2. Quadrature in the right-hand side Often, the integral equation is given in the form V g = f + V0 f , where
Z V0 f := 0
1
2 A0 ( · , t) log x( · ) − x(t) + B0 ( · , t) f (t) dt,
(8.4)
(8.5)
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347
for some 1-periodic smooth A0 , B0 , but with no additional condition on A0 . These problems arise in direct methods for BIE and also when Robin boundary conditions are considered. We propose here a very simple way of dealing with the evaluations of V0 f (zi ) by making the most of the work done for the coefficient matrix. Let Af (s, t) := A0 (s, t)f (t), and
Z
1
Vf g :=
Bf (s, t) := B0 (s, t)f (t)
2 Af ( · , t) log x( · ) − x(t) + Bf ( · , t) g(t) dt.
0
Obviously, V0 f = Vf 1. Moreover, if Af is the matrix given by Vf ψj (zi ), then we have ∆h V0 f = Af e, with e := (1, . . . , 1)T . We then apply the same kind of approximation of the matrix Af as we did for A, obtaining a new approximated matrix Bf , by following the same steps as in sections 3, 4. Then the method gbh =
N X
gbj ψj ∈ Vh ,
such that
Bb g = ∆h f + Bf e,
(b Ph )
j=1
gives a unique approximate solution to (8.4), for h small enough. Since we have done some approximation at the data, we will not be able to obtain the stability inequality of theorem 5.1. However, an inequality of the form kb gh k0 6 Ckgks for s big enough can be obtained. Theorem 8.2. For the modified method (b Ph ), the expansions and bounds of theorem 6.6 and corollary 6.7 hold. g be the coefficients of gh∗ and b gh . Then, by propositions 6.3 and Proof. Let g∗ and b 6.4, there exists a sequence {rk } ⊂ C ∞ such that M X g = (Af − Bf )e = hk ∆h rk + O∞ hM +1 . B g∗ − b k=m+1
We can therefore continue, with due adaptations, with the proof of theorem 6.6 to obtain the desired results. In the trivial situation when A0 ≡ 0, we can simply use the trapezoidal rule, which is asymptotically optimal for periodic integrals. This can be easily seen to be equivalent to applying our method the obvious choice C0 ≡ 0 and taking integer Pwith l points as nodes of L, i.e., Lu := k=−l ck u(k). Therefore, the analysis of this model is included in ours. Our analysis also includes the case when A0 (s, s) ≡ 0 (see numerical examples) and we are using piecewise constant functions, which allows us to take C0 ≡ 0, and therefore, to apply the trapezoidal rule for the right-hand side integrals.
348
9.
R. Celorrio, F.-J. Sayas / Full collocation for boundary integral equations
Numerical experiments
In this section we will illustrate two of the results obtained in preceding sections with some numerical examples. First, we show how Richardson extrapolation accelerates the convergence of the method when applied to an equation of the form V g = f plus a postprocessing like those of section 8.1. Secondly, we will show the nodal superconvergence of corollary 6.7 for an equation of the form V g = f + V0 f for which an exact solution is known. For the examples we take splines of degree m = 0 and the three-point rule (4.1). 9.1. Richardson extrapolation Let Γ be the ellipse given by
x(t) := 0.15 + 2 cos(2πt), 0.2 + sin(2πt) .
We consider the integral operator Z 1 K0 λ x( · ) − x(t) g(t) dt, V g :=
(9.1)
(9.2)
0
for some λ ∈ R, λ > 0, where K0 is a modified Bessel function of order 0 [1, section 9.6]. In fact, we can write for y > 0, K0 (y) = (ıπ/2)H0(1) (ıy), where H0(1) is the Hankel function of the first kind and order zero. The kernel of (9.2) can be decomposed in the usual form (2.1) with A(s, t) := −I0 (λ|x(s) − x(t)|)/2, with I0 another modified Bessel function related to the classical Bessel functions [1]. Since I0 is analytical and even, then A ∈ C ∞ (R2 ). The function B is also smooth and V satisfies the invertibility hypothesis, therefore, defining an isomorphism between H r and H r+1 for all r. Let Ω be the unbounded domain exterior to Γ. We consider the integral equation V g = K0 λ x( · ) . (9.3) The right-hand side function belongs to C ∞ . If g is the unique solution of (9.3), then the function u : Ω → R Z 1 K0 λ y − x(t) g(t) dt u(y) := 0
is the unique solution of the exterior boundary value problem 2 in Ω, −∆u + λ u = 0, on Γ, u(y) = K0 (λ|y|), ∂u (y) + λu(y) = o |y|−1/2 , as |y| → ∞. ∂r Therefore, u(y) = K0 (λ|y|) for all y ∈ Ω. A simple possibility for fully discretizing (9.3) would be taking C ≡ A. However, we will explore two other possibilities:
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349
(a) C(s, t) = A(s, s) ≡ −1/2; (b) C(s, t) = −1/(2 cosh(λ|x(s) − x(t)|)). In both cases C is 1-periodic, which was not necessary in our analysis, but can be used as an advantage in reducing the number of evaluations of C. Option (a) is especially simple and requires only the computation of the coefficients γk,0 for k > 0 with the aid of expansions (4.2). Option (b) has the advantage that when λ is somewhat large or the size of Γ is relevant, C(s, t) decreases rapidly as the parameters s, t correspond to distant points of Γ. This is also the case for the kernel function K0 (λ|x(s) − x(t)|) and, therefore, to compute a small number we take the sum of two small quantities, unlike in (a). In this case we have to compute γk,n for n = 0, 1, 2. For n > 1 we can derive expansions similar to those given for γk,0. As a numerical approximation we take uh (y) := Th gh∗ (y), with gh∗ the numerical solution to (9.3) and Th defined by (8.1) for F (y, t) := K0 (λ|y − x(t)|). For our examples we take λ = 1/5 and we test at the point y0 := (−4, 2) ∈ Ω. We remark that the asymptotic expansion of the error uh − u begins at h3 . Let N0 = 64 and Nj := (3/2)j N0 for j = 1, 2, 3, 4. We consider the approximations ui,0 := u1/Ni (y0 ),
i = 0, . . . , 4.
By Richardson extrapolation we define the approximations ui,j := ui,j−1 +
µ2+j (ui,j−1 − ui−1,j−1), 1 − µ2+j
1 6 j 6 i 6 4,
where µ = 2/3, destined to cancel successive powers of h in the asymptotic expansion of the error (cf. [16]). Let, finally, ei,j := |u(y 0 ) − ui,j |. In table 1 we give the errors ei,j (0 6 j 6 i 6 4) for the discretization given with option (a). The left column indicates the number of points in the finest grid used for computations in the corresponding row. The theoretical results state that ei,j 6 C(1/Ni )3+j and, therefore, log(ei,j /ei+1,j ) ≈ 3 + j. log(3/2) We also show these experimental convergence rates (E.C.R.). The top row shows the expected orders. Notice that the results confirm the expected orders of convergence Table 1 Errors and E.C.R. for option (a). 3 64 96 144 216 324
1.99E−06 5.96E−07 1.78E−07 5.30E−08 1.58E−08
2.97 2.98 2.99 2.99
4
5
6
9.76E−09 1.98E−09 3.94 6.22E−11 3.98E−10 3.96 8.44E−12 4.92 2.91E−13 7.94E−11 3.97 1.13E−12 4.95 2.66E−14 5.91 1.11E−15
350
R. Celorrio, F.-J. Sayas / Full collocation for boundary integral equations Table 2 Errors and E.C.R. for option (b). 3 64 96 144 216 324
2.15E−06 6.43E−07 1.92E−07 5.71E−08 6.70E−08
2.98 2.98 2.99 2.99
4
5
6
9.16E−09 1.85E−09 3.94 5.60E−11 3.72E−10 3.96 7.62E−12 4.92 2.85E−13 7.44E−11 3.97 1.03E−12 4.94 2.69E−14 5.83 2.00E−15
and show how not only the convergence rates are faster but also the numerical results improve as we apply extrapolation. Finally, table 2 represents the same results obtained by the discretization taking into account option (b). 9.2. Nodal superconvergence Let x = (x1 , x2 ), Γ and V be as in the previous example and consider the integral operator Z (x( · ) − x(t)) · x0 (t)⊥ λ 1 g(t) dt, V0 g := K1 λ x( · ) − x(t) π 0 |x(s) − x(t)| where (v1 , v2 )⊥ := (v2 , −v1 ) and K1 (u) = −K00 (u). It can be easily seen that V0 is an operator of the form (8.5) for some A0 and B0 . Furthermore, A0 (s, s) ≡ 0. Let f (t) := πeλx1 (t) . Then, the unique solution of V g = f + V0 f is g(t) = λx02 (t)eλx1 (t) . We remark that V0 is the integral operator related to the double layer potential for the Helmholtz equation (as V is that related to the single layer potential). We take λ = 1/5. For the full discretization of V we take C(s, t) ≡ −1/2, as in option (a) in the preceding example. For V0 we also take the simplest option, Cf (s, t) := A0 (s, s)f (s) ≡ 0. This fact implies that for the discretization of the right-hand side operator the second part is null, and thence we approximate by quadrature the first part, which involves a periodic continuous (but not smooth) function. Notice that
∗
C g − Qh g = max gi∗ − g(zi ) =: EN , h ∞ 16i6N
∗ ) (g1∗ , . . . , gN
where are the coefficients (in this case, also the nodal values) of Ch∗ g. In table 3 we give the errors EN for different values of N and use consecutive grids
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351
Table 3 Maximum nodal errors and E.C.R. N
M.N.E.
E.C.R.
16 32 64 128 256 512
1.38E−02 3.91E−03 1.03E−03 2.65E−04 6.70E−05 1.69E−05
1.82 1.92 1.96 1.98 1.99
to estimate the order of convergence, which was proven to equal 2. The numerical results confirm the theoretical ones. References [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972). [2] D. Arnold and W.L. Wendland, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985) 317–341. [3] R. Celorrio and F.-J. Sayas, Extrapolation techniques and the collocation method for a class of boundary integral equations, to appear in J. Austral. Math. Soc. Ser. B. [4] R. Celorrio and F.-J. Sayas, The Euler–Maclaurin formula in presence of a logarithmic singularity, BIT 39 (1999) 780–785. [5] M. Crouzeix and F.-J. Sayas, Asymptotic expansions of the error of spline Galerkin boundary element methods, Numer. Math. 78 (1998) 523–547. [6] G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind, Computing 25 (1980) 89–130. [7] W. McLean, Fully-discrete collocation methods for an integral equation of the first kind, J. Integral Equations Appl. 6 (1994) 537–571. [8] W. Rudin, Functional Analysis (MacGraw-Hill, New York, 1973). [9] J. Saranen, The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane, Numer. Math. 53 (1988) 499–512. [10] J. Saranen, On the effect of numerical quadrature in solving boundary integral equations, in: Notes on Numerical Fluid Mechanics, Vol. 21 (Vieweg, Braunschweig, 1988) pp. 196–209. [11] J. Saranen and W.L. Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 45 (1985) 91–108. [12] F.-J. Sayas, Asymptotic expansions of the error of some boundary element methods, Thesis doctoral, University of Zaragoza (1994). [13] F.-J. Sayas, Fully discrete Galerkin methods for systems of boundary integral equations, J. Comput. Appl. Math. 81 (1997) 311–331. [14] I.H. Sloan, Boundary element methods, in: Theory and Numerics of Ordinary and Partial Differential Equations, eds. M. Ainsworth et al. (Oxford Science, Oxford, 1995) pp. 138–180. [15] J.F. Steffensen, Interpolation, 2nd ed. (Chelsea, New York, 1950). [16] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, Berlin, 1972). [17] Y. Yan and I.H. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988) 549–579.
