Full asymptotics of spline Petrov–Galerkin methods for some periodic pseudodifferential equations

Advances in Computational Mathematics 14: 75–101, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Full asymptotics of spline Petrov–Galerkin methods forsome periodic pseudodifferential equations

Víctor Domínguez a and Francisco-Javier Sayas b

a Departamento Matemática Aplicada, Universidad de Zaragoza Edificio de Matemáticas,Campus Universitario, 50009 Zaragoza, Spain

E-mail: [email protected] Departamento Matemática Aplicada, Universidad de Zaragoza, Centro Politécnico Superior,

María de Luna s/n, 50015 Zaragoza, SpainE-mail: [email protected]

Received 12 February 2000; revised 15 November 2000; accepted 1 December 2000Communicated by Yuesheng Xu

In this paper we study the existence of a formal series expansion of the error of splinePetrov–Galerkin methods applied to a class of periodic pseudodifferential equations. Fromthis expansion we derive some new superconvergence results as well as alternative proofs ofalready known weak norm optimal convergence results. As part of the analysis the approxima-tion of integrals of smooth functions multiplied by splines by rectangular rules is analyzed indetail. Finally, some numerical experiments are given to illustrate the applicability of Richard-son extrapolation as a means of accelerating the convergence of the methods.

Keywords: splines, Petrov–Galerkin methods, boundary integral equations, Richardson ex-trapolation

AMS subject classification: 65R20

1. Introduction

This paper concerns itself with the numerical solution of a class of periodicpseudodifferential equations by means of Petrov–Galerkin methods with smoothestsplines as test and trial functions. The importance of pseudodifferential equations lieson the fact that boundary integral operators on smooth curves of the plane can be trans-formed into this kind of operators if a parameterization of the curve is given.

Petrov–Galerkin schemes for operator equations have been widely studied in anabstract setting even for unbounded operators [12]. Their interest ranges from the neces-sity of stabilizing Galerkin methods for some equations to the possibility of improvingconvergence. If, as it is our case, the operator is a compact perturbation of an invertibleoperator, it is clear that stability and convergence depend only on stability of the methodfor the reduced equation and on approximation properties of the sequence of discretespaces. In the particular case of equations of the second kind (compact perturbations ofthe identity) a vast literature on the subject has been developed, with an emphasis on

76 V. Domínguez, F.-J. Sayas / Full asymptotics of spline Petrov–Galerkin methods

how stability hinges on the duality pairings of trial and test spaces [6,7]. If one choosesa well conditioned basis for both spaces, the problem is the equivalent to having uni-form bounds for the Gram matrices and their inverses. Our paper deals with a wideset of equations (which includes equations of the second kind as a particular case) in aHilbert scale. This imposes that stability is obtained simultaneously in a range of norms,although it is again a question of bounds for discrete approximations of the principalpart of the operator. With smoothest splines as trial and test spaces, the type of boundsnecessary to derive stability is well known [14].

The main result of this work is the proof of existence of an asymptotic expansion(in integer powers of the discretization parameter h) of the error obtained by comparisonof the numerical solution with an optimal order projection onto the trial space. This firstexpansion is a false asymptotic series in the sense that all the coefficients depend againon the parameter. However, as a simple byproduct we obtain a proper expansion of theerror of linear postprocessings of the solution, as happens when we construct the solutionof the original boundary value problem (by a representation formula or a potential) oncethe associated boundary integral equation is solved.

It was already known that spline Petrov–Galerkin methods exhibit superconver-gence properties in weak norms depending on the balance between the degree of the testand trial spline spaces and the order of the operator. The asymptotic series obtained inthis work gives an alternative proof of this fact by showing that weakest norm supercon-vergence is tantamount to finding a non-zero term in the expansion. On the other hand,we obtain some point superconvergence results on points whose position relative to thediscretization grid is related to the roots of Bernoulli polynomials.

The key to these results lies in the appropriate comparison of the numerical so-lution with a Fourier spline projection Ddhu. The operator Ddh is a particular case oftrigonometric-spline Petrov–Galerkin discretization already studied in [2] and it is usedhere for the sake of asymptotic comparison, because of its optimal approximation prop-erties. To do this we apply some recent expansions obtained in [9]. The second key factleading to our analysis is the possibility of expanding some pseudodifferential operatorsin a formal series of simpler operators. The main difficulties are then the study of com-mutation properties of the auxiliary discrete operator Ddh with simple pseudodifferentialoperators and the coupling of theoretical (operator) and numerical expansions.

We have to remark that the scope of application of these results is a priori limitedto operators that can be expanded in the above mentioned formal series of operators ofinteger order. If one just demands a limited expansion, the numerical asymptotic se-ries turns out to be a sum of a finite number of error terms. However, we insist on thefact that for the full asymptotics of a method the whole of the formal series is relevant,and not only the principal part, on which stability relies. Compared to [16], where thecollocation method is worked out, this article presents some novelties. Because of notcomparing with the exact solution but with a projection onto the discrete space we areallowed to deal with the error series in itself and not with the action of an smoothingfunctional applied to the error. In addition to the advantages drawn from this, the inclu-sion of the effect of numerical integration will be possibly simpler in this frame. This

V. Domínguez, F.-J. Sayas / Full asymptotics of spline Petrov–Galerkin methods 77

work also widens preceding analysis of [8] in several ways. First, we include a muchvaster scope of operators and methods (Crouzeix and Sayas [8] only study logarithmicoperators with ‘constant coefficients’ and Galerkin methods). Also the comparison witha better discrete projection than interpolation gives advantages when working with weaknorms. Finally, since qualocation methods [4,20] can be considered as a partial applica-tion of numerical integration to Galerkin methods, the results of this paper can be takenas a first step towards an asymptotic analysis of that family of numerical schemes.

The practical application of having asymptotic expansions of the error are mani-fold: the use of Richardson extrapolation for acceleration of convergence and for a pos-teriori error estimation, the obtaining of an alternative viewpoint on superconvergenceand the new results on point convergence.

The paper is structured as follows. Section 2 presents the operator equations andnumerical methods. The Fourier spline projector Ddh is then introduced in section 3,where we also derive some asymptotic properties of the action of the basic monomialoperators on the error of this discrete operator. Following these results, the main work ofsection 4 is the full asymptotic expansion of the consistency error of the whole family ofspline Petrov–Galerkin methods, which is then applied in section 5 to give (under stabil-ity hypotheses) the results of this paper. Some technical questions used in section 4 havebeen treated separately in section 6, where we study the approximation of the integral ofthe product of a smooth function with a spline by displaced rectangular sums. In fact,what we obtain is a new kind of Euler–Maclaurin expansion when splines, placed in thesame grid as the composite integration rule is defined, act as a class of weights in theintegrals.

The possibilities of applying Richardson extrapolation to accelerate the conver-gence of the method as well as the pointwise convergence are illustrated by two differentnumerical examples. The first of them is a very simple oddly elliptic operator equation ofnegative order. The second one is a strongly elliptic equation of positive order, requiringsome additional numerical integration. The article is finished with an appendix wherewe show that two important classes of operators belong to the set of pseudodifferentialoperators for which the preceding analysis is valid.

Throughout the paper C,C ′, C ′′ denote constants independent of the discretizationparameter and of any other quantity that is multiplied by them, being possibly differentin each occurrence. The Landau symbol O is used in the standard way.

2. Operators and methods

We briefly introduce the space of periodic distributions and the sequence ofSobolev spaces. Let D be the set of 1-periodic infinitely differentiable functions

D := {ϕ : R → C | ϕ ∈ C∞(R), ϕ = ϕ(· + 1)},

endowed with the metrizable locally convex topology defined by the norms

‖ϕ‖∞,k := max0�l�k

(supx∈R

∣∣ϕ(l)(x)∣∣), k � 0.


The elements of its dual space D′ are referred to as periodic distributions. Given T ∈ D′we can define its Fourier coefficients

T (m) := 〈T , φ−m〉, m ∈ Z,

where 〈·, ·〉 denotes the duality product and φm := exp(2πmi·). It is well known thatany periodic distribution is univocally determined by its Fourier coefficients. Integrablefunctions f ∈ L1(0, 1) define periodic distributions in the usual sense

〈f, ϕ〉 :=∫ 1

0f (t)ϕ(t) dt, ϕ ∈ D.

In this case, functional and distributional Fourier coefficients coincide.We define the periodic Sobolev spaces

Hs :={u ∈ D′

∣∣∣ ∑m�=0

|m|2s∣∣ u(m)∣∣2 <∞},

for arbitrary s ∈ R. Then Hs is a Hilbert space with the inner product

(u, v)s = u(0)v(0)+∑m�=0

|m|2s u(m)v(m).

Clearly H 0 = L2(0, 1) and (·, ·)0 is the L2-product. For r > s, Hr is compactly embed-ded into Hs .

A linear map A :D′ → D′ such that A :Hs → Hs−n is bounded for all s ∈ R fora given value n ∈ Z is called a pseudodifferential operator (�do in short) of order (atmost) n. We will be interested in a particular set of pseudodifferential operators, definedin terms of some particular operators which we specify now.

For j ∈ Z we define the operator

Dju :=∑m�=0

(2mπ i)j u(m)φm, (1)

which is a �do of order j . Notice that DjDk = Dj+k for all j and k. Moreover, D1 = Dis the differentiation operator, D0u = u − u(0) for all u, and D−1 is a sort of inversedifferential operator since

D−1u(t) =∫ t

0

(u(x)− u(0)) dx.

On the other hand, we consider the Hilbert transform

Hu :=∑m�=0

sign(m)u(m)φm,

where

sign(m) ={

1, if m > 0,−1, if m < 0.


Obviously, H is a �do operator of order 0 and H2u = u− u(0) for all u.We say that A is an expandable pseudodifferential operator of order n and we write

A ∈ E(n) if there exists two sequences of functions (aj )nj=−∞, (bj )nj=−∞ ⊂ D such that

for allM integer

A =n∑

j=−MajD

j +n∑

j=−MbjHDj +KM, (2)

with KM a �do of order −M − 1. The set of all expandable operators will be denotedby E . Algebraic properties of this set, which are not needed for what follows, can befound in [10].

The aim of this work is the numerical solution of equations of the form

Au = f, (3)

where A ∈ E(n). This will be done by means of a standard Petrov–Galerkin schemewith smoothest splines on a uniform grid as test and trial spaces. Let N be a positiveinteger, h := 1/N and xi := ih for all i ∈ Z. The space of 1-periodic smoothest splinesof degree d � 1 over the grid {xi}i∈Z is defined by

Sdh := {u ∈ Cd−1 | u|[xi ,xi+1] ∈ Pd

},

where Pd is the set of polynomials of degree at most d. The space of periodic piecewiseconstant functions will be consequently denoted S0

h. It is well known Sdh ⊂ Hs for alls < d + 1/2.

The Petrov–Galerkin method for solving (3) is{uh ∈ Sdh,(Auh, rh)0 = (f, rh)0, ∀rh ∈ Sd ′

h .(4)

We restrict ourselves to the case d � n, so that Auh ∈ H 0 for all uh ∈ Sdh . The methodcould be also extended to cover the more general case where d + d ′ � n, understandingthe H 0 product as a duality product. However for our analysis it will be simpler to avoidthis generality.

The stability analysis of these methods has been studied in [15] and some previousworks. We return to the subject of stability and convergence in section 5.

3. A Fourier spline projection

In this section we define a family of projections onto the spline spaces by means ofa Fourier coefficient interpolation. We define the set of indices

&N :={µ

∣∣∣ − N2< µ � N

2

}.


Given u ∈ D′ we define Ddhu as the unique solution of the problem{Ddhu ∈ Sdh,Ddhu(µ) = u(µ), µ ∈ &N.

This operator is a particular case of Petrov–Galerkin projection when the trial spaceis Sdh , the test space is a set of trigonometric polynomials and the operator to which thisis applied is the identity (see [2]). Therefore, this operator yields optimal convergenceproperties: namely, for s � t � d + 1 and s < d + 1/2∥∥Ddhu− u∥∥

s� Cht−s‖u‖t , ∀u ∈ Ht. (5)

For k � 0, let Bk be the Bernoulli polynomial of degree k (see [1, chapter 23]). Wedenote by B k the 1-periodization of Bk. In other words,

B k(x +m) = Bk(x), ∀x ∈ [0, 1), ∀m ∈ Z.

We also introduce the constants

γ dk := − 1

k!( −(d + 1)

k − (d + 1)

).

Applying the same arguments leading to a similar result in [9], given in different norms,the following expansion can be easily proven,

Proposition 1. For all u ∈ HM+1 we have in the norm of H 0

Ddhu− u =M∑

k=d+1

hkγ dk B k(N ·)u(k) + O(hM+1)‖u‖M+1.

The next result studies the action of operators Dj and HDj , which act as monomialsin the expansions (2), over the numerical expansion of proposition 1. For conveniencewe introduce the functions

Ck := HB k.

Since B k(−t) = (−1)kB k(t) and H maps even functions into odd functions and vicev-ersa, then Ck(−t) = (−1)k+1Ck(t).

Proposition 2. Let j � d−1 andM � d+1−j . Then in the norm ofH 0, the followingexpansions hold for all u ∈ HM+1+j :

Dj(Ddhu− u)= M∑

k=d+1−jhkγ

d−jk B k(N ·)u(k+j) + O

(hM+1

)‖u‖M+1+j ,

HDj(Ddhu− u)= M∑

k=d+1−jhkγ

d−jk Ck(N ·) u(k+j) + O

(hM+1

)‖u‖M+1+j .


Proof. The first assertion follows readily by proposition 1 and the fact that

DjDdh = Dd−jh Dj .

From this expansion and the fact that H :H 0 → H 0 is bounded we obtain

HDj(Ddhu− u) =

M∑k=d+1−j

hkγd−jk H

(B k(N ·)u(k+j))+ O

(hM+1)‖u‖M+1+j .

Therefore once we prove that for the values of j and k involved, we have

hkH(B k(N ·)u(k+j)) = hkCk(N ·) u(k+j) + O

(hM+1)‖u‖M+1+j , (6)

the result follows.To do this, we consider the truncation operator for the Fourier series

SNv :=∑µ∈&N

v(µ)φµ,

which satisfies

‖SNv − v‖0 � Chr‖v‖r , (7)

for all r � 0 with C depending only on r. A simple computation shows that

B k(N ·) =∑m�=0

Bk(m)φmN,

where we have applied that Bk(0) = 0. Also H[B k(N ·)] = [HB k](N ·) = Ck(N ·).Therefore,

H(B k(N ·)SNv

)= H

( ∑µ∈&N

∑m�=0

B k(m)v(µ)φµ+mN)

=∑µ∈&N

∑m�=0

sign(µ+mN)B k(m)v(µ)φµ+mN

=( ∑µ∈&N

v(µ)φµ

)(∑m�=0

sign(mN)B k(m)φmN

)= SNv H

[B k(N ·)].

Hence, applying (7), we obtain

H(B k(N ·)v)= H

(B k(N ·)SNv

)+ O(hr)‖v‖r = Ck(N ·)SNv + O

(hr)‖v‖r

=Ck(N ·)v + O(hr)‖v‖r ,

and (6) follows with v = u(k+j) and r = M + 1 − k. �


4. Consistency error expansions

Let

Ln :={

n∑j=1

ajDj∣∣∣ aj ∈ D, ∀j

},

that is, the set of linear differential operator of order n with smooth periodic coefficientand without zero order term.

Proposition 3. Let A ∈ E(n) and d � n+1. Then there exist two sequences (Rk)k, (Tk)kwith Rk,Tk ∈ Ln+k satisfying the following property: for all M � d + 1 − n andu ∈ HM+n+1 the expansion

A(Ddhu− u) =

M∑k=d+1−n

hk(B k(N ·)Rku+ Ck(N ·)Tku

)+ O(hM+1

)‖u‖M+n+1

holds in H 0.

Proof. LetM � d + 1 − n. Since A ∈ E(n), we have

A(Ddhu−u

) =n∑

j=−M+d+1

ajDj(Ddhu−u

)+ n∑j=−M+d+1

bjHDj(Ddhu−u

)+KM(Ddhu−u),where KM is �do of order −M + d. Therefore by (5)∥∥KM(Ddhu− u)∥∥0 � C

∥∥Ddhu− u∥∥−M+d � C ′hM+1‖u‖d+1 � C ′′hM+1‖u‖M+n,

for all u ∈ HM+n. Now, by proposition 2 it follows that for u ∈ HM+n+1

A(Ddhu− u)= n∑

j=−M+d+1

aj

{M∑

k=d+1−jhkγ

d−jk Bk(N ·)u(j+k) + O

(hM+1

)‖u‖M+j+1

}

+n∑

j=−M+d+1

bj

{M∑

k=d+1−jhkγ

d−jk Ck(N ·)u(j+k) + O

(hM+1

)‖u‖M+j+1

}+ O

(hM+1)‖u‖M+n

=M∑

k=d+1−nhkB k(N ·)

{n∑

j=d+1−kγd−jk aju

(j+k)}

+M∑

k=d+1−nhkCk(N ·)

{n∑

j=d+1−kγd−jk bju

(j+k)}

+ O(hM+1)‖u‖M+n+1.


Defining for k � d + 1 − n the operators

Rk :=n+k∑j=d+1

γd+k−jk aj−kDj , Tk :=

n+k∑j=d+1

γd+k−jk bj−kDj

the proof is finished. �

Theorem 4. Let A ∈ E(n), d � n and d ′ � 0. Then there exists a sequence (Fk)k withFk ∈ Ln+k satisfying the following property: for all rh ∈ Sd ′

h and u ∈ HM+n+1

(A(Ddhu− u), rh)0 =

M∑k=d+d ′+2−n

hk(Fku, rh)0 + O(hM+1)‖u‖M+n+1‖rh‖0.

Proof. We first prove the theorem for d � n+1. Let u ∈ HM+n+1 and rh ∈ Sd ′h be fixed.

All the forthcoming bounds will be independent of them without explicit indication. Bythe previous proposition,

(A(Ddhu− u), rh)0 =

M∑k=d+1−n

hk(B k(N ·)Rku, rh

)0 +

M∑k=d+1−n

hk(Ck(N ·)Tku, rh

)0

+ O(hM+1

)‖u‖M+1+n‖rh‖0. (8)

On the one hand we study products of the form:

(B k(N ·)f, rh

)0 =

N∑i=1

∫ xi+1

xi

B k(Nx)f (x)rh(x) dx

= hN∑i=1

∫ i+1

i

B k(x)f (hx)rh(hx) dx

=∫ 1

0Bk(t)

[h

N∑i=1

f(h(i + t))rh(h(i + t))

]dt.

The term in brackets is a displaced rectangular sum to approximate (f, rh)0. The asymp-totics of this sum given in lemma 11 proves that

(B k(N ·)f, rh

)0 =∫ 1

0Bk(t)

[(f, rh)0 +

ν∑j=d ′+1

hjγ d′j Bj (t)(−1)j

(f (j), rh

)0

]dt

+ O(hν+1)‖f ‖ν+1‖rh‖0

=ν∑

j=d ′+1

hjγ d′j (−1)jαj,k

(f (j), rh

)0 + O

(hν+1

)‖f ‖ν+1‖rh‖0, (9)


where we have denoted

αj,k :=∫ 1

0Bj(t)Bk(t) dt,

and applied that the integral of Bk over the unit interval vanishes. Notice that by well-known symmetry properties of the Bernoulli polynomials, αj,k = 0 whenever j + k isodd.

Taking ν = M − k and f = Rku in (9) we obtain the expansion

(B k(N ·)Rku, rh

)0 =

M−k∑j=d ′+1

hjγ d′j (−1)jαj,k

(DjRku, rh

)0

+ O(hM−k+1)‖u‖M+n+1‖rh‖0, (10)

since Rk is a differential operator of order n+ k. This is done for k � M − (d ′ + 1). Forthe remaining terms we apply lemma 10 to obtain∣∣hk(B k(N ·)Rku, rh

)0

∣∣ � ChM+1‖u‖M+n+1‖rh‖0, M − d ′ � k � M. (11)

On the other hand, we define

βj,k :=∫ 1

0Bj(t)Ck(t) dt,

which vanishes when j+k is even. Then the same ideas prove that for k � M− (d ′ +1)

(Ck(N ·)Tku, rh

)0 =

M−k∑j=d ′+1

hjγ d′j (−1)jβj,k

(DjTku, rh

)0

+ O(hM−k+1

)‖u‖M+n+1‖rh‖0, (12)

whereas for the other values of k a bound similar to (11) holds.Finally consider the sequence of differential operators: for m � d + d ′ + 2 − n

Fm :=

m−(d ′+1)∑k=d+1−n

γ d′m−k(−1)m−kαm−k,kDm−kRk, if m even,

m−(d ′+1)∑k=d+1−n

γ d′m−k(−1)m−kβm−k,kDm−kTk, if m odd.

Then (8), (10)–(12) and the cancellation properties of the coefficients αj,k and βj,k provethe result.

The case d = n needs some simple adjustments. The first expansion of proposi-tion 2 holds true, whereas the second one has to be written as follows:

HDd(Ddhu− u)= hγ 0

1C1(N ·)SNu(d+1) +M∑k=2

hkγ 0k Ck(N ·) u(d+k)

+ O(hM+1)‖u‖M+1+d ,


because of the unboundedness of C1. We can then restate proposition 3 for d = n

substituting the first term of the sum by h(B1(N ·)R1u+C1(N ·)T1SNu). The remainderof the preceding proof can then be straightforwardly adapted. �

5. Full expansions of the error and some consequences

In this section we transform the consistency expansions of theorem 4 into globalerror expansions, comparing the numerical solution with the Fourier spline projection ofthe exact solution. This is first done in the natural norm of the method and then trans-formed into stronger norms. As byproducts we derive point superconvergence resultsand asymptotic expansions under the action of regularising operators. We first deal withthe stability of the method to move afterwards to convergence results.

Let A ∈ E(n) be invertible. Its principal part is given by

anDn + bnHDn.

We say that A is strongly elliptic if there exits ϕ ∈ D such that

Re(ϕ(x)

(an(x)+ bn(x)

))> 0, ∀x ∈ R,

Re((−1)nϕ(x)

(an(x)− bn(x)

))> 0, ∀x ∈ R.

Equivalent definitions of strong ellipticity, including one in Gårding’s inequality form,are given in [19]. Following already classical notations (see [20], for instance), we saythat A is oddly elliptic if there exists ϕ ∈ D such that

Re(ϕ(x)

(an(x)+ bn(x)

))> 0, ∀x ∈ R,

Re((−1)nϕ(x)

(an(x)− bn(x)

))< 0, ∀x ∈ R,

which is equivalent to HA being strongly elliptic.We now give necessary and sufficient conditions for the Petrov–Galerkin method

(PG in the sequel) to converge. The natural norm to study convergence and stability ofthe PG scheme is that of H(n+d−d ′)/2. To simplify some forthcoming expressions wedenote

α := d − d ′.

Stability of the method means the existence of a positive constant C independent of hsuch that

‖uh‖(n+α)/2 � C‖u‖(n+α)/2, ∀u ∈ H(n+α)/2,being uh the numerical solution (4) with f = Au as right hand side. It is well known thatstability is equivalent to convergence in the same norm and to a classical Céa estimate(cf. [13]). The Galerkin case (d = d ′) is well known [19,22]: stability is then equivalentto strong ellipticity. For the more general PG case, one can always transform the numer-ical scheme into a Galerkin method, where one applies the theory. We summarize thisresults in the following proposition.


Proposition 5. The PG method is stable if and only if one of the two following condi-tions holds:

(a) A is strongly elliptic and d and d ′ have the same parity;

(b) A is oddly elliptic and d and d ′ are of opposite parity.

In that case there exists β > 0 such that for h small enough the Babuška–Brezzi condi-tion is satisfied:

infuh∈Sdh

suprh∈Sd ′h

|(Auh, rh)0|‖uh‖(n+α)/2‖rh‖(n−α)/2 > β. (13)

In the sequel Gd,d′

h u denotes the numerical solution of the Petrov–Galerkinmethod (4) when f = Au. We first give an asymptotic expansion of the comparisonbetween Gd,d

′h u and Ddhu. From this point to the end of this section we will not ex-

plicitly indicate the regularity assumptions for the exact solution. This demands thedistinction of several cases and notations become cumbersome. Consequently, we as-sume that u ∈ D. Asymptotic series will be denoted in the usual way: we write that insome norm

ah ∼∞∑k=k0hkak,

when for allM ∥∥∥∥∥ah −M∑k=k0hkak

∥∥∥∥∥ � ChM+1.

Proposition 6. For all s < d + 1/2 the expansion

Ddhu−Gd,d ′h u ∼

∞∑k=d+d ′+2−n

hkGd,d ′h A−1Fku

holds in Hs . Moreover, the expansion is also valid in the L∞ norm of the dth orderderivatives.

Proof. We first show the result for s = (n + α)/2, i.e., in the natural stability norm.We distinguish two cases: n − α � 0 and n − α < 0. In the first case, since ‖rh‖0 �‖rh‖(n−α)/2 for all rh, by (13) and theorem 4 we obtain∥∥∥∥∥Ddhu−Gd,d ′

h u−M∑

k=d+d ′+2−nhkG

d,d ′h A−1Fku

∥∥∥∥∥(n+α)/2

� ChM+1‖u‖M+n+1. (14)

If n− α < 0, we have the inverse inequality

‖rh‖0 � Ch(n−α)/2‖rh‖(n−α)/2, ∀rh ∈ Sd ′h . (15)


We apply theorem 4 withM ′ such that M � M ′ + (n− α)/2 < M + 1. Proceeding asbefore and applying (15), we obtain∥∥∥∥∥Ddhu−Gd,d ′

h u−M∑

k=d+d ′+2−nhkG

d,d ′h A−1Fku−

M ′∑k=M+1

hkGd,d ′h A−1Fku

∥∥∥∥∥(n+α)/2

� ChM+1‖u‖M ′+n+1. (16)

To bound the addenda corresponding to indicesM + 1 � k � M ′ we apply stability andobtain∥∥Gd,d ′

h A−1Fku∥∥(n+α)/2 � C

∥∥A−1Fku∥∥(n+α)/2 � C ′‖u‖(n+α)/2+k � C ′′‖u‖M+α+1,

since A−1Fk is �do of order k. In case that n − α is even, this bound is attainable foru ∈ HM+α . These bounds and (16) prove the result.

The validity of the expansion in stronger norms can be derived with standard tech-niques from the inverse inequalities of the splines. See [8, theorem 2] for a similarresult. �

If s � n− d ′ − 1 and uh := Gd,d ′h u, proposition 6 and (5) show readily that

‖uh − u‖s � Chd+d ′+2−n.

This is the weak norm superconvergence phenomenon of PG methods which had alreadybeen observed in [15]. In fact, the result is optimal since writing k = d + d ′ + 2 − n,the nontrivial term A−1Fku appears with the kth power of h in the expansion. In strongernorms, the error is simply an approximation error, which cannot be improved since theapproximation of u by Ddhu is optimal. Furthermore, by induction we can derive thefollowing asymptotic series.

Corollary 7. For all u ∈ D, there exists a sequence of functions gk such that

Ddhu− uh ∼∞∑

k=d+d ′+2−nhkDdhgk,

in Hs with s < d + 1/2 and in the L∞ norm of the dth order derivatives.

Moreover, by comparison of Ddhu and u in some special points (see [9, corol-lary 3.2]) we can straightforwardly prove the following nodal superconvergence prop-erty.

Corollary 8. Let u be smooth enough. Then if d is even and d ′ � n

maxi

∣∣uh(xi)− u(xi)∣∣ � Chd+2. (17)


If d is odd, d ′ � n and ε is the only root of Bd+1 in (0, 1/2), then

maxi

∣∣uh(xi ± εh)− u(xi ± εh)∣∣ � Chd+2. (18)

To finish this section we show how the action of a linear regularising operatorwith values on a normed space affects proposition 6 by producing a proper error ex-pansion, where all the coefficients are independent of h. This kind of operators appearsin postprocessings of the solution of boundary integral equations, such as evaluation ofpotentials.

Corollary 9. Let T :Hν → X be linear and bounded linear for all ν, being X a normedspace. Then for every smooth u there exists a sequence of functions (vk)k such that in X

T (u− uh) ∼∞∑

k=d+d ′+2−nhkT vk.

Proof. This is straightforward consequence of proposition 6 and the fact that∥∥T (Ddhu− u)∥∥X

� ChM,

for allM because of (5). �

6. Approximation of some inner products

In this section we state and prove some technical results, which have already beenused in section 4, that study the asymptotic behavior of the following numerical quadra-ture: given f , rh ∈ Sdh and t ∈ R∫ 1

0f (s)rh(s) ds ≈ h

N∑i=1

f(h(i + t))rh(h(i + t)).

We thus have approached the inner product of a function by a spline by a displacedrectangular rule. Let

Lh(f, t) := hN∑i=1

f(h(i + t)).

Notice that Lh(f, ·) is a 1-periodic function.In the sequel we will be dealing with splines of degree d � 1. All the results are

valid for piecewise constant functions (d = 0) provided that we take the mean value ofrh at the breakpoints as rh(xi). However the proofs can be done with classical techniqueswithout any complication.


Lemma 10. Let rh ∈ Sdh and f ∈ Hν with 1/2 < ν � d + 1. Then uniformly int ∈ [0, 1]

Lh(f rh, t) =∫ 1

0f (s)rh(s) ds + O

(hν)‖f ‖ν‖rh‖0.

Proof. Following [4,9] we write rh ∈ Sdh in the form

rh =∑µ∈&N

rh(µ)ϕdµ,

where ϕµ := Ddhφµ. Moreover, we have that

ϕdµ =(

1 +1d(N ·, µN

))φµ, ∀µ ∈ &N

being (cf. [4])

1d(x, y) := yd+1∑m�=0

1

(m+ y)d+1φm(x). (19)

Since for all r > 1

sup−1/2�y�1/2

∑m�=0

1

|m+ y|r <∞, (20)

we have ∣∣1d(x, y)∣∣ � C|y|d+1, ∀(x, y) ∈ R × [−1/2, 1/2]. (21)

Notice first that

rh(h(i + t)) =

∑µ∈&N

rh(µ)

[1 +1d

(t,µ

N

)]φµ(h(i + t)).

Therefore we can decompose

Lh(f rh, t)=∑µ∈&N

rh(µ)

[1 +1d

(t,µ

N

)]f (−µ)

+∑µ∈&N

rh(µ)

[1 +1d

(t,µ

N

)]{∑m�=0

f (−µ+mN)φm(t)}, (22)

where we have used the easily verifiable identity

h

N∑i=1

f(h(i + t))φµ(h(i + t)) = f (−µ)+

∑m�=0

f (−µ+mN)φm(t).

We remark that all the series involved converge absolutely and uniformly in t .


On the one hand,∑µ∈&N

rh(µ)f (−µ) =∫ 1

0f (s)rh(s) ds + O

(hν)‖f ‖ν‖rh‖0, (23)

since ∥∥∥∥f −∑

−µ∈&Nf (µ)φµ

∥∥∥∥0

� Chν‖f ‖ν .

Secondly, we can use (21) and the fact that ν � d + 1 to bound∣∣∣∣ ∑µ∈&N

rh(µ)1d

(t,µ

N

)f (−µ)

∣∣∣∣�C ∑µ∈&N

| rh(µ)|∣∣∣∣µN∣∣∣∣d+1∣∣f (−µ)∣∣

�Chν∑µ∈&N

∣∣ rh(µ)∣∣|µ|ν∣∣f (−µ)∣∣ � Chν‖rh‖0‖f ‖ν .(24)

Finally, by (20),∣∣∣∣ ∑µ∈&N

rh(µ)

[1 +1d

(t,µ

N

)]{∑m�=0

f (−µ+mN)φm(t)}∣∣∣∣

� C‖rh‖0

[ ∑µ∈&N

∣∣∣∣∑m�=0

f (−µ+mN)φm(t)∣∣∣∣2]1/2

� C‖rh‖0

[ ∑µ∈&N

{∑m�=0

| − µ+mN |2ν∣∣f (−µ+mN)∣∣2}{∑m�=0

h2ν 1

|m− µ/N |2ν}]1/2

� Chν‖rh‖0‖f ‖ν . (25)

Gathering all the bounds we have obtained so far, the result follows readily. �

Lemma 11. Let f ∈ HM+1 and rh ∈ Sdh . Then

Lh(f rh, t)=∫ 1

0f (s)rh(s) ds +

M∑j=d+1

(−1)jhjγ dj Bj (t)∫ 1

0f (j)(s)rh(s) ds

+ O(hM+1)‖rh‖0‖f ‖M+1,

uniformly in t ∈ R.

Proof. Consider again decomposition (22) with 1d defined by (19). We first remarkthat the bounds (23) and (25) holds for ν = M + 1. Thus

Lh(f rh, t)=∫ 1

0f (s)rh(s) ds +

∑µ∈&N

rh(µ)1d

(t,µ

N

)f (−µ)

+ O(hM+1)‖rh‖0‖f ‖M+1.


We now recall the expansion [9, lemma 3.1]

1d(x, y) =M∑

j=d+1

γ dj (2π i)jBj (x)yj +GM(x, y),

where the remainder satisfies∣∣GM(x, y)∣∣ � C|y|M+1 ∀(x, y) ∈ R × [−1/2, 1/2].

Then

∑µ∈&N

rh(µ)1d

(t,µ

N

)f (−µ)

=M∑

j=d+1

hjγ dj Bj (t)

{ ∑µ∈&N

rh(µ)(2π iµ)j f (−µ)}

+∑µ∈&N

rh(µ)GM

(t,µ

N

)f (−µ)

=M∑

j=d+1

hjγ dj Bj (t)(−1)j{ ∑µ∈&N

rh(µ)f (j)(−µ)}

+ O(hM+1)‖f ‖M+1‖rh‖0,

proceeding as in (24). Finally applying (23) to f (j) with ν = M + 1 − j , the resultfollows. �

7. Numerical experiments

In this part we expose two examples corroborating our theoretical results. The firstone is a Petrov–Galerkin method for an oddly elliptic operator equation. The secondone is a Galerkin method for a strongly elliptic hypersingular equation. In this last case,numerical integration is necessary in order to fully discretize the problem. This is donein a way that preserves the theoretical properties of the method.

An oddly elliptic operator

Let A := D−1 + J where Ju := u(0). Then A is an invertible oddly elliptic �do oforder −1. Moreover, D−1 can be written as the convolution with B1

D−1u = −∫ 1

0B1(· − t)u(t) dt,


as can be easily derived from the Fourier expansion of the first Bernoulli polynomial.We want to solve numerically the equation Au = f with a PG method with S1

h and S0h as

trial and test space respectively:uh ∈ S1

h,

Juh J rh −∫ 1

0

∫ 1

0B1(s − t)uh(t)rh(s) dt ds =

∫ 1

0f (s)rh(s) ds, ∀rh ∈ S0

h.(26)

Methods where S0h acts as test space are called region methods (cf. [14, chapter 12]).

We take f (t) := exp(sin(2πt)) cos(2πt) + sin(2πt) so that the exact solution of theproblem is

u(t) = Df (t) = π exp(sin(2πt)

)(1 + cos(4πt)− 2 sin(2πt)

)+ 2π cos(2πt).

The discrete scheme (26) is equivalent to a linear system, where all the coefficientscan be calculated exactly. For our test we have chosen N = 8, 16, 32, 64, 128, 256and 512.

If uh denotes the numerical solution for h = 1/N , and ζ1 = 1/2 − √3/6, ζ2 =

1/2 + √3/6 are the two roots of B2, we compute the errors

e0h := max

i=0,...,N−1

∣∣uh(xi)− u(xi)∣∣ = O(h2), (27)

ej

h := maxi=0,...,N−1

∣∣uh(xi + ζjh)− u(xi + ζjh)∣∣ = O(h3), j = 1, 2. (28)

The bound in (28) follows from (18), whereas the expected convergence value in (27)can be inferred from the comparison with D1

hu in the knots (see [9, corollary 3.2]).Table 1 shows these errors together with estimated convergence rates (e.c.r.) obtainedby comparing consecutive errors in the table.

Given ϕ ∈ D the functional

T :Hν → C

u �→ (u, ϕ)0

Table 1Nodal errors and e.c.r. for the first example.

N e0h e.c.r. e1h e.c.r. e2h e.c.r.

8 2.983 0.947 0.96416 0.902 1.726 0.159 2.569 0.171 2.49832 0.222 2.025 1.925E−2 3.052 1.979E−2 3.10964 5.500E−2 2.010 2.462E−3 2.968 2.449E−3 3.014

128 1.373E−2 2.003 3.066E−4 3.054 3.059E−4 3.001256 3.430E−3 2.001 3.827E−5 3.001 3.830E−5 2.998512 8.578E−4 1.999 4.786E−6 3.000 4.786E−6 3.000


Table 2Errors after extrapolation.

N ε0h ε1h ε2h ε3h ε4h

8 4.88E−0216 2.08E−03 1.03E−0332 1.17E−04 1.41E−05 2.08E−0664 7.11E−06 2.10E−07 1.02E−08 2.03E−09

128 4.41E−07 3.25E−09 3.90E−11 6.50E−13 2.63E−12256 2.75E−08 5.06E−11 1.52E−13 6.96E−16 6.08E−17512 1.72E−09 7.89E−13 5.92E−16 6.93E−19 1.33E−20

Table 3E.c.r. for the errors in table 2.

4 6 8 10 12

4.5514.154 6.1964.039 6.067 7.6804.010 6.017 8.023 11.064.002 6.004 8.007 9.867 15.404.001 6.001 8.002 9.972 12.16

is bounded for all ν. Since in this example F2k+1 ≡ 0 for all k, the expansion of propo-sition 6 contains only even powers of h. Therefore, corollary 9 reads now

T u− T uh =M∑k=2

h2kT uk + O(hM+2

).

We take ϕ(t) = cos(4πt) and calculate T uh for the same values of h. We then applyRichardson extrapolation (cf. [21]) to give new approximations to T u as follows:

γ 0h := T uh,γj

h := 4j γ j−1h − γ j−1

2h

4j − 1, j = 1, . . . , 4.

Table 2 shows the errors εjh := |T u − γ jh | whereas table 3 gives the correspondingestimated convergence rates. Expected values of the order of the method are indicatedon top of this table.

A hypersingular equation

Let 6 ⊂ R2 be a bounded domain with smooth boundary 7, given by a regular

1-periodic parametric representation x : R → 7. Consider then the hypersingular equa-


tion

Wu(s) := |x′(s)|2π

f.p.∫ 1

0∂n(s)∂n(t) log

∣∣x(s)− x(t)∣∣ ∣∣x ′(t)

∣∣u(t) dt = f (s), (29)

where f.p. stands for the Hadamard finite part of the integral and ∂n(s) denotes the outernormal derivative at the point x(s). Notice that (cf. [5, section 6.6]) W = DVD with

Vu := 1

2π

∫ 1

0log∣∣x(·)− x(t)

∣∣u(t) dt = − 1

2iHD−1 + K0,

being K0 an integral operator with infinitely differentiable periodic kernel, and thereforea �do of order −∞. Hence W = −1/(2i)HD + K for a new integral operator K with thesame properties.

This integral equation appears in the double layer representation of the solution ofthe Neumann boundary value problem for the Laplace operator. Namely, if g :7 → R

and f (s) := |x ′(s)|g(x(s)), then any solution of (29) gives by means of the representa-tion formula

ω(y) :=∫ 1

0κ(s, y)u(s) ds, κ(s, y) := ∣∣x′(s)

∣∣∂n(s) log∣∣y − x(s)

∣∣ (30)

a solution to

1ω = 0, in R2 \ 7, ∂nω = g, on 7. (31)

We point out that (29) is solvable if and only if f (0) = 0, the solution being uniquemodulo constant functions (notice that W1 = 0). We aim to calculate approximatelythe solution with null integral, by applying the Galerkin scheme with S1

h,0 := {uh ∈S1h | uh(0) = 0} as trial-test spaces. It can be easily checked that the theory given in

previous sections holds for this new situation.Using that W = DVD, the Galerkin scheme can be formulated in the following way:uh ∈ S1

h,0,

− 1

2π

∫ 1

0

∫ 1

0log∣∣x(s)− x(t)

∣∣u′h(t)r

′h(s) dt ds =

∫ 1

0f (s)rh(s) ds ∀rh ∈ S1

h,0.

(32)To assemble a linear system for this problem once a basis of S1

h,0 is chosen, the use ofnumerical integration is required. We briefly explain how this can be done in a way thatpreserves all theoretical properties of the method. Let S0

h,0 be the space of elements ofS0h with null integral. Clearly, DS1

h,0 = S0h,0. A B-spline basis for both discrete spaces

can be constructed as follows. Let

µ0(x) :={

1, x ∈ [−1/2, 1/2],0, otherwise,

µ1(x) :=x + 1, x ∈ [−1, 0],

1 − x, x ∈ [0, 1],0, otherwise.


For i = 1, . . . , N we consider the 1-periodic functions such that

ψ0i = µ0

( · − xi − h/2h

), ψ1

i = µ1

( · − xih

), in (xi − 1/2, xi + 1/2).

Obviously {ψj2 − ψj1 , ψj3 − ψj1 , . . . , ψjn − ψj1 } is a basis of Sj0,h for j = 0, 1. Once thecoefficients

βi :=∫ 1

0f (t)ψ1

i (t) dt, i = 1, . . . , N,

αi,j := −∫ 1

0

∫ 1

0log∣∣x(s)− x(t)

∣∣ψ0i (s)ψ

0j (t) ds dt, i, j = 1, . . . , N,

are computed, the system corresponding to (32) is easily constructed.We first use a quadrature formula of degree 3 with µ1 as weight function to ap-

proximate

βi = h∫ 1

−1f (xi + ht)µ1(t) dt ≈ h

12

(f (xi−1)+ 10f (xi)+ f (xi+1)

).

The coefficients αi,j can be approximated using the singularity subtraction explainedin [18], adapted from [11]. The results from [18] can be adapted to show that the numer-ical solution to the modified system, say u∗

h, satisfies ‖u − u∗h‖−1 � Ch3, and the order

of the method is thus preserved. Moreover if u∗h =∑N

i=1 u∗i ψ

1i and we define (see (30))

ωh(y) := h

12

N∑i=1

u∗i

(κ(xi−1, y)+ 10κ(xi , y)+ κ(xi+1, y)

) ≈∫ 1

0κ(s, y)u∗

h(s) ds,

then we obtain an asymptotic expansion of the error

ω(y)− ωh(y) =M∑k=3

hkvk(y)+ O(hM+1

),

uniformly in compact sets of R2 \ 7. Finally the pointwise superconvergence is still

valid, namely (27) and (28) hold.In our experiments we have taken the ellipse x(t) = (x1(t), x2(t)) := (R cos 2πt,

sin 2πt) and

f (t) := 3x1(t)[x2

1 (t)− x22 (t)]− 6x1(t)x2(t)x

′1(t).

Then the exact solution of the hypersingular equation (29) satisfying u(0) = 0 is

u(t) = 1

2(1 + R)2[−1 + 2R + (1 + R) cos(4πt) sin(2πt)

].

In table 4 we show e0h and e1

h for different values of N and estimate the orders of conver-gence (R = 2).


On the other hand, we take y0 = (3, 4) and define a new set of approximations byextrapolation for the same values of N :

ω0h := ωh(y0),

ωj

h := (3/2)j+2ω

j−1h − ωj−1

3h/2

(3/2)j+2 − 1, j > 0.

Notice that the ratio of discretization level between consecutive grids is 3/2. Tables 5and 6 show the errors εjh = |ω(y0)− ωjh|. The exact potential has been computed usingthe composite trapezoidal rule with error estimation to obtain thirty digits of the exactvalue.

Table 4Pointwise error and e.c.r.

N e0h e.c.r. e1h e.c.r.

64 4.987E−2 3.196E−396 2.248E−2 1.965 9.590E−4 2.969

144 1.005E−2 1.987 2.860E−4 2.984216 4.482E−3 1.990 8.520E−5 2.987324 1.998E−3 1.992 2.533E−5 2.991486 8.899E−4 1.994 7.524E−6 2.994

Table 5Errors after extrapolation.

N ε0h

ε1h

ε2h

ε3h

ε4h

ε5h

64 1.13E−596 3.56E−6 2.97E−7

144 1.10E−6 5.97E−8 1.23E−9216 3.34E−7 1.19E−8 1.72E−10 1.17E−11324 1.01E−7 2.37E−9 2.36E−11 1.11E−12 9.02E−14486 3.01E−8 4.71E−10 3.20E−12 1.02E−13 5.71E−15 4.60E−16

Table 6E.c.r. for the errors in table 5.

3 4 5 6 7

2.8512.904 3.9592.937 3.971 4.8502.959 3.980 4.898 5.8102.973 3.986 4.936 5.871 6.806


Acknowledgements

Both authors are partially supported by DGES Project PB97/1013. The first authoris supported by a scholarship of DGA reference B48/98.

Appendix: some expandable operators

In this section we prove that some boundary integral operators can be written asexpandable pseudodifferential operators. We introduce the following notation: if A ∈E(n) has the expansion given by (2) we write the formal series

Aexp=

n∑j=−∞

ajDj +

n∑j=−∞

bjHDj .

Notice first that if K is an integral operator with infinitely differentiable kernel then K ∈ Eand K

exp= 0. Moreover, if g ∈ D′ satisfies | g(m)| � C|m|n for m �= 0 being n ∈ Z, thenconvolution with g, i.e.,

g ∗ u :=∑m∈Z

g(m)u(m)φm

is �do of order n.

A.1. Logarithmic operators

We now show that integral operators of the form

Vu :=∫ 1

0A(·, t) log

(sin2 π(· − t))u(t) dt,

where A ∈ C∞(R2) is periodic in both variables, are expandable. We first investigate theparticular case

Vku :=∫ 1

0sink(2π(· − t)) log

(sin2 π(· − t))u(t) dt.

Lemma A.1. Let k � 0. Then Vk ∈ E(−k − 1) and there exists a sequence of complexnumbers (ξk,j ) such that

Vkexp=

−k−1∑j=−∞

ξk,jHDj .

Moreover,

V0exp= −HD−1.


Proof. If fk(t) := sink(2πt) log(sin2 πt) then Vk is the operator of convolution withfk. Since

fk(m) = −sign(m)k!ikmk+1

k∏j=0

1

1 − ((k − 2j)/m), if |m| � k + 1,

it follows that |fk(m)| � C|m|−k−1 for m �= 0 and hence Vk is �do of order −k − 1.Simple expansions show that

fk(m)= −sign(m)k!ikmk+1

k∏j=0

[M−k−1∑r=0

(k − 2j

m

)r+ ((k − 2j)/m)M−k

1 − ((k − 2j)/m)

]

=M−k−1∑r=0

sign(m)ηk,rmr+k+1

+ RkM+1(m)

with |RkM+1(m)| � C|m|−M−1, for some ηk,r ∈ C. Then the operator of convolutionwith the function ∑

|m|>kRkM+1(m)φm

is �do of order −M − 1. This proves the result with ξk,j := (2π i)j ηk,j−k−1. �

Proposition A.2. In the hypotheses given above, V ∈ E(−1) and

Vexp=

−1∑k=−∞

bkHDk.

Proof. Consider first the following periodic version of the Taylor expansion. For allk � 1 we can write

sink(2πt) =∞∑j=kρj,kt

j

(notice that ρk,k �= 0 and that ρk,j = 0 if j + k is odd). Then we can construct an infiniteupper triangular matrix such that

σ1,1 σ1,2 . . .

σ2,2 . . .. . .

ρ1,1 ρ1,2 . . .

ρ2,2 . . .. . .

=

1

1

2!. . .

.


Obviously, we obtain σj,k = 0 if j + k is odd. We can then define a sequence ofdifferential operators

L0 := I, Lk :=k∑j=1

σj,kDj , k � 1

which serves to create a periodic Taylor expansion: for f ∈ D

f (t) =2M−1∑k=0

Lkf (s) sink(2π(t − s)) + R2M(t, s) sin2M

(π(t − s)), (A.1)

where R2M ∈ C∞. If we apply (A.1) to A(s, ·) and define

ck(s) := (−1)kLk[A(s, ·)](s)

we can split V in the form

Vu =2M−1∑k=0

ckVku+∫ 1

0R2M(·, t) sin2M

(π(· − t)) log

(sin2(π(· − t)))u(t) dt,

where R2M ∈ C∞. The remainder is an integral operator whose kernel has 2M − 1continuous derivatives plus weakly singular derivatives of order 2M. Hence this operatoris bounded fromHs toHs+2M for all s ∈ [−2M, 0]. The result then follows readily fromlemma A.1. �

It is well known that logarithmic kernel integral operators play a central role in thetheory of boundary integral equations. For example, see [3,17] and references therein.

A.2. Integro–differential operators with Cauchy kernels

Let x ∈ D be such that x(t) defines a simple closed curve in C and |x′(t)| �= 0 forall t and x(t) �= x(s) if t − s /∈ Z. We consider operators of the form

Cu(s) :=n∑j=0

aj (s)Dj u(s)+

n∑j=0

1

π ip.v.

∫ 1

0Cj(s, t)D

j u(s)x′(t)

x(t) − x(s) dt,

where aj , Cj are 1-periodic and infinitely differentiable and p.v. stands for the Cauchyprincipal value of the integral. For simplicity we denote D0u = u instead of its definitiongiven in (1). Operators like this have been considered in [3,17] and include an widevariety of boundary integral operators.

Proposition A.3. In the hypotheses given above, C ∈ E(n) and

Cexp=

n∑j=0

ajDj +

n∑j=0

bjHDj .


Proof. Let

H(s, t) :=

1

π

tan π(t − s)x(t) − x(s) , t − s /∈ Z,

1

x′(s), t − s ∈ Z.

Then H is smooth and 1-periodic. Moreover, we can write

1

π ip.v.

∫ 1

0Cj(·, t)v(t) x′(t)

x(t) − x(·) dt

= −i p.v.∫ 1

0Cj(·, t)v(t) cot

(π(t − ·)) dt =: Hj v

with Cj(s, t) := H(s, t)Cj (s, t)x′(t). It is straightforward that Hj

exp= bjH beingbj (s) := Cj(s, s). Hence C is expandable. �

References

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).[2] D.N. Arnold, A spline-trigonometric Galerkin method and an exponentially convergent boundary

integral method, Math. Comp. 41 (1983) 383–397.[3] D.N. Arnold and W.L. Wendland, On the asymptotic convergence of collocation methods, Math.

Comp. 41 (1983) 349–381.[4] G.A. Chandler and I.H. Sloan, Spline qualocation methods for boundary integral equations, Numer.

Math. 58 (1990) 537–567.[5] G. Chen and J. Zhou, Boundary Element Methods (Academic Press, New York, 1992).[6] Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind inte-

gral equations, SIAM J. Numer. Anal. 35 (1998) 406–434.[7] Z. Chen, Y. Xu and J. Zhao, The discrete Petrov–Galerkin method for weakly singular integral equa-

tions, J. Integral Equations Appl. 11 (1999) 1–35.[8] M. Crouzeix and F.-J. Sayas, Asymptotic expansions of the error of spline Galerkin boundary element

methods, Numer. Math. 78 (1998) 523–547.[9] V. Domínguez and F.-J. Sayas, Local expansions of periodic spline interpolation with some applica-

tions, Math. Nachr., to appear.[10] V. Domínguez, Asymptotic analysis of projection methods for boundary integral equations, Ph.D.

thesis, University of Zaragoza (2001).[11] 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.[12] M.A. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Ya.B. Rutitskin and V.Ya. Stetsenko, Approximate

Solution of Operator Equations (Wolters-Noordhoof Publishing, Groningen, 1972).[13] R. Kress, Linear Integral Equations, 2nd ed. (Springer-Verlag, Berlin, 1999).[14] S. Prössdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations

(Akademie Verlag, Berlin, 1990).[15] J. Saranen, Local error estimates for some Petrov–Galerkin methods applied to strongly elliptic equa-

tions on curves, Math. Comp. 48 (1987) 485–502.[16] J. Saranen, Extrapolation methods for spline collocation solutions of pseudodifferential equations on

curves, Numer. Math. 56 (1989) 385–407.


[17] J. Saranen and W.L. Wendland, On the asymptotic convergence of collocation methods with splinefunctions of even degree, Math. Comp. 45 (1985) 91–108.

[18] F.J. Sayas, Fully discrete Galerkin methods for systems of boundary integral equations, J. Comput.Appl. Math. 81 (1997) 311–331.

[19] G. Schmidt, The convergence of Galerkin and collocation methods with splines for pseudodifferentialequations on closed curves, Z. Anal. Anwendungen 3 (1984) 183–196.

[20] I.H. Sloan and W.L. Wendland, Spline qualocation methods for variable-coefficients elliptic equationson curves, Numer. Math. 83 (1999) 497–533.

[21] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, Berlin, 1972).[22] W. Wendland, Boundary element methods for elliptic problems, in: Mathematical Theory of Finite

and Boundary Element Methods, eds. A.H. Schatz, V. Thomée and W.L. Wendland (Birkhäuser, Basel,1990) pp. 219–276.

Full asymptotics of spline Petrov–Galerkin methods for some periodic pseudodifferential equations

Documents

Transcript of Full asymptotics of spline Petrov–Galerkin methods for some periodic pseudodifferential equations