Golberg [3] and Shu [6] follow the -essentially different ... · generalises the superconvergence theory of Shu [6], who considered a piecewise constant Galerkin method for the single

Theory and practice of superconvergence

based on a posteriori error estimation in a

Galerkin boundary element method

Jan H. Brandts

Mathematical Institute, Czech Academy of Sciences, Zitnd 25,

11 567 Praha 1, Czech Republic

Email: [email protected]

Abstract

In recent papers [2, 6] superconvergence results for Galerkin approxi-

mations for integral equations of the first kind have become available

m the case that piecewise polynomial spaces are used as test and trial

space. The superconvergence considered in those papers is based onthe approach of Richter [5] for equations of the second kind. In this

paper we will discuss this approach and compare it to superconver-

gence based on Sloan iteration [4, 7]. Furthermore, we will construct

a posteriori error estimators based on the superconvergence and testthem numerically, also with respect to adaptive refinement of themesh.

1 Introduction

1.1 Short overview

Following the results obtained in the second half of the seventies on

superconvergence for Galerkin discretisations of integral equations of

the second kind by Sloan [7, 4] and Richter [5], recently, similar resultshave become available for equations of the first kind. For example

Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X

334 Boundary Elements

Golberg [3] and Shu [6] follow the -essentially different- approaches

of Sloan and Richter respectively, but the situation for equations of

the first kind seems to be much more difficult, in particular when

piecewise polynomial approximation is considered. In [2] the author

generalises the superconvergence theory of Shu [6], who considered

a piecewise constant Galerkin method for the single layer potential.

This generalisation concerns a class of (even) pseudo-differential op-

erators of order (3 > 0 and also extends to piecewise linear approxima-

tions. Moreover, a (numerical) counter-example for superconvergence

in the piecewise quadratic approximation is given.

1.2 Goals and organisation of this paper

The goal of this paper is twofold. First, in Section 2, we show that

for equations of the second kind, the superconvergence of Richter

is implied by the superconvergence of Sloan. This is a motivation

to investigate the Richter type also for equations of the first kind,

since a wider range of settings might be open to superconvergence

of this type. Apart from that, it allows for easier and cheaper post-

processing mechanisms. We will then proceed to discuss Golberg's

approach [3] in estimating the error using the residual and comment

on the implications for piecewise polynomial spaces.Second, the superconvergence results of both papers [6, 2] are

used in the construction of a posteriori error estimators for the Galer-

kin approximation of the single layer potential equation. The post-

processors come directly from the standard finite element contextwhere they have been proved to be valuable tools in a posteriori

error estimation. The estimators are tested numerically, also for an

application that is not covered by the theory, i.e. for non-uniform

partitionings with respect to adaptive refinement. The results both

confirm as well as go beyond the underlying theory.

2 Two approaches to superconvergence

In this section we will recall two different approaches to the concept

of superconvergence in Galerkin boundary element methods. One ofthem, which is relatively well-known, is due to Sloan [3, 7] and the

other to Richter [5]. Both approaches lead to improved approxima-tions and to tools for a posteriori error estimation. We will distinguish

between integral equations of the first and of the second kind.


Boundary Elements 335

2.1 Equations of the second kind

2.1.1 Superconvergence of Richter type

Let 7f be a compact integral operator on ZA Consider the equationof the second kind

(7-#) = y (i)

where one is not an eigenvalue of # and / is a given function. The

Galerkin discretisation with discrete space W* consisting of all piece-

wise polynomials of degree k is as follows.

V^ G : < (/ - #)< , ^ > = < /, T/V, >. (2)

Denoting by P£ the.L^-projection on W^ and combining (1) and(2) gives

P^-<^ = P -< ). (3)

The left-hand side of (3) is the difference between the Z^-best

approximation of 0 and its Galerkin approximation and is therefore

a measure of the Z^-optimality of the Galerkin method. Whereas

both P^ and (^ have the same f/% convergence rate towards </;, one

can, under some smoothness assumptions on the partial derivatives

of the integral kernel of A\ prove that their mutual difference is of

higher order. We refer to the original paper by Richter [5] for detailsand proofs.

Definition 2.1 The Galerkin approximation is said to be supercon-vergent in the sense of Richter if

(6-0), (4)

where o is the Landau (lower case) order symbol.

There are two important consequences of Richter superconver-

gence. One is higher order convergence rates of the Galerkin approx-

imation in the Gauss points (indirectly caused by P£p( ) = p(g^) for

all p E W^ in the Gauss points g ). Second is, that post-processing

mechanisms for P^ can usually relatively easy be proved to be suc-

cessful for the Galerkin solutions as well. Such post-processors arewidely available and usually very easy to implement since they are

often based on some simple averaging or interpolation scheme. Wewill comment on this further on.



2.1.2 Superconvergence of Sloan type

For the Galerkin solution ^ the Sloan iterate S(0/>) is defined as

follows (Cf. 4, 7] for more details).

S(0h) = f + K4>h = (I - K)<j> + K<ph = <f>- K(4> - fa). (5)

Definition 2.2 The Galerkin approximation is said to be supercon-

vergent in the sense of Sloan if

|o = o(| - |o) (b-^0), (6)

where o is the Landau (lower case) order symbol.

So, Sloan superconvergence means that the post-processing mech-

anism in (5) gives an approximation with better approximation prop-

erties. This is in contrast to Richter superconvergence, which only

implies the possibility of successful post-processing the Galerkin so-

lution without having yet performed it. Since we already commented

on the availability of such post-processors, the difference is merely a

matter of (historically grown) terminology.

It is well-known that under some assumptions, Sloan supercon-

vergence indeed occurs. This is due to the following equality, which,

for h small enough, holds for all Wh G Wfc:

Here, the first operator on the right is the uniformly bounded discre-

tised inverse of 7 — K and the second operator KP^ — K has a normequal to the norm of its adjoint.

|A-pfc-A-|o^o = |(P£-/)A"o_o. (8)

Assuming that A * : L^ — > JFP is compact for some 0 < s < k -f 1 we

have

|(P% - 7)7Cu|o < C/ |7r?;|, < C/ |o (9)

Because point wise convergence on a compact set implies uniform con-vergence, this leads to the following theorem.

Theorem 2.3 (Ch.2 Sect. 3 of [4]) Assume that <p G H^^ and7C : 1 % -^ #* is compact for some 0 < s < A; + 1. Then tAe

iterate SX /J of the GaJerkm approximation (^ G M^ satisfies

| i. (10)



2.1.3 Relation between the two types

It is easily shown that when the Sloan iterate superconyerges, we also

haye superconyergence in the sense of Richter. This means that if a

cheap post-processor for P^<p is ayailable, it might be preferable to

use it instead of doing the more expensiye Sloan iteration.

Corollary 2.4 Under the assumptions of Th.2.3 there is supercon-vergejice in tJ]e sense of EicMer too.

|P - 0<C/2. '|0 i. (11)

Proof. Combine equations (3) and (5) to link together the two con-

cepts of superconyergence as follows.

P%o-0t = P%(0-SW). (12)

Taking Z/ norms on both sides and using that the f^ projection has

norm less than or equal to one proyes the statement. O

2.1.4 A note on a posteriori error estimation

It is instructiye to note the following. If 5%0/J conyerges to 0 faster

than <^, conyerges to /L the difference <=k(%) := |(ph - 5%( )|o can

be used to estimate the error 0 — 0/Jo asymptotically exact in the

following sense. The quantity 7;(A) is called (global) efficiency index.

for/i-,0. (13)

Using ecj.(5) we can rewrite the estimator as

(14)and note that it is actually the norm of the residual that is used

as an error estimator. It is exactly this property that Golberg in

his paper [3] tries to establish for equations of the first kind using

superconyergence of a generalised concept of Sloan iteration.

2.2 Equations of the first kind

An equation of the first kind is typically of the form



For example, the single-layer potential equation for solving a Laplace

problem with Dirichlet boundary conditions can be written in this

way. For such equations, the idea to write V — / — (/ — V) and

to try to apply the theory for equations of the second kind is not

often successful, since (/ — V) might lack the desired properties, in

particular compactness.

2.2.1 Generalised Sloan iteration

As investigated by Golberg in [3], the idea of splitting the operator

can be quite useful. If V can be written as

I/ = 14 -# (16)

with VQ bounded with bounded inverse, and with K compact, then a

generalised concept of Sloan iteration can be defined. Explicitly, let

(17)

be the generalised Sloan iterate, were the Galerkin approximation 0/>

is defined by

PfcV^ = PfcVo^, - PfcA^/i = Pfc/. (18)

For VQ — / the generalised Sloan iterate reduces to the definition(5) of Sloan iterate for equations of the second kind.

2.2.2 Another note on a posteriori error estimation

Substituting K = VQ — V in eq.(17) leads to

and assuming superconvergence of S((j>h) in the sense of Def.2.2 leads

also here to the observation that the norm of the difference 0^ — 5(0 )

can be used as an asymptotically exact estimator for the error. Thistime however, the quantity of interest is not the pure residual (as it

was in eq.(14)), but VQ~* applied to the residual. It is therefore ofinterest to find a splitting as in eq.(16) such that

• S((f>h) is superconvergent,

• the appropriate norm of V (f — V(f)h) is easy to evaluate.



Although this is not always easy to establish, for the single-layer

potential on a smooth curve for example, VQ could be taken as the

principle ("circular") part making VQ into an isometry H* —» /P+*

so that an appropriate choice of norms can, at least in some extent,

satisfy the second condition. However, to prove superconvergence of

the corresponding Sloan iterate, Golberg assumes that the operators

VQ and P^ commute. This means that although trigoniometric poly-

nomials can be used, the theory does not include the spaces Wfc of

all piecewise polynomials of degree /c, which are of particular interest

in applications in which singularities and stiff behaviour are likely tooccur.

2.2.3 Relation with Richter type superconvergence

The straight-forward generalisation of eq.(12) is the following. Set-

ting VQ = I gives back eq.(12).

This means that for equations of the first kind, superconvergence

in the sense of Sloan (Cf.Def.2.2) does not automatically result in

superconvergence in the sense of Richter (Cf.Def.2.1), as was the case

in Corollary 2.4. If however, we include Golberg's assumption that

VQ and P£ commute, then we get eq.(12) back from eq.(20), using

that VQ is bijective. And since this assumption was sufficient for

superconvergence of the generalised Sloan iterate, it also is for Richtersuperconvergence.

Remark 2.5 Assume that H and P^ commute. Then the Galerkinapproximation </>/> satisfies

P£( _ P^VrT^Kfih — PL^T*/, (21)

which is the Galerkin discretisation of the equation (I — V^K)<p =

VQ~*/ which is of the second kind. The generalised Sloan iterate (17)

equals the Sloan iterate (5) for this equation and its superconvergence

depends on V^K.

This remark suggests that the assumption that VQ and P£ com-

mute is actually stronger than one might think at first sight. For us,this was the main motivation to abandon the concept of Sloan iterate,

at least for where the single-layer potential equation is concerned, and

concentrate on Richter superconvergence.



2.2.4 Richter type superconvergence for equations of the

first kind

The paper [6] of Shu formed the basis for our investigations into

superconvergence of Richter type for integral equations of the first

kind. One of the main results in Shu's paper is the following.

Theorem 2.6 (Shu, [6]) Consider the single-layer potential equa-

Uon y^ = / on a smooth curve, [/sing piecewise constant approxi-

mations on uniform partitionings, and assuming that the solution 0

is smooth enough, it holds that

inV -<t>ho< Ch* (22)

In [2], the author studies higher order polynomial approximations.

Moreover, instead of limiting the theory to the single layer potential,

a class of even pseudo-differentia! operators of order 2/3 > 0 is consid-

ered. This leads to superconvergence results for the piecewise linear,

and a counter example for the piecewise quadratic case. Particu-

lar attention is paid to the smoothness requirements for the exact

solution. To be more explicit, the results are as follows.

Theorem 2.7 The piecewise constant (resp. linear) Galerkin ap-

proximation 0^, (o\) of the equation VQ = /, where the pseudo-differential operator VQ defined by

00 OO /

If<^= T <^%t, then 1/0 = 0o+

are superconvergent if the order 2(3 > 0. For m either zero or one,

if (6 E#\ 3>2^ +m

if (/) G \ 5 = 2^+77?

^^ if ^ i + < 2| 4- m

3 Numerical experiments

We will now present numerical examples that illustrate the supercon-

vergence behaviour. As our model problem we use the single layerpotential equation on the circle. Essentially this means approximat-

ing l/o^ = / with Vg as in (23) with /J = .



3.1 Smooth solution and equidistant partitionings

We consider the problem with exact solution

(Xz) = 24 cos(6?rz) + 16 cos(87rz). (24)

3.1.1 Piecewise constant approximation

The piecewise constant discrete solution is calculated for 2^,j =

3,•••,8 equidistant subintervals. A post-processed solution is cal-

culated by continuous piecewise linear interpolation on the averaged

jumps in the approximation. This leads to an a posteriori error esti-

mate on each interval.

OTAL EFFICIENCY INDEX i

n

r

OTAL EFFICIENCY INDEX ' D92934e*000

Lr _

0 200C 4000 6000 8000 '0000 12000

Figure 1. Local efficiency index for decreasing h.



The four pictures in Figure 1 give the local efficiency index, i.e. the es-

timated error divided by the exact error on the interval for 16, 32,128

and 256 subintervals. Below in Figure 2 we show the exact solution

and also the behaviour of the global efficiency index.

h * || eff. index.

8

16

32

64

128

256

0.945590

1.267805

1.092934

1.025410

1.006486

1.001565

0 2000 4000 6000 8000

Figure 2. Exact solution (right) and global efficiency index.

From this experiment we conclude the following. First, the error es-

timator works very well. The global efficiency index exhibits evenO(h^) behaviour instead of O(h). This can happen by realising that

the norm of the difference between post-processed approximation and

the approximation itself can, in theory, give the exact norm of the

error even though neither of them is equal to the exact solution. Sec-

ond, we see that also locally the error estimator works fine provided

that the element is not near an extreme value of the exact solution

(but also close to those extremes the behaviour is not dramatically

bad). Third, and this is quite interesting, it seems that asymptot-

ically, the error is -even locally- overestimated which might be of

importance in practice.

3.1.2 Piecewise linear approximation

We repeat the previous experiment but now with piecewise constantapproximation. The post-processing mechanism is taken from [1]. It

projects the two linear functions on two adjacent subintervals on the

quadratic functions on the union of those intervals, and takes the left

part as post-processed solution on the left interval. In Figure 3 we



show the dotted exact solution, the Galerkin solution and the post-

processed solution on the first Eve (of sixteen) intervals. In particular

on the right interval, one can distinguish very well between the linear

and the quadratic approximation.

Figure 3. Post-processing the piecewise linear approximation.

We continue to show the local efficiency indices in Figure 4 below,

for 8,16, 64 and 128 subintervals. Note that the behaviour is more orless the same in structure as in the piecewise constant case.

TOTAL EFFICIENCY INDEX I 349481etOO TOTAL EFFICIENCY ,ND£X 1 110500etOOO

n

\ rn| | _

n l i-^ U L^

! "

.LfL.



TOTAL EFFICIENCY INDEX i TOTAL EFFICIENCY INDEX.I 002B87e+000

I

partitioned unit inlen/al

Figure 4. Local efficiency indices for the piecewise linear

approximation.

To conclude this experiment we show in Figure 5 a logarithmic pic-

ture of the exact error \(j) — 0/Jo and the superconvergent quantity

|P 0 — (f>h\o- Left we present the global efficiency indices.

h-i || el

4

8

16

32

64

128

0

1

1

1

11

T.index.

8138

3495

1105

0227

0070

0029 SUPERCONVERGENCE

Figure 5. Superconvergence (right) and global efficiency index (left).

The slope of the first is two, indicating order In? behaviour, the slope

of the second is three, which stands for oder h? (as was proved in

Th.2.7).



The slope of the first is two, indicating order h^ behaviour, the slope

of the second is three, which stands for oder h^ (as was proved in

Th.2.7).

3.2 Nonsmooth solution and adaptive refinement

We consider the problem with exact solution

(25)

This function (which is displayed in Figure 4) is H (C) but not in

j£P(C) for some 5 > 1. Using piecewise constant approximations, we

concentrated on refinement. We refined according to the principle

that the mean value of the squared error (0 — 0 ) on each subin-

terval should be the same, and the total smaller than some given

tolerance. An interval that had too large an (estimated) error was

simply divided into two. We started with an equidistant partitioning

in four subintervals. The refinement went as follows. After two steps

in which all intervals were refined, only the middle four were refined.

Then followed three steps in which only the middle two were refined.

This resulted in a final grid with 2-2-4 + 4 + 2 + 2 + 2 = 26 subin-

tervals, of which 12 with length 2~^, six with length 2~' , two with

length 2~^. two with length 2~* and four with length 2~8.

As becomes clear from Figure 6, in which we show the local effi-

ciency indices for the last four steps, the local efficiency indices in themiddle of the domain still behave very well under refinement around

the midpoint. Again, the estimator behaves worst at the local ex-

trema of the exact solution.

Y INDEX I 1789456+00 DIAL EFFICIENCY INDEX '



TOTAL EFFICIENCY INDEX: 1 03295Se»000

TAT

2000 4000 6000 8000 10000 12000 14000 16000 18000

TOTAL EFFICIENCY INDEX 1 024481 etOOO

Figure 6. Local efficiency index during refinement.

Below in Figure 7 we show the exact solution (dotted line) and its

final approximation (stepfunction) in one picture. The vertical lines

are there to emphasize the final grid. On the right, we zoomed in on

the middle part of the left picture, and stretched it in the ^-direction

by a factor of about eight.

2000 4000 6000 8000 10000 12000 14000 16000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000

Figure 7. Exact solution (right) and global efficiency index.

4 Concluding remarks

We have seen that there exist successful ways of a posteriori esti-

mating the error in piecewise polynomial Galerkin boundary element

methods without iterating the discrete solution. Even in the case



that the partitionings are not uniform and the exact solution is non-

smooth and stiff", the numerical behaviour was good.

References

[1] J.H. Brandts. A note on uniform superconvergence for the Timo-

shenko beam. Mathematical Methods and Models in the Applied

1994.

[2] J.H. Brandts. Superconvergence of a Galerkin boundary element

method for the single layer potential using piecewise polynomial

approximations. Report Mathematisches Seminar, Christian Al-

brechts Universitdt zu Kiel, 1998.

[3] M.A. Golberg. Superconvergence and the use of the residual

as an error estimator in the BEM-I: Theoretical Development.

Submitted to Boundary Element Communications, 1997.

[4] M.A. Golberg (Ed.) Numerical solution of integral equations.

Volume 42 of: Mathematical concepts and methods in science

and engineering (Series editor: Angelo Miele), Plenum Press,

New York and London, 1990.

[5] G.R. Richter. Superconvergence of piecewise polynomial galerkin

approximations for Fredholm integral equations of the second

kind.

[6] H.Z. Shu. The superconvergence of a Galerkin collocation

method for first kind boundary integral equations on smooth

curves. Journal of Computational and Applied Mathematics,

[7] I.H. Sloan. Improvement by iteration for compact operator equa-

tions. Mathematics of Computation, 30:758-764, 1976.


Documents

Golberg [3] and Shu [6] follow the -essentially different ... · generalises the superconvergence theory of Shu [6], who considered a piecewise constant Galerkin method for the single