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Journal of Computational and Applied Mathematics 106 (1999) 307–316
www.elsevier.nl/locate/cam

Adaptive nite element–boundary element solution of
boundary value problems
O. Steinbach
Universitat Stuttgart, Mathematisches Institut A, Pfaenwaldring 57, D 70569 Stuttgart, Germany
Received 21 September 1998

Abstract
An adaptive nite element–boundary element algorithm is proposed to compute an approximate solution of a given
boundary value problem. The convergence in H 1 (
) is controlled by a boundary element based a-posteriori error estimator
from which an adaptive renement strategy is derived. Corresponding error estimates are given based on appropriate
c 1999 Elsevier Science B.V. All rights reserved.
boundary element error estimates in negative Sobolev norms.
Keywords: Boundary element; Finite element; Adaptivity

1. Introduction
For the solution of a homogeneous Dirichlet boundary value problem,
Lu(x) = 0

for x ∈
⊂ Rn (n = 2; 3);

u(x) = g(x) for x ∈

:= @


(1.1)

with an elliptic second order partial dierential operator L we consider a boundary element method,
where an approximate solution of Eq. (1.1) is described by a representation formula involving an
approximate boundary element solution. The (linear) interpolation of this solution denes a nite
element function within the bounded domain
. Using again the representation formula for the
solution and its partial derivatives we are able to control the error of this nite element–boundary
element solution and we can dene some adaptive strategy to construct a solution having an almost
minimal error in H 1 (
) for a xed boundary element solution. Besides an accurate computation of

a nite dimensional solution of Eq. (1.1) we can use the nal triangulation of our approach as an
initial mesh in nite element computations for more complicated problems as in Eq. (1.1), i.e., in the
case of partial dierential operators with non-constant coecients, or even as a boundary element
a-posteriori error estimator in nite element computations. Note that the consideration of Dirichlet
boundary conditions in Eq. (1.1) does not restrict the applicability of our method. For mixed boundary
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 7 3 - 4

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O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

value problems one may use any boundary element method to compute approximate solutions for the
unknown Cauchy data. Then one can apply the method proposed in this paper directly. Moreover,
the generalisation to inhomogeneous partial dierential equations in Eq. (1.1) will be straightforward.
The use of boundary element methods is the numerical solution of boundary integral equations
which are equivalent to the original boundary value problem, for an introduction, see e.g. [9].
In Section 2 we describe a direct boundary integral approach which is discretized by a Galerkin
method. Note that also qualocation or collocation schemes may be applied. For the solution of

mixed boundary value problems using boundary integral equations and boundary elements, see e.g.
[2,13]. If an approximate boundary element solution is determined, the solution of the original
boundary value problem can be computed inside
pointwise with a high accuracy due to available
error estimates in negative Sobolev norms [6]. However, in many applications one is interested
in a nite dimensional solution of the original problem to be used in a postprocessing such as
visualisation, computation of functionals including the solution or in nonlinear solution processes.
In Section 3 we give an adaptive strategy to compute such a solution, where we use a boundary
element based a-posteriori error estimator to dene an appropriate renement of the mesh. Note that
after computing the boundary element solution once, no further linear systems have to be solved.
Some considerations of the numerical amount of work are given in Section 4. Section 5 is devoted
to the numerical analysis of our method, i.e., to give corresponding error estimates. A numerical
example in Section 6 underlines the advantage of the proposed method even in a comparison with
an adaptive nite element computation.
2. Boundary element methods
If a fundamental solution U ∗ (x; y) of the partial dierential operator L in Eq. (1.1) is given, the
solution of the boundary value problem (1.1) can be described by the representation formula
u(x) =

Z

U ∗ (x; y)t(y) dsy −

Z

g(y)T ∗ (x; y) dsy

for x ∈
;

(2.1)

where t = Tu, T ∗ (x; y) = Ty U ∗ (x; y) and T is the conormal derivative operator according to L. For
x → , Eq. (2.1) gives the boundary integral equation
Z
1
(2.2)
(Vt)(x) = g(x) + g(y)T ∗ (x; y) dsy = : f(x) for x ∈
2
with the single layer potential operator

(Vt)(x) =

Z

U ∗ (x; y)t(y) dsy :

(2.3)

Note that V : H −1=2 ( ) → H 1=2 ( ) is bounded and satises a Gardings inequality [15]; for n = 2 we
assume diam
¡ 1 [5].
For a family of boundary triangulations h we consider trial spaces
Zh := span{’k }Nk=1 ⊂ H −1=2 ( )

(2.4)

of discontinuous splines of polynomial degree , e.g. of piecewise constant trial functions ( = 0).
Then the Galerkin variational formulation of Eq. (2.2) is to nd th ∈ Zh such that
hVth ; h iL2 ( ) = hf; h iL2 (

)

for all h ∈ Zh :

(2.5)

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309

Note that we may solve Eq. (2.5) for an adaptive rened family of triangulations h and corresponding trial spaces Zh up to some required accuracy. After a nal boundary element solution th is
computed, replacing in Eq. (2.1) t by th , an approximate solution of Eq. (1.1) is given by
uh (x) =

Z

∗

U (x; y)th (y) dsy −

Z

g(y)T ∗ (x; y) dsy

for x ∈
;

(2.6)

1
(2.7)
U ∗ (x; y)th (y) dsy + g(x) − g(y)T ∗ (x; y) dsy for x ∈ :
2
In the next section we will describe a nite dimensional approximation of Eq. (2.6) by using a nite
element interpolation with respect to an adaptive triangulation of
.
uh (x) =

Z

Z

3. An adaptive postprocessing algorithm
For the bounded domain
we consider a family of regular triangulations

H =

N

X

(3.1)


k

k=1

with an initial or coarse triangulation

H0 and M
nodal points xk . With respect to (3.1) we dene
the usual nite element trial space
1


W H = span{ k }M
k=1 ⊂ H (
)

(3.2)

of piecewise linear hat functions. Using Eq. (3.2) we dene an approximate nite element solution
of Eq. (1.1) by
ũ H (x) =

M

X

ũ k ·

k (x)

(3.3)

k=1

with coecients given by the approximate representation formulae (2.6) and (2.7),
ũ k = uh (xk ) for k = 1; : : : ; M
:

(3.4)

Note that ũ H is the linear interpolant for uh in W H . To controll the error of the approximate solution
(3.3) and to get a renement strategy we dene an approximate nite element error locally as
k ẽ H k H 1 (
k ) := k uh (x) − ũ H (x) k H 1 (
k )

for k = 1; : : : ; N


(3.5)

and rene all nite elements
k where
k ẽ H k H 1 (
k ) ¿ · max k ẽ H k H 1 (
l )
l=1;:::; N


(3.6)

is satised with some appropriate renement parameter . To avoid hanging nodes and to get a
family of regular triangulations, we may have to rene additional triangles as originally indicated
in Eq. (3.6). In two dimensions we use some standard renement rules as sketched in Fig. 1 (for
three hanging nodes per triangle), Fig. 2 (for two hanging nodes per triangle) and Fig. 3 (for one
hanging node per triangle). In the last two cases we have to distinguish if there is a hanging node
on the largest edge of the triangle or not.

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O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

Fig. 1. Renement strategy for three hanging nodes.

Fig. 2. Renement strategies for two hanging nodes.

Fig. 3. Renement strategies for one hanging node.

Note that we have to stop this renement strategy when some level of accuracy is reached which
depends clearly on the accuracy of the boundary element solution th .
Using the local error estimators (3.5) we are able to dene a global error estimator as
k ẽ H k H 1 (
) =

N

X
k=1

k ẽ H k 2H 1 (
k )

!1=2

:

(3.7)

4. Numerical complexity
The numerical complexity of the proposed algorithm consists of the part to solve the Galerkin
boundary element formulation (2.5), the computation of the nite element–boundary element solution
(3.3) by computing (3.4) via the representation formulae (2.6), (2.7) and the application of the local
error estimators (3.5).
Since the stiness matrix of the discrete single layer potential V in Eq. (2.5) is in general dense,
the Galerkin discretization of Eq. (2.5) as well as an iterative solution of the discrete linear system will require O(N 2 ) operations. Note that this can be reduced when using fast acceleration
techniques such as panel clustering [4]. The computation of the nite element–boundary element solution requires O(M
N ) operations. Note that one has to compute all nodal values only once since
no linear system has to be solved and the renement strategy is nested. Applying again the discrete

O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

311

representation formula (2.6) to compute the local error estimator (3.5) will cost O(N
N ) operations.
Therefore, the complexity of our nite element–boundary element algorithm can be estimated as
O(N (N + M
+ N
)):

(4.1)

An optimal nite element computation will cost O(M
[log M
]) operations in computing the solution
and O(N
) to get an error estimator. In our numerical example we will see that we can choose N
to be signicantly smaller than M
and N
, respectively. Hence we claim that the proposed method
is comparable to standard nite element methods when considering partial dierential equations with
constant coecients. Moreover, due to available error estimates in negative Sobolev norms, the
boundary element based error estimator may provide more accurate results as it will be seen from
the numerical example.

5. Error estimates
In this section we provide all required error estimates for the boundary element solution th of
Eq. (2.5) as well as error estimates according to Eqs. (3.3) and (3.5). First we note that in Zh there
holds the approximation property [8], i.e., for 6s6 + 1 and  ¡ 21 (n = 2); 60 (n = 3) we have
inf k  − h k H  ( ) 6c · h s− · k  k H s (

h ∈Zh

(5.1)

)

for all  ∈ H s ( ). Moreover, the solution th of the Galerkin formulation (2.5) is uniquely determined
[15] and satises the error estimate [5,6,9]
k t − th k H  ( ) 6c · h s− · k t k H s (

(5.2)

)

if t ∈ H s ( ) and −2−66s6+1;  ¡ 21 (n=2); 60 (n=3). Hence we get the maximal error
reduction of h2+3 when measuring the error in the H −2− ( ) norm, if the solution t is regular enough.
Note that the local bounds in Eq. (5.2) may dier when using interpolated boundary conditions gh
instead of g in the Galerkin formulation (2.5). Now it is straightforward that for x ∈
far enough
from the boundary we get the error estimate

Z


∗

|u(x) − uh (x)| 6  U (x; y)(t(y) − th (y)) dsy 

6 k U ∗ (x; · ) k H 2+ ( ) k t − th k H −2− (
6 c·h

s++2

)

· k t k H s( );

(5.3)

with t ∈ H s ( ); s6 + 1. Note that in the case when x is near to the boundary, i.e. dist(x; )6h2=3 ,
one can use other techniques to compute the solution uh (x) with high accuracy [10,12]. In a similar
manner as in Eq. (2.6) we can compute the partial derivatives in x ∈
by
@
@
uh (x) =
@xi
@xi

Z

U ∗ (x; y)th (y) dsy −

@
@xi

Z

g(y)T ∗ (x; y)dsy

providing similar error estimates as given in Eq. (5.3).
Now we are in a position to formulate the basic error estimates required.

(5.4)

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O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

Theorem 5.1. Let u ∈ H  (
); 62; be the solution of Eq. (1:1). For the nite element–boundary
element solution ũ H given by Eq. (3:3) there hold the pointwise error estimate for x ∈
k
− n2

|u(x) − ũ H (x)|6c · Hk

1

· |u|H  (
k ) + c · h ++ 2 · k t k H − 32 ( ) ;

(5.5)

and the local error estimates
n

1

k u − ũ H k L2 (
k ) 6c · Hk · |u|H  (
k ) + c · Hk2 · h ++ 2 · k t k H − 32 ( ) ;
n

1

k u − ũ H k H 1 (
k ) 6c · Hk−1 · |u|H  (
k ) + c · Hk2 · h ++ 2 · k t k H − 32 ( ) :

(5.6)

(5.7)

Proof. Using the pointwise error estimate (5.3) we get
|u(x) − ũ H (x)| 6 |uh (x) − ũ H (x)| + |u(x) − uh (x)|
6 |uh (x) − ũ H (x)| + c · h s++2 · k t k H s ( ) :
3

Since u ∈ H  (
) we have t ∈ H − 2 ( ) by applying the trace theorem and hence s6 − 23 . Since
ũ H is the linear interpolant of uh we can apply standard error estimates [1, Chapter 4] to derive Eq.
(5.5). Note that k uh k H  (
k ) can be bounded by k u k H  (
k ) due to denition (2.6). Taking the
square and integrating over
k gives Eq. (5.6). Using corresponding error estimates for 3 ũ H we
can derive Eq. (5.7).
From Eq. (5.7) we get directly the global estimate
n

1

k u − ũ H k H 1 (
k ) 6c · Hk−1 · |u|H  (
k ) + c · Hk2 · h++ 2 · k t k H − 32 ( ) :

(5.8)

As in the proof of Theorem 5.1 we get also
n

1

| k u − ũ H k H 1 (
k ) − k uh − ũ H k H 1 (
k ) |6c · H 2 · h++ 2 · k t k H − 32 ( ) ;

(5.9)

which provides that the error estimator (3.5) is equivalent to the error if h is suciently small.
6. Numerical example
As numerical example we consider for n = 2 the Dirichlet boundary value problem

u(x) = 0

for x ∈
;

u(x) = g(x) for x ∈ ;

(6.1)

where
is the L shaped domain as sketched in Fig. 4.
In Eq. (6.1) the given Dirichlet data are taken in such a way that the exact solution of the
boundary value problem (6.1) is given in polar coordinates by
2’
:
(6.2)
3
First we solve the Galerkin boundary integral formulation (2.5) starting from an initial triangulation
of N = 8 boundary elements. Using an a-posteriori error estimator as described in [11] we get a
hierarchy of adaptively rened boundary triangulations. All linear systems equivalent to Eq. (2.5)
2

u(x) = u(r; ’) = r 3 · sin

O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

313

Fig. 4. L shaped domain
and initial triangulation.
Table 1
H 1 (
) error of the nite–boundary element solution
Estimated

Exact

M


N


Absolute

Relative

Absolute

Relative

8
17
28
49
73
131
189
350
596
1095
2003

6
20
40
76
122
230
338
646
1116
2086
3868

1:66 − 1
1:14 − 1
8:26 − 2
5:96 − 2
4:52 − 2
3:34 − 2
2:72 − 2
1:97 − 2
1:49 − 2
1:09 − 2
8:09 − 3

3:04 − 1
2:08 − 1
1:51 − 1
1:09 − 1
8:26 − 2
6:09 − 2
4:97 − 2
3:60 − 2
2:72 − 2
2:00 − 2
1:48 − 2

1:66 − 1
1:14 − 1
8:27 − 2
5:96 − 2
4:53 − 2
3:34 − 2
2:73 − 2
1:97 − 2
1:49 − 2
1:10 − 2
8:21 − 3

3:05 − 1
2:08 − 1
1:51 − 1
1:09 − 1
8:27 − 2
6:11 − 2
4:98 − 2
3:59 − 2
2:72 − 2
2:01 − 2
1:50 − 2

are solved by a conjugate gradient method using a preconditioner as proposed in [14] which is well
suited for the adaptive renement case. In the example described here it was sucient to stop the
boundary element computation after using N = 86 boundary elements yielding a L2 error of
k t − th k L2 ( ) = 2:47 × 10−1 :
Using the Galerkin boundary element solution th we dene the nite element–boundary element
solution (3.3) rst on the initial domain triangulation as shown in Fig. 4 and then on the rened
triangulation when applying the renement criteria (3.6) with  = 0:3. In Table 1 we give both the
estimated error (3.7) and the exact error k u − uH k H 1 (
) using the exact solution given by Eq. (6.2)
for all levels of mesh renement.
For an assessment of our results we rst use a nite element computation on the nest nite
element–boundary element triangulation shown in Fig. 5 with M
= 2003 nodes getting an error of
k u − uh k H 1 (
) = 8:02 × 10−3 ;

k u − uh k H 1 (
)
= 1:46 × 10−2 :
k u k H 1 (
)

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O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

Fig. 5. FEM=BEM triangulation with 2003 nodes.

Hence, the nite element–boundary element solution we computed is closed to the nite element
solution minimizing the energy norm | · |H 1 (
) in W H .
Using the results given in [7] we can compare our results with an adaptive nite element computation based on an error estimator given in [3]. For the nal triangulation as shown in Fig. 6 with
M
= 1990 nodes and N
= 3904 volume elements the approximate nite element solution has an
error of
k u − uh k H 1 (
) = 1:22 × 10−2 ;

k u − uh k H 1 (
)
= 2:24 × 10−2 :
k u k H 1 (
)

Summarizing the numerical results we conclude that the nite element–boundary element method
proposed gives a triangulation providing some slightly better error results than an adaptive nite
element computation. Note that the nite element–boundary element solution was computed just
with 86 boundary elements only, and that the computation of the coecients is a postprocessing
without solving any linear system.
The proposed algorithm to dene a nite element–boundary element solution of a boundary value
problem in consideration and the boundary element based error estimator can be applied directly
to mixed boundary value problems and other partial dierential equations with constant coecients.
From this point of view there is also no restriction in the space dimension, however, adaptive
triangulations in three dimensions are still a complicated task.

O. Steinbach / Journal of Computational and Applied Mathematics 106 (1999) 307–316

315

Fig. 6. FEM triangulation with 1990 nodes.

Acknowledgements
Part of this work was done while the author was a Postdoctoral Research Associate at the Institute
for Scientic Computation (ISC), Texas A&M University, College Station. The nancial support of
the ISC is gratefully acknowledged. Special thanks are due to S. Tomov (ISC) for performing all
the nite element computations described in this paper.
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