A posteriori boundary element error estimation
Jang Jou, Jinn-Liang Liu∗
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan Received 27 January 1998; accepted 16 November 1998
Abstract
An a posteriori error estimator is presented for the boundary element method in a general framework. It is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators. The estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for Symm’s and hypersingular integral equations. Numerical results are also given to demonstrate the e ectiveness of the estimator for these equations. It can be used for adaptive h; p, and hp methods. c 1999 Elsevier Science B.V. All rights reserved.
MSC: 65N35; 65N15; 65R20; 65D07; 45L10
Keywords: Boundary integral equation; Boundary element method; A posteriori error estimate; Adaptive computation
1. Introduction
E ective and ecient a posteriori error estimators play a key role in adaptive numerical methods for boundary value problems. We introduce an error estimator for the boundary element method (BEM) applied to boundary integral equations (BIEs).
The estimator is motivated by the weak residual a posteriori error estimation developed mainly for partial di erential equations (PDEs) in connection with the nite element method (FEM). We refer to [19] for a general framework of the estimation and to [1–4,18–22] for further references on the application of such estimation in adaptive FEM and nite volume method (FVM). We nd that the approach is even more natural for BIEs since the residual is inherently in integral form rather than di erential form which entails speci c treatments of the jumps in the normal derivatives of the nite element solutions on the interfaces between elements. In fact, the various error estimators for FEM di er essentially in the way the jumps are handled. This does not appear to be an issue in
This work was supported by NSC under grant 87-2115-M-009-005, Taiwan.
∗Corresponding author. http:==www.math.nctu.edu.tw=˜ jinnliu.
E-mail address: janjou, jinnliu@math.nctu.edu.tw (J. Liu).
0377-0427/99/$ - see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S 0377-0427(99)00049-7
adaptive BEM; see [6–8,13,26,27]. Nevertheless, there is an intrinsic di erence between boundary integral operators and di erential operators; namely, di erential operators are local operators whereas boundary integral operators are nonlocal [26]. Certain localization concepts such as the in uence index of [26] and the augmented BEM of [13] have been introduced to accomodate this nonlocal property for the needed local and computable a posteriori error estimators.
Our approach is to construct local shape functions for the solution of local residual problems and then compute estimated errors in a localized energy norm which is induced by the diagonal (say, A) of the bilinear form (say, B) de ned by the variational BIEs.
The error estimator is rst presented in a general setting in Section 2 and then applied to Symm’s and hypersingular integral equations in Section 3. We brie y describe the main results in this article. The estimator for a computed solution, say uh, in some boundary element (BE) space Sh⊂ H is
obtained by solving element-by-element local problems in a complementary BE space Sc
h⊂ H. Here
H denotes some Sobolev function space to which the exact solution u belongs. The local problems use the same Galerkin formulation of approximation except that the right side of these problems is a residual of the approximate solution. It is shown here that, for both Symm’s and hypersingular integral equations in two space dimensions, the estimated error ˜e ∈ Sc
h satis es the estimate
C16 :=k ˜ekkekA B
6C2; (1.1)
where C1 and C2 are positive constants independent of the mesh size h; k · kA and k · kB are the
norms associated respectively with the bilinear forms A and B; e = u − uh is the exact error, and
is called the e ectivity index of the estimator.
A posteriori error estimates of the form as (1.1) are very important in practice since they are used to justify the e ectiveness of the resulting adaptive scheme. While this is not comprehensive, we compare our estimator (denoted by JL-estimator) to the estimators of [6–8] (CES-estimator), of [13] (FHK-estimator), and of [26,27] (WY-estimator) which are all based on various local postprocessing schemes on the residual error instead of solving local problems. For a better view in comparison, we summarize the main results of all estimators in Table 1 in which, for simplicity, we restrict to the following conditions: Symm’s and hypersingular BIEs in two space dimensions, unstructured mesh, and under the norms speci ed in the respective references.
The FHK-estimator is developed for the augmented Galerkin BEM described in [13] not for the standard Galerkin BEM. The augmented technique takes into account the behavior of the exact solu-tion near points of singularity. Hence, one has to have an a priori informasolu-tion about the singularities. Moreover, all other estimators are obtained by postprocessing the residual error (e.g., di erentiating
Table 1
Comparison of various estimators
Estimator Symm’s BIE Hypersingular BIE
WY-estimator [26] C16
CES-estimator [6] C16
FHK-estimator [13] C166C2
the residual) in some computable norm which often requires the data function (or equivalently the exact solution) to be smoother. Our approach, in contrast, does not incur the exact solution to be more regular than that is required by the standard a priori estimates. In other words, the estimator holds for the minimal regularity of the exact solution in the sense of Cea’s lemma [10]. The only unproven assumption that we make for our error analysis is the saturation assumption. This assump-tion is very moderate and natural since it essentially says that the approximate soluassump-tion in the larger BE space Sh⊕ Shc is a better approximation to the solution u than uh∈ Sh. This is generally true in
practice. If this assumption is replaced by other assumptions associated with higher regularity on the exact solution, one may be able to analyze the asymptotic exactness of the estimator such as that of [2] for FEM. We shall not consider this topic here.
Compared to the cost of computing the approximate solution uh, the cost of computing the
es-timated error ˜e is fractional since the complementary BE space Sc
h can be constructed using, for
instance, only one or two shape functions on each element. Consequently, we only have one or two equations in the solution of a local problem. In particular, if only one shape function is used for Sc
h, our estimator is then equivalent to the previous residual error estimators; see the numerical
ex-ample for a hypersingular integral equation presented in Section 4. Furthermore, since the estimated error is explicitly calculated there is no restriction on the choice of the norm used to measure the errors. In other words, whichever the norm appropriate for the approximate solution uh can also be
used for the estimated error ˜e. This can be very useful in practice when a more exible norm is needed for assessing the computed solution. We however only prove estimate (1.1) in the energy norm.
2. General framework of the estimator
Let be a bounded Lipschitz domain in Rn; n=2; 3, with boundary ˆ. Let be a closed or open
connected subset of ˆ. Let H be a Sobolev space equipped with the norm k · kH and be de ned on . Consider a general variational problem of the form: Find u ∈ H such that
B(u; v) = F(v) ∀v ∈ H: (2.1)
Assumption 1. Let F(·) be a continuous linear functional on H and B(·; ·) be a symmetric bilinear form on H × H such that there exist two positive constants 1 and 1 for which
B(u; v)6 1kukHkvkH ∀u; v ∈ H; (2.2)
B(u; u)¿ 1kuk2H ∀u ∈ H (2.3)
hold.
By Assumption 1, the energy norm k · kB induced by the bilinear form B(·; ·) is equivalent to the norm k · kH on H. Let Sh be a nite-dimensional subspace of H. The Galerkin approximation of
(2.1) is to nd uh∈ Sh such that
The Lax–Milgram theorem guarantees the existence and uniqueness of the solutions u and uh of
(2.1) and (2.4). Our main concern is to estimate the exact error e = u − uh. Let Th denote a nite
partition of associated with Sh. The partition Th is expressed as
Th= j : j = 1; 2; : : : ; N; N [ j=1 j= ; (2.5)
where the elements j are open disjoint intervals for n = 2 or triangles (or quadrangles or both) for
n=3 and j is the closure of j. The partition Th is not necessarily quasi-uniform. More speci cally,
let i denote the diameter of the inscribed circle for i∈ Th and let hi denote the diameter of the
element i. We assume there exists a positive constant independent of the mesh size h such that
6hi
i (2.6)
for all i∈ Th.
Based on the current partition, we shall construct another nite-dimensional subspace Sc h of H.
We call Sc
h a complementary space of Sh. In order to obtain a practical and ecient error estimator,
its construction is essential. For this, we require Sc
h to meet the following conditions.
Assumption 2. Assume the following conditions hold for Sc h: Sc h:= Shc( 1) ⊕ Shc( 2) ⊕ · · · ⊕ Shc( N); (2.7) Sh:= Sh⊕ Shc⊂ H; Sh6= ∅; Shc6= ∅; (2.8) ku − uhkB6ku − uhkB; ∈ [0; 1); (2.9) where Sc
h( i); i = 1; : : : ; N; denote subspaces whose basis functions have supports only in their
re-spective domain i; uh; is the approximate solution of (2:1) in the larger subspace Sh; and is a
constant independent of the mesh size h.
We do not explicitly compute uh. It is merely for the analysis of the estimator. We now observe
one of the most important properties that distinguishes boundary integral operators from di erential operators; namely, di erential operators are local operators whereas boundary integral operators are nonlocal. Translated into our setting, the di erence is that
B(wj; vi) 6= 0 ∀wj∈ Shc( j); vi∈ Shc( i); (2.10)
for BIEs while it always vanishes for PDEs. To treat this nonlocal property which con icts with the localization of a posteriori error estimation, we are led to consider an equivalent bilinear operator for B(·; ·). We de ne a new bilinear form A(·; ·) on Sh× Sh such that
A(w; v) =XN i=1 B(wi; vi) ∀w; v ∈ Sh; (2.11) where wi(x) = ( w(x) if x ∈ i; 0 otherwise (2.12)
and vi are similarly de ned for all i = 1; : : : ; N. Note that we can write w =Piwi and v =Pivi for
all w; v ∈ Sh. The bilinear form A(·; ·) is a block-diagonal form of B(·; ·) in Sh.
Assumption 3. Let A(·; ·) be an inner product on Sh× Sh and let there exist two positive constants
C1 and C2 (possibly depending on h) such that
C1kwkA6kwkB ∀w ∈ Sh; (2.13)
kwkB6C2kwkA ∀w ∈ Shc; (2.14)
|A(w; v)|6 kwkAkvkA; ∈ [0; 1); ∀w ∈ Sh; ∀v ∈ Shc; (2.15)
where the constant is independent of h and the norm k · kA is induced by A(·; ·).
Theorem 1. Let Assumptions 1–3 hold. Let u ∈ H and uh∈ Sh be the solutions of (2:1) and (2:4);
respectively. Then there exist unique solutions ˜ei∈ Shc( i); for all i = 1; : : : ; N; such that
B( ˜ei; vi) = F(vi) − B(uh; vi) ∀vi∈ Shc( i): (2.16)
Moreover; we have the estimate C1(1 − )
q
1 − 2kek
B6k ˜ekA6C2kekB; (2.17)
where e =u−uh is the exact error; ˜e=Pi ˜ei; and C1; C2; ∈ [0; 1) and ∈ [0; 1) are constants given
in Assumptions 2 and 3.
Proof. Assumption 1 ensures the uniqueness and existence of the solutions ˜ei∈ Shc( i); i = 1; : : : ; N;
of (2.16) as well as the solutions uh; e, and e satisfying, respectively,
B(uh; v) = F(v) ∀v ∈ Sh; (2.18)
B( e; v) = F(v) − B(uh; v) ∀v ∈ Sh; (2.19)
B(e; v) = F(v) − B(uh; v) ∀v ∈ H: (2.20)
Eqs. (2.18) and (2.19) imply e = uh− uh∈ Sh. By (2.11), (2.14), (2.16), and (2.20) we have, for
˜e =Pi ˜ei∈ Shc, k ˜ek2A =X i B( ˜ei; ˜ei) =X i B(e; ˜ei) = B(e; ˜e) = kekBk ˜ekB 6 C2kekBk ˜ekA
and hence the right-hand side of (2.17) holds. Since e − e = (u − uh) − (uh− uh) = u − uh;
we have, by (2.9) and a triangle inequality, (1 − )kekB6k ekB:
On the other hand, by (2.19) and (2.20), we have
k ek2B= B( e; e) = B(e; e)6kekBk ekB: Hence,
(1 − )kekB6k ekB6kekB: (2.21)
Let e = e1+ e2 such that e1∈ Sh and e2∈ Shc where e2 can also be written as e2=Pie2i; e2i∈ Shc( i).
Following (2.18),(2.4), (2.16) and (2.11), we have
k ek2B = B( e; e)
= B( e; e1) + B( e; e2) = B( e; e2) =X i B( e; e2i) =X i B( ei; e2i) = A( ˜e; e2) 6 k ˜ekAke2kA:
On the other hand, by (2.15) and (2.13),
k ek2A = A( e; e)
= A(e1; e1) + 2A(e1; e2) + A(e2; e2)
¿ ke1k2A+ ke2kA2− 2 ke1kAke2kA ¿ (1 − 2)ke 2k2A shows that C1 q 1 − 2k ek B6k ˜ekA
which, together with (2.21), implies the left-hand side of (2.17).
Note that (2.16) is a local problem since both trial and test functions all have the supports only in i. We therefore use
k ˜eikA= k ˜eikB= p
B( ˜ei; ˜ei)
as an error indicator for each element i, i = 1; : : : ; N. Consequently, the error estimator for the
approximate solution is de ned by
k ˜ekA= X
i
k ˜eik2A !1=2
and the e ectivity index is de ned by :=k ˜ekA
kekB:
Although the error estimator can also be de ned in the B norm, it is inecient since we then have to have a global calculation for the norm due to (2.10).
Remark 1. The rst paper using a formula similar to (2.16) for elliptic PDEs to develop an error estimator that we know of is by Adjerid and Flaherty in [1]. In [19], a general framework of this kind of error estimators is given for various types of variational problems in connection with FEM and FVM, while theoretical results are given in, e.g., [2,4,18,21,22]. Estimate (2.17) as well as its proof are slightly di erent from the previous works due to the fact of the nonlocal property (2.10). Inequality (2.9) is commonly used in these papers for error analysis. This saturation assumption is a very natural assumption since one expects that the approximate solution uh is in general a better
approximation to u than uh. The assumption consequently yields a minimal regularity for the exact
solution in H that is required to satisfy the optimal approximation for uh in the sense of Cea’s
lemma [10]. If, in particular, Sh consists of polynomials with degrees higher than that of Sh, one
can anticipate = (hr); r ¿ 0, which then asymptotically results in a better estimator according to
(2.17). If this assumption is replaced by other assumptions associated with higher regularity on the exact solution, asymptotic exactness of the estimator may be analyzed; see [2].
Remark 2. Let = sup{B(w; v)|w ∈ Sh; kwkB = 1; v ∈ Shc; kvkB = 1}. Then 61 and equals one
exactly if w and v are linearly dependent. Thus = 1 would contradict the complementarity of Sh
and Sc
h and the fact that both spaces are assumed to be nonempty. However, it is not clear that
is independent of the mesh size h. It should be noted that our use of (2.15) is closely related to that of the strengthened Cauchy–Schwarz inequality widely used in the analysis of iterative methods based on hierarchical bases [12].
3. Model problems
While the weak residual error estimators have been extensively studied for PDEs, it lacks evidence that they have been investigated for BIEs. We now show that Symm’s and hypersingular integral equations [6–8,13,14,16,23,25] indeed t into the framework by verifying inequalities (2.13)–(2.15) for these model problems.
Let the connect subset ⊆ ˆ be such that = ˆ if is closed and 6= ˆ if is open. As in [17], we de ne the Sobolev spaces
Ht( ˆ) = {u|
ˆ: u ∈ Ht+1=2(Rn)}; t ¿ 0;
H0( ˆ) = L2( ˆ)
and H−t( ˆ); t¿0, is the dual space of Ht( ˆ) with respect to the duality h·; ·i de ned by
hw; vi := Z
ˆwv ds ∀w ∈ H
We further de ne, for all t ∈ R, Ht( ) = {u| : u ∈ Ht( ˆ)};
ˆ
Ht( ) = {u ∈ Ht( ˆ) : supp u ⊆ }:
The duality properties are as follows:
(Ht( ))0= ˆH−t( ) and ( ˆHt( ))0= H−t( ):
For t ¿ 0, the norms in Ht( ˆ); Ht( ) and ˆHt( ) are de ned by
kukHt( ˆ)= inf {kvkHt+1=2(Rn): v|ˆ= u};
kukHt( )= inf {kvkHt( ˆ): v| = u};
kukHˆt( )= kukHt( ˆ):
For t ¡ 0, the norms are de ned by duality. Associated with Th, let Shp⊂ L2( ) denote the
nite-dimensional vector space of piecewise polynomials with degree p. 3.1. Symm’s integral equation
The Dirichlet problem for the Laplacian is related to the Symm’s integral equation
Vu(x) :=Z G(x; y)u(y) dsy= f(x); x ∈ ; (3.1)
where u is the unknown density, G(x; y) = −(1=2) ln |x − y| for n = 2 and G(x; y) = 1=(4|x − y|) for n = 3, and f is determined by some given Dirichlet data. We assume that f ∈ H1=2( ).
The operator
V : u ∈ ˆH−1=2( ) → f ∈ H1=2( )
is a Fredholm operator of index zero and is an isomorphism for n=3 or for n=2 if cap( ) 6= 1; see e.g., [8,9,11,16,24]. Here cap( ) denotes the capacity, or conformal radius, or trans nite diameter of . We therefore assume that, for positive de niteness, cap( ) ¡ 1 for n = 2 which can always be arranged by scaling, if necessary. Consequently, the bilinear form de ned by
B(u; v) := hVu; vi =Z v(x)Vu(x) dsx ∀u; v ∈ ˆH−1=2( ) (3.2)
is symmetric, continuous, and coercive on ˆH−1=2( ) × ˆH−1=2( ). Furthermore, the linear functional de ned by
F(v) := hf; vi =
Z
f(x)v(x) dsx ∀v ∈ ˆH−1=2( ) (3.3)
is continuous on H−1=2( ).
Symm’s equation is thus a special model problem of (2.1) and Assumption 1 is satis ed with H = ˆH−1=2( ) and k·kH=k·kHˆ−1=2( ). For the Galerkin approximation (2.4), we can use, for example,
Sh= Sh0 a space of piecewise constants. The choice of the complementary space Shc is quite exible
Symm’s integral equation. We rst show that the constant in (2.15) is independent of the mesh size h. Our analysis of the strengthened Cauchy–Schwarz inequality (2.15) follows closely to that of [3,12] for second-order elliptic PDEs. We rst cite a lemma from [3].
Lemma 1. Let (·; ·) and h·; ·i denote two inner products on a vector space X. Let k·k and |·| denote the corresponding norms. Suppose that there exist positive constants 1 and 2 such that
0 ¡ 16(z; z)hz; zi62 (3.4)
for all nonzero z ∈ X . For any nontrivial x; y ∈ X; let 1=kxkkyk(x; y) ; 2=hx; yi|x||y|: Then 1 2 2 (1 − 2 2)61 − 21: (3.5)
Lemma 2. Let the bilinear form B(·; ·) be deÿned by (3:2); the bilinear form A(·; ·) be deÿned by (2:11); the BE space Sh⊂ Shp⊂ L2( ); and the complementary space Shc⊂ Shq⊂ L2( ) be constructed
such that Assumption 2 holds. Then (2:15) holds for the constant ∈ [0; 1) independent of the mesh size h.
Proof. For w ∈ Sh and v ∈ Shc, let wi and vi be de ned by (2.12). Obviously, wi and vi∈ ˆH−1=2( ).
The proof of (2.15) can be reduced to an element-by-element estimate. On each i∈ Th, if i∈ [0; 1)
is independent of h such that
|B(wi; vi)|6 i p B(wi; wi) p B(vi; vi); (3.6) then |A(w; v)| 6X i |B(wi; vi)| 6X i i p B(wi; wi) p B(vi; vi) 6 X i B(wi; wi) !1=2 X i B(vi; vi) !1=2 = pA(w; w)pA(v; v) ∀w ∈ Sh; v ∈ Shc; where = max i i:
The proof can further be reduced to a reference element. For any element i, let i be an invertible
ane mapping
i: ∈ r 7→ i() = Qi + 0∈ i;
such that
i= i( r);
where r is the reference element. For n = 2, we take r to be an interval [ − a; a] with cap( r) ¡ 1
where a is a positive constant. For n = 3, since the partition Th is regular, the mapping has the
following property, see [10], i
hr||6|Qi|6
hi
r||; (3.7)
where hr and r are parameters of (2.6) in terms of the reference element. Let Sh;i and Sh;ic denote
the restrictions of Sh and Shc, respectively, to the element i. And let Sr and Src denote some xed
nite-dimensional spaces of polynomials de ned on the reference element r such that the mapping
i maps Sr onto Sh;i and Src onto Sh;ic . Using the change of variables, for each element i inequality
(3.6) becomes
B(wi; vi) = Ji2Br;i(wr;i; vr;i)
6 J2 i i
q
Br;i(wr;i; wr;i) q
Br;i(vr;i; vr;i)
= i q
J2
iBr;i(wr;i; wr;i) q
J2
iBr;i(vr;i; vr;i)
= i p
B(wi; wi) p
B(vi; vi);
where wr;i= wi◦ i∈ Sr; vr;i= vi◦ i∈ Src; Ji is the Jacobian of the mapping, and
Br;i(wr;i; vr;i) := Z r Z r −1 2 ln h i 2a| − | wr;i() ds vr;i() ds for n = 2,
Br;i(wr;i; vr;i) := Z r Z r 1 4|Qi( − )|wr;i() ds vr;i() ds
for n = 3. Clearly, Br;i(·; ·) de nes an inner product on Sr ⊕ Src. Since Br;i(·; ·) exhibits di erent
properties with respect to di erent kernels, the quantity i being independent of h is proved in two
separate cases of n=2 and 3. We rst prove for the case of n=2. Since hi6cap( ) ¡ 1, we assume
that hi is small enough such that hi=2a ¡ 1. Let
A1(wr; vr) := Z r Z r wr() ds vr() ds; A2(wr; vr) := Z r Z r −1 2 ln | − |wr() ds vr() ds
for all wr and vr in Sr⊕ Src. Then A1(·; ·) and A2(·; ·) are two inner products independent of h on
Sr⊕ Src. Note that Sr and Src are xed and linearly independent subspaces on the reference element r. Therefore, there exist two constants r;1 and r;2∈ [0; 1) independent of h such that, for j = 1; 2,
|Aj(wr; vr)|6 r;jAj(wr; wr)1=2Aj(vr; vr)1=2 ∀wr∈ Sr; vr∈ Src: (3.8)
We then have
|Br;i(wr;i; vr;i)|
= −12 ln2ahiA1(wr;i; vr;i) + A2(wr;i; vr;i) 6 −1 2 ln hi 2a
|A1(wr;i; vr;i)| + |A2(wr;i; vr;i)|
6−12 ln2ahi r;1A1(wr;i; wr;i)1=2A1(vr;i; vr;i)1=2+ r;2A2(wr;i; wr;i)1=2A2(vr;i; vr;i)1=2
6 r
−1
2 ln hi
2aA1(wr;i; wr;i) + A2(wr;i; wr;i)
1=2
×
−1
2 ln hi
2aA1(vr;i; vr;i) + A2(vr;i; vr;i)
1=2
= r q
Br;i(wr;i; wr;i) q
Br;i(vr;i; vr;i);
where r= max{ r;1; r; 2}. This concludes that i= r for all i = 1; 2; : : : ; N and they are independent
of h. For the case of n = 3, we de ne A3(wr; vr) := Z r Z r 1 4| − |wr() ds vr() ds
for all wr and vr in Sr ⊕ Src. Again, A3 is an inner product independent of h on Sr ⊕ Src with the
existence of the corresponding constant r; 3∈ [0; 1). That is, (3.8) holds for j = 3. For each element i, inequalities (3.7) imply that
1;i= hr i6 Br;i(z; z) A3(z; z)6 hr i = 2;i:
Therefore, with (2.6), Lemma 1 yields that 2 i 6 1 − 1;i 2;i 2 (1 − 2 r; 3) = 1 − ir hihr 2 (1 − 2 r; 3) 6 1 − hr r 2 (1 − 2 r; 3) ¡ 1
and that i is independent of h since ∈ (0; 1]; r=hr∈ (0; 1], and r;3∈ [0; 1) are all independent of
h. This thus completes the proof.
This lemma suggests that the construction of Sc
h is very exible. For example, the hierarchical
basis functions can be used in such a way that the shape functions of Sc
h are of the next higher
order than that of Sh. The error estimator can thus be used in all h-, p-, and hp-version BEM [23].
To justify estimate (2.17) for Symm’s equation, we need to further verify conditions (2.13) and (2.14). The following well-known lemma, see also, e.g., [5,6,15], is essential to establish these conditions and will be used for the next model problem as well. For simplicity, the lemma is restricted to two space dimensions only.
Lemma 3. Let be partitioned as in (2:5): Then; for t = 1
2 or t = −12 ; there exist two positive
constants C3 and C4 independent the number of sub-intervals N; i.e.; independent of the mesh
parameter h; such that C3 N X i=1 kuk2Ht(i)6kuk2Ht( ); (3.9) kuk2Hˆt( )6C4 N X i=1 kuk2Hˆt( i): (3.10)
Theorem 2. If all assumptions in Lemma 2 hold; then for Symm’s boundary integral equation (3:1) in two space dimensions we have estimate (2:17) with the constants ; C1; and C2 all independent
of h.
Proof. By Assumption 2, the basis functions for the complementary space Sc
h have supports in their
respective subintervals. Moreover, these functions are piecewise polynomials in L2( ). Evidently,
the spaces Sh and Shc so constructed make inequalities (3.9) and(3.10) hold for u ∈ Sh and u ∈ Shc,
respectively. Applying the equivalence of the norms k · kH and k · kB, inequalities (2.13) and (2.14) of Assumption 3 hold. By Lemma 2, the theorem is thus asserted.
3.2. Hypersingular integral equations
The Neumann problem for the Laplacian is related to the integral equation Wu(x) := −@n@
x Z
u(y)@G(x; y)@n
y dsy= f(x); x ∈ (3.11)
for the unknown displacement u on , where G(x; y) = −(1=2) ln|x − y| for n = 2 and G(x; y) = (1=4|x − y|) for n = 3, and f is determined by the given Neumann data. The integral in (3.11) is to be understood as a Hadamard nite-part integral. For simplicity, we speci cally consider the model problems used in [8], namely, the is a closed curve for n = 2 and is an open surface for n = 3.
Let H := {v ∈ H1=2( ): h1; vi = 0}; H0:= {g ∈ H−1=2( ): hg; 1i = 0} for n = 2 and H := {v ∈ ˆH1=2( )}; H0:= H1=2( ) for n = 3.
We de ne the bilinear form and the linear functional for (3.11) as
B(u; v) := hWu; vi ∀u; v ∈ H; (3.12)
F(v) := hf; vi ∀v ∈ H: (3.13)
The a priori theory presented in [8,11] suitable for our purposes is summarized in the following lemma for which the proof is therein referred.
Lemma 4. Assumption 1 holds for the variational problem (2:1) with the bilinear form and the linear functional given; respectively; by (3:12) and (3:13).
Note that the bilinear form can be written as B(u; v) =Z Z −12 ln(|x − y|)@u(y)@s
y dsy ! @v(x) @sx dsx (3.14) for n = 2 and B(u; v) =Z Z |x − y|1 3u(y) dsy v(x) dsx
for n = 3. Hence, the proof of the strengthened Cauchy–Schwarz inequality (2.15) for the hypersin-gular integral equations proceeds in a similar way as that for Symm’s integral equation.
Lemma 5. Let the bilinear form B(·; ·) be deÿned by (3:12); the bilinear form A(·; ·) be deÿned by (2:11); the BE space Sh⊂ Shp⊂ H; and the complementary space Shc⊂ Shq⊂ H be constructed such
that Assumption 2 holds. Then (2:15) holds for the constant ∈ [0; 1) independent of the mesh size h.
Moreover, with a similar proof of Theorem 2, the following theorem is thus a consequence of this lemma and Lemma 3.
Theorem 3. If all assumptions in Lemma 5 hold; then for the hypersingular integral equation (3:11) in two space dimensions we have the estimate (2:17) with the constants ; C1; and C2 all
Fig. 1. An adaptive algorithm.
Remark 3. The estimators developed in [8,13,26] are all analyzed on the bases of the dual norm H0
for the residual error f−Vuh or f−Wuh. The dual norm in general is not computable. Consequently,
their approaches require more regularity of the residual or equivalently more regularity of the exact solution u in order to measure the estimators in a higher and computable norm such as the L2( )
norm.
4. Numerical examples
Three objectives are considered for this section; namely, to justify the e ectiveness of the proposed estimator, to show the eciency of the resulting adaptive scheme, and to illustrate the complementary subspaces Sc
h. A standard h-version, adaptive algorithm based on the proposed error estimator is given
in Fig. 1.
The following two examples are related to the Laplacian
u = 0 in ; (4.1)
where the domain is an L-shaped polygon shown in Fig. 2. The boundary condition for (4.1) is
u = gD on (4.2)
for Symm’s integral equation and is @u
@n = gN on (4.3)
for the hypersingular integral equation. The functions gD and gN are chosen so that the corresponding
exact solutions are, in polar coordinates, u = r2=3sin 2=3 for (4.2) and u = r1=7sin =7 for (4.3); see
also [8].
Example 4.1 (Symm’s integral equation). As discussed above, one of the key ingredients of the present estimator is the construction of complementary subspaces Sc
h based on the current
Fig. 2.
For our experiments, the space Sh for Symm’s equation is given by piecewise constants. The
nodal points corresponding to piecewise constants are, in the sense of the average of quadrature rules, de ned at the middle points of elements. To satisfy assumption (2.8), we should avoid the shape functions of Sc
h that are de ned with nodal points being in the middle and that will generate
a constant function. Hence, we can, for example, construct Sc
h via the mapping of the shape functions r;1() = 12(1 − )2; ∈ [ − 1; 1];
r; 2() =12(1 + )2; ∈ [ − 1; 1]
(4.4) de ned on the reference element r= [ − 1; 1], i.e., Shc⊂ Sh2. Therefore, on each element, we have a
2 × 2 local system to be solved in Step 4.2 of the above adaptive algorithm. More speci cally, (2.16) leads to systems of 2 × 2 linear algebraic equations
Akek= bk; k = 1; : : : ; N; (4.5)
where the four entries of the matrix Ak and two entries of bk are given as
Ak(i; j) = Z k Z k −1 2 ln(|x − y|) k;j(y) dsy k;i(x) dsx; i; j = 1; 2; bk(i) = Z k f(x) k;i(x) dsx; i; j = 1; 2;
Fig. 2. Continued.
where k;i; i = 1; 2; are shape functions obtained by transforming the two basis functions (4.4) to
the element k. The solution of (4.5) then de nes an error indicator for that element as stated in
Step 4.3, namely,
˜ek= ˜ek;1 k;1+ ˜ek;2 k;2; ek= (ek;1; ek;2):
We then proceed to obtain the global error estimator in Step 5.
The mesh diagrams Figs. 2A–2G are showing a typical scenario of adaptive process as the esti-mator is capturing the point singularity at the origin. The estiesti-mator is very e ective as shown by the e ectivity indices in the last column in Tables 2A and 2B for both uniform and adaptive approaches, where the re nement factor is de ned in Step 6.2. Moreover, if the relative error was preset to, for instance, 1%, the uniform approach requires about 10 times elements of the adaptive approach. The adaptive method is clearly showing advantageous features for singularly behaved problems. Example 4.2 (A hypersingular integral equation). For our numerical experiments, we choose Sh=Sh1
a space of linear functions while the complementary space Sc
h is constructed, via the reference
element, by
Table 2A
Example 4.1 using uniform meshes = 0
N kekB r.e. 8 0.194 0.130 1.001 16 0.115 0.075 0.910 32 0.073 0.047 0.935 64 0.047 0.030 0.955 128 0.030 0.019 0.975 256 0.019 0.012 0.993 512 0.012 0.008 1.010 Table 2B
Example 4.1 using adaptive meshes = 0:1
N kekB r.e. 8 0.194 0.130 1.001 16 0.115 0.075 0.910 18 0.076 0.049 0.944 24 0.049 0.031 0.952 30 0.032 0.020 0.964 40 0.020 0.013 0.972 52 0.013 0.008 0.981
Again, it can be easily veri ed that Sc
h⊂ Sh2 satis es Assumption 2. On each element, (2.16)
corre-sponds to a single equation. Consequently, our estimator reduces to a residual-type error estimators developed in [8,26,13]. More speci cally, using the formulas in [23] to explicitly evaluate (3.14) for (4.6), we obtain B( i; i) = −21 Z i Z i ln|x − y|@ @si(y) y dsy ! @ i(x) @sx dsx = −21 Z 1 −1 Z 1 −1ln h i 2| − | d () d d ! d () d d =2 :
The error indicator for the element i is hence
i= p B( ˜ei; ˜ei) = r 2(F( i) − B(uh; i)); where the last term is a computable residual.
Using only one shape function (4.6) to de ne the complementary space Sc
h on each element, our
Table 3A
Example 4.2 using uniform meshes = 0
N kekB r.e. 8 0.138 0.463 0.830 16 0.124 0.380 0.820 32 0.112 0.316 0.828 64 0.101 0.265 0.842 128 0.090 0.224 0.856 256 0.081 0.191 0.871 512 0.073 0.163 0.886 Table 3B
Example 4.2 using adaptive meshes = 0:5
N kekB r.e. 8 0.138 0.463 0.830 10 0.128 0.392 0.822 12 0.117 0.331 0.849 14 0.106 0.280 0.865 16 0.097 0.240 0.882 18 0.088 0.206 0.898 20 0.080 0.179 0.915 22 0.073 0.157 0.932 24 0.067 0.138 0.949 26 0.062 0.122 0.965 28 0.057 0.109 0.981 30 0.053 0.099 0.997 Acknowledgements
The second author would like to express his gratitude to the Department of Mathematics, Texas A & M University, and especially to Professors Goong Chen and Jianxin Zhou, for a stimulating and enjoyable visit during which this work was undertaken.
References
[1] S. Adjerid, J.E. Flaherty, Second-order nite element approximations and a posteriori error estimation for two-dimensional parabolic systems, Numer. Math. 53 (1988) 183–198.
[2] S. Adjerid, J.E. Flaherty, Y.J. Wang, A posteriori error estimation with nite element methods of lines for one-dimensional parabolic systems, Numer. Math. 65 (1993) 1–21.
[3] R.E. Bank, Hierarchical bases and the nite element method, Acta Numerica 1996 (A. Iserles, ed.), Cambridge University Press (1996) 1–43.
[4] R.E. Bank, R.K. Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993) 921–935.
[5] T. Cao, Adaptive H- and H–R methods for symm’s integral equation, Comput. Methods Appl. Mech. Eng. 162 (1998) 1–17.
[6] C. Carstensen, An a posteriori error estimate for a rst-kind integral equation, Math. Comp. 66 (1997) 139–155. [7] C. Carstensen, D. Estep, P. Stephan, h-adaptive boundary element schemes, Comput. Mech. 15 (1995) 372–383. [8] C. Carstensen, E.P. Stephan, A posteriori error estimates for boundary element methods, Math. Comp. 64 (1995)
483–500.
[9] G. Chen, J. Zhou, Boundary Element Methods, Academic Press, San Diego, CA, 1992.
[10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978.
[11] M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988) 613–626.
[12] V. Eijkhout, P. Vassilevski, The role of the strengthened Cauchy–Buniakowskii–Schwarz inequality in multilevel methods, SIAM Rev. 33 (1991) 405–419.
[13] M. Feistauer, G.C. Hsiao, R.E. Kleinman, Asymptotic and a posteriori error estimates for boundary element solution of hypersingular integral equations, SIAM J. Numer. Anal. 33 (1996) 666–687.
[14] G.N. Gatica, G.C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2, Numer.
Math. 61 (1992) 171–214.
[15] N. Heuer, E.P. Stephan, T. Tran, Multilevel adaptive Schwarz method for the h–p version of the Galerkin boundary element method, Math. Comp. 67 (1998) 501–518.
[16] G.C. Hsiao, S. Zhang, Optimal order multigrid methods for solving exterior boundary value problems, SIAM J. Numer. Anal. 31 (1994) 680–694.
[17] J.L. Lions, E. Magenes, Non-Homogeneous Elliptic Boundary Value Problems and Applications, vol. I, Springer, Berlin, 1972.
[18] J.-L. Liu, Weak residual error estimates for symmetric positive systems, Numer. Funct. Anal. Optim. 14 (1993) 607–619.
[19] J.-L. Liu, On weak residual error estimation, SIAM J. Sci. Comput. 14 (1996) 1249–1268.
[20] J.-L. Liu, I.-J. Lin, M.-Z. Shih, R.-C. Chen, M.-C. Hsieh, Object oriented programming of adaptive nite element and nite volume methods, Appl. Numer. Math. 21 (1996) 439–467.
[21] J.-L. Liu, W.C. Rheinboldt, A posteriori nite element error estimators for inde nite elliptic boundary value problems, Numer. Funct. Anal. Optim. 15 (1994) 335–356.
[22] J.-L. Liu, W.C. Rheinboldt, A posteriori nite element error estimators for parametrized nonlinear boundary value problems, Numer. Funct. Anal. Optim. 17 (1996) 605–637.
[23] F.V. Postell, P. Stephan, On the h-,p- and h-adaptive versions of the boundary element method-numerical results, Comput. Methods Appl. Mech. Eng. 83 (1990) 69–89.
[24] I.H. Sloan, A. Spence, The Galerkin method for integral equations of the rst kind with logarithmic kernel: Theory, IMA J. Numer. Anal. 8 (1988) 105–122.
[25] W.L. Wendland, P. Stephan, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal. 112 (1990) 363–390.
[26] W.L. Wendland, D. Yu, Adaptive boundary element methods for strongly elliptic integral equations, Numer. Math. 53 (1988) 539–558.
[27] W.L. Wendland, D. Yu, A posteriori local error estimates of boundary element methods with some pseudo-di erential equations on closed curves, J. Comput. Math. 10 (1992) 273–289.
