Boundary-Hybrid Finite Elements and A Posteriori …an original finite element procedure, coined as...
Transcript of Boundary-Hybrid Finite Elements and A Posteriori …an original finite element procedure, coined as...
TICAM REPORT 96-26June 1996·
Boundary-Hybrid Finite Elements and A PosterioriError Estimation
Joseph M Maubach and Patrick J. Rabier
BOUNDARY-HYBRID FINITE ELEMENTS
AND A POSTERIORI ERROR ESTIMATION
BY
JOSEPH M. MAUBACH
AND
PATRICK J. RABIER
Department of Mathematics
University of Pittsburgh
Pittsburgh, PA 15260
ABSTRACT. Finite element methods that calculate the normal derivat.ive of the solution along
the mesh interfaces and recover the solution via local Neumann problems were introduced
about two decades ago by 1. Babuska, J.T. Oden and J.K. Lee for the treatment of the homo-
geneous Laplace equation and called "boundary-hybrid method". We revisit this approach for
general symmetric and positive definite elliptic equations with homogeneous boundary condi-
tions. The resulting approximation is nonconforming, and the corresponding error is orthog-
onal to all the conforming finite element subspaces. This crucial property shows immediately
how to derive an a posteriori error estimator for conforming finite element approximations via
Pythagoras' theorem. The investigation of this idea leads to a sound strategy for a posteriori
error analysis which gives conservative, yet accurate, estimates, is cheap for good conforming
approximations, and otherwise produces an enhanced solution at not significantly more than
the cost normally expected for such a result.
1. Introd uction.
In the second part of their paper "Mixed-Hybrid Finite Element Approximations of
Second-Order Elliptic Boundary Value Problems", Babuska, Oden and Lee [3], [4] discuss
e-mail: [email protected]
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an original finite element procedure, coined as "boundary hybrid method", to solve the
homogeneous Laplace equation
(1.1) {
6.u = 0
u=g
in n,on an,
where n c JR2 is a polygonal domain. Roughly, the procedure can be described as follows:
Consider a partition P of the domain n into polygonal elements. If the normal derivative
au / av of the solution is known across all the element interfaces, then the solution u can
(obviously) be recovered by solving the corresponding Neumann problem for the homoge-
neous Laplace equation over each element separately. A variational characterization of the
normal derivative au/ av is obtained in [4, Theorem 3.3]. Then, approximations of au / av,
e.g. by functions which are polynomials in each interface can be found by restricting
the variational formulation to an appropriate finite-dimensional space (standard Galerkin
procedure). To obtain u numerically, it suffices to solve the Neumann problem over each
element with auf av being replaced by its piecewise polynomial approximation. Of course,
the local Neumann problems need not be solved exactly, and in [4] this is exploited by
using higher degree harmonic polynomials within each element.
The specific form of the system (1.1) may have seemed especially important in light of
the fact that essentially no other elliptic problem has such an easily characterized dense
subset of solutions as the harmonic polynomials in two variables (a basis obtains by taking
the real and imaginary parts of (x + iy)n,n EN). However, with the advent of parallel
machines, any system of independent local problems can now be solved numerically at low
cost and with high accuracy and execution speed. Thus, the explicit knowledge of a special
set of solutions is no longer of significant help or importance.
In this paper, we give a variational characterization of the normal derivative of the
solution along element interfaces for general (symmetric) elliptic problems of the form
(1.2) {
- \7. A\7u + (J'U = f in n,u = 0 on an,
where D is an open bounded subset of JRn and f E L2(D). Here, "normal derivative"
must be understood as "A-normal derivative", i.e. au/avA := A\7u . v. Naturally, it is of
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importance that the quadratic form used for the characterization of au javA satisfies basic
properties of continuity and coercivity. In practice, this means that adequate care should
be given to the definition of its domain.
One available option is to use the characterization of auj avA towards the numerical
approximation of u, as was done in [4]for the problem (1.1). However, a fuller application
can be found to the a posteriori estimation of u - it when it E HJ (.0) is any approximation
of u. It is this second and less standard aspect that we have chosen to emphasize here.
vYithout going into technicalities, the general idea is to partition the domain n into
nonoverlap ping subdomains nJ( whose boundaries relative n (i.e. not including f := an)constitute the interface set fI. Denoting by A the space L2(fI), a first step consists in
constructing a suitable completion A of A (not surprisingly of H-1/2 type). The completion
A is problem-independent. Assumptions are made which ensure that the symmetric bilinear
form a(-,·) in HJ(n) associated with (1.2) is coercive. Given it E HJ(n), a it-dependent
quadratic form Q in A is constructed with the crucial property that
(1.3) a( u - it, u - it) = in[Q(-\),..\EA
and that the unique minimizer l of Q corresponds canonically to the A-normal derivative
of aujavA along the interface fI.
It turns out that the fact that l depends only upon u (i.e. not upon it) is of importance,
for after discretization of the space A, a minimization procedure of Q can be started with
a it-related initial guess -\0. If it is a good approximation of u, then Q(-\o) is expected to
fall below some prescribed tolerance T chosen to assess the quality of the approximation
it. If so, (1.3) shows that a(u - it,u - it) :::; T, and it is called a good approximation.
Otherwise, the minimization of Q(-\) by an iterative procedure produces a minimizing
sequence -\j,j ~ O. Once again from (1.3), it is called a good approximation if Q(Aj) ~ T
for some j, and the minimization procedure may stop. If, on the other hand, Q( Aj) > T
until the minimization procedure is complete, then it is (conservatively) deemed to be a
poor approximation. This means that a new approximate solution must be determined.
Remarkably, this can be accomplished with no further work, because the minimization of
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Q, now complete, also provides an enhanced solution if the discretization of the space A
has been properly chosen in the first place. This enhanced solution is not in HJ(n), as it
exhibits discontinuities along fI, but this is easily remedied if desired.
In summary, the error estimator described here has the two desirable features of always
providing an overestimate of the actual energy norm error and of yielding an enhanced
solution if the given approximation is not satisfactory. Furthermore, the workload IS,
loosely speaking, "inverse" proportional to the quality of the approximation.
All the numerical aspects involved in the minimization of Q are at least competitive
with those of traditional finite element methods (sparsity, size, storage, condition number,
etc ...). Of course, the fact that the domain of Q consists offunctions defined in the interface
f I is essential regarding size and storage: For instance, if n = 2 and discretization of A
is made via piecewise polynomials of degree k, the size of the system grows linearly with
respect to k instead of quadratically when a standard finite element method is used. Yet,
as we shall see, the latter do not produce more accurate solutions. Static condensation
reduces the problem to one of the same size as ours, but still requires much more storage
than our method for large k.
The local problems need to be solved just once at the beginning of the procedure and
not at each iteration as could be inferred from our initial comments. Not only this step
is evidently fully parallelizable, but even the minimization of Q lends itself perfectly to
elementwise parallel processing by the method recently introduced in [9] and [10].
Section 2 is devoted to some preliminaries including a definition of a space A_ which,
later (Section 4), will be shown "canonically" isomorphic to the space A mentioned earlier.
There are some notational technicalities due to the fact that normal derivatives along
sub domain interfaces are, of necessity, double-valued with one value being the negative
of the other. Our presentation bypasses the explicit use of oriented integrals (to avoid
mixing oriented and nonoriented integrals, and because no convenient symbol such as f is
standard in dimension greater than one) but this is a matter of taste and not a conceptual
difference.
In Section 3, we present the variational characterization of the A-normal derivative
along the interface fI and describe the error related to the discretization and Galerkin
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approximation. It is also shown that the estimator yields justifiable local error indicators,
although the upper bound property is lost at the local level.
Section 5 addresses the important issue of the accuracy of the error estimator in a
somewhat restricted setting nevertheless sufficient for most two-dimensional applications.
The results of Section 5 have the form of a priori estimates and show that both the error
estimator and the enhanced solution have optimal orders of accuracy. No "saturation
assumption" ([1], [6]) is needed. In addition, an unusual feature is highlighted: If the
partitioning of the open subset n is done according to a simple rule, the accuracy depends
only upon the interior regularity of the solution u and not upon its global regularity (the
latter being always limited by the presence of corners in .0).
When (7 = 0 in (1.2), the minimization of Q involves a linear constraint (more generally,
the same thing is true whenever (7 vanishes in a set with positive measure). The slight
modifications to the procedure needed in that case al'e examined in Section 6.
Section 7 presents numerical experiments.
2. Notation and preliminaries.
Let n c ]Rn be a bounded open subset with Lipschitz continuous boundary f. We
shall consider a "partition" P of n into open sub domains nK,l :::;J{ :::;N, satisfying a
number of technical conditions. These conditions always hold when P is built from a finite
element triangulation of n and the nK's are the elements or patches of elements of the
triangulation.
Specifically, we require thatN _ _
(H1) u .oK = n.K=I
(H2) .oK n nL = 0 if J{ #- L.
(H3) The boundary f K := anK is Lipschitz continuous.
It follows from (H3) that the n-dimensional Lebesgue measure of rK is zero, whence,
from (H1) and (H2):
(2.1)
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where all the integrals must be understood in the sense of the n-dimensional Lebesgue
measure.
It also follows from (H3) that the boundary f K may be equipped with a "canonical"
Lebesgue measure, denoted by µK. For 0:::; ](,L :::;N, set fo := f and
(2.2)
Cleary, fKL is a closed subset of both fK and fL, and hence fKL is both µl(- and µL-
measurable. As a result, fl(L may be equipped with both µK and µL measures. A natural
requirement now is
(H4) µl(lr = µLlr 'KL KL
and then (H4) appears as a compatibility condition for the set
(2.3) - Nf:= U fI<
/(=0
to be equipped with a measure µ defined by the condition
(2.4)
Since f I< is Lipschitz continuous, an outward unit normal vector VK exists µK- a.e.
(that is, µ- a.e.) in fl(. As a result of (2.4) and (H4), both VI< and VL exist µ- a.e. III
rl( L, and we shall require that
(H5) {1 ~ ](, L :::;N,]( =I- L} :::}VI<I = -VL1 .rKL rKL
If fKLM := fK n fL n fM, it follows at once from (H5) that VM = 0 in f/(LM if
1 ~ ](, L, J.V1 :::; Nand ](, Land M are distinct. Since VM is a unit vector µ- a.e. this
implies that µ(f K LM) = 0 in this case. Our next assumption is that this also holds when
either K, L or J.V1is 0, i.e. that
(H6) {O::;](, L, J.V1:::;N, J{ =I- L, J{ =I- lvI, L =I- J.V1} :::} µ(f K n f L n f M) = o.Finally, we shall assume that for 0 ~ J{ ::; N,
N(H7) fK = U fKL.
L=OL~I<
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Remark 2.1: It is not difficult to convince oneself that the hypotheses (H1) through (H7)
are not independent. We have not bothered with trying to find a minimal set of conditions
equivalent to (H1) - (H7) since the direct verification of (H1) - (H7) will never be an issue
in the applications we have in mind. 0
Let
(2.5) I:= {(K,L): 1:::; K,L:::; N,K # L,µ(fKL) > O},
and, for 1 :::;K :::;N,
(2.6) IK := {1 :::;L :::;N : (K,L) E I}.
Clearly, I is an index set for the interfaces f [(L, each being counted twice (once in f K
and once in f L), whereas IK labels (once) the interfaces contained in rK.
From (H6) and (H7), we have for 1 :::;K :::;N:
where the integrals are understood in the sense of the measure µ. Obviously, the integrals
corresponding to pairs (K, L) with µ(f]( L) = 0 in the right-hand side may be discarded,
so that
(2.7)
In (2.7), the term fr disappears whenever µ(fKo) = O.J1 1<0
Let fI be the disjoint union
I'I:= II fI<L,(K,L)EI
so that, in fI,fKL and fLK are different although they coincide as subsets of ]Rn. Ac-
cordingly, we set
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Elements of L2(rI) thus are collections ~ = (AKL) with AKL E L2(fKL). In general, the
functions AKL and ALK are different, despite the fact that they are both defined in the
same subset f K L of IRn.
There is a canonical identification
(2.8)N
L2(rI) ~ IIL2(f K \ f),K=l
obtained by associating with ~ = (AKL) E L2(rI) the collection (AK) defined by AKIr[(L
AKL,L E IK (this makes sense because of (H6),and AK is defined µ- a.e. in fK becauseN _
of (H7». The inverse map is given by (AK) E II L2(fK \ f) t---+ (AKL) E L2(fI) whereK=l
AKL := AK, .rKL
In future considerations, it will be convenient to use both the notations (AI<£) and
(AK) for elements of L2(rI) (depending upon whether the identification (2.8) is used).
No confusion should arise from this since the number of subscripts immediately identifies
which definition of L2(rI) is being used.
The norm of L2(I'I) is the natural one, namely
(2.9)
\Ve shall consider the spaces
(2.10)
with norm induced by t.he norm of HI(nl(), and the products
(2.11)N
HJ(P) := II VK,[(=1
N
HI(p) := II H1(nK)K=l
equipped with the product norm 1I·lh;p. Evidently, in (2.10) we have VK = HI(nK) when
µ(rKO) = o.
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A restriction map (trace) is defined:
(2.12)
which is of course linear and continuous for the norms of the spaces involved. On the other
hand, there are canonical embeddings
(2.13)
defined by v E HI (.0) ~ (VloK
)' Our first result gives a simple characterization of the
image of the embedding (2.13) via the trace operator (2.12) and the closed subspace of
L2(fI ):
(2.14)
Proposition 2.1. Let (VK) E H1 (P) and let v E L2(n) be defined by VIOK := VI(. Then,
v E H1(n) if and only if (VKI ) E A+ (where the identification (2.8) was used).rK\r
Proof. Ifv E COO(n) and VK:= vlo , it is obvious that VKI = VLI ' i.e. (VI(I ) EK rKL rKL rK\r
A+. By the denseness of COO(n) in H1(n), the continuity of the trace HI(P) --+ L2(fI)
and the closedness of A+ in L2(fI), this relation remains valid for v E HI(n).
Conversely, if (VI() E HI (P), V E L2(n) c 'D'(n) is defined by vloK
:= VI( and'P E'D(n),
we have
Since 'P = 0 in f, we find (see (2.7)
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and the terms in the right-hand side cancel out if (VI<I ) E A+ (i.e. VK = VKL in fgL)r[(\r
since VKi = -VLi in f 1';L from (H5). It follows that
I.e. avjfJxi is represented by the L2(n) function defined by aVI<jaxi in nK, 1 :::;J{ :::;N.
Thus, V E H1(n). 0
The natural complement of the space A+ in (2.14), that is
(2.15)
plays a crucial role in our subsequent considerations. The starting point is the remark that
every element A = (AK) E A_ (where the identification (2.8) was used) can be viewed as
an element of
(2.16)
(see (2.10) and (2.11) via
(2.17)
K
H-1(P) := (HJ(P»* = IIV;K=I
The (product) norm of H-1(P) will be denoted by II . II-l,P.If wE HJ(n), then w can be identified with the collection (wln[() E HJ(P) through the
embedding (2.13), so that (~, w) makes sense. Note that
(2.18)
because
- 1(A, w) = 0, Vw E Ho (n), VA E A_,
Nj j~ AK'wK = ~ AKwK,K=I r[(\r (K,L)EI r[(L
and all the terms in the right-hand side cancel out since 10 [( = w L in f K L (Proposition
2.1) whereas AI( = -)..L in f[{L if ~ E A_.
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Definition 2.1. The space A_ is the closure of A_ in H-I (P).
From Definition 2.1, every .:\ E A- is the limit in H-1(P) of a sequence from the space
A_. From (2.18) we obtain by continuity that
(2.19) (.:\,w) = 0, Vw E HJ(n), VA E A_.
(2.20)
equipped with the norm
(2.21)
Naturally, similar definitions can be made for the spaces HA(flK), 1:::; J(:::; N.
If VK E CCXJ(D.K), we may define the "A-normal" derivative
(2.22)
Proposition 2.2. (i) The mapping v E CCXJ(nK) ~ aVKjavt E L2(fK) has a unique
linear continuous extension from HA(nK) into V!(. Furthermore, Green's formula
(2.23)
holds for WK E VK.
(ii) Ifv E cCXJ(n) and VK:= vloK
' 1 ~ J(::; N, then the collection (avKjaVj~) can be
viewed as an element of H-1(P) via
(2.24) (aVK) N 1 aUK( av~ ' (WI(») := K~l a A WI(, V(WI() E HJ(P),!\ fJ( VI(
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and, in H-I(P), we have (avKlav~) E A_.
(iii) The mapping v E c=(n) -- (aVI\-!avt) E A_ C H-1(P) (where VK := vloK
) has
a unique linear continuous extension from H.4.(n) into A_ C H-1(P).
Proof. (i) This is essentially standard. Briefly, for VK E COO(nK) and by Green's formula,
we have
(2.25)
for WK E COO(nK), and hence for WK E H1(ng). In particular, (2.25) holds for WK E VK,
and it follows that
where AI, > 0 is a constant depending only upon .4,. As a result,
(2.26)
with II· IIr n denoting the norm of VI(.,K
The existence and uniqueness of a continuous extension of av K I avt as an element of
V'/( for VK E HA(nK) follows from (2.26) and the denseness of COO(n/\,) in HA(D.g). (For
the latter point, see Grisvard [8, p. 59] when .-l = I; the proof in the general case is
similar.) Next, (2.23) is just the extension of (2.25) by denseness and continuity.
(ii) The only point to be emphasized here is that, as an element of H-1(P), we have
((avKlav~» E A_. This follows from the fact that for (wg) E HJ(P), we have
I.e. ((aug I av~» and ((av/( I Bvt »lrK\r) coincide in H-1 (P), while aVI{ I av~ = -BVLI avt
in rKL,(K,L) E I, from (H5) and from \7v/( = \7vL = \7v in rKL when v E COO(n).
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N(iii) First, from (ii), the mapping v E c=(n) t-+ ((aV[(jav~» E IT L2(fK) has
K=lN
a unique linear continuous extension from HA(n) into IT V; := H-I(P). From (ii),K=l
(avKjaV~) E A_ when v E c=(n), and by using once again the denseness of c=(n) in
HA(n) it follows that (avKjaVp:> E A_. 0
Remark 2.2: Part (i) of Proposition 2.2 remains valid with VK replaced by HI (nK), andN
(2.17) also defines an embedding of A_ into IT HI(nK)*. Although (avKjaV~) can beK=I
Nviewed as an element of IT HI(nK)* via (2.24), it is not true that (aVK j av~) E A_ in
K=Ithis case, because the integrals in the right-hand side of (2.24) do not reduce to integral
Nin f K \ f. It follows that for v E HA(n), (aVl( j av~) E IT H1(nK)* cannot be viewed as
K=lN
an element of the closure of A_ in IT H1(ng)*. 01(=1
3. Application to a posteriori error estimation.
From now on, we assume that the matrix A. = (aij) E (C1(n»nxn is symmetric (aij =aji) and satisfies the uniform ellipticity condition
(3.1)
where a > 0 is a constant and 1·1 denote the euclidian norm. Given a function (7 E L=(n)
such that
(3.2) (7( x) ~ 0'0 > 0 a.e. in .0,
and a function f E L2(n), we consider the boundary value problem
(3.3) {
- \7. (AY'u) + (7U = f in n,u = 0 in r,
whose (unique) solution u E HJ(n) is characterized by
(3.4) a(u,v) = in fv, V v E HJ(n),
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where a(·, .) : HJ (.0) x HJ( n) -+ ]R is the bilinear symmetric form
(3.5) a(v,w):= in(A\7v)' \7w + in (7VW.
For 1 :::;J( :::; N, we consider the "sub domain" bilinear forms a!\-(·,·) : VI\" X VI\" -+ ]R
defined by
(3.6)
and their sum ap(-,') : HJ(P) x HJ(P) -+ ]R
(3.7)
Ifv,w E HJ(n) and VI(:= VlnK,WI(:= WinK' SO that v = (VI() and W = (WI() modulo
the embedding (2.13), then
(3.8)
by (2.1), (3.5) and (3.7).
ap(v,w) = a(v,w), V V,W E HJ(f2)
The bilinear forms aI«(·,·) are now used to solve "local" Neumann problems:
Lemma 3.1. Let ~ := (AI() ElL, so that AI( E V!(, 1 :::;J{ :::;N. Then, there is a unique
element ¢>(A):= (¢>I«(AI\"» E HJ(P) such that
(3.9)
In addition, there are constants C1 (P) > 0 and C2(P) > 0 independent of A such that
(3.10)
Proof. By (3.1) and (3.2) the bilinear forms aJ«(·,·) are VI(- coercive, whence the existence
and uniqueness of ¢>I«(AI(). The inequalities in (3.10) follow at once from the properties
14
It follows that if> K(>..K) in (3.9) can be viewed, formally at
of A and (7 und the definition of the norms of H6(P) and H-1 (P) through standard
elementary arguments. 0
Fix 1 :::;1(0 :::;N and let v E V(nKo), so that v](o = v, v]( = 0 for K # Ko. By (2.19),
()'Ko,VKo) = 0 if ~ = (AK) E A_. This shows that for general VKo E VKo, (AKo,VKo)
depends only upon V](oIrKo
least, as the solution of the boundary value problem
¢K(AK) = 0 in fl( n f,
a¢J«(AJ()javt = AK in f \ fJ(.
Lemma 3.2. Let ~ E A_ and let ¢(~) E HJ(P) be defined as in Lemma 3.1. Then
- 1ap(¢(A),V) = 0, \Iv E Ho(n).
Proof. Let ~ = (AJ(), so that ¢(~) = (¢]((>.J() and (3.9) holds. In particular, let v E
HJ(n) and set VJ( := vloK
' From (3.7) and (3.9), we have
_ N _
ap(¢(A),V) = ~ (AJ(,V]() = (A,v),1\=1
and (~, v) = 0 by (2.19). 0
Now, let it E H J (n) be an arbitrary element. In practice, it may be viewed as an
approximation of the solution u of (3.3) - (3.4). By the VJ(-ellipticity of the bilinear form
aJ«(·, .), we obtain
Lemma 3.3. There is a unique clement 13:= (13K) E HJ(P) such that
(3.11)
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Lemma 3.4. Let ~ E A_, and let ¢(~), {3 E HJ(P) be defined as in Lemmas 3.1 and 3.3,
respectively. vVe have
(3.12)
and hence
(3.13)
a1'(¢(~) + {3,v) = a(u - it, v), '<IvE Hci(n),
a1'(¢(~) + {3 - (u - tt),v) = 0, '<IvE Hci(n).
Proof. From Lemma 3.2, we already have a1'(¢(~),v) = 0 for A E A_, so that (3.12)
amounts to
a1'({3,v) = a(u - it, v), '<IvE Hci(n).
To see this, add up the identities (3.11) with VK := vlOK
to get
and a1'(it,v) = a(it,v) by (3.8), fnfv = a(u,v) by (3.4). Relation (3.13) follows from
(3.12) since a(u - it,v) = a1'(u - it,v) by (3.8). 0
To motivate our next result, recall the canonical embedding (2.13): HJ (n) '--+ HJ(P).
Let ~ E A_ and suppose that ¢(~) + {3 E HJ(n) (and not merely HJ(P». Then, by (3.8)
and (3.12) and the HJ(n)-ellipticity of the bilinear form a(·,·) it follows t.hat ¢(~) + {3 =
u - it. Lemma 3.5 below shows that this happens for one and only one A E A_.
Lemma 3.5. For A E A_, the following conditions are equivalent:
(i) ¢(~) + {3 E HJ(n) (hence ¢(~) + {3 = u - it from the above).
(ii) ~ = (aUK jav~), where UK := U!OK' 1:::;]{ :::;N.
Proof. As a preamble, we note that (3.3) implies that u E H A(n), so that (aUK j f)Vj~) E A_by Proposition 2.2 (iii). Also, by Proposition 2.2 (i), we have
16
From (3.3), \7. (AVUK) = (7U[( - f, and hence the above formula may be rewritten as
Together with (3.9) and (3.11), this yields
(3.14)
(i) ~ (ii). If ¢C)..) + 13 = U - U, then (¢(5..) + (3)II1K = UK - uK· As (¢(5..) + (3)II1K
¢K(A[() + 13K (by definition of ¢(5..) and (3), it follows from (3.14) that (AK - aau!f, VK) =VK
0, VVK E VK. Thus, AK = aUKlavt in Vi<,l :::;J( :::;N, so that 5.. = X:= (auKlavt)N _ _
in n Vk = H-1(P) (hence A = l in A_ c H-1(P».K=I(ii) ~ (i). If 5.. = (auKlav~), then AK = aUKlavt is in Vk,l:::;]{:::; N, and (3.14)
reads
By the VK-ellipticity of aK(', .), this implies ¢K(AK) + 13K = uK - UK, 1 :::;]{ :::;N, and
hence ¢(A) + 13= U -u. 0
vVe are now in a position to give our main result, that is, a variational characterization
of X := (aUK Iavt)· This characterization permits the calculation of (aUK IaVR) without
any knowledge of the solution u. Instead, U - ti, and hence u, can next be obtained by
solving the local problems (3.9) and (3.11) with AK = aUK IaVI~'
Theorem 3.1. The quadratic functional J := A_ ~ IR defined by
(3.15) -- 1 - - -J(A) := "2ap(¢(A), ¢(A» + ap(f3, ¢(A),
is continuous and coercive, and its unique minimizer is X := (au K Iavt). In addition
(3.16)a( u - u, u - u) = 2J(X) + ap(f3, (3) :::; 2J( 5..) + ap(f3, (3) =
ap(¢(5..) + 13,¢(5..) + (3), VA E A_.
17
Proof. The continuity and coercivity of J are immediate consequences of the inequalities
(3.10). In order to prove that its unique minimizer is none other than X:= (auKjaVf{),
let us first observe that
(3.17)
Since u - it E HJ(n), we have ap(cP(~) + /3 - (u - it),u - it) = 0 for ~ E A_ by (3.13)
in Lemma 3.4. Thus, by writing cP(~) + (3 = cP(~) + (3 - (u - it) + (u - it), we infer that
(3.18)ap(cP(~) + (3, cP(~) + (3) =
ap(cP(~) + /3 - (u - it), cP(~) + /3 - (u - it» + ap(u - it,u - it),
for every A E A_.
Altogether, relations (3.17) and (3.18) show that 2J(~) + ap((3,/3) ~ ap(u - it,u - it),
with equality if and only if cP(~) + (3 = u - 'u, i.e. if and only if ~ = X from Lemma 3.5. In
summary, for A E A_, we have
2J(~) + ap(/3, (3) ~ 2J(X) + ap((3, (3) = ap( u - it, u - ,u).
This implies at once that
and that (3.16) holds since ap( u - it, u - it) = a( u - it, u - it) by (3.8). 0
From Theorem 3.1, the "energy-norm" error a( u - it, u - it )1/2 can be calculated via
the minimization of the functional J in (3.15). It is noteworthy that ~ E A_ is (loosely
speaking) defined only in I'I := II fKL, aset with dimension n-1. Also, the inequality(K,L)EI
(3.16) shows that the quantity
(3.19)
is always an overestimate of the actual error a( u - it, u - it )1/2. Thus, the discretization of- -
the space A_ and the use of approximate minimizers in place of l, two inevitable features
18
of any numerical minimization procedure, will produce conservative estimates of the actual
error.
A strategy for a posteriori error estimation: The most natural way to use (3.19)
as an error estimator is first to substitute for A data readily available from the approxi-
mate solution it itself. In practical applications when it is a (conforming) finite element
approximation, the derivative ait KIavt is always defined in the space L2 (f J( ), hence in
L2(fJ( \ r). However, in general (aitKlav~) ~ A_, whence ~ := (a'ClKlavt-) cannot be
chosen in (3.19). This is easily remedied, e.g. by choosing ~ = (AK) where
(3.20) VL E IK.
Incidentally, the choice (3.20) corresponds to ~ being the projection of (ait K Iavt) onto
A_ relative to the orthogonal splitting
(3.21)
Other weighted averages are suggested in Ainsworth and aden [1].
The step consisting in substituting ~ as given in (3.20) into (3.19) involves only solving
local problems (specifically, (3.9) and (3.11»), which can be done in parallel and with high
accuracy, yet at a nominal cost. If the estimator (3.19) falls below a prescribed tolerance,
(3.16) ensures that the energy norm a( u - it, u - 'u) 1/2 does not exceed that same tolerance,
and therefore it may be called a good approximation of u (relative, of course, to the given
tolerance) .
Otherwise, a finite-dimensional subspace A~ of A_ may be chosen in such a way that
~ in (3.20) satisfies the condition ~ E A~, and an iterative minimization procedure of
.J - or, equivalently, (3.19) - over A~ may be launched, with initial guess >'0 = >.. This
produces a sequence >.j E A~. If for any j ~1 the quantity a'P(fjJ(~j) + /3, fjJ(>.j) + /3)1/2
falls below the prescribed tolerance, the very same arguments used above when j = 0 show
that a( u - it, u - £t )1/2 does the same thing, hence it is a good approximation of u, and the
minimization procedure need not be continued further.
19
If the minimization of j over A~ arrives to completion, thus exhibiting a minimizer-kl E A~, the approximate solution it will be declared unsatisfactory (this is, of course,
a conservative call). Accordingly, a better solution should be sought. Here, a significant-k
bonus is that it +</J(~ ) +13, which obtains at no extra cost, is already an enhanced solution
if the space A~ has been chosen large enough. In this respect, see Remark 5.1 later and
subsequent comments. In other words, the effort for the minimization of j has not been
wasted. The enhanced solution it +</J(ik) +13 is not in HJ(n), being discontinuous across the
interfaces f KL. Thus, the error estimation procedure described above cannot be repeated-k
by simply replacing it by it + </J(l ) +13. But it should be clear that an HJ(n)-interpolation
of it + </J(ik) + 13 can be implemented at nominal cost, thus providing a slightly modified
approximation itl E HJ(n) of u whose quality can be tested again if desired.
The strategy outlined above relies upon Galerkin approximation. This aspect is dis-
cussed in more detail in the next section. In particular, it will be seen that local problems
need not be solved at each step: This can be done once and for all at the beginning of the
procedure for a set of (local) basis functions. It will also appear that the estimator has
justifiable value regarding local error estimation, an attractive feature in some applications.
Another important aspect is the accuracy of the estimator: The fact that any choice
of .\ E A_ in (3.19) produces an overestimate for a(u - it,u - it)1/2 was used in a crucial
way, but it is also important that the resulting estimator does not provide exceedingly
conservative predictions. This issue is considered in Section 5 in a special case relevant in
most (two-dimensional) problems. There, we will show that the estimator may be made
\'ery accurate (in theory, this depends upon the choice of the sub domains nK) for solutions
having good interior regularity.
4. Galerkin approximation.
Due to the fact that the elements of the space A_ correspond to two functions in each
interface f KL, (K, L) E I (one being the negative of the other), the spaces A_ and A_ are
somewhat inconvenient to use. Our first task here will be to reformulate the minimization
problem of Theorem 3.1 in a more familiar setting.
We shall denote by fI the interface ofthe partition P, that is, the (non disjoint; compare
20
with fI in Section 2) union
(4.1) fI := U fKL.(K,L)EI
Observe that those f K L with µ(f K d = 0 have been neglected since they play no role in
functional spaces based upon the measure µ defined at the beginning of Section 2. Note
also that since (K,L) E I if and only if (L,K) E I, and since fKL = fLK, we have
(4.2)
where
(4.3)
We shall set
(4.4)
fI:= U fKL,(K,L)EI>
I>:= {(K,L) E I: K > L}.
A := L2(fI) ~ IT L2(fKL),(K,L)EI>
where the identification is made by restrictionj extension by 0 and patching. There is a
canonical (except for the choice of the numbering of .01, ... , nN) isomorphism
(4.5a)
given by
(4.5b)
where
(4.5c) {
AKL if L EIK and K > L,AI( =
IrIa _ ALI( if L E If{ and K < L.
21
vVe shall denote by i\ the completion of the space A equipped with the H-l(P)-norm
"ia the isomorphism (4.5). It is then trivial that the isomorphism (4.5) extends as an iso-
morphism of A onto A_. For A E A, we shall denote by ~ the element of A_ corresponding
to A through this isomorphism.
As a result, the functional j in (3.15) becomes a functional
(4.6) - -- 1 - - -A E A ~ J(>.) := J(>.) := 2ap( 4>(A), 4>(>.» + ap( 4>(A), ,8),
which is continuous and coercive (Theorem 3.1). Also by Theorem 3.1, the unique mini-
mizer lEA of J is such that X = (auK j avt ).For each pair (K,L) E I>, let (A1LhEN be an increasing sequence of finite-dimensional
00
subspaces such that U A1<L is dense in L2 (f KL)' Call Ak the product spacek=O
(4.7) A k := II A j(L C A(K,L)EI>
(see (4.4». (The space A~ at the end of Section 3 is just the image of Ak through the
isomorphism (4.5).) Then, U Ak is dense in A (see (4.1» and hence in A since (obviously)k=O
A is dense in A. It thus follows from the continuity and coercivity of the functional
J in (4.6) along with general properties of the Galerkin approximation that the unique
minimizer lEA of J satisfies the condition
(4.8) \ l' \ k· A-~ = 1m ~ In ,k--oo
where for k E N,lk E Ak is the unique solution of the minimization problem
(4.9)
For A, B E A, we shall set
(4.10) b(>.,B):= ap(4)(~),4>(B))
22
and
(4.11)
so that
(4.12)
Given (K, L) E I>, set
(4.13)
f(A) := -ap(!3, 4>(~»
J(>') = ~b(A, A) - f(A), V >. E A.2
and let (pkL)l<i<dk be a basis of AtL' The functions p}(L can be viewed in A:= L2(fI)- - KL
after extension by 0 outside f K L, and then A E Ak C A has a unique decomposition
(4.14)
for suitable coefficients ekL E JR. As a result, the minimization of the functional J in
(4.12) amounts to solving a linear system
(4.15)
with symmetric positive definite matrix B with coefficients b(P}(L,P}J) and right-hand side
T] with components f(pkL)'
From (4.10) and (4.11), the calculation of the coefficients of B and of the components
of T] requires the calculation of the quantities 4>(P}(L)'
Fix (Ko, Lo) E I> and let 1 :::;i :::;dtLo' According to (3.9) (and recalling the defini-
tion of the embedding (4.5)), we have 4>(jj}(oLo) = (¢>K(P}(oLol )) where 4>K(P}(oLol
) =rK rK
o if K =1= Ko, K =1= Lo (because P}(oLo = 0 in fI \ fKoLo) and 4>Ko(pkoLolr
) andKo
4>Lo (pkoLo ) are characterized byIrLo
(4.16)
23
and
(4.17)
From the above, <P(P}(oLo) viewed as a function in L2(n), has support in nKo U nLo for
every (Ko,Lo) E I>. It now follows from (4.10) and the definition of the bilinear form
ap(·,·) (see (3.6) and (3.7» that if (K,L),(I,J) EI>:
unless (nK U nL) n (n[ u .oJ) i= 0 (i.e. I = K or L, or J = K or L). This result expresses
the sparsity of the matrix B.
It is also of some practical importance to notice that, after <p(p~J) has been determined,
the calculation of the coefficients b(pk L' p~J) reduces to an integral along f K L. Indeed,
we have by (4.10) and (3.6), (3.7) and (3.9)
But since PkL = 0 outside fKL, only the values M = K and AI = L are actually involved
in the right-hand side, and hence, recalling the embedding (4.5),
(4.18)
As the calculation of b(p}\oL' p}(L) requires using <p(p}( L)' all the <p(p}( d, (K, L) E I>, must
be determined. Since b(·, .) is symmetric, the roles of pk L and p~J can be exchanged in
(4.18) (which is not obvious from merely looking at the right-hand side).
The evaluation of the components of the right-hand side 1] of (4.15), i.e. of f(pkL) =-ap(j3, <p(pkd) (see (4.11» can be done as follows: First, j3 may be calculated via (3.11),
and next f(P}(L) is obtained through a boundary integral along fKL since, by (3.6), (3.7)
and (3.9) (and the symmetry of ape, .»
(4.19)
24
Remark 4.1: Since a1'((3,¢>(i>}\d) = fnf¢>(i>}(d - a1'(it,¢>(i>i\L» by (3.11), and since
<!>(P}(L) has support in D.K U D.L, we get (with (K,L) E I»
a formula showing that (3 need not be calculated at this stage. However, (3 is needed to
recover U - ii. = <p(~) + (3 and hence any approximation <p(.A) + (3 of u - U. 0
Local error estimation: It often happens that the quality of the approximation U of u is
not assessed solely by the smallness of a( u - u, U - U )1/2 but also depends upon the locally
defined quantities aK( UK - Ul(, UK - UK )1/2. It follows at once that the convergence of
<p(~k) + 13 towards u - U in energy norm implies
(4.21)
for every :fixed 1 :::;K :::;N. Therefore, in spite of the fact that aK( <PK(.A ~ ) +13K, <PK(.A ~ ) +13K) need not represent an overestimate for aK( UK - UK, UK -UK), i.e. the global property
does not carry over to the sub domain level, nevertheless (4.21) reveals that a relationship
still exits between the local errors aK( UK - uK, UK - UK )1/2 and their calculable counter-
parts aK(<PK(.Aj() + (3K,<PK(.At) + 13K)1/2. Thus, although the latter can only be called
local "error indicators", (4.21) provides a reasonable justification of their use for the as-
sessment of the quality of U based upon local criteria. Many of the local error indicators
suggested in the literature are heuristic and do not possess a property similar to (4.21).
5. Accuracy of the estimator.
As in the previous section, we denote by ~ k the unique minimizer of the functional J
in the space Ak (see (4.7). From (4.6) and Theorem 3.1, the quantity
(5.1)
where Xk E A_ is associated with lk through the isomorphism (4.5), is an overestimate of
the energy norm error a( U - u, U - u)I /2. We shall now discuss the accuracy of (5.1) as an
25
estimator for a( u - it, u - U)1/2 under additional assumptions about the partition P and
the approximation spaces Ak. Our goal is to improve upon the general but vague result
lim 2J(lk) + ap((3, (3) = a( u - u, u - u),k-oo
by exhibiting orders of convergence.
We shall limit our discussion to the case when .0 and nK are polygons. More specifically,
each nK will be an element of a small patch of adjacent elements of a finite element mesh
(e.g. used for the calculation of u). As a result, the interfaces fKL are just some of the
edges of the finite element mesh which do not lie in the boundary f. Note when nK is a
patch of elements, that some of the edges of the initial mesh will not appear as interfaces
f K L. This remark will be important later on.
We shall assume that the number of interfaces f K L, LEI K, is uniformly bounded from
above irrespective of possible refinements of the mesh and subsequent redefinition of the
partition P. To avoid tedious additional technicalities, we shall make the rather strong
assumption that each subdomain .oK is regularly affine equivalent to one among finitely
many polygonal "reference" shapes chosen once and for all. Here, "regular" has its usual
geometric meaning in finite elements terminology. This assumption may be relaxed. Yet, as
it remains realistic even in cases where the meshes are locally refined, it seems appropriate
to include it for expository purposes.
For the space Aj(L' we choose
(5.2)
where for every integer m ;:::0,
(5.3) Pm(fKd := {polynomials of degree at most min fKL}.
The notation (5.2) is consistent with k[(L being a function of k for each pair (K,L) E I>.
We set
(5.4 ) hKL := length of fKL, hK:= diameter of .0[(,
26
and henceforth assume that the ratios hK /hKL, L E II{, are uniformly bounded.
Lemma 5.1. There is a constant C > 0 independent of the partition P, independent of
1 :::;J{ :::;N and independent ofvK E HI(nK) such that
(5.5)
where, with (7K := (7loK '
(5.6)
rK:= anK.
For x E nK, set aK(x) := (7K(<}-l(X».
Proof. Let <} : .oK -+ ]R2 be an affine transformation mapping nK onto one of the "ref-
erence:' polygonal shapes, here denoted by nK for convenience. Accordingly, we shall set
Evidently, (70 := mi~(7(x) :::;aK(x) :::;AI :=xEn
m~(7(x), so that the linear operator L: HI(nK) -+ L2(nK) defined by LVK := VK - vKxEn '
where vK := (JoK aI(VK)/(Jo/( aK) has IILII ~ 1 + M/[(70InKll/2]. Since LVK = 0 when
vK is constant, it follows from a standard variant of the Bramble-Hilbert lemma that
IVK - vK·lo 0 :::;CIVKII n" and hence that, K , K
(5.7)
where C > 0 is a constant depending only upon .oK, (70 and lV!.
If now VK E Hl(nK) and VK := VK 0 <}-l, then vj( = vK (see (5.6» and we have
IVK - vI(lo,r/( :::;Clh~PlvK - vK,lo,i'/( ([7, Theorem 3.1.2]) :::;C2h~PlvK - V](II/2,f'K ~
C3h~PllvK-vKII1,oK (Trace theorem) ~ C4h~PlvKll,nK (inequality (5.7»:::; C5h~PlvKiI,nK
([7, Theorem 3.1.2]). This completes the proof. 0
Lemma 5.2. There is a constant C > 0 independent of the partition P, independent of
1 :::; J{ :::;N and independent of AI{ E L2(f I{ \ f) such that (with (7K := (7loK
)
(5.8)
27
Proof. Recall that AK E L2(fJ( \f) defines an element of Vi- via (AK, VK) := IrK\r AKvK,
and that ¢>K(AK) is given by (3.9).
(5.9)aK(¢>K(AK),¢>K(AK) = r Ab:¢>K(AK) =irK
r Ah"(¢>K(>..Id - ¢>r...:(>..K)) + ¢>K(AK) r AK,irK irK
where ¢>K(>..K) E lR is defined by (5.6), i.e.
(5.10)
From (5.9), (5.10) and Lemma 5.1, it follows that
(5.11)
By letting VK = 1 in (3.9), we obtain InK (7K¢>K(AK) = (AK,l) = IrK AK- Thus,
¢>A"(>..K) = (IrK AK)/(inK (7K) from (5.10). Also, I¢>K(AK)!I.nK :::;CaK(¢>K(AK),¢>K(AK»I/2
where C > 0 is a constant depending only upon A and n. By substitution into (5.11), we
thus get
an inequality of the form X2 :::;XZ + y2 with X := aK(¢>K(AK),¢>J((>..J(»I/2, y :=
I IrK AKI/(inK (7J()1/2 and Z = Ch~PIAJ(lo,rK'This inequality implies X2 ~ Z2 + 2y2 ~ 2Z2 + 2y2, and (5.8) now follows from the
definition of X, Y and Z above (recall µ(fK n r) = 0 in this Case 1).
Case 2: VK =1= HI(nK). If so, each VJ( E VK vanishes in a subset of fK with positive
measure. By using an affine transformation q> : nK ---+ ]R2 mapping .oJ( onto one of the
28
reference polygonal shapes, and by proceeding as in the proof of Lemma 5.1, it is easily
seen that
Together with the choice VK = ¢>K(AJ() in (3.9) and by using once again the inequality
I¢>J«(AK )ll,nK :::;CaK( ¢>K(AJ(), ¢>K(AK »1/2 with C > 0 depending only upon A and .0,
this yields
an inequality stronger than (5.8). 0
Consistent with the notation used in Section 3, we shall set X := (au K / av::>, UK·-
U!OK' where of course u E HJ(n) continues to denote the unique solution of the equivalent
problems (3.3) and (3.4). The first remark is that since n is a polygon, the solution u
possesses the better regularity u E HS(n) for some 8 > 3/2 (8 ~ 2 if .0 is convex). For
this, we refer to Grisvard [8, Chapter 5]. As a result, the derivative aUK/avt is defined
in HS-3/2(fJ() and hence in L2(fJ(L),(K,L) E I. As we also know from Section 3,
.x E A_. Together with the above result that aUK/aV-R E L2(f](L), whence.x E L2(I'I),
this suggests that.x E A_. That it is indeed so is shown below (this is not obvious because
j\'_ is the completion of A_ for a norm weaker than the L2 norm).
Lemma 5.3. l E A_.
Proof. Since the splitting L2(I'I) = A+ EBA_ is orthogonal (as noticed in Section 3), it
suffices to show that
(5.12)
for every (v K) E A+, where )..J( := au K / avt. Every element (v K) E A+ corresponds to
a collection VJ(L := vJ(lr ' 1 ~ K ~ N,L E Il(, such that VKL = VLl(, and then (5.12)KL
may be rewritten as
(5.13)
29
where AK L := AKI .rKL
Since 'D(fKd is dense in L2(fKd, it suffices to prove (5.13) for VKL E 'D(fKd (and
VKL = vLId. But then, it is clear that v defined by VK = VKL in fKL can be extended
as a function, still denoted by v, of 'D(n), so that the left-hand side of (5.13) is just the
pairing (X, v). That (X, v) = 0 now follows from X E fL, V E 'D(n), and (2.19). 0
Recall that to A = (AKd(K,L)EI> E A, the isomorphism (4.5) associates ~ = (AKd E
A_, where for (K,L) E I and L > K,AKL is defined by AKL := -ALK. This notation,
along with lKL := lKI ,(K,L) E I, is used in our next lemma where the accuracyrKL
question begins to be addressed.
Lemma 5.4. There is a constant C > 0 independent of the partition P, independent of
kEN and independent of the solution u as well as of its approximation u E HJ(n) such
that
(5.14)
- k -k0:::; ap(4)(l ) + (3,¢(l ) + (3) - a(u - u,u - u):::;
c[ L; hKL inf{IAKL-lKLI~,rKL :AKLEA1<L]'(K,L)EI>
Proof. The first inequality in (5.14) is already known from (3.16). Next, u - u = 4>(X)+ (3,
so that for A E A we have
ap(4)(~) + (3, 4>(~) + (3) - a(u - U, u - u) = ap(¢(~) + (3, 4>(~) + (3)-
ap( ¢(X) + (3, ¢(X) + (3) = ap( ¢(~) - 4>(X),¢(~) + ¢(X) + 2(3).
Now, write ¢(~) + ¢(X) + 2{3 = ¢(~) - ¢(X) + 2(¢(X) + (3) = ¢(~) - ¢(X) + 2(u - u). By
substitution (and linearity of ¢), we find
(5.15)ap(¢(~) + (3, ¢(~) + (3) - a(u - it, u - u) =ap(4)(~ - X), ¢(~ - X) + 2(u - it» =
ap(¢(~ - X), ¢(~ - X»,because ap(4)(~ - X),u - it) = 0 from Lemma 3.2. By definition of lk, it follows from
(5.15) that
-k -k - - - -(5.16) ap(4)(~ )+{3,¢(l )+{3)-a(u-it,u-11)=inf{ap(¢(A-l),¢(A-~»:AEAk}.
30
Since X E A- by Lemma 5.3, whence ~ - X E A_ for every A E A k, we now infer from
(5.16) and Lemma 5.2 that
(5.17)
-k -ka1'(<p(~ ) + (3, <p(~ ) + (3) - a(u - u,u - u) :::;
C inf{K~/'KIAK - !'Kli,r",r+ (l",r (AK - !,K l) 2/ (L" (lK ) : A E Ak}.For each 1 :::;K ~ N, IAK -lKI~,rK\r splits as the sum LJi)AKL -ll<L15,rKL' Since
the ratios hK / hK L( LEI 1<) are uniformly bounded, (5.17) yields
(5.18)
For (K, L) E I>, let A~(L E A ~<L denote the best L 2 (f K L )-approximation of lK L in
Ak .J( L, l.e.
(5.19)
Since A~<L consists of the polynomials of degree at most kKL ~ 0 in fKL (see (5.2)
and (5.3», the difference A~<L - lK L is always orthogonal to the constant functions in
f1<L, i.e. JrKL ().~<L -ll<L) = O. Both this relation and (5.19) remain valid not only
for (K,L) E I> but also for every pair (K,L) E I since (L,K) E I> if (K,L) 1. I>, and
A~(L = -AttK,lKL = -~LK in this case. Thus, by majorizing the right-hand side of (5.18)
with the special choice AJ(L = ).~(L' the integrals JrJ(\r(A} -lK) vanish and the second
inequality in (5.14) follows at once from (5.19). 0
To complete with, we need an estimate for each infimum appearing in the right-hand
side of (5.14). The following special case of Lemma 4.5 in Babuska and Suri [5] will suffice:
Lemma 5.5. Let s ~ 0 be a real number. There is a constant C > 0 depending only
upon s such that for every integer m ~ 0 and every v E HS(f I( L), we have
(5.20)hmin(m+I,s)
inf{lv - wlo,rKL : w E Pm(fKd}:::; C ~_~ , ,\. Ilvlls.rKL·
31
Actually, in [5] Lemma 5.5 is proved in the two-dimensional case when f J( L is replaced
by a "nondegenerate" triangle or parallelogram. The one-dimensional case considered in
Lemma 5.5 is of course simpler (though not considerably). Also, curiously, the case m = 0
in (5.20) is not considered in [5] although it does not present any particular additional
difficulty.
Since .0 is a polygon, the relation u E Hr(n) should not be expected to hold with r ;::::3
irrespective of the smoothness of f. However, things go differently in any open subset w
ofn such that n \ w is an open neighborhood (in n) of the vertices of n, for then we have
For convenience, we shall refer to r as being the "interior regularity" of u (although it
extends to the boundary away from the vertices). vVeshall also assume that no interface
f J( L contains a vertex of n as an endpoint. This condition often prohibits choosing nJ(as an individual element near the vertices of n, but a small patch will always suffice to
comply with this requirement (Figure 5.1). The net result of this hypothesis is that
(5.21)
provided that A is smooth enough, e.g.
(5.22)
as we shall henceforth assume to fix ideas.
Theorem 5.1. There is a constant C > 0 depending upon the interior regularity r(;::::2) of
u, but independent of the partition P and independent ofu as well as of the approximation
U E HJ(n), such that
(5.23)a( u - u,u - u)I/2 ~ a1'( </>(}/) + (3, </>(~k) + (3)1/2 ~
h2 min(kI<L+I,r-I) II a 112a(u-u,u-u)I/2+C{ ~ J(L . uJ( P/2.(K,L)EI> (kJ(L + 1)2r-3 av~ 3 r
1\ r-2, KL
32
Proof. In Lemma 5.5, let s = r - 3/2,m = kKL and v = aUK/aV~. Then, the conclusion
follows from Lemma 5.4 and elementary manipulations. 0
-k -kTheorem 5.1 reveals the high accuracy of the estimator ap( </>(l ) + ,8, ¢>(l + (3)1/2 for
solutions U having good interior regularity as the finite element mesh is held fixed and k,
i.e. kKL for each pair (K, L) E I>, is increased. Naturally, the difficulties induced by the
lack of smoothness of u in the vicinity of the vertices ofn do not disappear completely since
the norms IIauK / av~ IIr- ~,rK L will usually be larger for interfaces f K L near the corners.
From (5.23), the remedy is to increase kKL along those (and neighboring) interfaces. In
contrast, refining the mesh creates interfaces even closer to the singularity along which
IIauK / av~ IIr- %,r K L increases.
Also, in practice, neither </>(~k) = (</>K(lt-» nor (3 = ((3K) are calculated exactly.
The numerical evaluation of </>K(lt-) and (3K requires solving the local problems (3.9)
and (3.11), respectively, over the sub domains nK, and large angles in nK will affect the
accuracy ,of the local calculations. Note that large angles in some .0K'S will exist if n itself
has such large angles because of the standing assumption that no interfaces f K L terminates
at a vertex of n. However, the accuracy question is easier and cheaper to handle at the
local rather than global level.
-k --kRemark 5.1: Since u - u - </>(l ) - (3 = ¢>(21- 21 ), the method of proof of Theorem 5.1
also yields
(5.24)
- k - k 1/2a( u - it - ¢>(l ) - (3,u - U - ¢>(l ) - (3) :::;
h2rnin(kKL+l,r-I) II a 112c ')' KL UK 1/2{(K,t)'EI> (kKL + 1)2r-3 av~ r-%,rKL} ,
-kl.e. U + ¢>(l ) + (3 is a highly accurate approximation of u as k is increased if u has good
interior regularity r. 0-k
It is clear from (5.24) that u + ¢>(l ) + (3 will provide an enhanced solution (in theory
at least) if the polynomial degrees k KLare large enough. Typically, if it is a piecewise
polynomial of degree at most m, then a(u - u, u - u)I/2 = O(hm) is best possible. It then
33
-kfollows from (5.24) that U + <p(l ) + (3 is theoretically a better approximation than u if
kK L ~ m for each pair (K, L) E I, assuming of course sufficient interior regularity of u.
6. The positive semi definite case.
In this section we investigate the case when (7 = 0 in (3.3). The only, but significant,
difference is that the equivalent problems (3.3) - (3.4) continue to have a unique solution u
while the Fredholm alternative holds for the local problems (3.9) or (3.11) for every index
1 ~ K:::; N such that VK = HI(nK), or, equivalently, µ(fK n r) = O.
The new issue here is that for those indices K, the bilinear form aK(-,') in (3.6) satisfies
aK(vK,wK) = 0 whenever VK or WK is a constant. As a result, the solvabiltiy of (3.9)
requires the extra compatibility condition
(6.1)
and then <PK(>.'[<)is determined modulo a constant. Likewise, (3.11) has solutions if and
only if
(6.2)
and (3[( is determined modulo a constant.
The above suggests modifying the definitions of <p(~), ~ E A_, and (3 = ((3K) so that
<1>( j,) and (3 always exist. A Iit tIe thinking leads to the choices
and
(6.4)
aK((3I(, VK) = inK fVK - In1KI (inK f) (inK VI() - aK(UI(, VK), VVK E HI(nK),
for all those indices K such that VI( = HI(nK). Otherwise, the definition of <PK(AK) and
(3K is unchanged (i.e. given by (3.9) and (3.11), respectively).
34
The existence of 4>K(AI() and /3K in (6.3) and (6.4) is not a problem because the right-
hand sides vanish when VK is constant in nK, but their uniqueness is true only in the space
HI (nK)/ po(nK). vVhen VK =I Hl(nl(), 4>K(Al() and /31{ are unique and Vl( / po(nld =
Vn-. Thus, 4>( ~) and /3 exist and are unique in the space
(6.5)
In this setting, 4> remains linear and continuous if HJ(P)/ Po(P) is equipped with the
quotient norm. In this respect, observe that ap( (v l( ), (w K » is unambiguously defined for
(UK), (WK) E HJ(P)/Po(P) since its value depends only upon the equivalence class of (VK)
and (WK).
It is readily seen that Lemma 3.2 is no longer valid with the new definition of 4>( ~) and
,8, and hence Lemma 3.4 also fails to hold. However, the following variant remains true:
Lemma 6.1. Let ~ E A_, and let ~ and f satisfy the compatibility conditions
(6.6)
for every index 1 ~ J( :::; N such that Vl( = HI (nl( ). Then,
(6.7)
and hence
ap(4)(~) + /3, v) = a(u - u, v), Vv E HJ(n),
ap(4)(~) + /3 - (u - u),v) = 0, Vv E HJ(n).
Proof. From (6.3), (6.4) and (6.6), we have
for every VK E HJ(P). If v E HJ(n) and VK := vlnK
, it follows from (2.19) that ap(4)(~)+
/3, v) = In fv - a( u, v) = a( u - u, v). 0
35
The relevance of the compatibility conditions (6.6) stems from the fact that they hold
when AK := aUK/aV~ and U E HJ(n) is the unique solution of the problem
-\7. (A\7u) = fin n.
\Vith this remark and Lemma 6.1, a routine check reveals that Lemma 3.5 and Theorem 3.1
remain valid if attention is confined to those A E A_ satisfying the compatibility conditions
(6.6) (with the obvious modification that 4>(5..) + (3 E HJ(n)/po(p) in part (i) of Lemma
3.5; this implies 4>(~) + (3 = u - u in HJ(n)/po(p». In particular, X:= (aUK/aV~) E A_appears to be the (unique) minimizer of the functional j in (3.15) in the affine subspace
of A- in which the compatibility conditions (6.6) hold. Thus, X is now characterized by a
constrained minimization problem. This is the only difference with the situation described
in Section 3.
The modifications to the Galerkin approximation procedure described in Section 4 are
obvious: Instead of (4.9), the functional J is now being minimized in the affine subspace
of the space Akin which (using (4.5))
(6.8)
The formula (4.16) (resp. (4.17)) remains unchanged when VJ(o =f. HI(nKo) (resp.
FLo =f. Hl(nLo»' If V](o = HI(n](o)' then (4.16) must be replaced by
(6.9)
aKo(4)Ko('pkoLolr ),0](0) = r pkoLoV](o - -ln1 1(1 PkOLO) (1 VKO)'
Ko JrKOLo ](0 rKoLo OKo
Likewise, if VLo = HI(nLo), then (4.17) becomes
The formulas (4.18) and (4.19) remain unchanged if 4>]«(A]() and 13K are always chosen
to have zero average in n](, and Remark 4.2 (i.e. formula (4.20» may be repeated under
36
the same condition. The constraints (6.8) take the form (see (4.14»
(6.11)
where, according to (4.5), PkL := -P~K if L > K.There are several ways of handling constrained problems involving a linear constraint,
but it must be kept in mind that the use of an iterative method is important to our error
estimation strategy. If the constraints (6.11) are written in the form C~ = <p, one option
is to solve the system
where 8 appears as a Lagrange multiplier. This nonsymmetric formulation is only positive
semidefinite, and hence the use of well-established iterative methods (GMRES, CGS [11],
GCGLS [2]) seems not to be fully justified. On the other hand, the constraints (6.11) have
a form that allows for the use of the new parallel procedure in [9]. (Although not of "local
type" in the sense of that paper, the constraints (6.11) retain just enough properties to
justify the method.)
Theorem 5.1 continues to hold provided that, now lk E Ak is the solution of the dis-
cretized constrained minimization problem. Most modifications of the arguments are obvi-
ous, but some additional remarks are in order regarding a couple of specific points. First,
Lemma 5.1 is obviously no longer valid since (7 = 0, and the statement of Lemma 5.2 must
also be modified: In fact, the simpler inequality
(6.12)
holds. \Vhen VK = HI(.0 K ), hence f K \ f = f K modulo a set with measure 0, this follows
from (6.3) since
and ¢>K(AK) - If/KI InK ¢>K(AK) has zero average in nK. Thus, by Lemma 5.1 with (7 = 1,
we obtain aK(¢>K(AK),¢>K(AK» ::; Ch~PIAKlo,rKI¢>K(AK)iI,nK' and (6.12) follows after
37
majorizing 14>1({A1()ll,nK by CaK(4)J({AK), 4>K{AJ())I/2 where C > 0 is a constant depend-
ing only upon A and n. When VI( =I HI(nK), the validity of (6.12) was already noticed
in the proof of Lemma 5.2, and the same arguments continue to work when (7 = o.The next point that deserves clarification is the use of Lemma 3.2 for the justification
of (5.15) in the proof of Lemma 5.4. Indeed, as noted earlier, Lemma 3.2 is no longer valid
and hence cannot be used to show that ap(4)(~ - X),u - u) = O. Actually, this relation
need not be true for arbitrary A E A, i.e. arbitrary ~ E A_, but it continues to hold
when the image ~ of A through the isomorphism (4.5) satisfies the constraints (6.6): If
VI( = HI (nJ(), we have
(6.13)
a I( ( 4> J( ( AJ( - lJ( ), UJ( - UJ() = (A J( - ~J(, UK - it K) - In1J(I (A J( - AK, 1) inK (u K - UJ() =
(AK -lK' UK - UJ(),
because (AI(,l) = - InK! = (lK,l) (recall that lI( := aUJ(/av"t satisfies (6.6». If
now VI( =I Hl(nI(), then (6.13) holds by definition of 4>K(AK). Thus, (6.13) holds for
1 :::;J( :::;N if the constraints (6.6) are satisfied, and hence ap(4)(~ - X),u - u) = 0 by
(2.19). Evidently, the choice ,\ = lk is possible since Xksatisfies (6.6), and this is all that
is needed to establish the inequality (simpler than (5.18»
(6.14)
-k -kap(4)(l ) + (3,4>(~ ) +(8) - a(u - U,U - u) :::;
Cinf {~ ~ hI<LI'\KL - ~I\-LI~,rKL : A E Ak, (6.6) hOldS}.I(=ILEIJ(
The key point now is that if A~(L E Aj(L is the best L2(fl(L)- approximation of lKL,
given by (5.19), then A~(L -lKL is orthogonal to the constant functions in fKL, i.e.
};r A~(L -lKL = O. This implies at once that if A~( is defined by ,\~ = A~L' thenKL IrK
IrJ(\r A~( = IrJ(\r lK = InK f· This means that (6.6) holds with A = ,\0 := (A~(). Thus,
the choice A = ,\0 can be used to obtain an upper bound for the right-hand side of (6.14).
Together with (5.19) this now yields the tighter upper bound
38
trivially equivalent to (5.14). In other words, Lemma 5.4 remains valid as stated, and the
arguments of the proof of Theorem 5.1 can be repeated.
-kRemark 6.1: Although 4>(l ) + f3 E HJ (P)/ Po(P) suffices for energy-norm error cal-
culation, a little extra work is needed to obtain an enhanced solution since, obviously,
the constant CK E lR added to any choice of 4>K{A}) + f3K when VK = Hl(nK) (and
CK = 0 otherwise) is important to the quality of (say) the L2(nK)-approximation of UK
by UK + 4>K(Aj() + f3K +CK. Since the solution u is continuous across the interfaces f KL, a
reasonable criterion for the selection of the constants CK is the minimization of the jumps
I4>K(A}) + f3K + CK - 4>L('\i) - f3L - CLI along these interfaces. It can be shown that
minimization relative to the L2(rz)-norm produces a solution 4>(5..k) + f3 + c(c = (CK»
such that lu - U - 4>(5..k) - f3 - clo,n = O(hlu - u - 4>(5..k) - f3ll,n). Details are omitted for
breviety. 0
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39