We consider the approximation of elliptic boundary value problems by conforming finite element methods. A model problem, the Poisson equation with Dirichlet boundary conditions, is used to examine the convergence behavior of flux defined on an internal boundary which splits the domain in two. A variational definition of flux, designed to satisfy local conservation laws, is shown to lead to improved rates of convergence

Brezzi, Franco

Hughes, T. J. R.

Suli, Endre

English

Mathematical Institute Eprints Archive

VARIATIONAL APPROXIMATION OF FLUX IN
CONFORMING FINITE ELEMENT METHODS FOR
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS:
A MODEL PROBLEM
FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE S

ULI
Abstrat. We onsider the approximation of ellipti boundary
value problems by onforming nite element methods. A model
problem, the Poisson equation with Dirihlet boundary onditions,
is used to examine the onvergene behavior of ux dened on an
internal boundary whih splits the domain in two. A variational
denition of ux, designed to satisfy loal onservation laws, is
shown to lead to improved rates of onvergene.
1. Introdution
In mathematial modeling of physial phenomena one frequently en-
ounters instanes when the primary quantity of interest is not the
analytial solution to the underlying partial dierential equation but
a linear funtional of the analytial solution to the equation; in suh
ases, solving the dierential equation onsidered is only an interme-
diate stage in the proess of omputing the main quantity of onern.
For example, in uid dynamis one may be interested in alulating the
lift and drag oeÆients of a body immersed in a visous inompress-
ible uid whose ow is governed by the Navier-Stokes equations. The
lift and drag oeÆients are dened as integrals, over the boundary
of the body, of the stress tensor omponents normal and tangential to
the ow, respetively. Similarly, in elastiity theory, the quantities of
prime interest, suh as the stress intensity fator, or the moments of a
shell or a plate, are derived quantities.
A further aspet of measurement problems of this kind is that, fre-
quently, the funtional under onsideration may be expressed in var-
ious forms whih are mutually equivalent at the ontinuous level but
result in very dierent approximations under disretization. Thus it
1991 Mathematis Subjet Classiation. 65N30, 65N15, 65N50.
Key words and phrases. Finite element methods, onservation, error estimates,
ux funtionals.
1
2 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE S

ULI
is important to selet the appropriate representation of the funtional
before formulating its disretization. This basi idea has been widely
exploited in strutural mehanis [1, 2, 3, 4℄ and heat ondution [14℄ to
post-proess nite element approximations, and more reently also in
the eld of omputational uid dynamis in the ontext of a posteriori
error estimation for lift and drag omputations (f. [9℄).
In this paper we shall be onerned with the nite element approxi-
mation of one partiular funtional: the diusive ux over an interfae.
Our aim is to show that the natural \variational denition" of the dis-
rete ux an provide a high order of auray in suitably dened dual
norms. In doing so, we shall not aim at generality. On the ontrary,
we will try to present the main idea on the simplest possible prob-
lem, in order to stress what we believe to be the ruial points and
instruments, avoiding tehnialities as muh as possible. We believe
however that more general results are true, and the diÆulty in their
proof is mainly of a tehnial nature. The related problem of error
analysis of the diusive ux approximation over the entire boundary of
the omputational domain has been onsidered by Barrett and Elliott
in [5℄.
The paper is strutured as follows. The model problem is desribed
in Setion 2, together with the denition of the proposed approximation
of the ux, while Setion 3 is devoted to the proof of our error estimate.
Throughout the paper we shall use the usual notation for Sobolev
spaes and their norms and seminorms. See, for example, [7℄, [11℄.
2. The model problem and the main result
2.1. The geometry of the problem. Let 
 be a retangle in the
plane with boundary 
. We shall suppose that 
 is split into two
disjoint open subdomains 

1
and 

2
by a straight line  . We do not
assume   to be parallel to one of the edges of 
; however, for the sake
of simpliity of the notation we shall require that   has equation x = 0.
2.2. The model problem. For a given f , smooth enough, we onsider
the following problem:

nd u 2 H
1
0
(
) suh that
  u = f in 
:
(2.1)
VARIATIONAL APPROXIMATION OF FLUX 3
PSfrag replaements
 




2


1
Figure 1. The domain 
 and the internal boundary  .
It is well known that (2.1) has a unique solution. We set V := H
1
0
(
),
and for u and v in V we let
a(u; v) :=
Z


ru  rv dx dy:(2.2)
Hene, the variational formulation of (2.1) is

nd u 2 V suh that
a(u; v) = (f; v) 8v 2 V;
(2.3)
where, as usual, (; ) represents the inner produt in L
2
(
).
2.3. The deomposition and the disrete problem. Let T
h
be a
deomposition of 
 into triangles whih is ompatible with the splitting
of 
 into 

1
and 

2
(this obviously means that eah triangle is a subset
of one of the two subdomains.) For k an integer  1 we onsider the
spae V
k
h
dened as
V
k
h
:= fv
h
2 C
0
(
) \H
1
0
(
); v
h
jT
2 P
k
8T 2 T
h
g;
that is the usual nite element spae of ontinuous pieewise polynomi-
als of degree k over the deomposition T
h
whih obey the homogeneous
Dirihlet boundary ondition on 
.
In tandem with (2.3) onsider the following disrete problem:

nd u
h
2 V
k
h
suh that
a(u
h
; v
h
) = (f; v
h
) 8v
h
2 V
k
h
:
(2.4)
4 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE S

ULI
It is well known (see, for instane, [7℄) that (2.4) has a unique solution
u
h
, and that the following error estimates hold, whenever the analytial
solution u has the neessary regularity:
jju  u
h
jj
r;

 C h
k+1 r
juj
k+1;

; r = 0; 1:(2.5)
2.4. The disrete ux, and the statement of the main result.
First of all, in analogy with (2.2) we introdue
a
i
(u ; v) :=
Z


i
ru  rv d
(2.6)
for i = 1; 2. We dene now the ontinuous ux from 

1
into 

2
through
the interfae   as
F
u
:= (
u
x
)
j 
(2.7)
where we assumed that 

1
is to the left of  , that is


1
= f(x; y) 2 
; x < 0g:
Denition (2.7) is to be understood in the pointwise sense (or a.e. if u
is not smooth enough.) Here, however, we shall be more interested in
the ux in the distributional sense. Therefore, we notie that for every
' 2 D( ) = C
1
0
( ) we have
hF
u
; 'i =
Z
 
u
x
' ds;(2.8)
where h ; i is the duality pairing between D( ) and its dual. By Green's
formula we have
hF
u
; 'i =
Z
 
u
x
' ds = a
1
(u ; ~')  
Z


1
f ~'d
(2.9)
for every ~' in H
1
(

1
) that has trae equal to ' on   and equal to zero
on the rest of 

1
.
In turn, the disrete ux F
u
h
will be dened as a linear mapping
ating from the spae

h
= (V
k
h
)
j 
(2.10)
into IR. More preisely, in agreement with (2.9) and with [12℄, [10℄, we
set, for every '
h
2 
h
:
hF
u
h
; '
h
i = a
1
(u
h
; ~'
h
)  
Z


1
f ~'
h
d
(2.11)
VARIATIONAL APPROXIMATION OF FLUX 5
where ~'
h
is any funtion in V
k
h
whose trae on   oinides with '
h
.
Denitions suh as (2.11) are fundamental to the development of loal
onservation laws; see [10℄.
For a given ' 2 D( ) we onsider now its interpolant '
I
2 
h
. Our
goal is to estimate the error in the ux; thus we onsider
hF
u
  F
u
h
; '
I
i:(2.12)
Our main result, to be proved in the next setion, is that there exists
a onstant C, independent of u, ', and h, suh that
jhF
u
  F
u
h
; '
I
ij  Ch
2k
juj
k+1;

k'k
k+1=2; 
:(2.13)
Remark. The estimate (2.13) is essentially an error estimate in the
spae H
 k 1=2
( ). As usual, we pay for the inrease in the order of on-
vergene by the weakness of the norm. On the other hand, it is known
in similar situations that estimates of this type (that is, with high or-
der in dual spaes) are the ruial ingredient for proving that suitable
postproessings of the disrete solution onverge, with the same high
order, in more reasonable spaes, for instane in L
2
( ). These postpro-
essors are typially onstruted through suitable loal averages that
are generally rather inexpensive to ompute. We refer, for instane, to
the lassial papers from the Cornell shool (see for instane [6℄, and
[13℄), and to the more reent approah of [8℄.
Remark. One may also onsider the possibility when 
 is separated
into the subdomains 

1
and 

2
by a general smooth urve   (instead of
a straight line as assumed here). However, for k > 1, this neessitates
the use of isoparametri elements, ompatible with  , in our deompo-
sition. Moreover, the presene of a urved interfae   would require the
use of an approximate urve  
h
for the denition of the approximate
ux. It is lear that the additional tehnial ompliations to deal with
this situation would be onsiderable.
3. The proof of the error estimate
3.1. The onstrution of  . For a given ' 2 D( ) we onstrut now
a suitable lifting  2 D(

1
) suh that
 vanishes on a strip around 

1
n  ;(3.1)

s
 
x
s
= ( 1)
s=2
'
(s)
on  ; for s even; 0  s  k;(3.2)
6 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE S

ULI

s
 
x
s
= 0 on  ; for s odd; 0  s  k;(3.3)
and there exists a onstant C, independent of ', suh that
jj jj
k+1;

1
 C jj'jj
k+1=2; 
:(3.4)
Here and below the '
(s)
denotes the derivative of ' of order s with
respet to the variable y along  . Notie that the boundary onditions
(3.2), (3.3) are ompatible with the existene of a ontinuous lifting
(that is, of one satisfying (3.4)). For instane, we an extend (by zero)
the funtion ' to the whole line x = 0, nd the lifting in the half-plane
fx < 0g aording to [11℄ (Chapter 1, Theorem 7.5) and then apply a
suitable ut-o.
We also point out expliitly that, thanks to our onstrution,
 2 H
k 1
0
(

1
):(3.5)
Indeed, for k = 2 we have
 =  
xx
+  
yy
=  
xx
+ '
(2)
= 0 on  :
If k = 3 we also have
( )
x
=  
xxx
+  
yyx
=  
xxx
+ ( 
x
)
yy
= 0 on  ;
and for k = 4 we an add
( )
xx
=  
xxxx
+  
yyxx
=  
xxxx
+ ( 
xx
)
yy
=  
xxxx
  '
(2)
yy
= 0 on  ;
and so on.
Remark. The onstrution of  is feasible in more general geome-
tries (and for more general operators) than the one onsidered here.
The relevant properties, as we shall see, are (3.4) and (3.5). However,
the onstrution would be tehnially muh more ompliated.
3.2. The onstrution of . We an now proeed to the onstrution
of the new auxiliary funtion . We set
p =   in 

1
; and p = 0 in 

2
;(3.6)
and we dene  as the solution of the following problem:

nd  2 H
1
0
(
) suh that
  = p in 
:
(3.7)
Now, from (3.5) we easily dedue that p 2 H
k 1
(
); moreover,
jjpjj
k 1;

 jjpjj
k 1;

1
 C jj jj
k+1;

1
:(3.8)
VARIATIONAL APPROXIMATION OF FLUX 7
We also reall that  in (3.1) vanishes in a strip near 

1
n . Hene
p has ompat support in 
, and, in partiular, it belongs to H
k 1
0
(
).
Therefore there exists a onstant (that we, again, all C), independent
of  , suh that
jjjj
k+1;

 C jjpjj
k 1;

:(3.9)
The regularity result (3.9) is well known, and its proof an be obtained
by the standard tehnique of reeting the problem in an odd way a
suitable number of times, and then using the internal regularity results
for the problem in the enlarged domain. The ruial point is that the
odd reetions of p (whih is the Laplaian of ) are still in H
k 1
of
the enlarged domain. This is true thanks to (3.5).
Using (3.9), (3.8), and (3.4) we then have immediately:
jjjj
k+1;

 C jj'jj
k+1=2; 
:(3.10)
Remark. As we an see, in order to have an auxiliary funtion 
satisfying (3.7) and (3.10) we ould just assume that 
 is suÆiently
smooth, provided that p (=   ) satises (3.5) and (3.8).
3.3. The error estimates. As stated in (2.13), we wish to estimate
the error in the nite element approximation of the ux:
hF
u
  F
u
h
; '
I
i:(3.11)
It is immediate to see that, taking as ~'
I
the interpolant  
I
of  (in
V
k
h
), and using (2.9) and (2.11) we get
hF
u
  F
u
h
; '
I
i = a
1
(u  u
h
; ~'
I
) = a
1
(u  u
h
;  
I
)
= a
1
(u  u
h
;  
I
   ) + a
1
(u  u
h
;  )
= I + II:
(3.12)
The estimate of I follows easily from (2.6), (2.5), usual interpolation
estimates, and (3.4):
I  C h
2k
juj
k+1;

j j
k+1;

1
 Ch
2k
juj
k+1;

k'k
k+1=2; 
:(3.13)
The estimate of II is also easy: using (2.6), integrating by parts, using
(3.6), (3.3) and then (3.7) we obtain rst
II =  
R


1
(u  u
h
) d
 =
R


1
(u  u
h
) p d

=  
R


(u  u
h
) d
 = a(u  u
h
; ):
(3.14)
Then, using (3.14), hoosing 
I
as the usual interpolant of  in V
k
h
,
and using Galerkin orthogonality, (2.5), interpolation estimates, and
(3.10), we obtain
8 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE S

ULI
II = a(u  u
h
; ) = a(u  u
h
;   
I
)
 Ch
2k
juj
k+1;

jj
k+1;

 Ch
2k
juj
k+1;

k'k
k+1=2; 
:
(3.15)
Now, from (3.12), (3.13), and (3.15) we easily onlude the proof of the
desired estimate (2.13).
Remark. In the partiular ase of k = 1 we see that the ruial
properties (3.5), (3.4), and hene (3.10) an easily be obtained under
muh more general assumptions. It is then lear that the extension of
our result to more general problems and geometries, for linear elements,
is trivial. The tehnial diÆulties would arise only for k > 1.
Referenes
[1℄ I. Babu

ska and A. Miller. The post proessing approah in the nite
element method, part 1: alulation of displaements, stresses and other
higher derivatives of the displaements. Internat. J. Numer. Methods. Engrg.,
34:1085{1109, 1984.
[2℄ I. Babu

ska and A. Miller. The post proessing approah in the nite
element method, part 2: the alulation of stress intensity fators. Internat. J.
Numer. Methods. Engrg., 34:1111{1129, 1984.
[3℄ I. Babu

ska and A. Miller. The post proessing approah in the nite
element method, part 3: a posteriori estimates and adaptive mesh seletion.
Internat. J. Numer. Methods. Engrg., 34:1131{1151, 1984.
[4℄ I. Babu

ska and M. Suri. The p and h-p versions of the nite element
method, basi priniples and properties. SIAM Review, 36(4):578{632, 1994.
[5℄ J. W. Barrett and C. M. Elliott. Total ux estimates for a nite element
approximation of ellipti equations. IMA J. Numer. Anal., 7:129{148, 1987.
[6℄ J.H. Bramble and A.H. Shatz. Higher order loal auray by averaging
in the nite element method, Mathematis of Computation. 31 (1977), 94-111.
[7℄ Ph.G. Ciarlet. The nite element methods for ellipti problems. North Hol-
land, Amsterdam, 1978.
[8℄ B. Cokburn, M. Luskin, C.-W. Shu and E. S

uli. Postproessing of the
disontinuous Galerkin nite element method. In: B. Cokburn, G. Karni-
adakis, and C.-W. Shu (Eds. Disontinuous Galerkin Finite Element Methods.
Leture Notes in Computational Siene and Engineering, Vol. 11. Springer{
Verlag, 2000; Extended version submitted to Mathematis of Computation.
(2000).
[9℄ M.B. Giles, M.G. Larson, M. Levenstam, and E. S

uli. Adaptive error
ontrol for nite element approximations of the lift and drag in a visous
ow. Tehnial Report NA-97/06. Oxford University Computing Laboratory,
Wolfson Building, Parks Road, Oxford OX1 3QD, 1997.
[10℄ T.J.R. Hughes, G. Engel, L. Mazzei, and M.G. Larson. The Con-
tinuous Galerkin Method is Loally Conservative. Journal of Computational
Physis. 163 (2000), 467-488.
VARIATIONAL APPROXIMATION OF FLUX 9
[11℄ J. L. Lions and E. Magenes. Problemes aux limites non homogenes et
appliations, Vol 1. Dunod, Paris, 1968.
[12℄ L. D. Marini and A. Quarteroni. A relaxation proedure for domain de-
omposition methods using nite elements. Numer. Math. 55 (1989), pp. 575-
598.
[13℄ L.B. Wahlbin. Superonvergene in Galerkin Finite Element Methods. Le-
ture Notes in Mathematis. 1605, Springer{Verlag, 1977.
[14℄ J. A. Wheeler. Simulation of heat transfer from a warm pipeline buried in
permafrost. Proeedings of the 74th National meeting of the Amerian Institute
of Chemial Engineering, 1973.
Frano Brezzi, Dipartimento di Matematia and I.A.N.-C.N.R., Via
Abbiategrasso 215, 27100 Pavia, Italy
E-mail address : brezzidragon.ian.pv.nr.it
Thomas J.R. Hughes, Division of Mehanis and Computation, Du-
rand Building, Stanford University, Stanford, California 94305, USA
E-mail address : hughesam-sun2.stanford.edu
Endre S

uli, University of Oxford,, Computing Laboratory, Numeri-
al Analysis Group, Wolfson Building, Parks Road, Oxford OX1 3QD,
England
E-mail address : Endre.Suliomlab.ox.a.uk


Variational approximation of flux in conforming finite element methods for elliptic partial differential equations: a model problem

We consider the approximation of elliptic boundary value problems by
conforming finite element methods. A model problem, the Poisson equation with
Dirichlet boundary conditions, is used to examine the convergence behavior of
flux defined on an internal boundary which splits the domain in two. A
variational definition of flux, designed to satisfy local conservation laws, is
shown to lead to improved rates of convergence

S?li, Endre

PUblication MAnagement

Variational approximation of flux in conforming finite element methods
for elliptic partial differential equations: a model problem.

Oxford University Research Archive

VARIATIONAL APPROXIMATION OF FLUX INCONFORMING FINITE ELEMENT METHODS FORELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS:A MODEL PROBLEMFRANCO BREZZI, T.J.R. HUGHES, AND ENDRE SULIAbstrat. We onsider the approximation of ellipti boundaryvalue problems by onforming nite element methods. A modelproblem, the Poisson equation with Dirihlet boundary onditions,is used to examine the onvergene behavior of ux dened on aninternal boundary whih splits the domain in two. A variationaldenition of ux, designed to satisfy loal onservation laws, isshown to lead to improved rates of onvergene.1. IntrodutionIn mathematial modeling of physial phenomena one frequently en-ounters instanes when the primary quantity of interest is not theanalytial solution to the underlying partial dierential equation buta linear funtional of the analytial solution to the equation; in suhases, solving the dierential equation onsidered is only an interme-diate stage in the proess of omputing the main quantity of onern.For example, in uid dynamis one may be interested in alulating thelift and drag oeÆients of a body immersed in a visous inompress-ible uid whose ow is governed by the Navier-Stokes equations. Thelift and drag oeÆients are dened as integrals, over the boundaryof the body, of the stress tensor omponents normal and tangential tothe ow, respetively. Similarly, in elastiity theory, the quantities ofprime interest, suh as the stress intensity fator, or the moments of ashell or a plate, are derived quantities.A further aspet of measurement problems of this kind is that, fre-quently, the funtional under onsideration may be expressed in var-ious forms whih are mutually equivalent at the ontinuous level butresult in very dierent approximations under disretization. Thus it1991 Mathematis Subjet Classiation. 65N30, 65N15, 65N50.Key words and phrases. Finite element methods, onservation, error estimates,ux funtionals. 12 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE SULIis important to selet the appropriate representation of the funtionalbefore formulating its disretization. This basi idea has been widelyexploited in strutural mehanis [1, 2, 3, 4℄ and heat ondution [14℄ topost-proess nite element approximations, and more reently also inthe eld of omputational uid dynamis in the ontext of a posteriorierror estimation for lift and drag omputations (f. [9℄).In this paper we shall be onerned with the nite element approxi-mation of one partiular funtional: the diusive ux over an interfae.Our aim is to show that the natural \variational denition" of the dis-rete ux an provide a high order of auray in suitably dened dualnorms. In doing so, we shall not aim at generality. On the ontrary,we will try to present the main idea on the simplest possible prob-lem, in order to stress what we believe to be the ruial points andinstruments, avoiding tehnialities as muh as possible. We believehowever that more general results are true, and the diÆulty in theirproof is mainly of a tehnial nature. The related problem of erroranalysis of the diusive ux approximation over the entire boundary ofthe omputational domain has been onsidered by Barrett and Elliottin [5℄.The paper is strutured as follows. The model problem is desribedin Setion 2, together with the denition of the proposed approximationof the ux, while Setion 3 is devoted to the proof of our error estimate.Throughout the paper we shall use the usual notation for Sobolevspaes and their norms and seminorms. See, for example, [7℄, [11℄.2. The model problem and the main result2.1. The geometry of the problem. Let  be a retangle in theplane with boundary . We shall suppose that  is split into twodisjoint open subdomains 1 and 2 by a straight line  . We do notassume   to be parallel to one of the edges of ; however, for the sakeof simpliity of the notation we shall require that   has equation x = 0.2.2. The model problem. For a given f , smooth enough, we onsiderthe following problem: nd u 2 H10 () suh that  u = f in :(2.1)VARIATIONAL APPROXIMATION OF FLUX 3PSfrag replaements  21Figure 1. The domain  and the internal boundary  .It is well known that (2.1) has a unique solution. We set V := H10 (),and for u and v in V we leta(u; v) := Zru  rv dx dy:(2.2)Hene, the variational formulation of (2.1) is nd u 2 V suh thata(u; v) = (f; v) 8v 2 V;(2.3)where, as usual, (; ) represents the inner produt in L2().2.3. The deomposition and the disrete problem. Let Th be adeomposition of  into triangles whih is ompatible with the splittingof  into 1 and 2 (this obviously means that eah triangle is a subsetof one of the two subdomains.) For k an integer  1 we onsider thespae V kh dened asV kh := fvh 2 C0() \H10 (); vhjT 2 Pk 8T 2 Thg;that is the usual nite element spae of ontinuous pieewise polynomi-als of degree k over the deomposition Th whih obey the homogeneousDirihlet boundary ondition on .In tandem with (2.3) onsider the following disrete problem: nd uh 2 V kh suh thata(uh; vh) = (f; vh) 8vh 2 V kh :(2.4)4 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE SULIIt is well known (see, for instane, [7℄) that (2.4) has a unique solutionuh, and that the following error estimates hold, whenever the analytialsolution u has the neessary regularity:jju  uhjjr;  C hk+1 rjujk+1;; r = 0; 1:(2.5)2.4. The disrete ux, and the statement of the main result.First of all, in analogy with (2.2) we introdueai(u ; v) := Zi ru  rv d(2.6)for i = 1; 2. We dene now the ontinuous ux from 1 into 2 throughthe interfae   as Fu := (ux )j (2.7)where we assumed that 1 is to the left of  , that is1 = f(x; y) 2 ; x < 0g:Denition (2.7) is to be understood in the pointwise sense (or a.e. if uis not smooth enough.) Here, however, we shall be more interested inthe ux in the distributional sense. Therefore, we notie that for every' 2 D( ) = C10 ( ) we havehFu; 'i = Z  ux ' ds;(2.8)where h ; i is the duality pairing between D( ) and its dual. By Green'sformula we havehFu; 'i = Z  ux ' ds = a1(u ; ~')   Z1 f ~'d(2.9)for every ~' in H1(1) that has trae equal to ' on   and equal to zeroon the rest of 1.In turn, the disrete ux Fuh will be dened as a linear mappingating from the spae h = (V kh )j (2.10)into IR. More preisely, in agreement with (2.9) and with [12℄, [10℄, weset, for every 'h 2 h :hFuh; 'hi = a1(uh ; ~'h)   Z1 f ~'h d(2.11)VARIATIONAL APPROXIMATION OF FLUX 5where ~'h is any funtion in V kh whose trae on   oinides with 'h.Denitions suh as (2.11) are fundamental to the development of loalonservation laws; see [10℄.For a given ' 2 D( ) we onsider now its interpolant 'I 2 h. Ourgoal is to estimate the error in the ux; thus we onsiderhFu   Fuh ; 'Ii:(2.12)Our main result, to be proved in the next setion, is that there existsa onstant C, independent of u, ', and h, suh thatjhFu   Fuh ; 'Iij  Ch2k jujk+1; k'kk+1=2; :(2.13)Remark. The estimate (2.13) is essentially an error estimate in thespae H k 1=2( ). As usual, we pay for the inrease in the order of on-vergene by the weakness of the norm. On the other hand, it is knownin similar situations that estimates of this type (that is, with high or-der in dual spaes) are the ruial ingredient for proving that suitablepostproessings of the disrete solution onverge, with the same highorder, in more reasonable spaes, for instane in L2( ). These postpro-essors are typially onstruted through suitable loal averages thatare generally rather inexpensive to ompute. We refer, for instane, tothe lassial papers from the Cornell shool (see for instane [6℄, and[13℄), and to the more reent approah of [8℄.Remark. One may also onsider the possibility when  is separatedinto the subdomains 1 and 2 by a general smooth urve   (instead ofa straight line as assumed here). However, for k > 1, this neessitatesthe use of isoparametri elements, ompatible with  , in our deompo-sition. Moreover, the presene of a urved interfae   would require theuse of an approximate urve  h for the denition of the approximateux. It is lear that the additional tehnial ompliations to deal withthis situation would be onsiderable.3. The proof of the error estimate3.1. The onstrution of  . For a given ' 2 D( ) we onstrut nowa suitable lifting  2 D(1) suh that vanishes on a strip around 1 n  ;(3.1) s xs = ( 1)s=2'(s) on  ; for s even; 0  s  k;(3.2)6 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE SULIs xs = 0 on  ; for s odd; 0  s  k;(3.3)and there exists a onstant C, independent of ', suh thatjj jjk+1;1  C jj'jjk+1=2; :(3.4)Here and below the '(s) denotes the derivative of ' of order s withrespet to the variable y along  . Notie that the boundary onditions(3.2), (3.3) are ompatible with the existene of a ontinuous lifting(that is, of one satisfying (3.4)). For instane, we an extend (by zero)the funtion ' to the whole line x = 0, nd the lifting in the half-planefx < 0g aording to [11℄ (Chapter 1, Theorem 7.5) and then apply asuitable ut-o.We also point out expliitly that, thanks to our onstrution, 2 Hk 10 (1):(3.5)Indeed, for k = 2 we have =  xx +  yy =  xx + '(2) = 0 on  :If k = 3 we also have( )x =  xxx +  yyx =  xxx + ( x)yy = 0 on  ;and for k = 4 we an add( )xx =  xxxx +  yyxx =  xxxx + ( xx)yy =  xxxx   '(2)yy = 0 on  ;and so on.Remark. The onstrution of  is feasible in more general geome-tries (and for more general operators) than the one onsidered here.The relevant properties, as we shall see, are (3.4) and (3.5). However,the onstrution would be tehnially muh more ompliated.3.2. The onstrution of . We an now proeed to the onstrutionof the new auxiliary funtion . We setp =   in 1; and p = 0 in 2;(3.6)and we dene  as the solution of the following problem: nd  2 H10 () suh that  = p in :(3.7)Now, from (3.5) we easily dedue that p 2 Hk 1(); moreover,jjpjjk 1;  jjpjjk 1;1  C jj jjk+1;1:(3.8)VARIATIONAL APPROXIMATION OF FLUX 7We also reall that  in (3.1) vanishes in a strip near 1 n . Henep has ompat support in , and, in partiular, it belongs to Hk 10 ().Therefore there exists a onstant (that we, again, all C), independentof  , suh that jjjjk+1;  C jjpjjk 1;:(3.9)The regularity result (3.9) is well known, and its proof an be obtainedby the standard tehnique of reeting the problem in an odd way asuitable number of times, and then using the internal regularity resultsfor the problem in the enlarged domain. The ruial point is that theodd reetions of p (whih is the Laplaian of ) are still in Hk 1 ofthe enlarged domain. This is true thanks to (3.5).Using (3.9), (3.8), and (3.4) we then have immediately:jjjjk+1;  C jj'jjk+1=2; :(3.10)Remark. As we an see, in order to have an auxiliary funtion satisfying (3.7) and (3.10) we ould just assume that  is suÆientlysmooth, provided that p (=   ) satises (3.5) and (3.8).3.3. The error estimates. As stated in (2.13), we wish to estimatethe error in the nite element approximation of the ux:hFu   Fuh ; 'Ii:(3.11)It is immediate to see that, taking as ~'I the interpolant  I of  (inV kh ), and using (2.9) and (2.11) we gethFu   Fuh ; 'Ii = a1(u  uh; ~'I) = a1(u  uh;  I)= a1(u  uh;  I    ) + a1(u  uh;  )= I + II:(3.12)The estimate of I follows easily from (2.6), (2.5), usual interpolationestimates, and (3.4):I  C h2k jujk+1; j jk+1;1  Ch2k jujk+1; k'kk+1=2; :(3.13)The estimate of II is also easy: using (2.6), integrating by parts, using(3.6), (3.3) and then (3.7) we obtain rstII =   R1(u  uh) d = R1(u  uh) p d=   R(u  uh) d = a(u  uh; ):(3.14)Then, using (3.14), hoosing I as the usual interpolant of  in V kh ,and using Galerkin orthogonality, (2.5), interpolation estimates, and(3.10), we obtain8 FRANCO BREZZI, T.J.R. HUGHES, AND ENDRE SULIII = a(u  uh; ) = a(u  uh;   I) Ch2k jujk+1; jjk+1;  Ch2k jujk+1; k'kk+1=2; :(3.15)Now, from (3.12), (3.13), and (3.15) we easily onlude the proof of thedesired estimate (2.13).Remark. In the partiular ase of k = 1 we see that the ruialproperties (3.5), (3.4), and hene (3.10) an easily be obtained undermuh more general assumptions. It is then lear that the extension ofour result to more general problems and geometries, for linear elements,is trivial. The tehnial diÆulties would arise only for k > 1.Referenes[1℄ I. Babuska and A. Miller. The post proessing approah in the niteelement method, part 1: alulation of displaements, stresses and otherhigher derivatives of the displaements. Internat. J. Numer. Methods. Engrg.,34:1085{1109, 1984.[2℄ I. Babuska and A. Miller. The post proessing approah in the niteelement method, part 2: the alulation of stress intensity fators. Internat. J.Numer. Methods. Engrg., 34:1111{1129, 1984.[3℄ I. Babuska and A. Miller. The post proessing approah in the niteelement method, part 3: a posteriori estimates and adaptive mesh seletion.Internat. J. Numer. Methods. Engrg., 34:1131{1151, 1984.[4℄ I. Babuska and M. Suri. The p and h-p versions of the nite elementmethod, basi priniples and properties. SIAM Review, 36(4):578{632, 1994.[5℄ J. W. Barrett and C. M. Elliott. Total ux estimates for a nite elementapproximation of ellipti equations. IMA J. Numer. Anal., 7:129{148, 1987.[6℄ J.H. Bramble and A.H. Shatz. Higher order loal auray by averagingin the nite element method, Mathematis of Computation. 31 (1977), 94-111.[7℄ Ph.G. Ciarlet. The nite element methods for ellipti problems. North Hol-land, Amsterdam, 1978.[8℄ B. Cokburn, M. Luskin, C.-W. Shu and E. Suli. Postproessing of thedisontinuous Galerkin nite element method. In: B. Cokburn, G. Karni-adakis, and C.-W. Shu (Eds. Disontinuous Galerkin Finite Element Methods.Leture Notes in Computational Siene and Engineering, Vol. 11. Springer{Verlag, 2000; Extended version submitted to Mathematis of Computation.(2000).[9℄ M.B. Giles, M.G. Larson, M. Levenstam, and E. Suli. Adaptive errorontrol for nite element approximations of the lift and drag in a visousow. Tehnial Report NA-97/06. Oxford University Computing Laboratory,Wolfson Building, Parks Road, Oxford OX1 3QD, 1997.[10℄ T.J.R. Hughes, G. Engel, L. Mazzei, and M.G. Larson. The Con-tinuous Galerkin Method is Loally Conservative. Journal of ComputationalPhysis. 163 (2000), 467-488.VARIATIONAL APPROXIMATION OF FLUX 9[11℄ J. L. Lions and E. Magenes. Problemes aux limites non homogenes etappliations, Vol 1. Dunod, Paris, 1968.[12℄ L. D. Marini and A. Quarteroni. A relaxation proedure for domain de-omposition methods using nite elements. Numer. Math. 55 (1989), pp. 575-598.[13℄ L.B. Wahlbin. Superonvergene in Galerkin Finite Element Methods. Le-ture Notes in Mathematis. 1605, Springer{Verlag, 1977.[14℄ J. A. Wheeler. Simulation of heat transfer from a warm pipeline buried inpermafrost. Proeedings of the 74th National meeting of the Amerian Instituteof Chemial Engineering, 1973.Frano Brezzi, Dipartimento di Matematia and I.A.N.-C.N.R., ViaAbbiategrasso 215, 27100 Pavia, ItalyE-mail address : brezzidragon.ian.pv.nr.itThomas J.R. Hughes, Division of Mehanis and Computation, Du-rand Building, Stanford University, Stanford, California 94305, USAE-mail address : hughesam-sun2.stanford.eduEndre Suli, University of Oxford,, Computing Laboratory, Numeri-al Analysis Group, Wolfson Building, Parks Road, Oxford OX1 3QD,EnglandE-mail address : Endre.Suliomlab.ox.a.uk

Variational approximation of flux in conforming finite element methods for elliptic partial differential equations: a model problem

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