ETNA
Electronic Transactions on Numerical Analysis.
Volume 18, pp. 42-48, 2004.
Copyright  2004, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
DISCRETE SOBOLEV AND POINCARÉ INEQUALITIES FOR PIECEWISE
POLYNOMIAL FUNCTIONS
SUSANNE C. BRENNER
Abstract. Discrete Sobolev and Poincaré inequalities are derived for piecewise polynomial functions on two
dimensional domains. These inequalities can be applied to classical nonconforming finite element methods and
discontinuous Galerkin methods.
Key words. discrete Sobolev inequality, discrete Poincaré inequality, piecewise polynomial functions, nonconforming, discontinuous Galerkin.
AMS subject classifications. 65N30.
Let
be a bounded polygonal domain,
be a family of quasi-uniform simplicial
or quadrilateral triangulations of indexed by the mesh size . To streamline the presentation, we first introduce the following notation concerning :
the generic subdomain in
quadrilateral,
is denoted by
, which is either a triangle or a convex
is the set of the interior edges of ,
is the set of the vertices of ,
is the set of the two endpoints of the edge ,
is the set of the three vertices of the triangle ,
is the set of the three edges of the triangle ,
is the set of the edges in sharing the common endpoint ! ,
is the set of the triangles or quadrilaterals in that share the point #" in
their closures,
is the set of the two triangles or quadrilaterals in sharing the common edge
$ ,
&% % is the area of the subdomain ,
&% % is the length of the edge ,
'( is the orthogonal projection operator that maps ) * + onto the space of constant
functions on ,
the jump of a function , across an edge $ is denoted by - ,/. .
Note that even though a jump can be measured in two ways that differ by a minus sign,
this ambiguity does not affect the statements in this paper because the jumps always appear
in squared terms.
Let be a nonnegative integer. In the case of a simplicial triangulation , we define
0
1 3254 ,67) * 8 +:9;,/< 2 ,>== ? A@>B * C+EDFGA IHKJ
where @ B C+ is the space of polynomials of total degree L0 restricted to the triangle . In
*
the case of a quadrilateral triangulation , there is a bilinear homeomorphism M ? 9KNOQPR
S
(1)
Received September 16, 2003. Accepted for publication November 14, 2003. Recommended by Yuri Kuznetsov.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. E-mail: brenner@math.sc.edu. This research was supported in part by the National Science Foundation under Grant No.
DMS-03-11790
42
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43
Discrete Sobolev and Poincaré Inequalities
N 2 * J + * J +
, , >M ?
N
L 0
, B * C+
L0
M?
1 2 4 ,6A) * 8+ 9;, < 2 , == ? $B * C+EDFGA H
(2)
In order to state the discrete Poincaré inequality for piecewise polynomial functions we
define the mean value 4 , H + of a function ,6 1
at a point ! " by
*
(3)
, < * + J
4 4 , H * Q+ 2 % %
?
where % % is the number of triangles or quadrilaterals in .
The goal of this paper is to establish the following inequalities, where we use the standard
notation for Sobolev spaces [7, 4] and the positive constant is independent of :
(4)
, L * % %+
, ? % % '( - ,/. ?
B * C+
onto any given (convex) quadrilateral
. We
from the unit square
denote by the space of functions on such that is a polynomial on whose
degree in each variable is
. In other words, is a polynomial of individual
degree
in the curvilinear coordinates on induced by . We then define
!#"$
%
)(+*
'&
-,
, (5)
for all
L *
%
,6 1 ;
, L *
(6)
#"$
C41#"C$
D
% %+
)(+*
'&
?
0
% %+
)(+*
&
-,
% %+
, ?
%, % ?
?
%, % ?
32
546
$87
- ,. * + @? BA
3;<>=
% % ' ( - ,. E
32
$ &
$ &
/
.
./6
and
$ &
32
:9
.0/1
and
546
$-7
- ,/. * Q+ ?
" . The inequalities (4)–(7)
for a given point for all , &1 such that 4 4 , H * + 2
generalize the well-known discrete Sobolev and Poincaré inequalities [2, 14, 16] for finite
element functions in
* 8+ .
To avoid the proliferation of constants, we will use the notation
to represent the
inequality L * constant+
, where the constant is independent of . The statement
is equivalent to
and
.
We begin by establishing the discrete Sobolev inequality (4) for piecewise constant functions with respect to a simplicial triangulation , i.e., for the case where 0 2
in (1) and consists of triangles. Let be the space of piecewise constant functions with respect to and let * 8+ bebythe @ finite element space associated with . We define a linear
map 9 OP
(8)
J ! * + * + 254 4 H H * + D
, (7)
4 #"C$
L *
D
)(+*
&
.
9
/
$ &
?
32
3;<
A
=
F
GIHKJ
G
C!J
GMHNJ
GMLMJ
JOH%G
P
S
P Q
RF
P
P Q
S
T
:T
T
UP
5
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44
!
* Q+O
L EMMA 1. Let
T
=
Susanne C. Brenner
&A . The following estimate holds 9
% % - . D
be a vertex of the triangle
* + * + S
T
H
?
321
T
4
$
P
T
C
Proof. From (3) and (8) we have
* Q+O * + * + 2 % % S
T
* Q+O
T
T
6
=
* Q+ ?
T
which together with the Cauchy-Schwarz inequality implies that
* +;O * + * Q+ L % % %%
- . 2 S
T
=
T
T
T
=
E
326
H
2 % , * Q+ %
, % % , * + %, % , * + % % % %
,
4
$
L
3;
/
.
$
T
4
D ,C7@
D ,C7@
D ,C7@
DF$ $
3;
T
321
In view of the elementary facts that
* +;O * + . ?
H
3;
L
$ * + J
* + J
* + J
J
the estimates below follow immediately from Lemma 1:
(9)
O O % O S
T
(10)
(11)
S
T
S
T
T
/
.
H
H
T
T
541
546
D
$
T
4
D
$
32
P
T
P
T
P
J
T
D
$
32
$
32
54+"C$
% 3T
H
T
% % - . %% - . %% - . +"C$
?
J
C
The following lemma establishes (4) for the special case of piecewise constant functions
with respect to a family of quasi-uniform simplicial triangulations.
L EMMA 2. The following inequality holds:
%% D
*
- . Proof. From the discrete Sobolev inequality [2, 14, 16] for @ finite element functions in
* 8+ , we have
(13)
* % %+ (12)
T
!#"$
H
% %+
)(+*
'&
:9
T
541#"$ &
32
F
T
S
T
!#"$
H
'&
)(+*
Combining (9)–(11) and (13) we find that
T
54
!#"$
H
T
O S
T
!+"C$ &
S
T
!+"C$
S
T
.
/ #"C$ $ A
T
P
ETNA
Kent State University
etna@mcs.kent.edu
% %
%%
* %
H
H
32
32
H
- . * % %+ - . * % %+ %%
%+ - . E
)(+*
T
E
'&
45
Discrete Sobolev and Poincaré Inequalities
4
T
9
$ &
546
S
)(+*
'&
3T
C41#"C$ &
T
/
.
$ &
O S
T
3T
.
/
$ ?
=
T
/ +"C$
.
T
$ &
)(+*
'&
$ A
546
32
0 2
The next lemma shows that the result in Lemma 2 is also valid for quadrilateral triangulations, i.e., (4) is valid for the case where
in (2).
L EMMA 3. The inequality 12 holds for piecewise constant functions T with respect to
a quasi-uniform family of quadrilateral triangulations.
Proof. Let Q be the family of simplicial triangulations obtained from
by adding the
two diagonals of each quadrilateral in . Then Q is a quasi-uniform family of simplicial
triangulations. Let T be an arbitrary piecewise constant function with respect to . Since T
is also a piecewise constant function with respect to Q , Lemma 2 implies that
* +
(14)
where Q
T
!+"C$
*
H
% %+
(#*
'&
9
T
% % ' ( - . J
541+"C$ &
2
is the set of interior edges of .
E
T
546
$ A
Q
The inequality (12) follows from (14) and the observation that
- . 2
DF Q
T
* +
We can now establish the general case of (4).
T HEOREM 4. The inequality 4 holds for a quasi-uniform family of simplicial or quadrilateral triangulations.
be arbitrary and be the
orthogonal projection of into the
Proof. Let
space of piecewise constant functions, i.e.,
, 1
(,
)
( ,I+== ? 2 % % ,
?
*
,
DFG7 The following estimate [7, 4] is well-known:
,$O ( ,
(15)
541
?
$
* diam C+ % , % ?
.
H
/
D , !1 $
Moreover it follows from (15) and a standard inverse estimate [7, 4] that
(16)
% %
,$O
( ,
? , ? DF, !1
, since - ( ,. is a constant, the trace theorem (with scaling) and (15) imply
For #"C$
H
0
that
% % - ( , . O ' ( - ,. 2
L % % - ( ,. O - ,. ( ,3O ,
%
%
? E
(17)
E
$
H
0 < 54
$
% % ' ( * - ( ,/. O- ,/. + ? % ( 3, O , % ? ?
E
4
.0/
54
$
$
4
$ &
.
/
$
%, % ?
.
H
< /
$ ETNA
Kent State University
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46
Susanne C. Brenner
Combining Lemma 2, Lemma 3, and (15)–(17) we have
, #"$
H
?
H
&
*
'&
$ &
/
.
?
H
/
.
H
,$O ( ,
%, % ?
%, % ?
* %
% %+
$ &
)(+*
:9
#"$ &
(, * % %+
* % %+
%+ % %
, ?
)(+*
'&
'&
9
?
/
.
541#"$ &
541+"C$ &
C46
$ A
$ A
546
541
E
$ &
32
E
32
32
+"C$
%% ( (, - ,/. % % ' ( - ,. , - ( ,/. O '( - ,. % % '( - ,. 9
)(+*
)(+*
'&
$
$ A 546
32
* 8+
As in the case of finite element functions belonging to F , the discrete Poincaré
inequality (6) follows from the discrete Sobolev inequality (4).
T HEOREM 5. The inequality 6 holds for a quasi-uniform family of simplicial or quadrilateral triangulations.
Proof. Let
be arbitrary and
* +
, A1 , " 2 % % ,
,
"
be the mean of over . From the Poincaré-Friedrichs inequality for piecewise F
[3] we have
,$O , " (18)
,$O , " #"$
H
H
.
?
which together with (4) yields
%, % ?
4 #"C$
*
'&
% %+
/
)(+*
9
$ &
?
E
32
/
.
&
H
'&
32
.
9
/
$ &
E
4 #"$
H
32
$ A
4
=
4
$ A $
* % %+ ?
On the other hand, since 4 , H H + 2
* , we have, by (3),
% , " % 2 = 4 , O , " H * Q+= 2 % % == (20) , "
, * Q+O
=
= =
)(+*
4
$
(19)
,$O , " ?
% % ' ( - , O ,/" . %, % ?
% % ' ( - ,. % % ' ( - ,. functions
,"
?
== L $, O , "
=
#"$ * +
The estimate (6) now follows from (18)–(20) and the triangle inequality.
R EMARK 6. The inequality 6 clearly remains valid if we replace the condition
by the more general condition that
-
4 , H H * + 2
< , < * Q+ 2 J
< 2 .
where the nonnegative weights < satisfy
?
We now turn to the alternative forms (5) and (7) of the discrete Sobolev and Poincaré
?
inequalities.
K
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47
Discrete Sobolev and Poincaré Inequalities
* +
* +
T HEOREM 7. The inequalities 5 and 7 hold for a quasi-uniform family of simplicial
or quadrilateral triangulations.
Proof. In view of (4) and (6), it suffices to show that
% % ' ( - ,. %, % ?
- ,/. * + D , !1
?
Let ,6!1
be arbitrary and let - ,/. A@ + agree with - ,. at the endpoints of . By the
*
trace theorem (with scaling), we have
'( - ,. O - ,. (22) % %
, < O , < ? % , < O , < % ? J
%
%
? where , < is either the polynomial in @ * C+ that agrees with , at the vertices of the triangle or the curvilinear polynomial in
* C+ that agrees with , at the vertices of the quadrilateral
.
(21)
546
$
32
.0/
H
=
4
$ &
.
/
$
< =
H
$
3; <
? 4
32
?
0
$ &
It follows from standard interpolation error estimates and inverse estimates [7, 4] that
(23)
% % , < O , < ?
4
$ &
%, < O , < % ?
/
.
$
* diam C+ % , % ?
Furthermore a direct calculation yields
% % ' ( - ,. L % % - ,. (24)
54
$
E
4
.
H
546
$
$
- ,/. * Q+ H
?
3; <
=
%, % ?
.
H
/
$ The estimate (21) follows from (22)–(24) and the triangle inequality.
The discrete Sobolev inequality (4) and the discrete Poincaré inequality (6) for piecewise polynomial functions can be applied to many classical nonconforming finite element
functions [9, 11, 12, 13, 10, 6, 5] that satisfy the weak continuity condition
2 - ,. 2 % % * '( - ,/. +
(25)
DF For such functions the inequalities (4) and (6) simplify to
, (26)
!+"C$
L *
'&
L *
'&
D
% %+
, 4 +"C$
%, % ? J
?
and
(27)
, ?
(#*
D
% %+
(#*
/
.
/
.
?
$
$
respectively.
On the other hand, it follows from the alternative inequalities (5) and (7) that (26) and
(27) are also valid for nonconforming finite element functions [17, 19] that do not satisfy the
weak continuity condition (25) but are continuous at the vertices of .
The inequalities (4) and (6) can also be applied to discontinuous Galerkin methods. Indeed, by dropping the orthogonal projection operator
in (4) and (6) we immediately
arrive at the inequalities
' ( (28)
, !#"$
L *
%
'&
% %+
)(+*
:9
?
0
, ?
.
/
% % - ,. $ &
32
54
$ A
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48
Susanne C. Brenner
and
%, % ?
% % - ,. ?
The sums in (28) and (29) involving the jumps of , now appear naturally in many discontin
(29)
, 541#"$
L *
%
'&
% %+
)(+*
./6
9
$ &
E
32
$ A
546
uous Galerkin methods [8, 1].
R EMARK 8. The discrete Sobolev and Poincaré inequalities for finite element functions
in F are useful for example in the analysis of the stability of finite element methods
for parabolic problems [16] and the analysis of various nonoverlapping domain decomposition methods [2, 15, 18, 4]. The inequalities 26 – 29 enable similar analyses to be carried
out for classical nonconforming finite element methods and discontinuous Galerkin methods.
* 8+
* + * +
)
REFERENCES
[1] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini, Unified analysis of discontinuous Galerkin methods
for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), no. 5, pp. 1749–1779.
[2] J.H. Bramble, J.E. Pasciak, and A.H. Schatz, The construction of preconditioners for elliptic problems by
substructuring, I, Math. Comp., 47 (1986), pp. 103–134.
functions, SIAM J. Numer. Anal., 41 (2003),
[3] S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise
pp. 306–324.
[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods Second Edition ,
Springer-Verlag, New York-Berlin-Heidelberg, 2002.
[5] Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen, and X. Ye, Nonconforming quadrilateral finite elements: a
correction, Calcolo, 37 (2000), pp. 253–254.
[6] Z. Cai, J. Douglas, Jr., and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations, Calcolo, 36 (1999), pp. 215–232.
[7] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[8] B. Cockburn, G.E. Karniadakis, and C.-W. Shu, eds., Discontinuous Galerkin Methods, Springer-Verlag,
Berlin-Heidelberg, 2000.
[9] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, RAIRO Anal. Numér., 7 (1973), pp. 33–75.
[10] J. Douglas, Jr., J.E. Santos, D. Sheen, and X. Ye, Nonconforming Galerkin methods based on quadrilateral
elements for second order elliptic problems, M AN Math. Model. Numer. Anal., 33 (1999), no. 4, pp.
747–770.
[11] M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on triangles, Internat. J.
Numer. Methods Engrg., 19 (1983), no. 4, pp. 505–520.
[12] H. Han, A finite element approximation of Navier-Stokes equations using nonconforming elements, J. Comput.
Math., 2 (1984), 77–88.
[13] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial
Differential Equations, 8 (1992), no. 2, pp. 97–111.
[14] A.H. Schatz, V. Thomée, and W.L. Wendland, Mathematical Theory of Finite and Boundary Element Methods,
Birkhäuser, Basel-Boston-Berlin, 1990.
[15] B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition, Cambridge University Press, Cambridge, 1996.
[16] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin-HeidelbergNew York, 1997.
[17] E.L. Wilson, R.L. Taylor, W. Doherty, and J. Ghaboussi, Incompatible displacement models, In S.J. Fenves,
N. Perrone, A.R. Robinson, and W.C. Schnobrich, eds., Numerical and Computer Methods in Structural
Mechanics, pp. 43–57. Academic Press, New York, 1973.
[18] J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Review, 40 (1998), pp.
857–914.
[19] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J.
Numer. Anal., 34 (1997), 640–663.