ETNA
Electronic Transactions on Numerical Analysis.
Volume 32, pp. 33-48, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ANALYSIS OF THE DGFEM FOR NONLINEAR CONVECTION-DIFFUSION
PROBLEMS
MILOSLAV FEISTAUER AND VÁCLAV KUČERA
Abstract. The paper is concerned with the analysis of error estimates of the discontinuous Galerkin finite
element method (DGFEM) for the numerical solution of nonstationary nonlinear convection-diffusion problems
equipped with Dirichlet boundary conditions. First, the case of nonlinear diffusion as well as nonlinear convection
is considered. Then, the optimal -error estimate is discussed in the case of nonlinear convection and linear
diffusion.
Key words. nonlinear nonstationary convection-diffusion problems, nonlinear convection, nonlinear diffusion,
discontinuous Galerkin finite element method, NIPG, SIPG and IIPG versions, optimal error estimates
AMS subject classifications. 65M60, 76M10
1. Introduction. In a number of complex problems from science and technology, it is
necessary to approximate nonlinear singularly perturbed systems in domains with a complex
geometry, whose solutions contain internal or boundary layers. A possible numerical method
for an efficient solution of such problems is the discontinuous Galerkin finite element method
(DGFEM). This technique uses piecewise polynomial approximations of the sought solution
on a finite element mesh without any requirement on the continuity between neighbouring
elements. It allows the construction of higher order schemes in a natural way and it is suitable for the approximation of discontinuous solutions of conservation laws, or solutions of
singularly perturbed convection-diffusion problems having steep gradients. This method uses
advantages of the finite element method and finite volume schemes with an approximate Riemann solver and can be applied to unstructured grids, of the kind which are generated for
most complex geometries.
The original DGFEM was introduced in [26] for the solution of a neutron transport linear equation and analyzed theoretically in [24] and later in [23]. Nearly simultaneously the
DGFE techniques were developed for the numerical solution of elliptic problems in [33] and
for space semidiscretization of parabolic problems in [1] and [16] , using the interior penalty
Galerkin methods. In these papers the symmetric approximation of the diffusion terms is
used, and therefore the method is called SIPG (symmetric interior penalty Galerkin). Quite
popular is the NIPG (nonsymmetric interior penalty Galerkin) method, which was first introduced in [28]. Theoretical analyses of various types of the DGFE method applied to elliptic
problems can be found, e.g., in [1], [2], [3], [21], and [29].
The DGFEM has been used in a number of applications. Let us mention conservation
laws in [7], [14], [22], compressible flow in [4], [5], [8], [10], [12], [18], [20], [32], and
convection-diffusion problems in time-dependent domains in [31]. A survey of DGFE methods, techniques, and some applications can be found in [6].
In the discretization of nonstationary problems, one often uses the space semidiscretization, also called the method of lines. In [9], [11], and [15] the problem with a nonlinear convection and linear diffusion was analyzed. The work in [27] is concerned with the DGFEM
applied to the solution of a parabolic problem with a nonlinear diffusion, equipped with the
Neumann boundary condition prescribed on the whole boundary.
Received November 29, 2007. Accepted for publication June 3, 2008. Published online on November 18, 2008.
Recommended
by B. Heinrich.
Charles University Prague, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic
(feist@karlin.mff.cuni.cz, vaclav.kucera@email.cz).
33
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34
M. FEISTAUER AND V. KUČERA
In this paper, we shall analyze the DGFEM applied to nonstationary convection-diffusion
problems with nonlinear convection as well as diffusion and Dirichlet conditions on nonconforming meshes. Further, we discuss here the optimal -error estimates.
Let us note that we consider a problem in two space dimensions for simplicity only. All
results presented here can be extended to more space dimensions.
2. Formulations of the problem. First we shall introduce the formulations of the continuous and discrete problems.
2.1. Continuous problem. Let be a bounded polygonal domain with a Lipschitzcontinuous boundary and "! . We shall deal with the following initial-boundary value
problem: find #%$'&)(+*,,-./!1023546 , such that
7# : ; A < B#
*D1EGFIHJB#LK#M :ON 0PEQR& ( 0
<
98
9
C
>< =@?
(2.2)
#TS UVXW9YGZ\[ (9] *#^30
(2.3)
#X/C_02!`5*a# Z /C90bCdcfe
N
Z $fj4g be given functions, and
Let $3& ( 4g , # ^ $3h-,/!i0L3%4g and #
?
let A < clk , md*jn'0io , be prescribed Lipschitz-continuous fluxes. Without the loss of
generality we suppose that A < /!`p*q!i0rms*hnt0io . We assume that the function J satisfies the
(2.1)
conditions
(2.4)
(2.5)
JO$'4vu J Z 0LJ r? w 0x!zy{J Z y|J ? y}0
S J/# ? T~JB# €‚S "fS # ? ~# Sƒ0…„# ? 02# c%†e
This problem represents a simplified model of heat and mass transfer problems, where
nonlinear convection as well as diffusion appear. As an example we can mention the system
of compressible Navier-Stokes equations; see, e.g., [17].
The application of techniques from [30] allows to prove the existence and uniqueness of
a weak solution of the above problem.
2.2. Discretization. Let ‡iˆ be a partition of the closure into a finite number of closed
triangles, whose interiors are mutually disjoint.
Lebesgue measure),
7‘ For any ‰ cŠ‡ ˆ , we set S ‰"S*Œ‹z€ Žt /‰ (two
‘f–'—dimensional
˜t7‘
’
*
1
D
E
t
Ž
‹
“
‰
, the diameter of ‰ , and
’
*
I
‹
•
Ž
”
. We consider an index set
™
™
Ršp›O*œž–t!1¤ 0\n'0ro10\e\e€e Ÿ , such that all elements of ‡ ˆ are numbered by indices from , i.e.,
‡ ˆ * œ‰¢¡£Ÿ¡ . If two elements ‰¢¡202‰¦¥sc+‡ ˆ share a common face, which by definition has
to be a linear segment, we call them neighbours and set §T¡ƒ¥%*¨‰©¡«ª“9‰z¥ and ¬9/§@¡­¥•*
™
™i´
‹z€Ž' ? §_¡ƒ¥a* length of §@¡­¥ . For ®¯c we define m`B®°¯*±œ³²c
‰z¥ is a neighbour of ‰¢¡°Ÿ .
The boundary is formed by a finite number of faces of elements ‰µ¡ adjacent to . We
™€¸
denote all these
boundary faces by ¶¥ , where ²·c
vš¯¹±*bœ'~¯nt0€~ºo»0€e\e\e Ÿ and set
¼ B®°*½œ¾²¿c ™ ¸ ´ ¶ ¥ is a face of ‰ ¡ Ÿ'0r§ ¡ƒ¥ *¨¶ ¥ for ‰ ¡ c‡1ˆ , such that ¶ ¥ ·‰ ¡ 0À²{c ™ ¸ .
If ‰ ¡ is not adjacent to , we set ¼ B®°5*lÁ . Furthermore we set ¶B®°5*m`/®> ¼ /®° . We can
see that
m'/®°«ª ¼ B®°5*lÁ»0b‰©¡_*
¥
'– à Ä
Y ¡G]
§@¡­¥t0x‰©¡1ªµ*
¥
Ɩ à Å
Y ¡]
§_¡ƒ¥te
By Ç¡­¥ we denote the unit outer normal to 9‰È¡ on the face §@¡­¥ . In our case, Çp¡­¥ is constant
along §_¡ƒ¥ .
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DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
35
Over ‡1ˆ we define the broken Sobolev space
ɓÊ
´ ‘
É+Ê
f0r‡1ˆ'*Rœ€Ë Ë S c
/‰_„‰jc%‡1ˆ»Ÿ`0
equipped with the seminorm
For Ë©c
É ?
2? Ñ
;
—˜
‘ Ï
S ËS Ì†Í YGV[ ] *½Î t– ¤ S ËS Ì Í Y 5
e
£] Ð
¡
Àf02‡1ˆ‚ we set
σÓ
‘ Ï
ËS Ò * trace of ËS on § ¡­¥ 0
Ô ÏƒÓ
σÓ
Ó/Ï
n
Ë‚Õ Ò *
BËS Ò : ËS Ò , average of traces of Ë on §@¡ƒ¥t0
o σÓ
σÓ
ÓÏ
u Ë w Ò *aËS Ò ~ËS Ò , jump of traces of Ë on § ¡ƒ¥ e
Finally, we define the space of discontinuous piecewise polynomial functions
?
´ ‘
¶«ˆ*,¶@Ö [ ¹ f0r‡1ˆ'×* œ€Ë ËS c%Ø Ö / ‰_„‰jc‡1ˆ1Ÿ`0
where Ø /‰ is the space of all polynomials on ‰ of degree Ù .
Ö
In order to approximate the diffusion terms, we introduce the following forms defined
É
tf0r‡1ˆ' :
for #X02Ú¿c
Û ˆ /#@0rÚ* ;
–t¤†Ü ‘5Ï J/#2K#¢Ý€K¦Ú¦¬tC
¡
;
;
;
;
Ô
Ô
~ –t¤ –
ÏßÓ J/#2Ks#ÕT݀Çࡃ¥'u Ú w ¬`¶~¿á –t¤ –
ÏßÓ J/#2KÚÕ5ÝÇp¡­¥'u # w ¬‚¶
Ü
Ü
¡ ¥ < Y ¡] Ò
¡ ¥ < Y ¡G] Ò
¥Þ¡
¥Þ9¡
;
;
;
;
~ –t¤ –ÆÅ Ü ÏƒÓ JB#LK#ÈÝÇ¡­¥Ú¦¬‚¶+~¿á –t¤ –ÆÅ Ü ÏƒÓ KJB#°ÚÝÇ¡ƒ¥€#3¬‚¶
¡ ¥ Y ¡G] Ò
¡ ¥ Y ¡G] Ò
â
and
;
;
ˆ B#X02Ú\B8L* Ü N B8L°Ú¦¬'C©~Oá t– ¤ –ÆÅ Ü ƒÏ Ó J/#2K¦Ú+݀Çࡃ¥€#^ãB8L»¬‚¶
V
¡ ¥ Y ¡] Ò
;
: ;
–t¤ –ÆÅ Ü ƒÏ Ó«ä # ^ B8L° Ú¦¬‚¶Te
¡ ¥ Y ¡G] Ò
Further, we define the interior and boundary penalty
å
ˆi/#@0rÚT5*
;
;
;
: ;
–t¤ – < Ü Ò ÏßÓ ä u # w u Ú w ¬‚¶
–t¤ –ÆÅ Ü Ò ÏßÓ ä #Ú¦¬`¶Te
¡ ¥ Y ¡]
¡ ¥ Y ¡]
¥Þ¡
The weight ä is defined by
ä SÒ
where k
æ
! is a suitable constant.
σÓ
*lkæd畬9/§ ¡­¥ 0
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36
M. FEISTAUER AND V. KUČERA
Taking áv*ènt0¯! and ~¯n , we obtain the symmetric (SIPG), incomplete (IIPG) and
nonsymmetric (NIPG) variants of the approximation of the diffusion terms.
Finally, we define the convective form
é
; Ú
Ï
‘
t– ¤ Ü °< =@? A < B # 9C < ¬'C
¡
;
;
σÓ
/Ó Ï
ƒÏ Ó
É
:
–t¤ –'Ä Ü Ò ÏƒÓ / #TS Ò 02#TS Ò L0 Ç ¡­¥ °ÚS Ò `¬ ¶e
¡ ¥ Y ¡G]
ˆ /#@0rÚ5*q~
É
;
Here is the so-called numerical flux with properties specified later.
Now we can introduce the discrete problem (space semidiscretization with continuous
time, also called the method of lines).
D EFINITION 2.1. We say that #«ˆ is a DGFE solution of the convection-diffusion problem (2.1)–(2.3), if
?
#ˆ¦cdk 2u !i0L
¬
b)
B# ˆ â B8L¾0rÚ ˆ ¬'8
* ˆ B# ˆ 0rÚ
Z
c) #ˆ7/!'5*# ˆ 0
a)
(2.6)
Z
w ´ ¶_ˆt¾0
: é ˆ B# ˆ /8L¾0rÚ ˆ : J Z å ˆ B# ˆ B8L02Ú ˆ : Û ˆ B# ˆ B8L¾0rÚ ˆ ˆ \B8L¾0…„Ú ˆ c¶ ˆ 09„8×c./!i0L30
Z
where # ˆ is an ¶ ˆ approximation of the initial condition # and J Z ê! is a constant from
assumption (2.4).
The discrete problem (2.6) is equivalent to a large system of ordinary differential equations. If we apply a suitable ODE solver, we get a fully discrete problem. However, this
subject lies outside the framework of this paper. Here we shall be concerned with the analysis
of the semidiscrete problem (2.6).
3. Error analysis.
É
3.1. Some assumptions. We assume that the numerical flux has the following properties: É
´
(H1) B#X0LË90LÇp is defined in -ãë ? , where ë ? *RœÇ,cTì S Ç)St*ênƟ , and is Lipschitzcontinuous with respect to #@0»Ë :
É
É
É
S B #X0LË90LÇT~
B # 02Ë L0 Çp\S1"k†í2S #¢~# S : S Ës~Ë S 0
#X0»Ë70i# 01Ë cµ†0àÇ,c%ë ? e
B#X0LË90LÇp is consistent:
(H2)
É
(H3)
É
B#X0L#X0LÇp5*
; ì
A < B#‚î < 0b#%c%†0×Ça*ê/î ? 0\e€e\e€0Lî ì àcµë ? e
<>=X?
B#X0LË90LÇp is conservative:
É
É
/#@02Ë702Çp5*q~ /Ë702#@0€~Çà0b#@0»ËÈcµ†0àÇ,cë ? e
We shall assume that the weak solution # of problem (2.1)–(2.3) is regular, namely
(3.1)
?
9#
´2É
c% H /!i0L3 Ö › M 0
98
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37
DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
´rÉ
?
where Ùdï n denotes the given degree of approximation. Then #“c%k+Hru !10L w
Ö › rM and
# satisfies (2.1)–(2.3) pointwise. It is possible to show that the regular solution satisfies the
identity
(3.2)
é
å
¬
â
B#TB8L02ڈ` : ˆi/#X/8L¾02ڈ` : J Z iˆ /#X/8L¾02ڈ‚ : Û ˆ1B#TB8L02ڈ‚
¬'8
* ˆ BX
# 02Ú ˆ \B8L0…„Ú ˆ %
c ¶ ˆ 0 for a. a. 8†cO/!i0L3e
To treat the nonlinear diffusion terms, we need one more regularity assumption on the
solution # of the continuous problem:
ð\K#X/8L\ð í7ñ YGV ] k†ò for a. a. 8†c./!i0L3e

–
Let us consider a system œž‡iˆ‚Ÿ ˆ YGZ\[ ˆó ] , Z ô! , of partitions of the domain with the
following properties:
–
(A1) The system œž‡1ˆ1Ÿ ˆ YßZ€[ ˆó ] is regular: there exists a constant k ( {! , such that
7‘


õ ‘ "k†(×0…„‰jc‡ ˆ 0…„ c.!10 Z e
(3.3)
(A2) There exists a constant kà^
{! , such that
i‘5Ï


™
ak×^¬9/§@¡­¥ž0…„9®5c 0…„»²zcd¶/®>0è„ O
c /!i0 Z ¾e
3.2. Some auxiliary results. First we state some results necessary for our analysis; see,
e.g., [9].
L EMMA 3.1 (Multiplicative trace inequality). There exists a constant kpö÷{! indepen
dent of 02‰+0LË , such that
(3.4)
 ?
‘
‘
ð¾Ëð í1ø YùU ‘ ] ak†ö"H³ð¾Ëð í ø Y ] S ËS Ìàú Y ] : ‘ ¹ ð¾Ëð í1ø Y ‘ ] MÆ0


É ?
„‰jc‡1ˆi0Ë¢c
/ ‰¾0 cO/i! 0 Z ¾ e
L EMMA 3.2 (Inverse inequality). There exists a constant k
such that
(3.5)
¤

! independent of 20 ‰+0LË ,
ž¤  ?


‘
‘
S ËS Ì ú Y ] ak ‘ ¹ ðËð í ø Y ] 0…„‰jcµ‡ ˆ 0Ë¢cØ Ö ‰“¾0 c!10 Z e
Now, for Ë©c%×Æ we denote by û
ˆ Ë the ' -projection of Ë on ¶ ˆ :
û3ˆtË©cd¶«ˆ10üûfˆÆË~“Ë90iڈ'5*,!
„ãÚTˆIc¶«ˆie
‘
‘
Obviously, if ‰ýc¿‡iˆ , then the function ûãˆÆËS is the ×t‰“ -projection of ËS on Ø ‰“ .
Ö
Let þ©cOuGnt0/Ù w be an integer. It is possible to show, cf., e.g., [19, Lemma 4.1], that the operator
û3ˆ has the following properties.

L EMMA 3.3. There exists a constant kÿ¿! independent of 0r‰d0LË , such that
9Ê ?
‘
‘
ð\û3ˆtË)~“Ëð í ø Y ] k†ÿ ‘ › S ËS Ì Í ‚ú Y ] 0
(3.6)
 Ê
‘
‘
S û3ˆtËs~ËS Ìú Y ] ak×ÿ ‘ S ËS Ì Í ‚ú Y ] 0
(3.7)
 Ê ?
‘
‘
S û3ˆtËs~ËS Ì ø Y ] ak×ÿ ‘ ¹ S ËS Ì†Í `ú Y ] 0
(3.8)


É Ê ?
for all Ë©c
› /‰ , ‰ c ˆ and c.!10 Z , where þÈc.ußn'0/Ù w is an integer.
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38
M. FEISTAUER AND V. KUČERA

In what follows, we shall denote by k a generic constant independent of 02‰ , and/or
other quantities attaining different values
É
?in different places.
We set /8L*û ˆ #X/8L1~©#TB8L×c
Ö › Àf02‡ ˆ for a. a. 8×cO!10Lf . Then (3.4)–(3.8) yield
the following estimates.

3.4. There exists a constant kŒj! independent of 0r‰ , such that for all
 L EMMA

c./!i0 Z  ?
—€˜
Ž`ð«ð í1ø YGV«[ ] "k Ö › S #TS Ì`ú YGV ] 0

—€˜
†S S Ìú YGV«[ ] "k Ö S #S Ì ‚ú YGV ] 0
 ?
—˜
×S «S Ì ø YV[ ] k Ö ¹ S ËS Ì
‚ú YV ] 0
 ? 9# Di k Ö › 0
78 78 Ì ‚ú YV ]

å
ž ˆ7 0 »"k Ö7S #TS Ì
`ú YGV ] e
3.3. Properties of the convective form. We use the following notation:
?2Ñ
å
:
€
—
˜
ˆ s03 Ð
ðð\^
¿*ŠÎ
S S Ìàú YGV«[ ]
e
o
É ?
It is possible to show that ð†Ý`ð€^
is a norm in
f0r‡ ˆ .
é
Now, we shall be concerned with the properties of the form ˆ . Under assumptions
é
(H1)–(H3) and (A1)–(A2) the convective form ˆ is Lipschitz-continuous in the following
sense.


É ?
L EMMA 3.5. Let #@
0 #_ 0Lː c
f02‡ ˆ and cô/!i0 Z . Then there exists a constant
kô"! independent of #X0 #@ 0LË90 , such that
?LÑ
é
é
å
—€˜
S ˆi/#@02˂T~ ˆi#X 0LË»\S»k
ˆi/Ë702˂ : S ËS Ìú YGV«[ ] ?2Ñ
;
i‘Ï
‘5Ï e
- Î ð¾#I
~ # ð í ø YV ] :
¾
ð
I
#
~
T
#
ð
í YU ]
Ð
–t¤
¡
n
From this and results from Section 3.2 we obtain
L EMMA 3.6. Let # be the solution of the continuous problem, #_ˆ the solution of the
discrete problem
 and let
 * #«ˆµ~lûfˆÆ# c±¶«ˆ . Then there exists a constant kP ! ,
independent of c!10 Z , such that
é
é
 ?
S ˆ B#X0tT~ ˆ /# ˆ 0t€S»kIðið^ H Ö › S #TS Ì`ú YGV ] : ð ið í ø YGV ] M e
â is the coercivity of the discrete problem (2.6).
3.4. Coercivity. An important question
L EMMA 3.7 (Nonsymmetric case). Let kpæ q! . Then for the nonsymmetric diffusion
â
â
form Û ˆ and nonsymmetric right hand side ˆ (i.e., á *q~¯n ), we have
Û ˆ B# ˆ 0tX~ Û ˆ B#X0tT~
where
Rï"J
(3.9)
Z S iS Ìàú YGV«[ —€˜ ] 0
ˆ B# ˆ 0t :
ˆ B#X0t5* : ë©0

 ?
S ëÈS‚ak Î oÆJ ? 
~ J Z Ö S #S Ì ‚ú GY V ] : Ö › S #TS Ì `ú YGV ] : ð ið í ø YGV ] Ð
?2Ñ
å
—€˜
- H S iS Ìàú YGV«[ ] : ˆ »0t M e
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DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
39
L EMMA 3.8 (Symmetric case). Let
k æ
ï Î
¤
¤
J ? kàö°n : k 0
ZJ Ð
â
where k ö and k are the constants from Lemmas 3.1 and 3.2, respectively. Then for the
symmetric diffusion form Û ˆ and symmetric right hand side ˆ (i.e., á,*ên ), we have
â
â
ˆ /# ˆ 0t :
Û ˆ B# ˆ 0tX~ Û ˆ B#X0tT~
ˆ B#X0t5* : ë¢0
where
å
J Z
H S iS Ìú YGV[ —˜ ] ~ ˆ »0t M 0
o

 ?
S ëµS‚ak.ΫoÆJ ? ~J Z Ö S #S Ì ‚ú YGV ] : Ö › S #TS Ì `ú YGV ] : ð ið í ø YGV ] Ð
?LÑ
å
—˜
- H S iS Ì ú YGV[ ] : ˆ »0t M e
Rï
L EMMA 3.9 (Incomplete case). Let
k æ
ï"o Î
¤
J ? kàö°n : k 0
J Z Ð
â
¤
where k ö and k are the constants from Lemmas 3.1 and 3.2, respectively. Then for the
incomplete diffusion form Û ˆ and incomplete right hand side ˆ , we have
â
Û ˆ B# ˆ 0tX~ Û ˆ B#X0tT~
â
ˆ /# ˆ 0t :
ˆ B#X0t5* : ë¢0
where
å
J Z
H S iS Ìú YGV[ —˜ ] ~ ˆ »0t M 0
o

 ?
S ëµS‚ak.ΫoÆJ ? ~J Z Ö S #S Ì ‚ú YGV ] : Ö › S #TS Ì `ú YGV ] : ð ið í ø YGV ] Ð
?LÑ
å
—˜
- H S iS Ì ú YGV[ ] : ˆ »0t M e
Rï
The proofs of Lemmas 3.7–3.9 are rather technical. Therefore, we shall give here only a
sketch of the proof of Lemma 3.7.
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â ČERA
M. FEISTAUER AND V. KU
40
Proof of Lemma 3.7. We break down Û
?
ä B #X0t5*
ä ä"!
ä%$
ä"&
;
ˆ7B#X0t_~
ˆi/#@0t into individual terms by setting
J B#LK#¢ÝK ¬'C_0
–t¤ Ü ‘ Ï ;
;
Ô
B#X0t5*ê~ –t¤ –
ÏƒÓ JB#LK#ÕTÝ\Ç¡ƒ¥'u w ¬`¶T0
Ü
Ò
¡ ¥ < Y ¡]
¥Þ9¡
;
;
Ô
B#X0t5* –t¤ –
K tÕ5Ý\Ç¡­¥'u # w ¬‚¶T0
ÏƒÓ JB#L#
Ü
¡ ¥ < Y ¡G] Ò
¥Þ9¡
;
;
B#X0 t5*ê~ –t¤ –ÆÅ Ü ÏßÓ J×B#LK#©Ý\Ç ¡ƒ¥ ¬`¶0
¡ ¥ Y ¡] Ò
;
;
B#X0 t5* –t¤ –ÆÅ Ü ÏßÓ JB#L#
K ¯ÝÇ¡ƒ¥€#3¬‚¶d~
¡ ¥ Y ¡] Ò
;
;
~ –t¤ –ÆÅ Ü ÏƒÓ J/#2#
K ¯ÝÇ¡­¥€#^µ¬`¶Te
¡ ¥ Y ¡] Ò
¡
â
Therefore
Û ˆ /# ˆ 0tX~ Û ˆ /#@0tT~
â
ˆ B# ˆ 0t :
ˆ /#@0t5*
;
&
¡ =@?
ä ¡ /# ˆ 0tT~ ä ¡ B# [ t2
and we shall treat these terms separately:
1) First term:
;
?
?
ä B # ˆ 0tT~ ä B#X0t5* t– ¤ Ü ‘ Ï H J×B# ˆ 2K# ˆ
¡
;
* –t¤ Ü ‘ Ï Î H J×B# ˆ 2Ks# ˆ ~J/# ˆ LK¦û ˆ # M :
¡
: H J/#2K¦û ˆ #©~JB#LK# M ݀K#5¬'CÈ*
ä
Ð
~J/#2Ks# M ÝK ¬'C
H J/# ˆ LK¦û ˆ #¢~“J/#2K¦û ˆ # M
?
?
?
? : ä : ä
!
and, using (2.4), we estimate
? ;
—€˜
ä ? * t– ¤ Ü ‘5Ï JB# ˆ LK#ã݀K#¬tCdï"J Z S iS Ìú YGV«[ ] e
¡
Further, by (2.4), (2.5), the Cauchy inequality, (3.3) and Lemma 3.4, we get
?
S ä tS *
;
: H J/#ˆ`T~JB# M K# ݀K#5¬'C
Ð
–t¤ Ü ‘5Ï Î H JB#ˆ'~“J/# M K#
¡


—€˜
 Î HrBJ ? ~J Z : †
k×ò €M k Ö S #S ̂ú YGV ] : †
k×òpð'ið í1ø YGV ] Ð S iS Ì ú YGV«[ ] e
Finally, (2.4), the Cauchy inequality and Lemma 3.4 imply that
?
S ä tS *

!
;
¡
;
¡
–t¤ Ü ‘5Ï JB# H K¦û ˆ #©~OK# M ݞK#T¬'C
SS K#iSL¬tCd"J ? k
–t¤ Ü ‘ Ï J ? S K «

—˜
Ö S #TS Ì`ú YGV ] S iS Ì ú YGV«[ ] e
ETNA
Kent State University
etna@mcs.kent.edu
41
DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
2) Second term:
ä B#ˆ10tT~
;
*R~ –t¤
¡
;
ä /#@0t
Ó ( H J/# ˆ LK# ˆ ~JB# ˆ LK¦û ˆ # M
Ü Ïƒ)
–
¥ < Y ¡ ] Ò
¥Þ9¡
: HBJ/#ˆ‚LK¦ûfˆÆ#©~J/#2Kûfˆ'#M
: HBJ/#2Kûfˆ'#©~“J×B#LK#M *†Ý\Ç ¡­¥ u w ¬‚¶O* ? : : 0
ä
ä ä
!
where
;
ä ? *R~
;
Ô
–t¤ – < Ü Ò ÏßÓ J/#ˆ‚2K t ÕÝ\Ç ¡­¥ u w ¬‚¶Te
¡ ¥ Y ¡]
¥Þ¡
We do not estimate ä ? , since it will cancel out a similar term in the following. After applying
(2.4) and (2.5), we get
S ä S'*
;
;
–t¤ – < Ü Ò ÏƒÓ+
¡
¥ Y ¡G]
¥Þ9¡
BJ ? ~J Z ;

Î t– ¤
k æ
¡
: fð\K#Tð í ñ YGV ]
kæ
H/J/# ˆ T~JB#LMžK# : HJB# ˆ X~“J/#2MžKs#-,.Ý\Ç¡ƒ¥'u w ¬`¶
 ‘ Ï
;
Î t– ¤
¡
SS K «
SS í ø YU
7‘5Ï
‘ Ï
?LÑ
]Ð
å
?LÑ
ˆ7»0t ‘ Ï
SGS # ˆ ~“#SS í1ø YU 5
]Ð
?2Ñ
å
?2Ñ
ˆ »0t e
The multiplicative trace inequality implies that
2? Ñ
 
å
S ä S»"k Î HrBJ ? ~J Z : †
k×ò M Ö S #S ̂ú YGV ] : †
k×òpSS iSGS í ø YV ] Ð ˆi»0t and
L? Ñ

å
S ä S»{J ? k ÖiS #S Ì ‚ú YGV ] iˆ »0t e
!
3) Third term:
ä"! B # ˆ
;
* –t¤
¡
0tX~ ä"! B#X0t
;
Ô
Ô
– < Ü Ò ÏßÓ J/#ˆ‚LK#tÕTÝ€Ç ¡­¥ u #ˆ w ~ J/#2K#tÕÝ\Ç ¡ƒ¥ u # w ‚¬ ¶
¥ Y ¡]
¥Þ¡
;
;
* t– ¤ –
Ü ÏßÓ ( J/#ˆ`LK# * ÝÇ ¡ƒ¥ u #ˆã~.û3ˆt# w
¡ ¥ < Y ¡] Ò
¥Þ¡
: H J/# ˆ T~JB# M K# * ݀ǡ­¥`u û ˆ # w : J/#2K# * Ý\Çࡃ¥`u û ˆ ©
# ~# w ¬`¶
(
(
:
:
* ä ?! ä ! ä ! e
!
By (3.4), we get
ä"?! *
;
;
Ô
J B# ˆ LK#tÕTÝ\Ç¡ƒ¥'u w ¬`¶O*ê~ ä ? e
–t¤ – < Ü Ò ÏƒÓ ¡ ¥ Y G¡ ]
¥Þ9¡
ETNA
Kent State University
etna@mcs.kent.edu
42
M. FEISTAUER AND V. KUČERA
Due to the regularity condition (3.3), the function
u û3ˆt# w *ôu w . We get the estimate:
# is continuous and, thus, u # w *
;
–t¤ – < Ü Ò ƒÏ Ó+ HJ/#ˆ`T~JB#LMK#.,¿ÝÇ ¡ƒ¥ u w ¬`¶
¡
¥ Y ¡]
¥Þ9¡
€— ˜ 
 B J ? ~“J Z LkIS iS Ìàú YGV«[ ] Ö S #TS Ì `ú YGV ] e
S ä ! tS *
Finally, we have
;
! and
—˜ 
S ä ! 1S |J ? kIS iS Ìú YGV[ ] Ö S #S Ì ‚ú YV ] e
!
4) Fourth term:
;
–t¤ Ɩ Å Ü Ò ÏßÓ H J×B# ˆ 2Ks# ˆ ~JB#LK# M ÝÇp¡ƒ¥/¬`¶
¡ ¥ Y ¡]
;
;
*ê~ –t¤ –ÆÅ Ü ßÏ Ó H J/#ˆ‚LK# : H J /#ˆ`T~JB# M K¦û3ˆ'# : JB#LK# M Ý\Ç ¡­¥ ¬‚¶
Ò
¡ ¥ Y ¡]
* ä ?$ : ä $ : ä $
!
and these terms can be treated similarly as ä ? 0 ä and ä to obtain
!
;
;
ä%?$ *ê~ –t¤ –ÆÅ Ü ÏßÓ J×B# ˆ 2K ¯Ý\Çࡃ0¥ ¬‚¶T0
¡ ¥ Y ¡] Ò
ä $ B # ˆ 0tT~ ä $ B #X0t5*ê~
;
L? Ñ
 
å
S ä $ »S k Î HrBJ ? ~“J Z : ×
k†ò M Ö1S #TS Ì`ú YGV ] : ×
k†òSGS iSS í1ø YGV ] Ð ˆ7»0t 0
?2Ñ

å
S ä $ S»{J ? k 7Ö S #S Ì ‚ú YV ] 7ˆ »0t e
!
5) Fifth term:
;
–t¤ Ɩ Å Ü Ò ÏƒÓ / JB# ˆ LK#ãÝÇ¡­¥€# ˆ
¡ ¥ Y ¡]
~J/#2K ãÝÇp¡ƒ¥\#©~ H J×B# ˆ T~J/# M K#fÝ\Ç¡ƒ¥\#9^ M ¬‚¶
;
;
* –t¤ –ÆÅ Ü ÏƒÓ BJ×B# ˆ 2#
K fÝ\Çࡃ0¥ ¡ ¥ Y ¡G] Ò
: H JB# ˆ ~“J/# M K#ãÝÇp¡ƒ¥ H û ˆ #¢~“#^ M : JB#LK#ãÝÇp¡ƒ¥/ M ¬`¶
* ä"?& : "ä & : "ä & e
!
We have ä ?& *R~ ä $? and
;
;
K ãÝÇ¡ƒ0¥ †¬‚¶T0
ä & * –t¤ –ÆÅ Ü ÏƒÓ H J×B# ˆ X~J/# M #
"
¡ ¥ Y ¡G] Ò
ä"& B# ˆ 0tX~ ä"& /#@0t5*
since
yield
and
;
# * # ^ on . This, the Cauchy inequality and the multiplicative trace inequality
—˜ 
S ä"& S» /J ? ~J Z LkIS iS Ìú YGV[ ] Ö S #S Ì ‚ú YV ] 0
—˜ 
S ä"& S1|J ? kIS iS Ì ú YGV[ ] Ö S #S Ì ‚ú YV ] e
!
The use of the derived inequalities gives us the sought estimates (3.9).
ETNA
Kent State University
etna@mcs.kent.edu
DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
43
3.5. Main result. On the basis of the above analysis, we get the apriori error estimate
for the problem with the nonlinear convection and diffusion.
T HEOREM 3.10. Let assumptions (H1)–(H3) and (A1)–(A2) be satisfied and let the
constant kæ satisfy the assumption from Lemmas 3.7, 3.8 and 3.9 corresponding to the
NIPG, SIPG and IIPG version, respectively. Let # be the exact solution of problem (2.1)–(2.3)
satisfying the regularity conditions (3.1) and (3.3) and let # ˆ be the approximate solution
Z
Z
defined by (2.6)with # ˆ *,û ˆ # . Then the error 1 ˆ *#©~“# ˆ satisfies the estimate
2
(3.10)
å
: J Z
– 3
4
H¾S 1 ˆ 6«\S Ì ú YGV«[ — ˜ ] : ˆ 71 ˆ 6«01 ˆ 6«LLM@¬86
2 ‹zZ€ŽÆ[ ” (.5 ð'1 ˆ /8L\ð í ø YGV ]
o Ü Z

ak Ö 0

with a constant kê! independent of .
Proof. From (2.6), (3.2) and estimates in Sections 3.2–3.4 it follows that
å
n J Z
ð'1B8L\ð í ø YGV ] :
H³S iS Ìàú YGV«[ —€˜ ] : ˆi»0tLM
o 98
o
?LÑ :
å
—˜  ?
:
k 9 H ˆ »0t S iS Ì ú YGV[ ] M7H Ö › S #S Ì ‚ú YV ] : 'ð ið í ø YGV ] M
:  Ö › ? S 9#«ç•98\S ̂ú YGV ðið ø YGV : J Z  Ö S #TS Ì
`ú YGV å ˆ »0t ?2Ñ í
]
]
]
: Î oÆJ ? ~J Z  Ö S #TS Ì `ú YGV
]
:  Ö › ? S #S ̂ú YV : 'ð ið ø YGV H S iS Ì ú YGV[ —˜ : å ˆ »0t M4; e
í
]Ð
]
]
The application of Young’s inequality gives us
ð 1/8L\ð íiø YGV ] : J Z H S iS Ì ú YGV«[ —€˜
98
J Z å
H ˆ »0t : S iS Ìàú YGV«[ —€˜ ] M :

o
: n H  Ö › : J  Ö : oÆJ ?
Z
J Z
:  Ö › S 9#çÆ98\S Ì ‚ú YGV ] ; e
After integrating from ! to 8×c.u !10L
]
: å ˆ »0t M
n
ð ið í ø YV ]
'
J Z Ð


~J Z Ö : Ö › `M S #TS Ì `ú GY V ]
k:9zΫn :
w and noticing that 1/!`*# Zˆ ~.û ˆ # Z ,
* ! , we obtain
2
å
ð1/8L\ð í ø YGV ] : J Z Ü H S 16€S Ìú YGV[ —˜ ] : ˆ
Z2
n
n
ak 9¦Î n :
ð16€ð í ø YGV ] ¬86 :
J Z Ð Ü Z
J Z
: o•J ? ~“J Z :  M S #X 6€S :
Ì ‚ú YV ] ¬86
160162 M ¬86
2



Ö Ü H : J Z
Z 2
Ö › Ü S 9#çÆ98\S Ì YGV ] ¬86<;àe
Z
ETNA
Kent State University
etna@mcs.kent.edu
44
M. FEISTAUER AND V. KUČERA
Now the application of Gronwall’s lemma implies that
2
å
ð1/8L\ð í ø YGV ] : J Z Ü H S 16«€S Ìú YGV[ —˜ ] : ˆ 16«016«2 M ¬6
Z
 J Z : n n  :

k Ö
H J Z : o•J ? ~J Z : M
Î
J Z
J Z

-ºð#Tð í ø YßZ€[ %( = Ì ‚ú YGV ]/] : ð\7#«ç•98\ð í ø YßZ€[ (%= Ì `ú YGV ]/] Ð
J Z : n
-z\” > Î k
8 Ð 0P8×cOu !102 w e
J Z
: , the above estimate and estimates from Lemma 3.4 yield the sought
Finally, since 1 ˆ *?
result.
É ?
-norm,
4. Optimal error estimates. The error estimate (3.10) is optimal in the ×'
but suboptimal in the / -norm. Our goal is to derive an optimal error estimate in the
× -norm. It was carried out in [13] under the following assumptions.
Assumptions (B):
@
diffusion is linear, i.e., JB#5*BA{! ,
@
the discrete diffusion form Û ˆ is symmetric (i.e., we consider the SIPG version of
the discrete problem),
@
the polygonal domain is convex,


@
the meshes ‡ ˆ 0
c!10 Z , are conforming with standard properties from the finite
element method (i.e., without hanging nodes),
@
the exact solution # of problem (2.1)–(2.3) satisfies the regularity condition (3.1),
@
conditions (H1)–(H3) and (A1) are satisfied.
The derivation of the †µ/׀ -optimal error estimate was carried out with the aid of the
Aubin-Nitsche technique based on the use of the elliptic dual problem considered for each
C
c%ÆÀ :
(4.1)
É
F±c
Then the weak solution
such that
~ED#F|*
in f0GFfS UV
C
*!ie
' and there exists a constant k6h! , independent of C ,
ðFãð Ì ø YGV ] akIð C ð í ø YGV ] e
(4.2)
We shall give here a sketch of the proof of the optimal error estimate. Let us set

Û ˆi)0LË» :
fˆ9)02˂*

A
å
É
ˆ1)0LË»¾0H)0LË¢c
f0r‡1ˆ'e
Now, for each c·/!i0 Z and 8%c·u !i0L w we define the function #
“ ˆ -projection” of #TB8L on ¶ ˆ , i. e. a function satifying the conditions
(4.3)
# B8Làc%¶ ˆ 0
ãˆ


ðIàð\^
|ak
# ˆ B 8LL as the
ˆ /# B8L02Ú ˆ 5*? ˆ /#X/8L¾0rÚ ˆ Œ„fÚ ˆ cd¶ ˆ 0
and set I* #µ~|# . The use of the coercivity of the form
3.2–3.4 allow us to prove the estimates
(4.4)
/8L)£*
Ö S #TS Ì `ú YGV ] 0
ð'I 2 ð ^
"ak
and estimates from sections
Ö S # 2 S Ì ‚ú YGV ] 0
ETNA
Kent State University
etna@mcs.kent.edu
45
DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS


for all cO/!i0 Z .
It is necessary to derive optimal ×t -estimates of I and I 2  .
L EMMA 4.1. There exists a constant kô"! such that for all

ð'I†ð í ø YGV ] "k
(4.5)
?
Ö › S #TS Ì `ú YGV ] 0 and ðI
2

c./!10 Z  ?
ð í ø GY V ] ak Ö › S # 2 S Ì ‚ú YGV ] e
Proof. We have
ðI†ð í ø YGV ] *
K
É
C
I×0 e
– J.>
C
í1ø YV ] ð ð í ø YGV ]
Let Fac
tÀ be the solution of problem (4.1) for C c%×Æ satisfying (4.2) and let F ˆ be
the piecewise linear Lagrange interpolant of the function F . Obviously F ˆ cdkz ª©¶ ˆ and
F5ˆ7S UV *a! . Thus, we have
n
—˜  k  S FfS ð'FO~LF ˆ ð Ì ø YGV ] e
^ * o S FO~MF ˆ S Ìú YV[ ] É
The assumption that Fac
t implies that
ƒ
Ï
Ó
ÏßÓ
™
u F w Ò *!)*ôu K#F w Ò
„®5c 0 ²Ic%m'/®°¾e
Now, using (4.1) and Green’s theorem, we find after some calculations that
n
I0 C 5
*
Further, the symmetry of A
ˆ Fº0IXe
ˆ and (4.3) give
fˆiFˆ70IX5*3ˆ7I0F5ˆ'5*3ˆ7B#È~“#
0.F5ˆ`5*!i0
and thus
I×0 C 5
*
n
A
fˆiF¿~MFˆi0.IT¾e
Moreover, due to the properties of F and F׈ ,
F¿~MF ˆ €S Ò
and
σÓ
*a!
„3®c
ßÏ Ó
u F|~LF5ˆ w Ò ,
* !
„f®5c
™
™
0²Ic ¼ B®°¾0
0²Icdm`B®°¾e
After some calculation, we get
I0 C 5
*
;
¡
–t¤ Ü ‘Ï KµF¿~LF5ˆ`1K#IsD»C
;
~
;
¬9/§@¡­¥•
–t¤ ÓON/ÓOUtPRQƒÏ ÏTS Ü Ò ÏƒÓ Î à
k æ Ð
¡
?2Ñ
Ô
KµF|~LF ˆ LÕTÝ\Çࡃ¥Î
L? Ñ
;
¬9/§_¡ƒ¥ž
k æ
K F¿~MF ˆ X
È
Ý\Ç¡ƒ¥ Î
Î
Ð
–t¤ Ɩ Å Ü Ò ÏƒÓ
k
æ
¬
9
/§ ¡ƒ¥
¡ ¥ Y ¡G]
—˜
 S F¿~MF5ˆ7S Ìàú YGV ] S IàS Ìú YGV[ ]
?2Ñ
Z
;
i

5
‘
Ï
n
å Y
:
‘Ï
ˆ
€
ð
µ
K
O
F
M
~
F
€
ð
ˆ I×0 IX2
ù
Y
U
?2Ñ
1
í
ø
X
]
W
t
–
¤
k æ V ¡
~
;
?2Ñ
k æ
9¬ /§ ¡­¥ Ð
?LÑ
Ð
?2Ñ
e
u I w Di¶
IDi¶
ETNA
Kent State University
etna@mcs.kent.edu
46
M. FEISTAUER AND V. KUČERA
According to the multiplicative trace inequality, (4.2) and (4.4), we can write
;
n
2? Ñ
k æ V
–t¤
¡
akIðI†ð ^
Moreover,
i‘5Ï
V
;
¡
‘ Ï
ð€KµFO~MF ˆ €ð í1ø YùU X] W
 ‘ Ï
‘ Ï
5
–t¤ S FfS Ì ø Y ] W
?2Ñ
?2Ñ
"k
å
2? Ñ
ˆ I×0ITL ?
Ö › S #TS Ì `ú GY V ] ð C ð íiø YGV ] e


—€˜
S F|~LF ˆ S Ì ú YV ] S I†S Ì ú YGV«[ ] ak S FfS Ì ø YGV ] ðIàð^
 ?
ak Ö › S #S Ì ‚ú YGV ] ð C ð í ø YGV ] e
Combining the previous estimates, we find that
I×0 C †
"k

?
Ö › S #TS Ì
`ú YGV ] ð C ð í1ø YGV ] e
Hence,
ðI†ð í ø YGV ] *
C
 ?
I0 ak Ö › S #S Ì ‚ú YGV ] 0
– J.>
C
í1ø YV ] ð ð í ø YGV ]
K
which completes the proof of the first estimate in (4.5).
In the derivation of the estimate of the norm ð'I 2 ð í ø YGV ] we proceed similarly as above,
using the differentiation of identity (4.3) with respect to 8 .
Finally, we come to the optimal àµ/ -error estimate.
T HEOREM 4.2. Let assumptions (B) be fulfilled. Then the error 1tˆ*ô#µ~O#ˆ satisfies
the estimate
ð/1 ˆ ð í ñ GY Z\[ ("= í ø YGV ]B] k


?
֛ 0
with a constant k {! independent of .
Proof. Let # be defined by (4.3) and I*#¢~# , 6%*# ~.#ˆ . Then 1•ˆ*a#È~“#ˆs*
: 6 . Let us subtract (3.2) from (2.6, b), substitute 6“cd¶_ˆ for ڈ and use the relation
I
Î
6B8L
n D
0.6B8L Ð *
ð6B8L€ð í ø YGV ] e
98
o D»8
Then, we get
(4.6)
n D
ð'6B8L\ð í ø YGV ] : ˆ 6T/8L¾0[6T/8LL
o D18
é
é
*êu ˆ /#X/ 8L¾0.6B8L2T~ ˆ /# ˆ B8L0.6B8L2 w ~"I 2 B 8L0\6B8L2¾0
because ˆ B#TB8L~a# B8L¾0\6T/8LLµ*j! . The first right-hand side term can be estimated by
Lemma 3.6 and Young’s inequality (we omit the argument 8 ):
é
ˆ B#X06«X~
é
A
ˆ B# ˆ 06†
o
k
ð'6ð ^
:
A
 Y ?]
Ö › S #TS Ì`ú YGV ] : ð 65ð í ø YGV ] e
For the second term of the right-hand side of (4.6), by the Cauchy and Young’s inequalities
and Lemma 4.1, we have
SGI 2 0 6«\S»
o
n
k
 Y ?] 2
Ö › S # S Ì
‚ú YGV ] : ð 65ð í1ø YGV ] e
ETNA
Kent State University
etna@mcs.kent.edu
DGFEM FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
Finally, the coercivity of
side of (4.6).
Hence, we find that
(4.7)
D
ð65ð íiø YGV
»D 8
?

k Y Ö ›
]
ãˆ
47
following from Lemma 3.8 gives the estimate of the left-hand
: iA ð65ð ^
:
: k Ϋn : n ð'65ð ] Î n S #TS Ì `ú YGV ] S # 2 S Ì ‚ú YV ] Ð
í ø YV ] e
Ð
A
A
Now the proof is concluded in a standard way by the integration of (4.7) from ! to 8†c.u !102 w ,
the estimate of ð6!'\ð í ø YGV ] , and the application of Gronwall’s lemma.
5. Conclusion. In the paper we derived error estimates for the NIPG, SIPG and IIPG
versions of the DGFEM applied to the numerical solution of nonstationary convection-diffusion
problems with nonlinear convection as well as diffusion, equipped with Dirichlet boundary
condition. The computational grids can be nonconforming with hanging nodes. Further, we
discussed optimal †/׀ -error estimates of the SIPG method in the case of a linear diffusion and nonlinear convection, with the Dirichlet boundary condition on the whole boundary,
over conforming meshes.
There are still some open problems:
@
optimal †/׀ -error estimates in the case of a nonlinear diffusion,
@
derivation of optimal / -error estimates over nonconforming meshes,
@
the analysis of optimal error estimates in the case of a nonconvex polygonal domain
and mixed Dirichlet-Neumann boundary conditions under a realistic regularity
of the exact solutions of the nonstationary initial-boundary value problem and the
elliptic dual problem.
Acknowledgments. This work is a part of the research project financed by the Ministry
of Education of the Czech Republic No. MSM 0021620839. It was also partially supported
by the grant No. 201/08/0012 of the Grant agency of the Czech Republic. The research of
V. Kučera was supported by the project LC06052 of the Ministry of Education of the Czech
Republic (Jindřich Nečas Center for Mathematical Modelling) and the grant No. 48607 of the
Grant Agency of the Charles University in Prague.
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ETNA
Kent State University
etna@mcs.kent.edu
48
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