A priori error estimation for the dual mixed finite element method

Apriorierrorestimation
forthedualmixedfiniteelementmethod
oftheelastodynamicprobleminapolygonaldomain,I

L.BOULAAJINE,
∗
M.FARHLOUL
†
andL.PAQUET
‡

Abstract
Inthispaperweanalyzeanewdualmixedformulationoftheelastodynamic
systeminpolygonaldomains.Inthisformulationthesymmetryofthestraintensor
isrelaxedbytherotationalofthedisplacement.Forthetimediscretizationofthis
newdualmixedformulation,weuseanexplicitscheme.Aftertheanalysisofstability
ofthefullydiscretescheme,
L
∞
intime,
L
2
inspaceapriorierrorestimatesare
derivedfortheapproximationofthedisplacement,thestrain,thepressureandthe
rotational.Numericalexperimentsconfirmourtheoreticalpredictions.
MSC:65M60;65M15;65M50
Keywords:Sobolevspaces,elastodynamic,dualmixedfiniteelement,Newmarkscheme,
Lagrangemultiplier,hybridformulation,errorestimation.

1Introduction
Thepurposeofthispaperistheanalysisofafiniteelementmethodforapproximating
thelinearelastodynamicsystemusinganewdualmixedformulationforthediscretization
inthespatialvariablesandanexplicitNewmarkschemeforthediscretizationintime.The
explicitNewmarkschemeisshowntobestableunderanappropriateCFLcondition.The
analysisofanimplicitNewmarkschemewillbepresentedin[2].
Theanalysisofapriorierrorestimatesforthemixedfiniteelementmethodofasecond
orderhyperbolicsysteminregulardomainsusingsymmetricapproximationsofthestress
wasinitiatedin[1,16]seealso[15].Buttoourknowledgeasimilaranalysisforthedual
mixedformulationofthelinearelastodynamicsysteminnonregulardomains,introducing
∗
Universite´deValenciennesetduHainautCambre´sis,MACS,ISTV,F-59313-ValenciennesCedex9,
France,e-mail:lboulaaj@univ-valenciennes.fr
†
Universite´deMoncton,De´partementdeMathe´matiquesetdeStatistique,Moncton,N.B.,E1A3E9,
Canada,e-mail:mohamed.farhloul@umoncton.ca
‡
Universite´deValenciennesetduHainautCambre´sis,MACS,ISTV,F-59313-ValenciennesCedex9,
France,e-mail:Luc.Paquet@univ-valenciennes.fr.

1

asanewunknownthestraintensor,wasnotyetdone.Thereforethegoalofthispaper
istomakethisanalysis.Apriorierrorestimatesareprovedfortheapproximationofthe
displacement,thestrain,thepressureandtherotational,firstlyforthesemi-discretized
solutionandthenforthecompletelydiscretizedsolutionbytheexplicitNewmarkscheme
inthetimevariable.
Overthelasttwodecadestherehasbeenconsiderableinterestintheareaofmixed
finiteelementdiscretizationsofthecorrespondingstationaryproblem,i.e.thesystemof
linearelasticity;letusquote,forexample,[10,3,8,9].Themaindifficultyappearingin
thisproblemisfindingawaytotakeintoaccountthesymmetryofthestraintensor.In
ourapproach,thesymmetryofthestraintensorisrelaxedbyaLagrangemultiplier,which
isnothingelsethantherotation.
Theoutlineofthispaperisasfollows:section2definessomenotation,presentsthe
modelevolutionproblemweshallconsiderandrecalltwocomparisonresultsconcerning
continuousanddiscreteGronwall’sinequalities.Insection3,wedefinethenewdual
mixedformulationofthemodelevolutionproblem.Section4isdevotedtosomeregularity
resultsofthesolutionofourelastodynamicsystemintermsofweightedSobolevspaces.
Insection5,weintroducethesemi-discretemixedformulationandprovetheexistence
anduniquenessofthesolutionforthisformulationandrecallsomeresultsconcerningthe
inf-supandcoercivityconditions.Then,undersomeadequaterefinementrulesofmeshes,
weestablishsomeerrorestimatesonsomeinterpolationoperatorsandweproveaninverse
inequalityforthedivergenceoperator.Insubsection5.1.1,wederivesomeerrorestimates
betweentheexactsolutionofthemixedproblemandthesolutionoftheellipticprojection
problem,whichwillbeusedinsubsection5.1.2toderivetheerrorestimatesbetween
theexactandthesemi-discretesolution.Section6isconcernedwiththefullydiscrete
finiteelementscheme:existenceanduniquenessofthesolutionofthefullydiscretized
problem,stabilityanalysisandapriorierrorestimatesbetweentheexactsolutionand
itsfullydiscreteapproximationfortheexplicitscheme.Theproofoftheerrorestimates
restontheintroductionofanauxiliaryproblem:theellipticprojectionproblem.The
numericalexperimentsofsection7confirmourtheoreticalpredictions.Insection8we
presentconclusions.

2Preliminariesandnotations

2.1Themodelproblem
LetusfixaboundedplanedomainΩwithapolygonalboundary.Moreprecisely,we
assumethatΩisasimplyconnecteddomainandthatitsboundaryΓistheunionofa
finitenumberoflinearsegmentsΓ¯
j
,1
≤
j
≤
n
e
(Γ
j
isassumedtobeanopensegment).
Wealsofixapartitionof
{
1
,
2
,
∙∙∙
,n
e
}
intotwosubsets
I
N
and
I
D
.TheunionΓ
D
ofthe
Γ
j
,
j
runningover
I
D
,isthepartoftheboundaryΓ,whereweassumezerodisplacement
field.TheunionΓ
N
,oftheΓ
j
,
j
∈
I
N
,isthepartoftheboundaryΓwhereweassume
zerotractionfield.

2

InthisdomainΩ,weconsiderisotropicelastichomogeneousmaterial.Let
u
=(
u
1
,u
2
)
bethedisplacementfieldand
f
=(
f
1
,f
2
)
∈
[
L
2
(Ω)]
2
thebodyforceperunitofmass.Thus
thedisplacementfield
u
=(
u
1
,u
2
)satisfiesthefollowingequations:
u
tt
−
div
σ
s
(
u
)=
f
in[0
,T
]
×
Ω
,


u
=0on[0
,T
]
×
Γ
D
,
(2.1)
σ
s
(
u
)
.n
=0on[0
,T
]
×
Γ
N
,


u
(0
,.
)=
u
0
inΩ
,
u
t
(0
,.
)=
u
1
inΩ
,
where
u
0
and
u
1
aretheinitialconditionsondisplacementsandvelocities.
n
denotesthe
unitoutwardnormalfieldalongΓ.Thestresstensor
σ
s
(
u
)isdefinedby
(2.2)
σ
s
(
u
):=2
µ²
(
u
)+
λ
tr
²
(
u
)
δ.
Thepositiveconstants
µ
and
λ
arecalledtheLame´coefficients.Weassumethat
(2.3)(
λ,µ
)
∈
[
λ
0
,λ
1
]
×
[
µ
1
,µ
2
]
erehw0
<µ
1
<µ
2
and0
<λ
0
<λ
1
.

Asusual,
²
(
u
)denotesthelinearizedstraintensor(
i.e.,²
(
u
)=
1
(
r
u
+(
r
u
)
T
)and
δ
the
2identitytensor.
Forreasonsofsimplicityinourtheoreticalanalysis,wehavechosenhomogeneousbound-
aryconditionsonbothDirichletandNeumannboundaries.Theextensiontononhomo-
geneousboundaryconditionsisdonewithoutdifficulty.Letusnotethatnumericaltests
(seesection7)aremadeunderthenonhomogeneoussurfacetraction.Inthesequel,we
willusethefollowingnotation.For
τ
=(
τ
ij
)
∈
[
H
(
div
;Ω)]
2
,wedenoteby
¶µdiv
(
τ
)=
∂τ
11
+
∂τ
12
,∂τ
21
+
∂τ
22
,
∂x
1
∂x
2
∂x
1
∂x
2
as
(
τ
)=
τ
21
−
τ
12
.
For
v
=(
v
1
,v
2
)
∈
[
H
1
(Ω)]
2
,werecallthat
v∂v∂rot
v
=
2
−
1
.
∂x
1
∂x
2
Asusual,wedenoteby
L
2
(
.
)theLebesguespaceofsquareintegrablefunctionsandby
H
s
(
.
)
,s
≥
0,thestandardSobolevspaces.Theusualnormandseminormof
H
s
(
D
)are
denotedby
||
.
||
s,D
and
|
.
|
s,D
.Theinnerproductin[
L
2
(Ω)]
2
willbewritten(
.,.
).If
σ
=(
σ
ij
)
,τ
=(
τ
ij
)
∈
[
L
2
(Ω)]
2
×
2
,thenwedenoteby
ZXσ
:
τ
=
σ
ij
τ
ij
and(
σ,τ
)=
σ
:
τdx.
Ωj,i3

WenowintroducetheHilbertspace
[
H
Γ1
D
(Ω)]
2
:=
{
v
∈
[
H
1
(Ω)]
2
;
v
|
Γ
D
=0
}
.
Finally,inordertoavoidexcessiveuseofconstants,weusethefollowingnotation:
a
.
b
standfor
a
≤
cb
,withpositiveconstants
c
independentof
a,b
,
h
andΔ
t
.

2.2Gronwall’sinequalities
Inthissection,werecalltwocomparisonresults[19],whichwillbeusefulinthestability
andconvergenceanalysisofourproblem.Let