A priori error estimation for the dual mixed finite element method
39 pages
English

A priori error estimation for the dual mixed finite element method

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Niveau: Supérieur, Licence, Bac+2
A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, I L. BOULAAJINE ?, M. FARHLOUL †and L. PAQUET ‡ Abstract In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains. In this formulation the symmetry of the strain tensor is relaxed by the rotational of the displacement. For the time discretization of this new dual mixed formulation, we use an explicit scheme. After the analysis of stability of the fully discrete scheme, L∞ in time, L2 in space a priori error estimates are derived for the approximation of the displacement, the strain, the pressure and the rotational. Numerical experiments confirm our theoretical predictions. MSC: 65M60; 65M15; 65M50 Keywords: Sobolev spaces, elastodynamic, dual mixed finite element, Newmark scheme, Lagrange multiplier, hybrid formulation, error estimation. 1 Introduction The purpose of this paper is the analysis of a finite element method for approximating the linear elastodynamic system using a new dual mixed formulation for the discretization in the spatial variables and an explicit Newmark scheme for the discretization in time. The explicit Newmark scheme is shown to be stable under an appropriate CFL condition. The analysis of an implicit Newmark scheme will be presented in [2]. The analysis of a priori error estimates for the mixed finite element method of a second order hyperbolic system in regular domains using symmetric approximations of the stress was initiated in [1, 16] see also [15].

  • univ-valenciennes

  • ∂?12 ∂x2

  • mixed formulation

  • ?universite de valenciennes et du hainaut cambresis

  • fully discrete

  • problem

  • ?21 ?

  • error estimates

  • displacement field


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Publié par
Nombre de lectures 9
Langue English

Exrait

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

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

1
]
×
[
µ
1

2
]
erehw0

1

2
and0

0

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
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
φ
(
.
)

0suchthat
φ
t
(
t
)

ρφ
(
t
)+
η
(
t
)for
0

t

T
,where
ρ

0issomeconstantand
η
(
.
)

0,
η

L
1
([0
,T
]).Then
¶µZT(2.4)
φ
(
t
)

e
ρT
φ
(0)+
η
(
s
)
ds,

t

[0
,T
]
.
0Let(
k
n
)
n

0
,(
p
n
)
n

0
twonon-negativesequencesbegiven,
g
0

0givenalsoandletus
supposethatthesequence(
φ
n
)
n

0
satisfies:

φ
0

g
0
,
(2.5)
n
X

1
n
X

1

φ
n

g
0
+
p
s
+
k
s
φ
s
,

n

1
.
s
=0
s
=0
nehT)6.2(

µ
n
X

1
¶µ
n
X

1

φ
n

g
0
+
p
s
exp
k
s
,

n

1
.
s
=0
s
=0

3Thedualmixedformulation
Introducingasnewunknowns:

:=2
µ²
(
u
)
,p
:=

λ
div
(
u
)and
ω
:=
rot
(
u
)
,
2andthespaces:
(3.1)Σ
0
:=
{
(
τ,q
)

[
L
2
(Ω)]
2
×
2
×
L
2
(Ω);
div
(
τ


)

[
L
2
(Ω)]
2
,
(
τ


)
.n
=0onΓ
N
}
,

)2.3(

M
:=
{
(v

)

[
L
2
(Ω)]
2
×
L
2
(Ω)
}
,
4

westatethedualmixedformulationforourmodelhyperbolicequation(2.1):find(
σ
(
.
)
,p
(
.
))

L
2
([0
,T
];Σ
0
),
u
(
.
)

H
2
([0
,T
];[
L
2
(Ω)]
2
)and
ω
(
.
)

L
2
([0
,T
];
L
2
(Ω))suchthatforall(
τ,q
)

Σ
0
,forall(
v,θ
)

M
andfora.e.
t

[0
,T
],wehave
11(
σ
(
t
)

)+(
p
(
t
)
,q
)+(
div
(
τ


)
,u
(
t
))+(
as
(
τ
)

(
t
))=0
,
λµ2(3.3)(
u
tt
(
t
)
,v
)

(
div
(
σ
(
t
)

p
(
t
)
δ
)
,v
)

(
as
(
σ
(
t
))

)

(
f
(
t
)
,v
)=0
,
u
(0)=
u
0
,u
t
(0)=
u
1
.
Weconcludethissectionbyintroducingsomenotations.Weset
σ
=(
σ,p
)

=(
τ,q
)
,u
=(
u,ω
)
,v
=(
v,θ
)
,
∼∼∼∼11(3.4)
a
(

σ,τ

):=2
µ
(
σ,τ
)+
λ
(
p,q
)
,


σ,τ


Σ
0
,
(3.5)
b
(
τ,v
):=(
div
(
τ


)
,v
)+(
as
(
τ
)

)
,

τ

Σ
0
,

v

[
L
2
(Ω)]
2
×
L
2
(Ω)
.
∼∼∼∼Withthesenotations,themixedformulation(3.3)mayberewritten:find

σ
(
.
)=(
σ
(
.
)
,p
(
.
))

L
2
([0
,T
];Σ
0
)and
u
(
.
)=(
u
(
.
)

(
.
))

H
2
([0
,T
];[
L
2
(Ω)]
2
)
×
L
2
([0
,T
];
L
2
(Ω))suchthat
∼u
(0)=
u
0
,u
t
(0)=
u
1
andfora.e.
t

[0
,T
]:

a
(

σ
(
t
)


)+
b
(
τ

,

u
(
t
))=0
,

τ


Σ
0
,
)6.3(
b
(
σ
(
t
)
,v
)+(
F
(
t
)
,v
)=(
u
tt
(
t
)
,v
)
,

v

[
L
2
(Ω)]
2
×
L
2
(Ω)
,
∼∼∼∼where(
F
(
t
)
,

v
):=(
f
(
t
)
,v
).
4Regularityofthesolutions
Let
u

L
2
(0
,T
;[
H
Γ1
(Ω)]
2
)suchthat
du

L
2
(0
,T
;[
L
2
(Ω)]
2
),bethesolutionof(2.1).We
tdDconsidertheLame´operatordefinedby
L
:=

µ
Δ

(
λ
+
µ
)
r
div
.
Thus,equivalently
u
istheweaksolutionoftheproblem
u
tt
+
Lu
=
f
in[0
,T
]
×
Ω
,


u
=0on[0
,T
]
×
Γ
D
,
(4.1)
σ
s
(
u
)
.n
=0on[0
,T
]
×
Γ
N
,

u
(0
,.
)=
u
0
inΩ
,

u
t
(0
,.
)=
u
1
inΩ
.
5

Itiswellknown(see[10]or[13,14,5])thattheweaksolutionofthecorrespondingLame´
systemof(4.1)presentsvertexsingularities.Todescribethem,weneedtointroducethe
followingnotations:
Definition4.1
Let
S
j
(1

j

n
e
)
bethevertexofourpolygonaldomain
Ω
atthe
intersectionofthesides
Γ
j
and
Γ
j
+1

n
e
+1
:=Γ
1
)
.Letusdenoteby
ω
j
themeasureofthe
angleatthevertex
S
j
.Bythecharacteristicequationassociatedtothevertex
S
j
,wemean
thetranscendentalequationinthecomplexvariable
α
:
·
λ
+
µ
¸
2
(4.2)sin
2
(
αω
j
)=
α
2
sin
2
ω
j
,
µ3+λif
S
j
isavertexofDirichlettypei.e.
j,j
+1

I
D
,
(4.3)sin
2
(
αω
j
)=
α
2
sin
2
ω
j
,
if
S
j
isavertexofNeumanntypei.e.
j,j
+1

I
N
,
2
(
λ
+2
µ
)
2

(
λ
+
µ
)
2
α
2
sin
2
ω
j
(4.4)sin(
αω
j
)=(
λ
+
µ
)(
λ
+3
µ
)
,
if
S
j
isavertexofmixedtypei.e.
j

I
D
,j
+1

I
N
or
j

I
N
,j
+1

I
D
.
Definition4.2
Foranyscalarfunction
φ

C
0
(Ω)
suchthat
φ
(
x
)
>
0
forevery
x

Ω¯
\{
S
1
,S
2
,
∙∙∙
,S
n
e
}
andany
m,k

IN,wedefine
H
φm,k
(Ω)=
{
v

H
m
(Ω)

H
lomc
+
k
(Ω);
φD
β
v

L
2
(Ω)
,

β

IN
2
suchthat
m<
|
β
|≤
m
+
k
}
.
H
φm,k
(Ω)
isaHilbertspaceequippedwiththenorm:
µX¶
1
/
2
||
v
||
m,k
;
φ,
Ω
=
||
v
||
2
m,
Ω
+
||
φD
β
v
||
02
,
Ω
.
m<
|
β
|≤
m
+
k
Onthisspace,wealsodefinethesemi-norm:
µX¶
1
/
2
2β|
v
|
m,k
;
φ,
Ω
=
||
φDv
||
0
,
Ω
.
|
β
|
=
m
+
k
Weconsideralsothespaces
L
2
(0
,T
;
H
φm,k
(Ω))
endowedwiththenorm:
µZ
T

1
/
2
||
v
||
L
2
(
H
m,k
)
=
||
v
||
2
m,k
;
φ,
Ω
dt,
φ0and
L

(0
,T
;
H
φm,k
(Ω))
endowedwiththenorm
||
v
||
L

(
H
φm,k
)
=
ess
sup
k
v
(
t
)
k
m,k
;
φ,
Ω
.
T≤t≤06

Letusset
ξ
=min
j
=1
,...,n
e
ξ
j
where
ξ
j
=inf
{
Reα
j,k
;
Reα
j,k
>
0
}
,
kwhere
α
j,k
issolutionoftheappropriatetrancendantalequationappearingindefinition
4.1.By([10],lemma2.2),
ξ>
21
.Letuspicksome
α

]1

ξ,
1
/
2[if
ξ

1
,
andletustake
α
=0if
ξ>
1.
Nowwecangivethefollowingregularityresult:
Proposition4.3
Letussupposethattheappropriatecharacteristicequationamong4.2-
4.4foreachvertexof
Ω
hasnorootontheverticalline
Reα
=1
inthecomplexplane.
Let
φ

C
0
(Ω)
,likeaboveindefinition4.2,suchthat
φ
(
x
)=
r
j
(
x
)
α
inaneighborhood
ofthevertex
S
j
ofthepolygonaldomain
Ω
forevery
j
=1
,
∙∙∙
,n
e
where
r
j
(
x
)=
|
x

S
j
|
(
|
.
|
meansEuclidiannorm
)
.
Letussupposethat:

f

H
3
(0
,T
;[
L
2
(Ω)]
2
)
,
(4.5)
u
0
,u
1
,f
(0)

Lu
0
,f
t
(0)

Lu
1

[
H
Γ1
D
(Ω)]
2
,
f
tt
(0)

Lf
(0)+
L
2
u
0

[
L
2
(Ω)]
2
.
Then
u

C
(0
,T
;[
H
φ
1
,
1
(Ω)]
2

[
H
Γ1
(Ω)]
2
)
and
u
tt

L
2
(0
,T
;[
H
φ
1
,
1
(Ω)]
2

[
H
Γ1
(Ω)]
2
)
.
DDConsequently
σ

L

(0
,T
;[
H
φ
0
,
1
(Ω)]
2
×
2
)
,
p

L

(0
,T
;
H
φ
0
,
1
(Ω))
and
ω

L

(0
,T
;
H
φ
0
,
1
(Ω))
.
Moreover
σ
tt

L
2
(0
,T
;[
H
φ
0
,
1
(Ω)]
2
×
2
)
,
p
tt

L
2
(0
,T
;
H
φ
0
,
1
(Ω))
and
ω
tt

L
2
(0
,T
;
H
φ
0
,
1
(Ω))
.
Proof:
AccordingtoTheorem30.1p.442-443of[20]wehave
u

H
3
(0
,T
;[
H
Γ1
D
(Ω)]
2
)and
u
(4)

L
2
(0
,T
;[
L
2
(Ω)]
2
).Inparticular
u
tt

L
2
(0
,T
;[
H
Γ1
D
(Ω)]
2
)and(
u
tt
)
tt

L
2
(0
,T
;[
L
2
(Ω)]
2
).
Knowingthat
Lu
=
f

u
tt
,wehave
Lu
tt
=
f
tt

(
u
tt
)
tt

L
2
(0
,T
;[
L
2
(Ω)]
2
).Thus
u
tt

L
2
(0
,T
;[
H
Γ1
D
(Ω)]
2
)and
Lu
tt

L
2
(0
,T
;[
L
2
(Ω)]
2
).Thatis
u
tt

L
2
(0
,T
;
D
(
L
))
1,1where
D
(
L
)denotesthedomainoftheLame´operator.But
D
(
L
)
,

[
H
φ
(Ω)]
2
by
adaptingcorollary2.4p.326of[10].Thus
u
tt

L
2
(0
,T
;[
H
φ
1
,
1
(Ω)]
2
)andconsequently
σ
tt

L
2
(0
,T
;[
H
φ
0
,
1
(Ω)]
2
×
2
),
p
tt

L
2
(0
,T
;
H
φ
0
,
1
(Ω)),
ω
tt

L
2
(0
,T
;
H
φ
0
,
1
(Ω)).Onthe
otherhand
u
t

H
2
(0
,T
;[
H
Γ1
D
(Ω)]
2
)and
Lu
t
=
f
t

u
ttt

L
2
(0
,T
;[
L
2
(Ω)]
2
).Sothat
u
t

L
2
(0
,T
;
D
(
L
)).Andhence,
(4.6)
u
t

L
2
(0
,T
;[
H
φ
1
,
1
(Ω)]
2
)
.
Similarly,wehave
u

H
2
(0
,T
;[
H
Γ1
(Ω)]
2
)and
Lu
=
f

u
tt

L
2
(0
,T
;[
L
2
(Ω)]
2
),sothat
Du

L
2
(0
,T
;
D
(
L
)),andhencealso
u

L
2
(0
,T
;[
H
φ
1
,
1
(Ω)]
2
).Fromthisand(4.6)weget
u

C
(0
,T
;[
H
φ
1
,
1
(Ω)]
2
)

L

(0
,T
;[
H
φ
1
,
1
(Ω)]
2
)
.
Thus
σ

L

(0
,T
;[
H
φ
0
,
1
(Ω)]
2
×
2
),
p

L

(0
,T
;
H
φ
0
,
1
(Ω))and
ω

L

(0
,T
;
H
φ
0
,
1
(Ω)).More-
over
u,u
t

L
2
(0
,T
;[
H
Γ1
D
(Ω)]
2
)implies
u

C
(0
,T
;[
H
Γ1
D
(Ω)]
2
).

7

Proposition4.4
Letussupposethattheappropriatecharacteristicequationamong4.2-
4.4foreachvertexof
Ω
hasnorootontheverticalline
Reα
=2
inthecomplexplane.
Let
φ

C
0
(Ω)
likeinProposition4.3Letussupposethat:
f

H
6
(0
,T
;[
L
2
(Ω)]
2
)
f
(4)

L
2
(0
,T
;[
H
1
(Ω)]
2
)
21
u
0
,u
1
,f
(0)

Lu
0

[
H
Γ
D
(Ω)]
f
(1)
(0)

Lu
1

[
H
Γ1
D
(Ω)]
2
(4.7)
f
(2)
(0)

Lf
(0)+
L
2
u
0

[
H
Γ1
D
(Ω)]
2
f
(3)
(0)

Lf
(1)
(0)+
L
2
u
1

[
H
Γ1
(Ω)]
2
D
f
(4)
(0)

Lf
(2)
(0)+
L
2
f
(0)

L
3
u
0

[
H
Γ1
D
(Ω)]
2

f
(5)
(0)

Lf
(3)
(0)+
L
2
f
(1)
(0)

L
3
u
1

[
L
2
(Ω)]
2
Then
σ
tttt

L
2
(0
,T
;[
H
φ
0
,
1
(Ω)]
2
×
2
)
,
p
tttt

L
2
(0
,T
;
H
φ
0
,
1
(Ω))
,
ω
tttt

L
2
(0
,T
;
H
φ
0
,
1
(Ω))
and
u
tttt

L
2
(0
,T
;[
H
φ
1
,
2
(Ω)]
2
)

C
(0
,T
;[
H
Γ1
D
(Ω)]
2
)
.
Proof:
ByoncemoreTheorem30.1p.442-443of[20],itfollowsthat
u

H
6
(0
,T
;[
H
Γ1
D
(Ω)]
2
).
Bytheequation
Lu
(4)
=
f
(4)

u
(6)
andthehypothesis
f
(4)

L
2
(0
,T
;[
H
1
(Ω)]
2
),itfollows
bycorollary2.4p.326of[10]that
u
(4)

L
2
(0
,T
;[
H
φ
1
,
2
(Ω)]
2
).Thisimpliestheabove
assertions.

5Thesemi-discretemixedformulation
WeassumethatΩisdiscretizedbyaregularfamilyoftriangulations(
T
h
)
h>
0
inthesense
of[4].If
T
∈T
h
,thenwedenoteby
h
T
itsdiameter.Byabuseofnotation([4],remark17.1
p.131),
h
denotesalsomax
T
∈T
h
h
T
(therealmeaningof
h
isindicatedbythecontext).
WeintroducethefinitedimensionalsubspacesΣ
0
,h
and
V
h
×
W
h
ofΣ
0
and
M
respectively
definedby
(5.1)Σ
0
,h
:=
{
(
τ
h
,q
h
)

Σ
0
;

T
∈T
h
:
q
h
|
T

IP
1
(
T
)and
τ
h
|
T

[IP
1
(
T
)]
2
×
2

[
R
Curl
b
T
]
2
}
2(5.2)
V
h
×
W
h
:=
{
(
v
h

h
)

M
;

T
∈T
h
:
v
h
|
T

[IP
0
(
T
)]and
θ
h
|
T

IP
1
(
T
)
}
.
Notethatby
τ
h
|
T

[IP
1
(
T
)]
2
×
2

[
R
Curl
b
T
]
2
,wemeanthatthereexistspolynomialson
T
ofdegree

1:
p
11

IP
1
(
T
)
,p
12

IP
1
(
T
)
,p
21

IP
1
(
T
)
,p
22

IP
1
(
T
)andtworeal
numbers
α
1

2
suchthat
·
p
+
α
∂b
T
p

α
∂b
T
¸
τ
h
|
T
=
111
∂∂bx
T
2
121
∂∂bx
T
1
,
p
21
+
α
2
∂x
2
p
22

α
2
∂x
1
where
b
T
denotesthebubblefunctionfortheactualtriangularelement
T
definedby
b
T
=27
λ
1
λ
2
λ
3
.
8

λ
1

2

3
denotethebarycentriccoordinateson
T
.Nowweintroducethefollowingsemi-
discretizedproblem:Find(
σ
h
(
.
)
,p
h
(
.
))

L
2
([0
,T
];Σ
0
,h
),
u
h
(
.
)

H
2
([0
,T
];
V
h
)and
ω
h
(
.
)

L
2
([0
,T
];
W
h
)suchthatforall(
τ
h
,q
h
)

Σ
0
,h
,forall(
v
h

h
)

V
h
×
W
h
andfora.e.
t

[0
,T
],wehave:
11(
σ
h
(
t
)

h
)+(
p
h
(
t
)
,q
h
)+(
div
(
τ
h

q
h
δ
)
,u
h
(
t
))+(
as
(
τ
h
)

h
(
t
))=0
,
λµ2(5.3)(
u
h,tt
(
t
)
,v
h
)

(
div
(
σ
h
(
t
)

p
h
(
t
)
δ
)
,v
h
)

(
as
(
σ
h
(
t
))

h
)

(
f
(
t
)
,v
h
)=0
,
u
h
(0)=
u
0
,h
,u
h,t
(0)=
u
1
,h
.
Wemaythinkto
u
0
,h
and
u
1
,h
asapproximationsin
V
h
of
u
0
and
u
1
respectively.
Theinitialconditions
u
0
,h
and
u
1
,h
willbespecifiedlater.Withthenotations(3.4)and
(3.5),thesemi-discretizedproblem(5.3)mayberewritten:Find
σ
(
.
)=(
σ
h
(
.
)
,p
h
(
.
))

∼hL
2
([0
,T
];Σ
0
,h
)and
u
(
.
)=(
u
h
(
.
)

h
(
.
))

H
2
([0
,T
];
V
h
)
×
L
2
([0
,T
];
W
h
)suchthatfora.e.
∼ht

[0
,T
],wehave:
a
(

σ
(
t
)


)+
b
(
τ

,

u
(
t
))=0
,

τ

=(
τ
h
,q
h
)

Σ
0
,h
,

hhhhh
(5.4)
b
(

σ
h
(
t
)
,

v
h
)+(
F
(
t
)
,

v
h
)=(
u
h,tt
(
t
)
,v
h
)
,


v
h
=(
v
h

h
)

V
h
×
W
h
,
u
h
(0)=
u
0
,h
,u
h,t
(0)=
u
1
,h
.
Theexistenceanduniquenessofasolution((
σ
h
(
.
)
,p
h
(
.
))
,
(
u
h
(
.
)

h
(
.
)))of(5.3)orequiv-
alentlyto(5.4)areshowninthefollowinglemma:
Lemma5.1
Asolution
((
σ
h
(
.
)
,p
h
(
.
))
,
(
u
h
(
.
)

h
(
.
)))
of(5.3)orequivalentlyto(5.4)exists
andisunique.
Proof:
Thefirstandthesecondequationoftheevolutionproblem(5.4)canberewritten
fora.e.
t

[0
,T
]as

a
(
σ
(
t
)

)+
b
(
τ,u
(
t
))=0
,

τ
=(
τ
h
,q
h
)

Σ
0
,h
,

h

h

h

h

h
(5.5)


b
(

σ
(
t
)
,

v
)=

(
f
(
t
)

u
h,tt
(
t
)
,v
h
)
,


v
=(
v
h

h
)

V
h
×
W
h
.
hhhWemaythinkthesolution(
σ
(
t
)
,u
(
t
))

Σ
0
,h
×
(
V
h
×
W
h
)of(5.5),forafixedtime,asa
∼∼hhsolutionofthestationaryproblem:find(
σ,u
)

Σ
0
,h
×
(
V
h
×
W
h
)solutionof
∼∼hh
a
(
σ,τ
)+
b
(
τ,u
)=0
,

τ
=(
τ
h
,q
h
)

Σ
0
,h
,

h

h

h

h

h
(5.6)


b
(

σ,

v
)=(
g,v
h
)
,


v
=(
v
h

h
)

V
h
×
W
h
,
hhh9

where
g
=

(
f
(
t
)

u
h,tt
(
t
))

[
L
2
(Ω)]
2
.Weconsiderthepairofoperators(
S
h
,T
h
)defined
yb(
S
h
,T
h
):[
L
2
(Ω)]
2
−→
Σ
0
,h
×
(
V
h
×
W
h
)
g
7−→
(

σ,

u
)
.
hhTheevolutionproblem(5.5)canberewrittenas
¢¡
u
(
t
)=

T
h
P
h
0
f
(
t
)

d
2
u
2
h
(
t
)
,
td∼
h
¡
d
2
u
¢


σ
(
t
)=

S
h
P
h
0
f
(
t
)

dt
2
h
(
t
)
,
hwhere
P
h
0
istheL
2
-orthogonalprojectionfrom[
L
2
(Ω)]
2
onto
V
h
.
Inparticular
¡
0
d
2
u
h
¢
(5.7)
u
h
(
t
)=

T
h,
1
P
h
f
(
t
)

dt
2
(
t
)
.
RLetusshowthattheoperator
T
h,
1
|
V
h
:
V
h

V
h
isinvertible.Supposethat
Ω
g.T
h,
1
gdx
=
0.Thenfrom(5.6)weget
a
(
σ,σ
)=0i.e.
∼∼hhZZ1
|
σ
h
|
2
dx
+1
|
p
h
|
2
dx
=0
.
λµ2ΩΩHence
σ
h
=0and
p
h
=0i.e.
(5.8)
σ
=0
.
∼hBythefirstequationof(5.6),itnowfollowsthat:
b
(
τ

,

u
)=0
,

τ


Σ
0
,h
.
hhhTheinf-supinequality(5.10)yields

u
h
=(
u
h

h
)=0
.
RThusinparticular,if
Ω
g.
R
T
h,
1
gdx
=0,then
T
h,
1
g
=0
.
Now,if
g
h

V
h
and
Ω
g
h
.T
h,
1
g
h
dx
=0,thenby(5.8)wehave

σ
=0andbythe
hsecondequationof(5.6),wehave(
g
h
,v
h
)=0
,

v
h

V
h
.
Thus
.0=ghFinally,wehaveprovedthat
T
h,
1
|
V
h
:
V
h
−→
V
h
isinjective,thusinvertible.From(5.7)
follows:
2udh(5.9)

P
h
0
f
(
t
)+
2
(
t
)=(
T
h,
1
|
V
h
)

1
(
u
h
(
t
))
.
td01

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