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AdvancedNumericalMethodsand

TheirApplicationstoIndustrialProblems

AdaptiveFiniteElementMethods

LectureNotes

SummerSchool

YerevanStateUniversity

Yerevan,Armenia

2004

AlfredSchmidt,ArsenNarimanyan

CenterforIndustrialMathematics

UniversityofBremen

Bremen,Germany

www.math.uni-bremen.de/zetem/

1Introduction,motivation1

2Mathematicalmodeling5

2.1Density,

3Functionalanalysisbackground15

5Finiteelementapproximation25

8Meshrenementandcoarsening51

9Adaptivestrategiesforellipticproblems63

10Aspectsofecientimplementation70

11Errorestimatesviadualtechniques77

12Parabolicproblems-heatequation81

14Adaptivemethodsforparabolicproblems93

15TheStefanproblemofphasetransition98

16Thecontinuouscastingproblem107

16.4RobinIn

16.5DirichletIn

owCondition..........................115

References150

iii

1Introduction,motivation

1.1Introduction

knowledgeoftheprocess. ofEulerandLagrange.

ThestudyofPDEscontainstwomainaspects:

uniquenessofsolutions,

2.NumericalapproximationofPDEs.

methodsveryfastandproductive. modelequations. 1 resolutionofdierentscales. behaviourofobjectsisobservable. theproblem.

1.Meteorology:

2.CivilEngineering:

mechanics. nalin 2

3.Biology:

care. insects,orpetridishes. averagingoverindividuals. 3 thetreatmentwithPDEs.

4.TracFlow:

astoavoidthetracjamsontheroads.

Figure1.4:Trac

ow issimilartogas 4

2Mathematicalmodeling

2.1Density,

ux,andconservation uxand aconservationlaw. twosimpleexamples.

SugarinCoee(Concentration)

Temperatureinaspoon

orinapot(HeatDensity) smallvolumeelementcontainingP. dV

P=(x,y,z)

VolumeElementVContainingP

(V;t)=MassinVattimet jVj

Vtobecomesmallerandsmaller.

5 idea.

Analogoustomassdensityonecandene

Electricalchargedensity,

Populationdensity(biologicalorganisms),

movement.

So,wewanttodenethe

owvector( ux)q(P;t)inapointPattimettobetherate ux canbealsodenedthroughalimitingprocess. 6 uid owinatube. ux(orthe uid t Pt t+t uxrate uxvectorin dFn P dFn P

Thenthe

uxinthedirectionnFisdenedas jFjt ux q(P;t)n.Herethe ow inthepointPattimet. 7 aconservationofenergy(e.g.heat)inarod. xx12 dVdVdV xx1221...N elementsV1;V2;:::;VN.Thus E [x1;x2](t)=NX i=1E(Vi;t)=NX i=1jVijE(Vi;t) jVij|{z}

Energydensitye(Vi;t)

ForjVj!0weget

E [x1;x2](t)=Z x2 x

1e(x;t)dx

expressedinthefollowingway

ConservationLaw:

Energy

inx1 inx2 8 x1xq(x

2q(x, t)21

, t) Thusd dtE[x1;x2](t)=q(x1;t)q(x2;t) q(x2;t)q(x1;t)=Z x2 x 1@ @xq(x;t)dx andfromequation d dtE[x1;x2](t)=@@tZ x2 x

1e(x;t)dx=Z

x2 x

1@@te(x;t)dx

follows Z x2 x 1@ @te(x;t)dx=Z x2 x

1@@xq(x;t)dx

or Zx2 x 1 @te(x;t)+@@xq(x;t) dx=0 calculuscourse,theintegrandmustvanish: @te(x;t)+@@xq(x;t)=0 forallpointsxandforanytimet.

ConservationEquation:

@te(x;t)=@@xq(x;t) 9 rateofchangeofenergy=spacechangeofthe ux+Production @te(x;t)=@@xq(x;t)+f(x;t)

Chemistry:

Biology:

o),

TracFlow:

Drivinginandoutoftheroad,

Conservationlawinndimensions:

owoverallfaces.Using uxvector,qi. derivativesofthe @te(x;t)=nX i=1@@xiq i(x;t)+f(x;t); 10 @te(x;t)=divq(x;t)+f(x;t)

2.2PDEsasamodelingtool

e(x;t)=cpT(x;t)

Theheat

owsfrom owthroughahomogeneous ow, @T=@xi.e. q(x;t)=k@ @xT(x;t) tion c p@T @t=k@2T@x2+f

Thetemperaturebalancingthroughtheheat

owwithoutsourcesisillustratedinthe nextgure. Time 11 e(x;t)andheat q(x;t)=0 B B @k@ @x1T(x;t) k@ @x2T(x;t) k@ @x3T(x;t)1 C C A =krT(x;t) isequal to e(x;t)=cpT(x;t) c p@T @t=div(krT)+f c p@T @t=kT+f

Poissonequation:Imaginewehavesome

uidwhichisincompressibleandirrotational inadomain

R3andwewouldliketofollowthe

owofthe uid.Thisisanoftenmet probleminseveralareasof uidmechanics.Assumethatthe uidhasadensity(x;t) andthe (2.1)tdiv(v)=Q

Thedenitionofanincompressible

owyieldsthatthedensityisconstant,soweget (2.2)divv=Q=:=~Q

Nowweusethefactthatthe

(2.3)v=ru: (2.4)u=~Q; ascalarequationforthepotentialofthe ow.Onceuisdeterminedbysolvingthe 12 boundaryorthe uxacrosstheboundary,etc.). problem. mathematicalformulation.Givenacontainer

Rnwithliquid(water)withatem-

Figure2.1:MeltingoficeintheWater

beformulatedinthefollowingway: given:

Rn,0andT0=T0(x)(x2

).Atatimet>0, iscomposedoftwosub- domains sand compute: s;ltheheat equationisfullled (2.5)c@T @tdiv(krT)=f ineachofthephases,butmaydierbetweenthem.

T=0ont;

13 ofinterfacethat moveswithvelocityv,anddenotebyqltheheat ux(perunitsurface)contributedby theliquidphaseandbyqstheheat orreleasedatarateLvnd isequalto(qlnqsn)d surfaced ,weobtain (qlnqsn)d =Lvnd

Thisyields(dividingbothsidesbyd

andusingtheFourierlawfortheheat ux)the

Lvn=k@T

@njlk@T@njsont:

Onthexedboundary@

(2.6)T=TD

Asforinitialconditions,wehave

T(x;0)=T0(x)x2

(2.7) (0)=0(2.8) themodel.Typicalquestionsare: 14

3Functionalanalysisbackground

3.1BanachspacesandHilbertspaces

Let(X;d)beametricalspace.

d(xk;xl)!0fork;l!1 toalimitinX. inducedmetriciscalledaBanachspace. Let

2Rnbeaopenandboundedset.

C m;( )tobethesubspaceofCm( aconstanthsuchthat jDf(x)Df(y)jhjxyjx;y2 for0m.ThefunctionsfromthespaceCm;( )arecalledHoldercontinuousand

Lipschitzcontinuousforthecase=1.

)isthenaBanachspace withnormgivenby kfkCm;( )=kfkCm( )+max0jjmsup x;y2 ;x6=yjDf(x)Df(y)jjxyj kxkX=p (x;x)X;x2X 15 kx0kX0=supfjx0(x)j:kxkX1g: andX0. thatx0canberepresentedas x

0(y)=(x;y)forally2X:

Inthiscase

kx0kX0=kxkX itsnormeddual.

3.2BasicconceptsofLebesguespaces

Let denotebyLp( (3.1)Lp( ):=u:kukLp( )<1; wherethenormkukLp( )isdenedinthefollowingway:for1p<1 (3.2)kukLp( ):=0 @Z ju(x)jpdx1 A1 p andforp=1weset (3.3)kukL1( ):=esssup x2 fju(x)jg:

TheelementsofLp(

)whichareequalalmost everywhereon

Theorem3.2.For1p;q1,1

p+1q=1andu;v2Lp( );w2Lq( )wehave ku+vkLp( )kukLp( )+kvkLp( );Minkowski'sinequality; Z ju(x)w(x)jdxkukLp( )kwkLq( )Holder'sinequality; Z ju(x)w(x)jdxkukL2( )kwkL2( )Schwarz'inequality: 16 bibliography.

Theorem3.3.Assume1pq1.Then

1.Lp( )isaBanachspace.

2.ifu2Lq(

),thenu2Lp( )and (3.4)kukLp( )(volume )(1 p1q)kukLq( (3.5)Lq( ),!Lp(

4.ifp<1thenLp(

)isseparable. 5.Lp( )isre exiveifandonlyif13.3Weakderivatives Let beadomaininRn.DenotebyC1 0( )thesubsetofC1functionswithcompact supportin .BysupportofthefunctionudenedonacompactK wemean suppu= fx2K:u(x)6=0g; andifsuppu thenwesaythatuhasacompactsupport. ,ifitisdenedon .The loc( 17

Denition3.6.Thefunctionu2L1

loc(quotesdbs_dbs8.pdfusesText_14

[PDF] [PDF] Advanced Numerical Methods and Their Applications to Industrial

AdvancedNumericalMethodsand

TheirApplicationstoIndustrialProblems

AdaptiveFiniteElementMethods

LectureNotes

SummerSchool

YerevanStateUniversity

Yerevan,Armenia

AlfredSchmidt,ArsenNarimanyan

CenterforIndustrialMathematics

UniversityofBremen

Bremen,Germany

Contents

1Introduction,motivation1

2Mathematicalmodeling5

2.1Density,

3Functionalanalysisbackground15

5Finiteelementapproximation25

8Meshrenementandcoarsening51

9Adaptivestrategiesforellipticproblems63

10Aspectsofecientimplementation70

11Errorestimatesviadualtechniques77

12Parabolicproblems-heatequation81

14Adaptivemethodsforparabolicproblems93

15TheStefanproblemofphasetransition98

16Thecontinuouscastingproblem107

16.4RobinIn

16.5DirichletIn

References150

1Introduction,motivation

1.1Introduction

ThestudyofPDEscontainstwomainaspects:

2.NumericalapproximationofPDEs.

1.Meteorology:

2.CivilEngineering:

3.Biology:

4.TracFlow:

Figure1.4:Trac

2Mathematicalmodeling

2.1Density,

SugarinCoee(Concentration)

Temperatureinaspoon

P=(x,y,z)

VolumeElementVContainingP

Vtobecomesmallerandsmaller.

Analogoustomassdensityonecandene

Electricalchargedensity,

Populationdensity(biologicalorganisms),

So,wewanttodenethe

Thenthe

Energydensitye(Vi;t)

ForjVj!0weget

1e(x;t)dx

ConservationLaw:

Energy

2q(x, t)21

1e(x;t)dx=Z

1@@te(x;t)dx

1@@xq(x;t)dx

ConservationEquation:

Chemistry:

Biology:

TracFlow:

Drivinginandoutoftheroad,

Conservationlawinndimensions:

2.2PDEsasamodelingtool

Theheat

Thetemperaturebalancingthroughtheheat

Poissonequation:Imaginewehavesome

R3andwewouldliketofollowthe

Thedenitionofanincompressible

Nowweusethefactthatthe

Rnwithliquid(water)withatem-

Figure2.1:MeltingoficeintheWater

Rn,0andT0=T0(x)(x2

T=0ont;

Thisyields(dividingbothsidesbyd

Lvn=k@T

Onthexedboundary@

Asforinitialconditions,wehave