Discontinuous Galerkin finite element methods for a forward-backward heat equation

Abstract

A space-time finite element method is introduced to solve a model forward-backward heat equation. The scheme uses the discontinuous Galerkin method for the time discretization. An error analysis for the scheme is presented.

Full text

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DiscontinuousGalerkinFiniteElementMethods fora
Forw ard?Backw ardHeat Equation
DonaldA?Frenc h
?
Departmen tofMathematicalSciences? Univ ersityofCincinnati?POBo x???????Cincinnati? OH?????
January ??????
Abstract
A space?time?nite elementmethod isintroduced tosolvea modelforward?backwardheat
equation?TheschemeusesthediscontinuousGalerkinmethodfor the timediscretization? An
error analysisfor the schemeispresented?
Introduction?
Considerthefollowingforward?backwardinitial?boundaryvalueproblem??ndu?u?x?t? given
f?f?x?t? suchthat
xu
t
? u
xx
?fin ?????????????????
withboundaryconditions?BC?
u?? on?
?
??
?
???
andinitial?postconditions ?IC?PC?
u?? on?
?
?
??
?
?
???
where
?
?
? f???t??t??????g??
?
?f????t??t??????g??
?
?fx?x??????g??
?
?fx?x???????g?
and?inparticular?
?
?
isatt???and?
?
?
isatt???Wewilloftenusethenotation?????????
Thisequationarisesinelectronscattering?see?B????uid?ownearb oundarylayers?see?GM???
andrandomaccelerationofaparticle?see?FR??Therehavebeenseveralstudiesofthepartial
di?erentialequation?see?AFJK?foralonglistofreferencesonthisproblem??Numericalwork
?
ResearchsupportedinpartbytheTaftFoundationattheUniversityofCincinnatithroughtheirGrants?in?aid?
?

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includestheinvestigationofa?nitedi?erenceschemein?VK? andmetho dsusing thetransfor?
mationof???toa?rstordersystem carriedout in?AL??? Wewillinvestigateadiscon tinuous
Galerkin?dG???niteelement?fe?schemeapplieddirectlyto????????ThedGmethodisatechnique
toderivetime?stepping schemes for evolutionproblems ?see?J??? However?theproblem???????
doesnotlenditselfeasilytoatime?stepapproachduetothecouplingoftheforwardandbackward
parts?Althoughthefocusofthispaperisonacoercivityestimatewewillindicateinthelast
sectionaviableapproachthatcouldbeusedto?ndanapproximationwithtime?stepping?
ThedGmethodweintroducewillbesimilartothosein?J?forparabolicproblemsexceptthe
dataweintroduceoneachtimeslabwilloriginatefrom?pasttime?forx??and?futuretime?
forx??and?thus?alignitselfwiththeforward?backwardnatureoftheproblem?Ourmainresult
onauniformgridwherethere?nementsbyparameterh??inspaceandtimeareidenticaland
theapproximationspacesconsistofpiecewisepolynomialsofdegree?pand??is
?
?
?
?
?e
?x
?
?
?
?
L
?
???
?D?e??Ch
p
?p??????????????
whereeistheerrorandDisameasureofthejumpsintheerrorthatoccuratthetimenodes?
Ourgeneraltheoremwillalsoincludethecasewherespaceandtimere?nementscouldbedi?erent
ascouldthedegreesofthepiecewisepolynomialfunctions?
Wewillusethefollowingnotation?ForadomainAwede?netheL
?
innerproduct
?w?v?
A
?
Z
A
wvdAandnormjjwjj
?
A
??w?w?
A
?
Aweakformulationwhichisusedtomotivatethe?niteelementschemecanbefoundbymultiplying
???byv?H
?
?
???andintegratingover??ThesetH
m
?A?forsomedomainAistheSobolevspace
withfunctionsthathavemderivativesinL
?
?A??ThespaceH
?
?
?A?isfunctionsinH
?
?A?which
havetracezeroonthe?A?Integration?by?partsandtheBCare
?
usedtorewritetheterminvolving
u
xx
?Thesolution?u?u???t??of???????thensatis?esforeacht??????
?xu
t
?v?
?
??u
x
?v
x
?
?
??f?v???v?H
?
?
??????
wherewenoteu???t??H
?
?
????Bysettingv?uwehave
?
?
d
dt
?xu?u?
?
?ku
x
k
?
?
??f?u??
De?ning
kwk
????
?max
v?H
?
?
???
?w?v?
kv
x
k
?
weobtain?usingthearithmetic?geometricmeaninequality
ab?
?
?
a
?
?
?
??
b
?
???
?

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with???that
d
dt
?xu
?
???
?
?ku
x
k
?
?
?kfk
?
????
?
In tegrationwithrespect totover??? ??gives
kx
???
uk
?
?
?
?
?kjxj
???
uk
?
?
?
?
?ku
x
k
?
?
?
Z
?
?
kfk
?
????
dt????
Thisisafundamentalaprioriestimatefortheproblemandonewewillmimicinouranalysis?The
estimate???showsthatif???????hasasolutionitwillbeunique?Wewillassumethatthereisa
uniquesolutionwhichisasregularasrequiredbytheanalysis?
Throughoutthispap erCwilldenoteagenericpositiveconstant?notnecessarilythesameat
di?erentoccurences?WewillusethePoincar?einequality?
kuk
?
?Cku
x
k
?
foru?H
?
?
???????
inthesucceedingsections?
ApproximationMethod?
Inthissectionweintroducethespace?time?niteelementschemethatusesthedGmethodforthe
timediscretization?
We?rstspecifythegrids?Leth????M???andx
j
?jhwherej???????????????M????
Letk???Nandt
j
?jkforj?????????N?AlsoletI
n
??t
n??
?t
n
??Wehavekeptthegrids
uniformentirelyforsimplicity?weexpectourresultswouldstillholdforquasi?uniformmeshes?
Wenowturntothede?nitionoftheapproximationspaces?Letp???LetX
p
h
bethespace
ofcontinuouspiecewisepolynomialsofdegree?ponthespatialgridwhicharezeroat???Let
q???LetD
q
k
bepiecewisepolynomialsofdegree?qwhereq??onthetemporalgrid??Note
thatfunctionsinD
q
k
arenotnecessarilycontinuousatt?t
j
?Alsode?ne
S
n
???I
n
?Z
n
?
?lim
t?t
n
?t?t
n
Z???t??Z
n
?
?lim
t?t
n
?t?t
n
Z???t??and?Z
n
??Z
n
?
?Z
n
?
?
LetS
hk
bethecollectionoffunctions?V?inX
p
h
?D
q
k
suchthatthefollowingholds
V
?
?
??on?
?
andV
N
?
??on?
?
????
WiththesenotationsthedGmethodisde?nedbythefollowingvariationalproblem??ndU?S
hk
suchthat
B
n
?U?????f???
S
n
???X
p
h
?P
q
?I
n
?????
where
B
n
?U?????xU
t
???
S
n
??U
x
??
x
?
S
n
??x?U
n??
???
n??
?
?
?
?
??x?U
n
???
n
?
?
?
?
?
?

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Ifwelet
B?w?v??
N
X
n??
B
n
?w?v?
thenproblem????canberestatedasfollows??ndU?S
hk
suchthat
B?U?????f???
?
???S
hk
?????
Notethatthe solutionandtest spacesarethesameand?nitedimensional?Onceweestablish
uniquenesstheexistencepropertywillfollow?
Beforederivingourstabilityestimatesforthescheme????wewilldescribeourassumptionson
theapproximationspaces?see?C???Wewillusetheinverseproperty
k?
t
k
I
n
?Ck
??
k?k
I
n
???P
r
?I
n
??????
WeassumethereexistsanapproximationoperatorA
hk
whichmapssmoothfunctionsintoS
hk
and
satis?es
N
X
n??
?
?
?
?
?
?
d
dt
?
?
?
d
dx
?
r
?I?A
hk
?v
?
?
?
?
?
?
S
n
?C
?
k
q????
?h
p?r??
?
?
????
where??????r?????and
k?I?A
hk
?v???t
n
?k
?
?Ch
p??
n???????????N?????
Wenowshowthatthebilinearform?B?w?v?ispositive de?niteoveraclassoffunctions?see?F?
forarelatedargument??Let
D?v??
?
N??
X
n??
?
?
?
jxj
???
?v
n
?
?
?
?
?
?
?
?
?
?
?
jxj
???
v
?
?
?
?
?
?
?
?
?
?
?
jxj
???
v
N
?
?
?
?
?
?
Lemma?Supposev?v
x
?L
?
????vmayhavejumpsintimeatlevels
?
t
n
?v
t
?L
?
?S
n
?forn?
????????N?v
?
?
??on?
?
?
?andv
N
?
??on?
?
?
then
B?v?v??kv
x
k
?
?
?
?
?
D?v?????
?Sincethejumps??v
n
??aredi?erencesonvonecanmakeaninterpretationofDinvolvingtime
derivativesofv??
Proof?Wewillstartwiththeleftsideof????usingthenotationv
n
?v???t
n
??
B
n
?v?v???xv
t
?v?
S
n
?kv
x
k
?
S
n
?
?
x?v
n??
??v
n??
?
?
?
?
?
?
x?v
n
??v
n
?
?
?
?

Page 5

?
?
?
h
?
xv
n
?
?v
n
?
?
?
?
?
xv
n??
?
?v
n??
?
?
?
i
?
?
?
xv
n??
?
?v
n??
?
?
?
?
?
?
xv
n??
?
?v
n??
?
?
?
?
?
?
h
?
xv
n
?
?v
n
?
?
?
?
?
?
xv
n
?
?v
n
?
?
?
?
i
?kv
x
k
?
S
n
?
?
?
?
?
?
xv
n
?
?v
n
?
?
?
?
?
?
xv
n
?
?v
n
?
?
?
?
?
?
?
?
?
xv
n
?
?v
n
?
?
?
?
?
?
?
?
?
xv
n??
?
?v
n??
?
?
?
?
?
?
xv
n??
?
?v
n??
?
?
?
?
?
?
?
?
?
xv
n??
?
?v
n??
?
?
?
?
?kv
x
k
?
S
n
?
Summingovernfrom?toNweobtain
B?v?v??kv
x
k
?
?
?
?
?
?
?
?
?
jxj
???
v
N
?
?
?
?
?
?
?
?
?
?
jxj
???
v
?
?
?
?
?
?
?
?
?
?
?
N??
X
n??
?
jxj??v
n
?
?
?
??v
n
?
v
n
?
???
?
?
?
?
?
?
N
X
n??
?
x?
?
v
n??
?
?
?
??v
n??
?
v
n??
?
???
?
?
?
?
?
?
N??
X
n??
?
x?v
n
?
?
?
??
?
?
?
?
?
?
N
X
n??
?
jxj?v
n??
?
?
?
??
?
?
?
Shiftingtheindicesinthesumsweobtain
B?v?v??kv
x
k
?
?
?
?
?
?
?
?
?
jxj
???
v
N
?
?
?
?
?
?
?
?
?
?
jxj
???
v
?
?
?
?
?
?
?
?
?
?
?
N??
X
n??
?
jxj
?
?v
n
?
?
?
??v
n
?
v
n
?
??v
n
?
?
?
?
??
?
?
?
?
?
?
N
X
n??
?
x??v
n??
?
?
?
??v
n??
?
v
n??
?
??v
n??
?
?
?
???
?
?
?
?
Theestimate????followsfromthisbycompletingthesquaresinthelasttwosummations??
Wenowobservethattheproblemde?nedby????hasatmostonesolution?Ifthereweretwo
solutionsandZ?S
hk
wastheirdi?erencethen
B?Z????????S
hk
?
Letting??Zwehavefrom ????that
??B?Z?Z??kZ
x
k
?
?
?
?
?
D?Z?
ApplyingthePoincar?einequality?????weconcludethatkZk
?
??showingthereisonlyonesolution?
Sincethenumberofdegreesoffreedominthesolutionandtestspacesisthesamewealsoknow
thereexistsasolution?
ErrorEstimate?
InthissectionwedevelopanestimateoftheaccuracyofthedGscheme????assumingthesolution
?