Optimal a priori error estimates of parabolic optimal control problems with a moving point control

0.25 MB·

by  Dmitriy Leykekhman

Optimal a priori error estimates of parabolic
optimal control problems with a moving point
control
DmitriyLeykekhmanandBorisVexler
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2

r
a
M

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AbstractInthispaperweconsideraparabolicoptimalcontrolproblemwithaDirac

] typecontrolwithmovingpointsourceintwo spacedimensions.We discretizethe
A
problemwithpiecewiseconstantfunctionsintimeandcontinuouspiecewiselinear
N
finite elements in space. For this discretization we show optimal order of conver-
. gencewithrespecttothetimeandthespacediscretizationparametersmodulosome
h
t logarithmicterms.Erroranalysisforthesameproblemwascarriedoutintherecent
a
paper[17],however,theanalysistherecontainsaseriousflaw.Oneofthemaingoals
m
ofthispaperistoprovidethecorrectproof.Themainingredientsofouranalysisare
[
theglobalandlocalerrorestimatesonacurve,thathaveanindependentinterest.

2
v
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4 1 Introduction
0
3
0 Inthispaperweprovidenumericalanalysisforthefollowingoptimalcontrolprob-
. lem:
1
1 T α T
0 minJ(q,u):= ku(t)−uˆ(t)k2 dt+ |q(t)|2dt (1)
7 q,u 2 0 L2(Ω) 2 0
Z Z
1
subjecttothesecondorderparabolicequation
:
v
i u(t,x)−∆u(t,x)=q(t)δ , (t,x)∈I×Ω, (2a)
X t γ(t)
u(t,x)=0, (t,x)∈I×∂Ω, (2b)
r
a
u(0,x)=0, x∈Ω (2c)
DmitriyLeykekhman
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA, e-mail:
[email protected]
BorisVexler
Technical University of Munich, Chair of Optimal Control, Center for Mathematical Sciences,
Boltzmannstraße3,85748GarchingbyMunich,Germany,e-mail:[email protected]
1
2 DmitriyLeykekhmanandBorisVexler
andsubjecttopointwisecontrolconstraints
q ≤q(t)≤q a.e.inI. (3)
a b
HereI=(0,T),Ω⊂R2 isaconvexpolygonaldomainandδ istheDiracdelta
γ(t)
functionatpointx =γ(t)ateacht.Wewillassume:
t
Assumption1 • γ∈C1(I¯)andmaxt∈I¯|γ′(t)|≤Cγ.
Assumption2 • γ(t)⊂Ω ⊂⊂Ω ,foranyt∈I,withΩ ⊂⊂Ω.
0 1 1
The parameter α is assumed to be positive and the desired state uˆ fulfills uˆ ∈
L2(I;L∞(Ω)). The control bounds q ,q ∈R∪{±∞} fulfill q <q . The precise
a b a b
functional-analyticsettingisdiscussedinthenextsection.
For the discretization,we consider the standard continuouspiecewise linear fi-
nite elements in space and piecewise constant discontinuous Galerkin method in
time. This is a special case (r=0, s=1) of so called dG(r)cG(s) discretization,
seee.g.[14]fortheanalysisofthemethodforparabolicproblemsande.g.[25,26]
forerrorestimatesinthecontextofoptimalcontrolproblems.Throughout,wewill
denotebyhthespatialmeshsizeandbyk thesizeoftimesteps,seeSection3for
details.
Themainresultofthepaperisthefollowing.
Theorem1.Letq¯beoptimalcontrolfortheproblem(1)-(2)andq¯ betheoptimal
kh
dG(0)cG(1) solution. Then there exists a constantC independentof h and k such
that
kq¯−q¯ k ≤C |lnh|3(k+h2)+C |lnh|k kq¯k +kuˆk .
kh L2(I) γ L2(I) L2(I;L∞(Ω))
(cid:0) (cid:1)(cid:16) (cid:17)
We would also like to point out that in addition to the optimal order estimate,
modulologarithmicterms, our analysis does notrequire any relationshipbetween
thesizesofthespacediscretizationhandthetimestepsk.
The problem with fixed location of the point source (i.e. with δ (x) for some
x0
fixed x ∈Ω) starting with the work of Lions [23], was investigated in a number
0
of publications,see [2, 3, 10, 12, 28] for the continuousproblemand[16, 21, 22]
for the finite element approximation and error estimates. There is also a closely
relatedproblemofmeasuredvaluedcontrols,whichreceivedalotofattentionlately
[5,6,7,8,20].
TheproblemwithmovingDiracwasconsideredin[9,27]onacontinuouslevel.
The erroranalysis was carried outin the recentpaper [17]. However,the analysis
there contains a serious flaw. The last inequality in the estimate (3.33) in [17] is
not correct. One of the main goals of this paper is to provide the correct proof.
The main ingredientsof our analysis are the globaland local error estimates on a
curve,Theorem2andTheorem3,respectively.Theseresultsarenewandhavean
independentinterest.
Throughout the paper we use the usual notation for Lebesgue and Sobolev
spaces. We denoteby(·,·)Ω theinnerproductin L2(Ω) andby (·,·)I˜×Ω the inner
productinL2(I˜×Ω)foranysubintervalI˜⊂I.
Parabolicoptimalcontrolproblemswithamovingpointcontrol 3
Therestofthepaperisorganizedasfollows.InSection2wediscussthefunc-
tionalanalyticsettingoftheproblem,statetheoptimalitysystemandproveregular-
ityresultsforthestateandfortheadjointstate.InSection3weestablishimportant
global and local best approximationresults along the curve for the heat equation.
FinallyinSection4weproveourmainresult.
2 Optimal control problem andregularity
Inordertostatethefunctionalanalyticsettingfortheoptimalcontrolproblem,we
firstintroducetheauxiliaryproblem
v(t,x)−∆v(t,x)= f(t,x), (t,x)∈I×Ω,
t
v(t,x)=0, (t,x)∈I×∂Ω, (4)
v(0,x)=0, x∈Ω,
witharight-handside f ∈L2(I;Lp(Ω))forsome1<p<∞.Thisequationpossesses
auniquesolution
v∈L2(I;H1(Ω))∩H1(I;H−1(Ω)).
0
DuetotheconvexityofthepolygonaldomainΩthesolutionvpossessesanaddi-
tionalregularityfor p=2:
v∈L2(I;H2(Ω)∩H1(Ω))∩H1(I;L2(Ω)),
0
withthecorrespondingestimate
kvk +kvk ≤Ckfk , (5)
L2(I;H2(Ω)) t L2(I;L2(Ω)) L2(I;L2(Ω))
see, e.g.,[15].From theSobolevembeddingH2(Ω)֒→W1,s(Ω) foranys<∞in
twospacedimensionsandthepreviouslemmawecanestablishthefollowingresult
fors>2,
kvk ≤Cskvk ≤Cskfk . (6)
L2(I;W1,s(Ω)) L2(I;H2(Ω)) L2(I;L2(Ω))
The exact form of the constant can be traced, for example, from the proof of [1,
Thm.10.8].Inaddition,thereholdsthefollowingregularityresult(see[21]).
Lemma1.If f ∈L2(I;Lp(Ω))foranarbitrary p>1,thenv∈L2(I;C(Ω))and
kvk ≤C kfk ,
L2(I;C(Ω)) p L2(I;Lp(Ω))
whereC ∼ 1 ,as p→1.
p p−1
Wewillalsoneedthefollowinglocalregularityresult(see[21]).
Lemma2.LetΩ ⊂⊂Ω ⊂⊂Ω and f ∈L2(I;L2(Ω))∩L2(I;Lp(Ω )) for some
0 1 1
2≤p<∞.Thenv∈L2(I;W2,p(Ω ))∩H1(I;Lp(Ω ))andthereexistsaconstantC
0 0
4 DmitriyLeykekhmanandBorisVexler
independentof psuchthat
kvk +kvk ≤Cp(kfk +kfk ).
t L2(I;Lp(Ω0)) L2(I;W2,p(Ω0)) L2(I;Lp(Ω1)) L2(I;L2(Ω))
To introduce a weak solution of the state equation (2) we use the method of
transposition,(cf.[24]).Foragivencontrolq∈Q=L2(I)wedenotebyu=u(q)∈
L2(I;Lp(Ω)) with 2≤ p<∞ a weak solution of (2), if for allϕ∈L2(I;Lp′(Ω))
with 1+ 1 =1thereholds
p p′
hu,ϕiL2(I;Lp(Ω)),L2(I;Lp′(Ω))= w(t,γ(t))q(t)dt,
I
Z
where w∈L2(I;W2,p′(Ω)∩H1(Ω))∩H1(I;Lp′(Ω)) is the weak solution of the
0
adjointequation
−w(t,x)−∆w(t,x)=ϕ(t,x), (t,x)∈I×Ω,
t
w(t,x)=0, (t,x)∈I×∂Ω, (7)
w(T,x)=0, x∈Ω.
Theexistenceofthisweaksolutionu=u(q)followsbydualityusingtheembedding
L2(I;W2,p′(Ω))֒→L2(I;C(Ω))forp′>1.UsingLemma1wecanproveadditional
regularityforthestatevariableu=u(q).
Proposition2.1 Without lose of generality we assume 2 ≤ p <∞. Let q∈Q =
L2(I) be given and u=u(q) be the solution of the state equation (2). Then u∈
L2(I;Lp(Ω)) for any p <∞ and the following estimate holds for p→∞ with a
constantCindependentof p,
kuk ≤Cpkqk .
L2(I;Lp(Ω)) L2(I)
Proof. Toestablishtheresultweuseadualityargument.Thereholds
1 1
kuk = sup (u,ϕ) , where + =1.
L2(I;Lp(Ω)) I×Ω p p′
kϕkL2(I;Lp′(Ω))=1
Let w be the solution to (7) forϕ∈L2(I;Lp′(Ω)) with kϕkL2(I;Lp′(Ω)) =1. From
Lemma1,w∈L2(I;C(Ω))andthefollowingestimateholds
C C
kwkL2(I;C(Ω))≤ p′−1kϕkL2(I;Lp′(Ω))= p′−1 ≤Cp, as p→∞.
Thus,
Parabolicoptimalcontrolproblemswithamovingpointcontrol 5
kuk = sup (u,ϕ)
L2(I;Lp(Ω)) I×Ω
kϕkL2(I;Lp′(Ω))=1
= q(t)w(t,γ(t))dt≤kqk kwk ≤Cpkqk .
L2(I) L2(I;C(Ω)) L2(I)
I
Z
Remark1.WewouldliketonotethattheaboveregularityrequiresonlyAssumption
2onγ.Higherregularityofγisneededforoptimalordererrorestimatesonly.
Afurtherregularityresultforthestateequationfollowsfrom[13].
Proposition2.2 Let q∈Q=L2(I) be given and u=u(q) be the solution of the
stateequation(2).Thenforeach1<s<2thereholds
u∈L2(I;W1,s(Ω)) and u ∈L2(I;W−1,s(Ω)).
0 t
Moreover,thestateufulfillsthefollowingweakformulation
hu,ϕi+(∇u,∇ϕ)= q(t)ϕ(t,γ(t))dt forall ϕ∈L2(I;W1,s′(Ω)),
t 0
I
Z
where 1 + 1 = 1 and h·,·i is the duality product between L2(I;W−1,s(Ω)) and
s′ s
L2(I;W1,s′(Ω)).
0
Proof. For s<2 we have s′ >2 and thereforeW1,s′(Ω) is embedded intoC(Ω¯).
0
Thereforetheright-handsideq(t)δ ofthestateequationcanbeidentifiedwithan
γ(t)
elementin L2(I;W−1,s(Ω)). Using the result from[13, Theorem5.1]on maximal
parabolic regularity and exploiting the fact that −∆: W1,s(Ω)→W−1,s(Ω) is an
0
isomorphism,see[19],weobtain
u∈L2(I;W1,s(Ω)) and u ∈L2(I;W−1,s(Ω)).
0 t
Given the above regularity the corresponding weak formulation is fulfilled by a
standarddensityargument.
Asthenextstepweintroducethereducedcostfunctional j: Q→Ronthecon-
trolspaceQ=L2(I)by
j(q)=J(q,u(q)),
whereJ isthecostfunctionin(1)andu(q)istheweaksolutionofthestateequa-
tion (2) as defined above. The optimal control problem can then be equivalently
reformulatedas
min j(q), q∈Q , (8)
ad
wherethesetofadmissiblecontrolsisdefinedaccordingto(3)by
Q ={q∈Q|q ≤q(t)≤q a.e.inI}. (9)
ad a b
By standardargumentsthis optimizationproblempossesses a uniquesolution q¯∈
Q=L2(I)withthecorrespondingstateu¯=u(q¯)∈L2(I;Lp(Ω))forall p<∞,see
6 DmitriyLeykekhmanandBorisVexler
Proposition 2.1 for the regularity of u¯. Due to the fact, that this optimal control
problem is convex, the solution q¯ is equivalently characterized by the optimality
condition
j′(q¯)(∂q−q¯)≥0 forall∂q∈Q . (10)
ad
The(directional)derivative j′(q)(∂q)forgivenq,∂q∈Qcanbeexpressedas
j′(q)(∂q)= (αq(t)+z(t,γ(t)))∂q(t)dt,
I
Z
wherez=z(q)isthesolutionoftheadjointequation
−z(t,x)−∆z(t,x)=u(t,x)−uˆ(t,x), (t,x)∈I×Ω, (11a)
t
z(t,x)=0, (t,x)∈I×∂Ω, (11b)
z(T,x)=0, x∈Ω, (11c)
andu=u(q)ontheright-handsideof(11a)isthesolutionofthestateequation(2).
Theadjointsolution,whichcorrespondstotheoptimalcontrolq¯isdenotedbyz¯=
z(q¯).
The optimality condition (10) is a variational inequality, which can be equiva-
lentlyformulatedusingtheprojection
P : Q→Q , P (q)(t)=min q ,max(q ,q(t)) .
Qad ad Qad b a
Theresultingconditionreads: (cid:0) (cid:1)
1
q¯(t)=P − z¯(t,γ(t)) . (12)
Qad α
(cid:18) (cid:19)
Inthenextpropositionweprovideregularityresultsforthesolutionoftheadjoint
equation.
Proposition2.3 Letq∈Qbegiven,letu=u(q)bethecorrespondingstatefulfill-
ing(2)andletz=z(q)bethecorrespondingadjointstatefulfilling(11).Then,
(a)z∈L2(I;H2(Ω)∩H1(Ω))∩H1(I;L2(Ω))andthefollowingestimateholds
0
k∇2zk +kzk ≤C(kqk +kuˆk ).
L2(I;L2(Ω)) t L2(I;L2(Ω)) L2(I) L2(I;L2(Ω))
(b) IfΩ ⊂⊂Ω, thenz∈L2(I;W2,p(Ω ))∩H1(I;Lp(Ω )) for all2≤ p<∞and
0 0 0
thefollowingestimateholds
k∇2zk +kzk ≤Cp2(kqk +kuˆk ).
L2(I;Lp(Ω0)) t L2(I;Lp(Ω0)) L2(I) L2(I;L∞(Ω))
Proof. (a) Theright-handsideoftheadjointequationfulfillsu−uˆ∈L2(I;Lp(Ω))
forall1<p<∞,seeProposition2.1.DuetotheconvexityofthedomainΩwe
directlyobtainz∈L2(I;H2(Ω)∩H1(Ω))∩H1(I;L2(Ω))andtheestimate
0
Parabolicoptimalcontrolproblemswithamovingpointcontrol 7
k∇2zk +kzk ≤Cku−uˆk .
L2(I;L2(Ω)) t L2(I;L2(Ω)) L2(I;L2(Ω))
TheresultfromProposition2.1leadsdirectlytothefirstestimate.
(b)FromLemma2for p≥2wehave
k∇2zk +kzk ≤Cpku−uˆk .
L2(I;Lp(Ω0)) t L2(I;Lp(Ω0)) L2(I;Lp(Ω))
Hence,bythetriangleinequalityandProposition2.1weobtain
ku−uˆk ≤C pkqk +kuˆk .
L2(I;Lp(Ω)) L2(I) L2(I;L∞(Ω))
(cid:16) (cid:17)
Thatcompletestheproof.
3 Discretization andthe bestapproximationtype results
3.1 Space-timediscretizationand notation
Fordiscretizationoftheproblemundertheconsiderationweintroduceapartitions
of I =[0,T] into subintervalsI =(t ,t ] of length k =t −t , where 0=
m m−1 m m m m−1
t <t <···<t <t =T.Weassumethat
0 1 M−1 M
k ≤κk , m=1,...,M−1, forsome κ>0. (13)
m+1 m
The maximaltime step is denotedby k=max k . The semidiscrete space X0 of
m m k
piecewiseconstantfunctionsintimeisdefinedby
X0={v ∈L2(I;H1(Ω)): v | ∈P (I ;H1(Ω)), m=1,2,...,M},
k k 0 k Im 0 m 0
where P (I;V) is the space of constant functions in time with values in Banach
0
spaceV.WewillemploythefollowingnotationforfunctionsinX0
k
v+= lim v(t +ε):=v , v−= lim v(t −ε)=v(t ):=v , [v] =v+−v−.
m m m+1 m m m m m m m
ε→0+ ε→0+
(14)
LetT denoteaquasi-uniformtriangulationofΩwithameshsizeh,i.e.,T =
{τ}isapartitionofΩintotrianglesτofdiameterh suchthatforh=max h ,
τ τ τ
diam(τ)≤h≤C|τ|12, ∀τ∈T
hold.LetV bethesetofallfunctionsinH1(Ω)thatarelinearoneachτ,i.e.V is
h 0 h
theusualspaceofcontinuouspiecewiselinearfinite elements.We willrequirethe
modifiedCle´mentinterpolanti : L1(Ω)→V andtheL2-projectionP : L2(Ω)→
h h h
V definedby
h
(P v,χ) =(v,χ) , ∀χ∈V . (15)
h Ω Ω h
8 DmitriyLeykekhmanandBorisVexler
Toobtainthefullydiscreteapproximationweconsiderthespace-timefiniteelement
space
X0,1={v ∈X0: v | ∈P (I ;V ), m=1,2,...,M}. (16)
k,h kh k kh Im 0 m h
Wewillalsoneedthefollowingsemidiscreteprojectionπ :C(I¯;H1(Ω))→X0de-
k 0 k
finedby
πv| =v(t ), m=1,2,...,M, (17)
k Im m
andthefullydiscreteprojectionπ :C(I¯;L1(Ω))→X0,1definedbyπ =i π.
kh k,h kh h k
TointroducethedG(0)cG(1)discretizationwedefinethefollowingbilinearform
M M
B(v,ϕ)= ∑hv,ϕi +(∇v,∇ϕ) + ∑([v] ,ϕ+ ) +(v+,ϕ+) , (18)
t Im×Ω I×Ω m−1 m−1 Ω 0 0 Ω
m=1 m=2
whereh·,·i isthedualityproductbetweenL2(I ;W−1,s(Ω))andL2(I ;W1,s′(Ω)).
Im×Ω m m 0
Wenote,thatthefirstsumvanishesforv∈X0.Rearrangingtheterms,weobtainan
k
equivalent(dual)expressionforB:
M M−1
B(v,ϕ)=− ∑hv,ϕi +(∇v,∇ϕ) − ∑(v−,[ϕ] ) +(v−,ϕ−) . (19)
t Im×Ω I×Ω m k m Ω M M Ω
m=1 m=1
Inthetwofollowingtheoremsweestablishglobalandlocalbestapproximation
type results along the curve for the error between the solution v of the auxiliary
equation(4)anditsdG(0)cG(1)approximationv ∈X0,1definedas
kh k,h
B(v ,ϕ )=(f,ϕ ) forall ϕ ∈X0,1. (20)
kh kh kh I×Ω kh k,h
Since dG(0)cG(1) method is a consistent discretization we have the following
Galerkinorthogonalityrelation:
B(v−v ,ϕ )=0 forall ϕ ∈X0,1.
kh kh kh k,h
3.2 Discretizationofthecurve andthe weight function
To definefully discrete optimizationproblemwe will also requirea discretization
ofthecurveγ.Wedefineγ =πγby
k k
γ| =γ(t ):=γ ∈Ω , m=1,2,...,M, (21)
k Im m k,m 0
i.e., γ is a piecewise constant approximation of γ. Next we introduce a weight
k
function
σ(t,x)= |x−γ(t)|2+h2 (22)
q
andadiscretepiecewiseconstantintimeapproximation
Parabolicoptimalcontrolproblemswithamovingpointcontrol 9
σ(t,x)= |x−γ(t)|2+h2. (23)
k k
q
Define
σ :=σ| =σ(t ,x)=σ(t ,x). (24)
k,m k Im k m m
Onecaneasilycheckthatσandσ satisfythefollowingpropertiesforany(t,x)∈
k
I×Ω,
kσ−1(t,·)kL2(Ω),kσk−1(t,·)kL2(Ω)≤C|lnh|21, t∈I¯, (25a)
|∇σ(t,x)|, |∇σ(t,x)|≤C, (25b)
k
|∇2σ(t,x)|≤C|σ−1(t,x)|, (25c)
k k
|σ(t,x)|≤|∇σ(t,x)|·|γ′(t)|≤CC , (25d)
t γ
maxσ(x,t)≤Cminσ(x,t), ∀τ∈T. (25e)
x∈τ x∈τ
3.3 Globalerrorestimatealong thecurve
Inthissectionweprovethefollowingglobalapproximationresult.
Theorem2 (Global best approximation). Assume v and v satisfy (4) and (20)
kh
respectively.ThenthereexistsaconstantCindependentofkandhsuchthatforany
1≤p≤∞,
|(v−v )(t,γ(t))|2dt≤C|lnh|2×
kh k
I
Z
χ∈inXf0,1 kv−χk2L2(I;L∞(Ω))+h−4pkπkv−χk2L2(I;Lp(Ω)) .
k,h(cid:16) (cid:17)
Proof. To establish the result we use a duality argument. First, we introduce a
smoothed Delta function, which we will denote by δ˜ . This function on each I
γk m
isdefinedasδ˜ andsupportedinonecell,whichwedenotebyτ0,i.e.
γk,m m
(χ,δ˜ ) =χ(γ )=χ(γ(t )), ∀χ∈P1(τ0), m=1,2,...,M.
γk,m τm0 k,m m m
Inadditionwealsohave(see[31, Appendix])
kδ˜γkkWps(Ω)≤Ch−s−2(1−1p), 1≤p≤∞, s=0,1. (26)
Thusinparticularkδ˜γkkL1(Ω)≤C,kδ˜γkkL2(Ω)≤Ch−1,andkδ˜γkkL∞(Ω)≤Ch−2.
Wedefinegtobeasolutiontothefollowingbackwardparabolicproblem
10 DmitriyLeykekhmanandBorisVexler
−g(t,x)−∆g(t,x)=v (t,γ(t))δ˜ (x), (t,x)∈I×Ω,
t kh k γk
g(t,x)=0, (t,x)∈I×∂Ω, (27)
g(T,x)=0, x∈Ω.
Thereholds
M
v (t,γ(t))δ˜ (x)ϕ (t,x)dtdx= ∑ v (t,γ(t)) δ˜ (x)ϕ (t,x)dx dt
kh k γk kh kh k γk kh
ZI×Ω m=1ZIm (cid:18)ZΩ (cid:19)
M
= ∑ v (t,γ(t))ϕ (t,γ(t))dt
kh k kh k
m=1ZIm
= v (t,γ(t))ϕ (t,γ(t))dt.
kh k kh k
ZI
Letg ∈X0,1bedG(0)cG(1)solutiondefinedby
kh k,h
B(ϕ ,g )=(v (t,γ(t))δ˜ ,ϕ ) , ∀ϕ ∈X0,1. (28)
kh kh kh k γk kh I×Ω kh k,h
ThenusingthatdG(0)cG(1)methodisconsistent,wehave
T
|v (t,γ(t))|2dt=B(v ,g )=B(v,g )
kh k kh kh kh
0
Z (29)
M
=(∇v,∇g ) − ∑(v ,[g ] ) ,
kh I×Ω m kh m Ω
m=1
wherewehaveusedthedualexpression(19)forthebilinearformBandthefactthat
thelasttermin(19)canbeincludedinthesumbysettingg =0anddefining
kh,M+1
consequently[g ] =−g .Thefirstsumin(19)vanishesduetog ∈X0,1.For
kh M kh,M kh k,h
eacht, integrating by parts elementwise and using that g is linear in the spacial
kh
variable,bytheHo¨lder’sinequalitywehave
1
(∇v,∇gkh)Ω= 2∑(v,[[∂ngkh]])∂τ≤CkvkL∞(Ω)∑k[[∂ngkh]]kL1(∂τ), (30)
τ τ
where[[∂g ]]denotesthejumpsofthenormalderivativesacrosstheelementfaces.
n kh
FromLemma2.4in[29]wehave
∑k[[∂ngkh]]kL1(∂τ)≤C|lnh|21 kσk∆hgkhkL2(Ω)+k∇gkhkL2(Ω) ,
τ
(cid:16) (cid:17)
where∆ :V →V isthediscreteLaplaceoperator,definedby
h h h
−(∆ v ,χ) =(∇v ,∇χ) , ∀χ∈V .
h h Ω h Ω h
Toestimatetheterminvolvingthejumpsin(29),wefirstusetheHo¨lder’sinequality
andtheinverseestimatetoobtain