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105Finiteelementapproximationsoftopologicalderivatives
29
10501Familyoffiniteelements
InΩweconsiderafamilyoftriangulations{Th}h>0.WitheachelementT∈Th,
weassociatetwoparametersp(T)andσ(T),wherep(T)denotesthediameter
ofthesetT,andσ(T)isthediameterofthelargestballcontainedinT.Weset
h=maxT∈T
hp(T).Wemakethefollowingassumptionsonthetriangulations.
(H10)Regularityassumption:thereexistsσ>0suchthat
p(T)
σ(T)
≤σforT∈Th
andh>0.
(H11)Inverseassumption:thereexistsp>0suchthat
p(T)
h
≤pforT∈Thand
h>0.
(H12)WedenotebyΩh=∪T∈T
hTthedomainobtainedbyatriangulation,with
ΩhitsinteriorandwithaΩhitsboundary.ThenweassumethatthevertexesofTh
placedontheboundaryaΩhbelongsalsotoaΩ.
Considerthespaces:
Vh={vh∈C(Ω):vh|T∈P1(T)forT∈Thandvh=0inΩ\Ωh}
and
Wh={vh∈C(Ωh):vh|T∈P1(T)forT∈Th},
whereP1(T)isthespaceofpolynomialsofdegree1onT.Vhisavectorsubspace
ofH1
0(Ω)andWhisasubspaceofH1(Ω).
10502
Numericalsolutionofsemilinearproblem
ByvirtueoftheassumptionF∈C0,1(Ω×R)andbythemeanvaluetheorem,we
deducethefollowinglocalLipschitzcondition:forallM>0thereexistscM>0
suchthat:
|F(x,v1)−F(x,v2)|≤cM|v1−v2|,
x∈Ω,|v1|,|v2|≤M.
(1.49)
UsingclassicalargumentswecandeducefromthemonotonicityofF(x,.)and
(1.49),theexistenceofauniquesolutionto(1.13)inH1
0(Ω)∩C(Ω);werefer
to[39]fortheboundednessofthesolution.DuetotheconvexityofΩ,wecan
deducethatthesolutionisinH2(Ω)(see[13]).
Theweakformulationoftheequation(1.13)isthefollowing:
a(v,z)=(F(x,v),z)L2(Ω),z∈H
0(Ω),
1
where:
a(v,z)=∫
Ω
∇v(x)∇z(x)dx.
(1.50)
