Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations

0.2 MB·English

by  Guido Kanschat

Multigrid methods for Hdiv-conforming
5
discontinuous Galerkin methods for the Stokes
1
0
2 equations

n
a
J G. Kanschat∗ Y. Mao†

4
2 March 17,2014

]
A
N
Abstract
.
h AmultigridmethodfortheStokessystemdiscretizedwithanHdiv-conforming
t discontinuousGalerkinmethodispresented. Itactsonthecombinedvelocityand
a
m pressurespacesandthusdoesnotneedaSchurcomplementapproximation. The
smoothers used are of overlapping Schwarz type and employ a local Helmholtz
[
decomposition.Additionally,weusethefactthatthediscretizationprovidesnested

1 divergence free subspaces. We present the convergence analysis and numerical
v evidence that convergence rates arenot only independent of mesh size, but also
1 reasonablysmall.
2
0
6 1 Introduction
0
1. TheefficientsolutionoftheStokesequationsisanimportantstepinthedevelop-
0 mentoffastflowsolvers. Inthispaperwepresentanalysisandnumericalresults
5 for amultigridmethod withsubspace correctionsmoother, whichperformsvery
1 efficientlyondivergence-conformingdiscretizationswithinteriorpenalty. Weob-
: tainconvergenceratesfortheStokesproblemwhicharecomparabletothosefor
v
theLaplacian.
i
X Multigrid methods are known to be the most efficient preconditioners and
r solversfordiffusionproblems. Nevertheless,forStokesequations,thedivergence
a constraint makesthesolutionprocessmorecomplicated. Atypicalsolutionem-
ploystheuseofblockpreconditioners,e.g.[13,22,27,28],buttheirdisadvantage
is,thattheirperformanceislimitedbytheinf-supconstantoftheproblem. This
could be avoided, if the multigrid method operated on the divergence free sub-
spacedirectly, andthuswouldnothavetodealwiththesaddlepoint problemat
all. Suchmethodshavebeendevelopedindifferentcontextandhaveprovenvery
successfulasreportedforinstancebyHiptmair[18]forMaxwellequationsandby
Scho¨berl[33]forincompressibleelasticitywithreducedintegration.
∗[email protected], Interdisziplina¨res Zentrum fu¨r Wissenschaftliches Rechnen
(IWR),Universita¨tHeidelberg,ImNeuenheimerFeld368,69120Heidelberg,Germany
†[email protected],DepartmentofMathematics,TexasA&MUniversity,3368TAMU,College
Station,TX77843,USA
1
The main ingredients into such a method are a smoother which operates on
thedivergencefreesubspaceandagridtransferoperatorfromcoarsetofinemesh
which maps the coarse divergence free subspace into the fine one. The second
objectivecanbeachievedbyusingamixedfiniteelementdiscretizationforwhich
theweakly divergence freefunctions arepoint-wisedivergence free. For sucha
discretization, thenatural finiteelement embedding operator fromcoarse tofine
meshdoesnotincreasethedivergenceofafunction. Discretizationsofthistype
areavailable,suchasforinstanceinScottandVogelius[36,39],Neilanandcoau-
thors[14,16]andZhang[41,42].Here,wefocusonthedivergenceconformingdis-
continuousGalerkin(DivDG)methodofCockburn,Kanschat,andScho¨tzau[11]
duetoitssimplicity.
FollowingtheapproachbyScho¨berl[33],inordertostudysmoothersforthe
Stokes equations, we first consider a problem on the velocity space only with
penaltyforthedivergence. Thisleadstoasingularlyperturbedproblemwithan
operatorwithalargekernel.Whenitcomestosmoothersforsuchoperators,there
aretwobasicoptions. Oneapproach istosmooththekernel spaceexplicitly, as
proposedforinstancebyHiptmair[18]andXuin[19].Theotheroptionwaspre-
sentedbyArnold,Falk,andWintherin[2,3]andsmoothensthekernelimplicitly,
whileneveremployinganexplicitdescriptionofit.
Wefollowtheimplicitapproachandusethesamedomaindecompositionprin-
ciple (i.e additive and multiplicative Schwarz methods and vertex patches), but
insteadoftheMaxwellordivergencedominatedmassmatrixasin[2,3]applyitto
theDivDGStokesdiscretization.Then,weprovetheconvergenceofthemultigrid
methodwithvariableV-cyclealgorithmforthesingularlyperturbedproblem.The
second pillar we rest on is the equivalence between singularly perturbed, diver-
gence dominated ellipticforms and mixed formulations established by Scho¨berl
in[32,33]. Thisequivalenceallowsustoapplythesmoothertoamixedformula-
tion of nearly incompressible elasticity and then to proceed to the Stokes limit.
As far as we know, the combination of these techniques has not been applied
the DivDG method in [11]. Since our analysis is based on domain decompo-
sition, fundamental results are also drawn from the seminal paper by Feng and
Karakashian [15]ondomain decomposition for discontinuous Galerkinmethods
forellipticproblems.
There is a close relation between our technique and the smoother suggested
byVankain[38]fortheMACscheme: theMACschemecanbeconsideredthe
lowest order case of the DivDG methods (see [23]). In this case, the subspace
decomposition structure of Vanka smoother corresponds to Neumann problems
on cells, while our smoother is based on Dirichlet problems for vertex patches.
GeneralizationsoftheVankasmootherhavebeenappliedsuccessfullytodifferent
other discretizations albeit their velocity-pressure spaces are not matched in the
senseof(2)(seeforinstance[37,40]andliteraturecitedthere).
Recently, analternativepreconditioning methodfor Stokesdiscretizationsof
the same type as here has been introduced in [5] by Ayuso et al. Their method
is based on auxiliary spaces introduced by Hiptmair and Xu in [19]. The exact
sequencepropertyofthedivergence-conforming velocityelementplaysacrucial
role as in our scheme, but their preconditioner uses a multigrid method for the
biharmonic problem to solve the Stokes problem. As a consequence, it is not
possible tousethepreconditioning method forno-slipboundary conditions. On
theotherhand,ithasbeendemonstratedin[26]thatthemultigridmethodherecan
be lifted to the biharmonic problem, providing an efficient method for clamped
2
boundaryconditions.
The paper isorganized as follows. InSection2 wepresent the model prob-
lemandtheDGdiscretization. Themultigridmethodanddomaindecomposition
smootherarederivedinSection3.Section4isdevotedtotheconvergenceanalysis
ofourpreconditioningtechniquewiththemanresultinTheorem1onpage9.The
paperconcludeswithnumericalexperimentsinSection5.
2 The Stokes problem and its discretization
WeconsiderdiscretizationsoftheStokesequations
u + p = f in Ω,
−△ ∇
u = 0 in Ω, (1)
∇·
u = uB on ∂Ω,
with no-slip boundary conditions on a bounded and convex domain Ω Rd
⊂
withdimensiond = 2,3. ThenaturalsolutionspacesforthisproblemareV =
H1(Ω;Rd)forthevelocityuandthespaceofmeanvaluefreesquareintegrable
0
functionsQ = L2(Ω)forthepressurep,although wepoint outthatotherwell-
0
posedboundaryconditionsdonotposeaproblem.
In order to obtain a finite element discretization, we partition the domain Ω
intoahierarchyofmeshes T ofparallelogramandparallelepipedcells
ℓ ℓ=0,...,L
{ }
intwoandthreedimensions,respectively.Inviewofmultilevelmethods,theindex
ℓreferstothemeshleveldefinedasfollows: letacoarsemeshT begiven. The
0
meshhierarchyisdefinedrecursively,suchthatthecellsofT areobtainedby
ℓ+1
splittingeachcellofT into2dcongruentchildren(refinement).Thesemeshesare
ℓ
nestedinthesensethateverycellofT isequaltotheunionofitsfourchildren.
ℓ
Wedefinethemeshsizeh asthemaximumofthediametersofthecellsofT .
ℓ ℓ
Duetotherefinementprocess,wehaveh =2−ℓh .
ℓ 0
Byconstruction, thesemeshesareconforminginthesensethateveryfaceof
acelliseitherattheboundaryorawholefaceofanothercell;nevertheless,local
refinementandhangingnodesdonotposeaparticularproblem,sincetheycanbe
treatedfollowing[20,21]. ByF wedenotetheset of allfacesof themeshT ,
ℓ ℓ
whichiscomposedofthesetofinteriorfacesFi andthesetofallboundaryfaces
ℓ
F∂.
ℓ
WeintroduceashorthandnotationforintegralformsonT andonF by
ℓ ℓ
(φ,ψ) = φ ψdx, φ,ψ = φ ψds,
Tℓ TX∈TℓZT ⊙ (cid:10) (cid:11)Fℓ FX∈FℓZF ⊙
1 1
2 2
φ = φ2dx , φ = φ2ds ,
(cid:13) (cid:13)Tℓ (cid:18)TX∈TℓZT| | (cid:19) (cid:13) (cid:13)Fℓ (cid:18)FX∈FℓZF | | (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
Thepoint-wisemultiplicationoperatorφ ψreferstotheproductφψ,thescalar
⊙
productφ ψandthedoublecontractionφ :ψforscalar,vectorandtensorargu-
·
ments,respectively.Themodulus φ =√φ φisdefinedaccordingly.
Inordertodiscretize(1)onthe| m|eshT ,⊙wechoosediscretesubspacesX =
ℓ ℓ
V Q ,whereQ Q.Following[11],weemploydiscretesubspacesV ofthe
ℓ ℓ ℓ ℓ
× ⊂
3
spaceHdiv(Ω),where
0
Hdiv(Ω)= v L2(Ω;Rd) v L2(Ω) ,
∈ ∇· ∈
H0div(Ω)=(cid:8)v∈Hdiv(Ω) v·(cid:12)(cid:12)n=0 on∂Ω(cid:9).
Here, we choose the well-kn(cid:8)own Raviart–T(cid:12)homas space [30(cid:9)], but we point out
(cid:12)
thatanypairofvelocityspacesV andpressurespacesQ isadmissible,ifthekey
ℓ ℓ
relation
V =Q (2)
ℓ ℓ
∇·
holds.ThedetailsofconstructingtheRaviart–Thomasspacefollow.
Each cell T Tℓ can be obtained asthe image of a linear mapping ΨT of
thereferencecell∈T = [0,1]d. Onthereferencecell, wedefinetwopolynomial
spaces:first,Q ,thespaceofpolynomialsindvariables,suchthatthedegreewith
k
respecttoeachvaribabledoesnotexceedk.Second,weconsiderthevectorvalued
spaceofRavibart–ThomaspolynomialsVk = Qdk+xQk. PolynomialspacesVT
andQT onthemeshcellT areobtainedbythepull-backunderthemappingΨT
(seeforinstance[4]). Thepolynomialdbegreekbisarbitbrary,butchosenuniformly
onthewholemesh.Thus,wewillomittheindexkfromnowon. Concludingthis
construction,weobtainthefiniteelementspaces
Vℓ = v∈H0div(Ω) ∀T ∈Tℓ :v|T ∈VT ,
Qℓ =(cid:8)q∈L20(Ω) ∀(cid:12)(cid:12)T ∈Tℓ :q|T ∈QT (cid:9).
(cid:8) (cid:12) (cid:9)
2.1 Discontinuous Galerk(cid:12)indiscretization
WhilethefactthatV isasubspaceofHdiv(Ω)impliescontinuityofthenormal
ℓ 0
componentofitsfunctionsacrossinterfacesbetweencells,thisisnottruefortan-
gentialcomponents.Thus,V H1(Ω;Rd),anditcannotbeusedimmediatelyto
ℓ
6⊂
discretize(1). Wefollowtheexampleinforinstance[11,24,25]andapplyaDG
formulationtothediscretizationoftheellipticoperator. Here,wefocusonthein-
teriorpenaltymethod[1,29].LetT andT betwomeshcellswithajointfaceF,
1 2
andletu andu bethetracesofafunctionuonF fromT andT ,respectively.
1 2 1 2
OnthisfaceF,weintroducetheaveragingoperator
u +u
u = 1 2. (3)
{{ }} 2
Inthisnotation,theinteriorpenaltybilinearformreads
aℓ(u,v)=(∇u,∇v)Tℓ +4 σL{{u⊗n}},{{v⊗n}} Fiℓ
2 u , n(cid:10) v 2 v , (cid:11)n u (4)
− {{∇ }} {{ ⊗ }} Fiℓ − {{∇ }} {{ ⊗ }} Fiℓ
+2(cid:10)σLu,v F∂ℓ − ∂n(cid:11)u,v F∂ℓ(cid:10)− ∂nv,u F∂ℓ. (cid:11)
Theoperator“ ”denote(cid:10)stheKr(cid:11)onecke(cid:10)rprodu(cid:11)ctoftw(cid:10)ovecto(cid:11)rs. Wenotethatthe
term4 u n⊗: v n actuallydenotestheproductofthejumpsofuandv.
{{ ⊗ }} {{ ⊗ }}
Thediscreteweakformulationof(1)readsnow:find(u ,p ) V Q ,such
ℓ ℓ ℓ ℓ
∈ ×
thatforalltestfunctionsv V andq Q thereholds
ℓ ℓ ℓ ℓ
∈ ∈
u v
ℓ , ℓ a (u ,v )+(p , v ) (q , u )= (v ,q ) (f,v ).
Aℓ pℓ qℓ ≡ ℓ ℓ ℓ ℓ ∇· ℓ − ℓ ∇· ℓ F ℓ ℓ ≡ ℓ
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(5)
4
Discussionontheexistenceanduniquenessofsuchsolutionscanbefoundfor
instancein [11,12,17,24]. Here,wesummarize,thatissymmetric. IfσLissuf-
ficientlylarge,theforma (.,.)ispositivedefiniteindependentlyofthemultigrid
ℓ
levelℓ [0,L],andthatthuswecandefineanormonV by
ℓ
∈
v = a (v ,v ). (6)
ℓ Vℓ ℓ ℓ ℓ
(cid:13) (cid:13) p
Inordertoobtainoptimalcon(cid:13)verg(cid:13)enceresultsandtosatisfyProposition2.2below,
σL is chosen as σ/hL, where hL is mesh size on the finest level L and σ is a
positiveconstantdependingonthepolynomialdegree.Bythischoice,thebilinear
formsonlowerlevelsareinheritedfromfinerlevelsinthesense,that
aℓ(uℓ,vℓ)=aL(uℓ,vℓ), uℓ,vℓ Vℓ. (7)
∀ ∈
Aparticularfeatureofthismethodis(see[10,11]),thatthesolutionu isin
ℓ
thedivergencefreesubspace
V0 = v V v =0 , (8)
ℓ ℓ ∈ ℓ ∇· ℓ
wherethedivergenceconditionis(cid:8)tobeund(cid:12)erstoodin(cid:9)thestrongsense.
(cid:12)
Proposition 0.1 (Inf-sup condition). For any pressure function q Q , there
ℓ
∈
existsavelocityfunctionv V ,satisfying
ℓ
∈
(q, v)
inf sup ∇· γℓ >0 (9)
q∈Qℓv∈Vℓ v Vℓ q Qℓ ≥
(cid:13) (cid:13) (cid:13) (cid:13)
whereγ = c hL = c√2ℓ−L a(cid:13)nd(cid:13)cis(cid:13)a(cid:13)constantindependentofthemultigrid
ℓ hℓ
levelℓ. q
Proof. Theproofofthispropositioncanbefoundin[35,Section6.4]. Indeed,a
different result is proven there, with γ 1/k, where k is the polynomial de-
ℓ
≈
gree in the hp-method. Thorough study of the proof though reveals, that this
k-dependence is due to the penalty parameter of the form σ k2/h . In our
ℓ ℓ
≈
case,thepenaltyparameterdependsonthefinemesh,notonh ,suchthatσ
ℓ ℓ
(hℓ/hL)/hℓ, and that the role of the k2 in the penalty is taken by the facto≈r
hℓ/hL.
Foranyu V ,weconsiderthefollowingdiscreteHelmholtzdecomposition:
ℓ
∈
u=u0+u⊥ (10)
whereu0 V0isthedivergencefreepartandu⊥belongstoitsa (.,.)-orthogonal
∈ ℓ ℓ
complement. Forfunctionsfromthiscomplementholdstheestimate:
Lemma0.1. Letu⊥ V bea (.,.)-orthogonaltoV0,thatis,
∈ ℓ ℓ ℓ
a (u⊥,v)=0 v V0.
ℓ ∀ ∈ ℓ
Then,thereisaconstantα>0suchthat
α u⊥ 2 a (u⊥,u⊥) 1 u⊥ 2, (11)
d2 ∇· ≤ ℓ ≤ γ ∇·
ℓ
(cid:13) (cid:13) (cid:13) (cid:13)
γ istheinf-supcons(cid:13)tantfrom(cid:13)inequality(9). (cid:13) (cid:13)
ℓ
5
Proof. Ontheleftside,wealreadyarguedabovethatσL ischosenlargeenough
such that a (.,.) is uniformly positive definite. Thus, we have with a positive
ℓ
constantα
αk∇u⊥k2Tℓ ≤aℓ(u⊥,u⊥).
Butthen,
d2
u⊥, u⊥ d2 u⊥, u⊥ a (u⊥,u⊥),
ℓ
∇· ∇· Ω ≤ ∇ ∇ Tℓ ≤ α
(cid:16) (cid:17) (cid:16) (cid:17)
Ontherightside,letq = u⊥. Thenq Q dueto(2). From(9),weconclude
ℓ
∇· ∈
that there isu V such that u = q and u 1/γ q . On theother
∈ ℓ ∇· Vℓ ≤ ℓk k
hand,u⊥ istheerroroftheorthogonalprojecti(cid:13)on(cid:13)intoVℓ0. Thus,u⊥ mustbethe
elementwithminimalnorm,andinparticular (cid:13)u⊥(cid:13) u .
Vℓ ≤ Vℓ
(cid:13) (cid:13) (cid:13) (cid:13)
2.2 The nearly incompressiblep(cid:13)ro(cid:13)blem(cid:13) (cid:13)
Wearegoing toprove convergence uniform withrespect totherefinement level
ℓ of the proposed multigrid method for the Stokes problem by deviating twice.
First, we provide estimates robust with respect to the parameter ε of the nearly
incompressible problem: find (u ,p ) V Q such that for all (v ,q )
ℓ ℓ ℓ ℓ ℓ ℓ
∈ × ∈
V Q thereholds
ℓ ℓ
×
u v
ℓ , ℓ +ε(p ,q )= (v ,q ). (12)
Aℓ pℓ qℓ ℓ ℓ F ℓ ℓ
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Thisproblemisconnectedwiththesimplerpenaltybilinearform(seeforin-
stancealso[17])
A (u ,v ) a (u ,v )+ε−1( u , v ) (13)
ℓ,ε ℓ ℓ ℓ ℓ ℓ ℓ ℓ
≡ ∇· ∇·
andthesingularlyperturbed,ellipticproblem: findu V suchthatforallv
ℓ ℓ ℓ
∈ ∈
V thereholds
ℓ
A (u ,v )=(f,v ). (14)
ℓ,ε ℓ ℓ ℓ
Lemma0.2. Let(um,pm)bethesolutionto(12)anduebethesolutionto(14).
Then,if (2)holds,thefollowingequationsholdtrue:
um =ue, and εpm= um = ue.
∇· ∇·
Proof. Testing(12)withv =0andq Q yields
ℓ ℓ ℓ
∈
( um,qℓ)+ε(pm,qℓ)=0 qℓ Qℓ.
− ∇· ∀ ∈
Dueto(2),thistranslatestothepoint-wiseequalityεpm = um. Substituting
∇·
pmin(12)andtestingwiththepair(vℓ, vℓ),whichispossibleagaindueto(2),
∇·
weobtainthatumsolves(14).
Ifontheotherhanduesolves(14),weintroducepe = 1ε∇·ue,whichtranslates
to
( ue,qℓ)+ε(pe,qℓ)=0 qℓ Qℓ,
− ∇· ∀ ∈
corresponding to (12) tested with (0,q ). On the other hand, (12) tested with
ℓ
(vℓ,0) is obtained directly from (14) substituting pe. Thus, the equivalence is
proven.
6
Inordertohelpkeepingthenotationseparate,weadoptthefollowingconven-
tion:thesubscriptεisdroppedwhereverpossible.Furthermore,curlylettersrefer
tothemixedform,whilestraightcapitalsrefertooperatorsonthevelocityspace
only.Thus:
a (u,v)thevectorvaluedinteriorpenaltyform
ℓ
A (u,v)theformofthesingularlyperturbed,ellipticproblem(14)
ℓ
u v
, themixedbilinearform(12)
Aℓ p q
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Similarly, capital letters like in R for the smoother (26) refer to the singularly
ℓ
perturbed,ellipticproblem,while isthecorrespondingsymbolfortheStokes
ℓ
R
smoother(24). Additionally,weassociateoperatorswithbilinearformsusingthe
samesymbol:
Aℓ,ε:Vℓ Vℓ (Aℓ,εu,v)=Aℓ,ε(u,v)=Aℓ(u,v)=AL(u,v) u,v Vℓ
→ ∀ ∈
ℓ,ε :Xℓ Xℓ ( ℓ,εx,y)= ℓ,ε(x,y)= ℓ(x,y)= L(x,y) x,y Xℓ
A → A A A A ∀ ∈
3 Multigrid method
InSection2,weintroducedhierarchiesofmeshes T . Duetothenestednessof
ℓ
{ }
mesh cells, the finiteelement spaces associated withthesemeshes arenested as
well:
V0 V1 ... VL,
⊂ ⊂ ⊂
Q0 Q1 ... QL.
⊂ ⊂ ⊂
X0 =V0 Q0 X1 ... VL QL =XL.
× ⊂ ⊂ ⊂ ×
Thisrelationalsoextendstothedivergencefreesubspaces,seeforinstance[26]:
V0 V0 ... V0. (15)
0 ⊂ 1 ⊂ ⊂ L
Thenestednessofthespacesimpliesthatthereisasequenceofnaturalinjec-
tions :X X oftheform (v ,q )=(I v ,I q ),suchthat
ℓ ℓ ℓ+1 ℓ ℓ ℓ ℓ,u ℓ ℓ,p ℓ
I → I
I : V V , I :Q Q , (16)
ℓ,u ℓ ℓ+1 ℓ,p ℓ ℓ+1
→ →
I :V0 V0 . (17)
ℓ,u ℓ → ℓ+1
TheL2-projectionfromX X isdefinedby t(v ,q ) = (It v ,It q )
ℓ+1 → ℓ Iℓ ℓ ℓ ℓ,u ℓ ℓ,p ℓ
with
v It v ,w =0 w V q It q ,r =0 r Q .
ℓ+1− ℓ,u ℓ+1 ℓ ∀ ℓ ∈ ℓ ℓ+1− ℓ,p ℓ+1 ℓ ∀ ℓ∈ ℓ
(18)
(cid:0) (cid:1) (cid:0) (cid:1)
The -orthogonalprojection ℓfrom(VL QL) (Vℓ Qℓ)isdefinedby
A P × → ×
u v u v
AL Pℓ p , qℓℓ =AL p , qℓℓ (19)
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
for all (u,p) (VL QL),(vℓ,qℓ) Vℓ Qℓ. Similarly, TheA-orthogonal
∈ × ∈ ×
projectionPℓfromVL Vℓisdefinedby
→
AL(Pℓu,vℓ)=AL(u,vℓ) (20)
forallu VL,vℓ Vℓ.
∈ ∈
7
3.1 The V-cyclealgorithm
Inthissubsection wedefineV-cyclemultigridpreconditioners andB for
ℓ,ε ℓ,ε
B
theoperators andA , respectively. Forsimplicityofthepresentation, we
ℓ,ε ℓ,ε
A
droptheindexε.
First, we define the action of the multigrid preconditioner : X X
ℓ ℓ ℓ
B →
recursivelyasthemultigridV-cyclewithm(ℓ) 1pre-andpost-smoothingsteps.
Let beasuitablesmoother. Let = −1.≥Forℓ 1,definetheactionof
Rℓ B0 A0 ≥ Bℓ
onavector =(f ,g )by
ℓ ℓ ℓ
L
1. Pre-smoothing:beginwith(u ,p )=(0,0)andlet
0 0
ui = ui−1 + ui−1 i=1,...,m(ℓ), (21a)
(cid:18)pi(cid:19) (cid:18)pi−1(cid:19) Rℓ(cid:18)Lℓ−Aℓ(cid:18)pi−1(cid:19)(cid:19)
2. Coarsegridcorrection:
u u u
m(ℓ)+1 = m(ℓ) + t m(ℓ) , (21b)
(cid:18)pm(ℓ)+1(cid:19) (cid:18)pm(ℓ)(cid:19) Bℓ−1Iℓ−1(cid:18)Lℓ−Aℓ(cid:18)pm(ℓ)(cid:19)(cid:19)
3. Post-smoothing:
ui = ui−1 + ui−1 , i=m(ℓ)+2,...,2m(ℓ)+1
pi pi−1 Rℓ Lℓ−Aℓ pi−1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
(21c)
4. Assign:
u
= 2m(ℓ)+1 (21d)
BℓLℓ (cid:18)p2m(ℓ)+1(cid:19)
We distinguish between the standard and variable V-cyclealgorithms by the
choice
m(L) standardV-cycle,
m(ℓ)=
(m(L)2L−ℓ variableV-cycle,
wherethenumberm(L)ofsmoothingstepsonthefinestlevelisafreeparameter.
Wereferto astheV-cyclepreconditionerof .Theiteration
L L
B A
u u u
(cid:18)pkk++11(cid:19)=(cid:18)pkk(cid:19)+BL(cid:18)LL−AL(cid:18)pkk(cid:19)(cid:19) (22)
istheV-cycleiteration.
ThedefinitionofthepreconditionerB :V V fortheellipticoperatorA
ℓ ℓ ℓ ℓ
→
followsthesameconcept,butdroppingthepressurevariables.
3.2 Overlapping Schwarz smoothers
Inthissubsection, wedefineaclassofsmoothing operators based onasub-
ℓ
R
space decomposition of the space X . Let be the set of vertices in the tri-
ℓ ℓ
angulation T , and let T be theset of celNls inT sharing thevertex υ. They
ℓ ℓ,υ ℓ
formatriangulationwithN(N >0)subdomainsorpatcheswhichwedenoteby
Ω N .
{ ℓ,υ}υ=1
8
ThesubspaceX =V Q consistsofthefunctionsinX withsupport
ℓ,υ ℓ,υ ℓ,υ ℓ
×
inΩ .Notethatthisimplieshomogeneousslipboundaryconditionson∂Ω for
ℓ,υ ℓ,υ
thevelocitysubspaceV andzeromeanvalueonΩ forthepressuresubspace
ℓ,υ ℓ,υ
Q .TheRitzprojection :X X isdefinedbytheequation
ℓ,υ ℓ,υ ℓ ℓ,υ
P →
u v u v v
ℓ , ℓ,υ = ℓ , ℓ,υ ℓ,υ X .
Aℓ Pℓ,υ pℓ qℓ,υ Aℓ pℓ qℓ,υ ∀ qℓ,υ ∈ ℓ,υ
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
(23)
Notethateachcellbelongstonotmorethanfour(eightin3D)patchesT ,one
ℓ,υ
foreachofitsvertices.
ThenwedefinetheadditiveSchwarzsmoother
=η −1 (24)
Rℓ Pℓ,υAℓ
υX∈Nℓ
whereη (0,1]isascalingfactor, isL2symmetricandpositivedefinite.
ℓ
∈ R
Similarly,wedefinesmoothersofthesingularlyperturbedellipticoperatorA ,
ℓ
namely,P :V V isdefinedas
ℓ,υ ℓ ℓ,υ
→
A (P u ,v )=A (u ,v ) v V , (25)
ℓ ℓ,υ ℓ ℓ,υ ℓ ℓ ℓ,υ ℓ,υ ℓ,υ
∀ ∈
andtheadditiveSchwarzsmootheris
R =η P A−1. (26)
ℓ ℓ,υ ℓ
υX∈Nℓ
4 Convergence analysis
In this section, we provide a proof of the convergence for the variable V-cycle
iteration with additive Schwarz preconditioning method. Our proof is based on
theassumptionthatthedomainΩisboundedandconvex,whichwillbeomitted
forsimplicityinthestatementoffollowingtheoremsandpropositions. Ourmain
resultis:
Theorem1. Themultileveliteration fortheStokesproblem(5)with
L L
I −B A
thevariableV-cycleoperator definedinSection3.1employingthesmoother
L
B
definedinequation(24)withsuitablysmallscalingfactorη isacontraction
ℓ
R
withcontractionnumberindependentofthemeshlevelL.
Proof. First,weconsiderthenearlyincompressibleproblem(12). Forthisweak
formulation,wehavebyTheorem3,thatthemultigridmethod is
L,ε L,ε
I −B A
equivalenttothemethodI BL,εAL,εappliedtothesingularlyperturbedprob-
−
lem(14)inthevelocityspace.
ConvergenceofthemultileveliterationI BL,εAL,εisshowninTheorem2
−
forallε > 0withacontractionnumberδ < 1independentofLandε. Thus,by
Theorem 3,thesameholdsfor L,ε L,εwithpositiveε.
I−B A
Finally,in(12)wecanletεconvergetozero. Thelimityieldsthewell-posed
Stokesproblem (5), and sincethecontractionnumber δ isindependent of ε, we
obtain uniform convergence with respect to the mesh level L in the limit ε
→
0.
Thetheoremsandlemmasofthefollowingsubsectionsservetoestablishthe
buildingblocksoftheproofofTheorem1.
9
4.1 The singularly perturbed problem
Theorem2. LetR bethesmootherdefinedin(26)withsuitablysmallscalingfac-
ℓ
torη.Then,themultileveliterationI BLALwiththevariableV-cycleoperator
−
BLdefinedinSection3.1isacontractionwithcontractionnumberindependentof
themeshlevelLandtheparameterε.
Theproofofthistheoremispostponedtopage12andrelieson
Proposition2.1. IfR satisfiestheconditions:
ℓ
AL (I RℓAℓ)w,w 0, w Vℓ (27)
− ≥ ∀ ∈
and (cid:0) (cid:1)
(Rℓ−1[I−Pℓ−1]w,[I−Pℓ−1]w)≤βℓAL([I−Pℓ−1]w,[I−Pℓ−1]w), ∀w∈Vℓ
(28)
whereβ =O( 1 )isdefinedinequation(55)below.Then
ℓ γℓ
0 AL (I BℓAℓ)w,w δAL(w,w), w Vℓ (29)
≤ − ≤ ∀ ∈
whereδ= Cˆ andC(cid:0)ˆaredefinedinLe(cid:1)mma2.3.
1+Cˆ
Proof. Inthecaseofself-adjointoperatorsA whichareinheritedfromacommon
ℓ
bilinearforma(.,.),thispropositionwouldbepartofthestandardmultigridtheory
ifβ wereconstant.Itsproofcanbeadaptedfromsimilartheoremsin[2,8,9].We
ℓ
willprovetheversionneededhereintheappendix.
In the remainder of this section we use several propositions and lemmas to
establishoursmoother R satisfiestheassumptions ofProposition2.1. Foru
ℓ
∈
(I P )wwitharbitraryw V ,itfollowsfromthediscreteHelmholtzdecom-
ℓ−1 ℓ
− ∈
positioninSection2andtheprojectionoperatorP inSection3.2thatuadmits
ℓ,υ
alocaldiscreteHelmholtzdecomposition
uυ =u0υ+u⊥υ (30)
Lemma 2.1. Given L2-symmetric positive definite R defined in 3.2 and sym-
ℓ
metricpositivedefiniteAL(, )definedin(13),thereexistsaconstantη (0,1]
· · ∈
independentofℓsuchthat
η(Rℓ−1u,u)= inf AL(uυ,uυ) (31)
Σuυυ∈uυVℓ=,υuυX∈Nℓ
Proof. Thefollowingproofcanbefoundin[2]fortheL2-innerproductinsteadof
a (.,.).Wecopyitheretoascertainthatitdoesnotdependontheactualstructure
ℓ
oftheoperatorALsinceitispurelyalgebraic. Thus,itappliestotheoperatorAL
inthispaperasitappliestothedifferentoperatorappliedthere.Recallthat
R =η P A−1 =η P A−1. (32)
ℓ ℓ,υ ℓ ℓ,υ L
υX∈Nℓ υX∈Nℓ
From
u= uυ (33)
υX∈Nℓ
10