Convergence of Finite Volume MPFA O type Schemes for Heterogeneous Anisotropic Diffusion Problems on

Leo Agelas

Description

Convergence of Finite Volume MPFA O type Schemes for Heterogeneous Anisotropic Diffusion Problems on General Meshes L. Agelas† † IFP Energies Nouvelles, 1 et 4 avenue de Bois-Preau, 92852 Rueil Malmaison, France Tel.: Fax: C. Guichard? ? IFP Energies Nouvelles, 1 et 4 avenue de Bois-Preau, 92852 Rueil Malmaison, France Tel.: Fax: R. Masson‡ ‡ IFP Energies Nouvelles, 1 et 4 avenue de Bois-Preau, 92852 Rueil Malmaison, France Tel.: Fax: Abstract In this paper we prove the convergence of the finite volume MultiPoint Flux Approximation (MPFA) O scheme for anisotropic and heterogeneous diffusion problems, under a local coercivity condition which can be easily checked numerically. Our framework is based on a discrete hybrid varia- tional formulation which generalizes the usual construction of the MPFA O scheme. The novel feature of our framework is that it holds for general polygonal and polyhedral meshes as well as for L∞ diffusion coefficients, which is essential in many practical applications. Key words : Finite Volume, MPFA, Convergence Analysis, Diffusion Equation, Full Tensor, Anisotropy, Heterogeneities, General Meshes

• scheme family


• piecewise constant


• t? has


• cell unknowns


• any given


• finite volumes


• diffusion tensor


• using specific


• volume discretization


• scheme

Sujets

• Masson
• Agelas
• Guichard
• Step function
• Diffusion MRI

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

Extrait

ConvergenceofFiniteVolumeMPFAOtypeSchemes
forHeterogeneousAnisotropicDiffusionProblemson
GeneralMeshes

L.Agelas
†
†
IFPEnergiesNouvelles,1et4avenuedeBois-Pre´au,92852RueilMalmaison,France
Tel.:[phone]
Fax:[phone]
[e-mail]

C.Guichard
⋆
⋆
IFPEnergiesNouvelles,1et4avenuedeBois-Pre´au,92852RueilMalmaison,France
Tel.:[phone]
Fax:[phone]
[e-mail]

R.Masson
‡
‡
IFPEnergiesNouvelles,1et4avenuedeBois-Pre´au,92852RueilMalmaison,France
Tel.:[phone]
Fax:[phone]
[e-mail]

Abstract
InthispaperweprovetheconvergenceofthefinitevolumeMultiPoint
FluxApproximation(MPFA)Oschemeforanisotropicandheterogeneous
diffusionproblems,underalocalcoercivityconditionwhichcanbeeasily
checkednumerically.Ourframeworkisbasedonadiscretehybridvaria-
tionalformulationwhichgeneralizestheusualconstructionoftheMPFA
Oscheme.Thenovelfeatureofourframeworkisthatitholdsforgeneral
polygonalandpolyhedralmeshesaswellasfor
L
∞
diffusioncoefficients,
whichisessentialinmanypracticalapplications.
Keywords
:FiniteVolume,MPFA,ConvergenceAnalysis,Diffusion
Equation,FullTensor,Anisotropy,Heterogeneities,GeneralMeshes

ConvergenceanalysisoftheMPFAOscheme

1Introduction
Inthispaper,weconsiderthesecondorderellipticequation
div(
−
Λ
∇
u
)=
f
inΩ
,
(1)
u
=0on
∂
Ω
,
whereΩisanopenboundedconnectedpolygonalsubsetof
R
d
,
d
∈
N
∗
,and
f
∈
L
2
(Ω).ItisassumedinthefollowingthatΛisameasurablefunctionfrom
Ωtothesetofsquare
d
-dimensionalmatrices
M
d
(
R
)suchthatfora.e.(almost
every)
x
∈
Ω,Λ(
x
)issymmetricanditseigenvaluesareintheinterval[
α
(
x
)
,β
(
x
)]
with
α,β
∈
L
∞
(Ω),and0
<α
0
≤
α
(
x
)
≤
β
(
x
)
≤
β
0
.Itresultsthatthereexistsa
uniqueweaksolutionto(1)in
H
01
(Ω)denotedby
u
¯inthefollowingofthispaper.
TheMultiPointFluxApproximation(MPFA)Omethodisacellcenteredfinite
volumediscretizationofsuchsecondorderellipticequationsdescribedforexample
in[1]and[8].Itisawidelyusedschemeintheoilindustryforthediscretization
ofdiffusionfluxesinmultiphaseDarcyporousmediaflowmodels(seeforexample
[13],[14],and[18]).
Let
σ
beanyinteriorfaceofthemeshsharedbythetwocells
K
and
L
,and
n
K,σ
itsnormalvectoroutward
K
.Cellcenteredfinitevolumeschemesusethecell
unknowns
u
M
foreachcell
M
ofthemeshasdegrees
R
offreedom.Theyaimto
buildconservativeapproximations
F
K,σ
ofthefluxes
−
σ
Λ
∇
u

n
K,σ
dσ
aslinear
combinationsofthecellunknowns
u
M
usingneighbouringcells
M
ofthecells
K
or
L
.Thefluxesareconservativeinthesensethat
F
K,σ
+
F
L,σ
=0.
ThemainassetsoftheMPFAOschemearetoderiveaconsistentapproximation
ofthefluxesongeneralmeshes,andtobeadaptedtodiscontinuousanisotropicdif-
fusioncoefficientsinthesensethatitreproducescellwiselinearsolutionsforcellwise
constantdiffusiontensors.Forthatpurpose,itsconstructionusesinadditiontothe
cellunknowns,theintermediatesubfaceunknowns
u
σs
foreachface(edgein2D)
σ
ofthemeshandeachvertex
s
oftheface
σ
.Roughlyspeaking,assumingthateach
vertex
s
ofanycell
K
issharedbyexactly
d
facesofthecell
K
,subfluxes
F
Ks,σ
are
builtusingacellwiseconstantdiffusioncoefficientandalinearapproximationof
u
oneachcell
K
sharedby
s
.Then,theintermediateunknownsareeliminatedbythe
fluxcontinuityequationsoneachfacearoundthevertex
s
,andtheapproximateflux
F
K,σ
isthesumofthesubfluxesovertheverticesoftheface
σ
.Ageneralizationof
thisconstructionisproposedin[13]forgeneralpolyhedralmeshes.
RecentpapershavestudiedtheconvergenceoftheMPFAOscheme.In[17],[3],
[15],theconvergenceoftheschemeisobtainedonquadrilateralmeshes.Theproofs
arebasedonequivalencesoftheMPFAOschemetomixedfiniteelementmethods
usingspecificquadraturerules.Theconvergenceoftheschemeisobtainedprovided
thatasquare
d
-dimensionalmatrixdefinedlocallyforeachcellandeachvertexof
thecell,dependingbothonthedistortionofcellandonthecelldiffusiontensor,
isuniformlypositivedefinite.Thisanalysisconfirmsthenumericalexperiments
showingthatthecoercivityandconvergenceoftheschemeislostinthecasesof
strongdistortionofthemeshand/oranisotropyofthediffusiontensor.

InternationalJournalonFiniteVolumes

2

ConvergenceanalysisoftheMPFAOscheme

ThefirstconvergenceproofoftheMPFAOschemeongeneralpolygonaland
polyhedralmeshesisintroducedin[6].Theconvergenceanalysisholdsforfairly
generalmeshesin2Dand3D,fordiffusiontensorswithminimalregularityincluding
discontinuousdiffusioncoefficientswhichareessentialinoilindustryapplications,
andforminimalregularityassumptionsonthesolution.Moreover,itcoverstheall
familyofMPFAOschemesforarbitrarychoicesofthecellcenters,ofthesocalled
continuitypoints,andofthesubfaces.
Adifferentapproachispresentedin[20]basedonsymmetricandnonsymmetric
mimeticfinitedifferenceschemesusingsubfacesunknowns.Thesymmetricversion
ofthisschemehasalsobeenindependentlyintroducedin[19]intwodimensions.As
shownin[16]whichdevelopsasimilaranalysis,thenonsymmetricversionofthis
mimeticfinitedifferenceschemematcheswiththeMPFAOschemefamily.Error
estimatesarederivedin[20]ongeneralpolygonalandpolyhedralmeshesundera
localcoercivitycriteriaandforpiecewiseregulardiffusiontensors.

In[6],itisassumedthatforeachcell
κ
andeachvertex
s
ofthecell,thenumber
offacesofthecell
κ
sharingthevertex
s
isequaltothespacedimension
d
.This
paperpresentsageneralizationoftheMPFAOschemetopolyhedralmeshesnon
satisfyingthislatterassumptionandextendstheconvergenceanalysispresentedin
[6].Italsodetailstheproofsonlysketchedin[6].

Inthispaper,following[6],adiscretehybridvariationalformulationisintroduced
usingtheframeworkdescribedin[12],[11].Itinvolvesthedefinitionoftwopiecewise
constantgradientsandstabilitytermsusingresidualsofthesecondgradient.The
firstgradienthasaweakconvergencepropertyandisfixedintheconstruction.
Thesecondoneisassumedtobeconsistentinthesensethatitisexactonlinear
functions.Forusualmeshessuchthateachvertexofanycell
K
issharedbyexactly
d
facesofthecell
K
,thestabilitytermsarevanishingandourdiscretevariational
formulationwillbeshowntobeequivalenttotheusualMPFAOscheme.
Moreover,itprovidesageneralizationoftheOschemeonmoregeneralpolyhe-
dralcells.
Asufficientlocalconditionforthecoercivityoftheschemeisderivedwhichwill
yieldexistence,anduniquenessofthesolution.Underthiscoercivitycondition,and
auniformstabilityassumptionfortheconsistentgradient,theconvergenceofthe
schemeincludingthecaseof
L
∞
diffusioncoefficientscanbeproved.
Thispaperisoutlinedasfollows.Section2describesthediscreteframeworkin-
cludingthedefinitionofthefinitevolumediscretizationofthedomain,thedegrees
offreedomandthediscretefunctionspaceswiththeirassociatedinnerproductsand
norms.Section3isdevotedtothedefinitionofageneralframeworkforMPFAO
typeschemesbasedonahybridvariationalformulationandthedefinitionoftwo
piecewiseconstantgradients.Section4provesthewell-posednessofthefinitevol-
umeschemeunderasufficientcoercivityconditioninvolvingcomputationslocalto
eachnodeofthemeshanddependingonthegeometryandonthediffusiontensor
anisotropy.Theconvergenceoftheschemeisprovedundertheabovecoercivityas-
sumption,usualshaperegularityassumptions,andauniformstabilityassumption
fortheconsistentgradientinsection5for
L
∞
diffusiontensor.Insection6,two

InternationalJournalonFiniteVolumes

3

ConvergenceanalysisoftheMPFAOscheme

examplesofconstructionoftheconsistentgradientarediscussed.Thefirstcon-
structionallowsustoderiveastrongerbutsimplercoercivityconditioninvolving
thecoercivityofa
d
-dimensionalmatrixforeachvertex
s
ofeachcell
K
.Onthe
otherhandthisconstructiondoesnotholdfornon-matchingmeshes.Thesecond
exampleisbasedontheconsistentgradientintroducedin[13].Section7isdevoted
tonumericalexamplesin2Dand3D.
√dotproduct
id
=1
x
i
y
i
,andby
|
x
|
thenorm
x

x
.Thenotations
λ
max
(
M
)and
Notations:
In
P
thefollowing,foranyvectors