A Discontinuous Coarse Spaces (DCS) Algorithm for Cell Centered Finite Volumes based Domain Decomposition Methods: the DCS-RJMin algorithm

0.27 MB·

by  Kévin Santugini-Repiquet

A Discontinuous Coarse Spaces (DCS)
Algorithm for Cell Centered Finite Volumes
4 based Domain Decomposition Methods: the
1
0
DCS-RJMin algorithm
2

n
a Ke´vinSantugini∗
J

4
January15,2014
1

]
A
N Abstract
h. Inthispaper,weintroduceanewcoarsespacealgorithm,the“Discontinuous
t CoarseSpaceRobin JumpMinimizer”(DCS-RJMin), tobeused inconjunction
a
withone-leveldomaindecompositionmethods(DDM).Thisnewalgorithmmakes
m
useofDiscontinuousCoarseSpaces(DCS),andisdesignedforDDMthatnaturally
[ producediscontinuousiteratessuchasOptimizedSchwarzMethods(OSM).This

algorithmissuitablebothatthecontinuouslevelandforcell-centeredfinitevolume
1
discretizations. Atthecontinuouslevel,weprove,undersomeconditionsonthe
v
3 parametersofthealgorithm,thatthedifferencebetweentwoconsecutiveiterates
1 goesto0.Wealsoprovidenumericalresultsillustratingtheconvergencebehavior
1 oftheDCS-RJMinalgorithm.
3 Keywords:discontinuouscoarsespace,optimizedSchwarzmethod.
.
1
0
1 Introduction
4
1
: Due to the ever increasing parallelism in modern computers, and the ever increasing
v
affordability of massively parallel calculators, it is of utmost importance to develop
i
X
algorithmsthatarenotonlyparallelbutscalable.Inthispaper,weareinterestedinDo-
r mainDecompositionMethods(DDM)whichareonewaytoparallelizethenumerical
a
resolutionofPartialDifferentialEquations(PDE).
In Domain Decomposition Methods, the whole domain is subdivided in several
subdomainsandacomputationunitisassignedtoeachsubdomain. Inthispaper, we
onlyconsidernon-overlappingdomaindecompositions.Thenumericalsolutionisthen
computedinparallelinsideeachsubdomainwithartificialboundaryconditions. Then,
subdomainsexchangeinformationbetweeneachother. Thisprocessisreapplieduntil
convergence. Inpractice,suchascheme,callediterativeDDM,shouldbeaccelerated
∗Universite´ Bordeaux, IMB, CNRS UMR5251, MC2, INRIA Bordeaux - Sud-Ouest
Kevin.Santuginimath.u-bordeaux1.fr
1
usingKrylovmethods. However,forthepurposeofanalyzinganalgorithm,itcanbe
interestingtoworkdirectlywiththeiterativealgorithmitselfasKrylovaccelerationis
soefficientitcanhideawaysmalldesignproblemsinthealgorithm.
Inone-levelDDM,onlyneighboringsubdomainsexchangeinformation.Mostclas-
sicalDDMareone-level. Whileone-levelDDMcanbeveryefficientandconvergein
a few iterations, they are not scalable: convergence can never occur before informa-
tionhaspropagatedbetweenthetwofurthestapartsubdomains. I.E.,aonelevelDDM
must iterate at least as many times as the diameter of the connectivity graph of the
domain decomposition. Typically, if N is the number of subdomains, this means at
√
leastO(N)iterationsforone-dimensionalproblems,O( N)fortwo-dimensionalones
√
andO(3N)forthree-dimensionalones. ForDDMtobescalable,somekindofglobal
informationexchangeisneeded. Thetraditionalapproachtoachievesuchglobalinfor-
mationexchangeisaddingacoarsespacetoapre-existingone-levelDDM.
To the author knowledge, the first use of coarse spaces in Domain Decomposi-
tionMethodscanbefoundin[16]. Becausecoarsespacesenableglobalinformation
exchange, scalability becomes possible. Well known methods with coarse spaces are
the two-level Additive Schwarz method [3], the FETI method [13], and the balanc-
ingNeumann-Neumannmethods[12,4,14]. Coarsespacesarealsoanactiveareaof
research, see [2, 15] for high contrast problems. It is not trivial to add an effective
coarsespacetoone-levelDDMthatproducediscontinuousiteratessuchasOptimized
SchwarzMethods,see[6,7],and[5,chap.5].
In[9],theauthorsintroducedtheideaofusingdiscontinuouscoarsespaces. Since
manyDDMalgorithmsproducediscontinuousiterates,theuseofdiscontinuouscoarse
corrections is needed to correct the discontinuities between subdomains. In that pro-
ceeding,onepossiblealgorithm,theDCS-DMNV(DiscontinuousCoarseSpaceDirich-
letMinimizerNeumannVariational),wasdescribedatthecontinuouslevelandatthe
discrete level for Finite Element Methods on a non-overlapping Domain Decompo-
sition. In [17], a similar method, the DCS-DGLC algorithm was proposed. Both
theDCS-DMNVandtheDCS-DGLCarewellsuitedtofiniteelementdiscretizations.
Also, a similar approach was proposed in [8] for Restricted Additive Schwarz(RAS),
anoverlappingDDM,
ItwasprovenrecentlythattheproofofconvergenceforSchwarzfoundin[11,1]
canbeextendedtotheDiscreteOptimizedSchwarzalgorithmwithcellcenteredfinite
volumemethods,see[10]. Itwouldbeinterestingtohaveadiscontinuouscorsespace
algorithmthatissuitedtocellcenteredfinitevolumes.Unfortunately,neithertheDCS-
DMNV algorithm nor the DCS-DGLC algorithm are practical for cell centered-finite
volumemethods: thestiffnessmatrixnecessarytocomputethecoarsecorrectionisn’t
assparseasonewouldintuitivelybelieve. Inthispaper, ourmaingoalistodescribe
one family of algorithms making use of discontinuous coarse spaces but suitable for
cellcenteredfinitevolumesdiscretizations.
In §2, we briefly recall the motivations behind the use of discontinuous coarse
space. In§3,wepresenttheDCS-RJMinalgorithm. In§4,weprovethatundersome
conditionsonthealgorithmparameter,theL2-normofthedifferencebetweentwocon-
secutiveiteratesgoestozero. Finally,wepresentnumericalresultsin§5.
2
2 OptimizedSchwarzandDiscontinuousCoarseSpaces
Let’sconsiderapolygonaldomainΩinR2. Asasimpletestcase,wewishtosolve
ηu−(cid:52)u= f inΩ,
u=0on∂Ω.
Withoutacoarsespace,theOptimizedSchwarzMethodisdefinedas
Algorithm2.1(CoarselessOSM). 1. Set u0 to either the null function or to the
i
coarsesolution.
2. Untilconvergence
(a) Setun+1astheuniquesolutionto
i
ηun+1−(cid:52)un+1= f inΩ,
i i i
∂un+1 ∂un
i +pun+1= j +punon∂Ω ∩∂Ω ,
∂nnn i ∂nnn j i j
i i
un+1=0on∂Ω ∩∂Ω.
i
In practical applications, such an algorithm should be accelerated using Krylov
methods. However,studyingtheiterative(Richardson)versioncangivemathematical
insightontheconvergencespeedoftheKrylovacceleratedalgorithm.
ThemainshortcomingofthecoarselessOptimizedSchwarzmethodsistheabsence
ofdirectcommunicationbetweendistantsubdomains. Togetascalablealgorithm,one
canuseacoarsespace. AgeneralversionofacoarsespacemethodfortheOSMis
Algorithm2.2(GenericOSMwithcoarsespace). 1. Setu0toeitherthenullfunc-
i
tionortothecoarsesolution.
2. Untilconvergence
(a) Setun+1astheuniquesolutionto
i
ηun+1/2−(cid:52)un+1= f inΩ,
i i i
∂un+1/2 ∂un
i +pun+1/2= j +punon∂Ω ∩∂Ω ,
∂nnn i ∂nnn j i j
i i
un+1/2=0on∂Ω ∩∂Ω.
i
(b) Compute in some way a coarse corrector Un+1 belonging to the coarse
spaceX,thenset
un+1=un+1/2+Un+1.
MoreimportantthanthealgorithmusedtocomputethecoarsecorrectionUn+1 is
thechoiceofanadequatecoarsespaceitself. Theideaspresentedin[9]stillapply. In
particular,thecoarsespaceshouldcontaindiscontinuousfunctionsandthediscontinu-
ities of the coarse corrector should be located at the interfaces between subdomains.
3
Forthesereasons,wesupposethewholedomainΩismeshedbyeitheracoarsetrian-
gularmeshoracartesianmeshT andweuseeachcoarsecellofT asasubdomain
H H
Ω of Ω. The optimal theoretical coarse space A is the set of all functions that are
i
solutions to the homogenous equation inside each subdomain: for linear problems,
the errors made by any iterate are guaranteed to belong to that space. With an ade-
quatealgorithmtocomputeUn+1,thecoarsespaceA givesaconvergenceinasingle
coarseiteration. Unfortunatelythiscompletecoarsespaceisonlypracticalforonedi-
mensional problems as it is of infinite dimension in higher dimensions. One should
thereforechooseafinitedimensionalsubsetX ofA.
d
ThechoiceofthecoarsespaceX isprimordial. Itshouldhaveadimensionthatis
d
asmallmultipleofthenumberofsubdomains. TochooseX ,oneonlyneedtochoose
d
boundary conditions on every subdomain, then fill the interior of each subdomain by
solvingthehomogenousequationineachsubdomain. Inthispaper,wehavenottried
tooptimizeX andforthesakeofsimplicityhavechosenX asthesetofallfunctions
d d
in A with linear Dirichlet boundary conditions on each interface between any two
adjacentsubdomains.
3 The DCS-RJMin Algorithm
WenowdescribetheDCS-RobinJumpMinimizeralgorithm:
Algorithm3.1(DCS-RJMin).
Set p>0andq>0andX afinitedimensionalsubspaceofA.
d
Setu0toeither0ortothecoarsespacesolution.
UntilConvergence
1. Setun+21 astheuniquesolutionto
ηun+21 −(cid:52)un+12 = f inΩi,
∂uni+12 +pun+12 = ∂unj +punon∂Ω ∩∂Ω ,
∂ννν i ∂ννν j i j
ij ij
u =0on∂Ω ∩∂Ω .
i i j
2. SetUn+1inX astheuniquecoarsefunctionthatminimizes
d
∑N ∑ (cid:13)(cid:13)(cid:13)∂(uin+12 +Uin+1)+q(un+21 +Un+1)
i=1j∈N(i)(cid:13) ∂νννi i i
−∂(unj+21 +Ujn+1)−q(un+21 +Un+1)(cid:13)(cid:13)(cid:13)2 ,
∂νννi j j (cid:13)L2(∂Ωi∩∂Ωj)
whereννν istheoutwardnormaltosubdomainΩ andN (i)thesetofall jsuch
i i
thatΩ andΩ areadjacent.
j i
3. Setun+1:=un+1/2+Un+1.
4
4 Partial “Convergence” results for DCS-RJMin
Wedon’thaveacompleteconvergencetheoremfortheDCS-RJMinalgorithm. How-
ever, we can prove the iterates of the DCS-RJMin algorithm are close to converging
when p=q:
Proposition 4.1. If q = p. Then, the iterates produced by the DCS-RJMin algo-
rithm3.1satisfylim (cid:107)un+1/2−un(cid:107) =0.
n→+∞ i i L2
Proof. Let u be the mono-domain solution, set en =un−u, then, following Lions
i i i
energyestimates[11],
(cid:90) (cid:90)
η |en+1/2−en|2Dxxx+ |∇(en+1/2−en)|2Dxxx
i i i i
Ωi Ωi
(cid:90) ∂(en+1/2−en)
= i i ·(en+1/2−en)
∂Ωi ∂ννν i i
1 (cid:32)(cid:90) ∂(en+1/2−en) ∂(en+1/2−en) (cid:33)
= | i i +p(en+1/2−en)|2−| i i −p(en+1/2−en)|2
4p ∂Ωi ∂ννν i i ∂ννν i i
1 (cid:32)(cid:90) ∂(en+1/2−en) (cid:90) ∂(en+1/2−en) (cid:33)
= | i i +p(en+1/2−en)|2− | i i −p(en+1/2−en)|2
4p ∂Ωi ∂ννν i i ∂Ωi ∂ννν i i
1 (cid:18) (cid:90) (cid:12)(cid:12)(cid:18)∂en (cid:19) (cid:32)∂en) (cid:33)(cid:12)(cid:12)2
= ∑ (cid:12) j +pen − i +pen (cid:12)
4p j ∂Ωi∩∂Ωj(cid:12)(cid:12) ∂νννi j ∂νννi i (cid:12)(cid:12)
−∑(cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:32)∂eni+1/2 −pen+1/2(cid:33)−∂enj+1/2) −pen+1/2(cid:12)(cid:12)(cid:12)(cid:12)2(cid:19)
j ∂Ωi∩∂Ωj(cid:12)(cid:12) ∂νννi i ∂νννi j (cid:12)(cid:12)
WesumtheaboveequalityoverallsubdomainsΩ andget
i
(cid:90) (cid:90)
η∑ |en+1/2−en|2Dxxx+ |∇(en+1/2−en)|2Dxxx=
i i i i
i Ωi Ωi
=(∑i,j)41p(cid:90)Γij(cid:12)(cid:12)(cid:12)(cid:12)(cid:20)∂∂νeννni +pen(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)2−(cid:90)Γij(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34)∂e∂n+ννν1i/2 +pen+1/2(cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2,
where [·] represents a jump across the interface. Since the coarse step of the DCS-
RJMinalgorithmminimizestheRobinJumps,wehave
(cid:90) (cid:90)
η∑ |en+1/2−en|2Dxxx+ |∇(en+1/2−en)|2Dxxx≤
i i i i
i Ωi Ωi
≤ ∑ 1 (cid:32)(cid:90) (cid:12)(cid:12)(cid:12)(cid:20)∂en +pen(cid:21)(cid:12)(cid:12)(cid:12)2−(cid:90) (cid:12)(cid:12)(cid:12)(cid:20)∂en+1 +pen+1(cid:21)(cid:12)(cid:12)(cid:12)2(cid:33).
(i,j)4p Γij(cid:12) ∂νννi (cid:12) Γij(cid:12) ∂νννi (cid:12)
Summingovern≥0yieldsthestatedresult.
5
5 Numerical Results
Figure 1: Convergence for OSM and DCS-RJmin with Ω=[0,4]2, f(x,y)=0 and
randominitialboundaryconditions. Plottinglog((cid:107)e (cid:107) /(cid:107)e (cid:107) ).
50 ∞ 0 ∞
We have implemented the DCS-RJMin algorithm in C++ for cell-centered finite
volumes on a cartesian grid. We chose Ω=]0,4[×]0,4[, η =0 and iterated directly
ontheerrorsbychoosing f =0. WeinitializedtheRobinboundaryconditionsatthe
interfaces between subdomains at random and performed multiple runs of the DCS-
RJMinalgorithmforvariousvaluesof p,qandofthenumberofsubdomains. Wehad
pvaryfrom1.0to20.0with0.5incrementsandqtakesthefollowingvaluespm×10pe
with p in {1.0,2.0,4.0,8.0} and p in {0,1}. We consider 2×2, 4×4, 6×6 and
m e
8×8 subdomains. There are always 20×20 cells per subdomains. In Figure 1, we
plotlog((cid:107)e (cid:107) /(cid:107)e (cid:107) )asafunctionof pforvariousvaluesofq. First,wenoticethat
50 ∞ 0 ∞
for each value of q, the convergence deteriorates above a certain p . In fact, for low
q
values of q and high values of p, the iterates diverge. For two different values of q,
the curves are very close when p is smaller than both p . We also notice than even
q
thoughwecouldonlyproveProposition4.1forthecase p=q,weobservenumerical
convergence even when p(cid:54)=q. In fact p=q is not the numerical optimum. This is
to be expected at the intuitive level: for a theoretical proof of convergence, we want
the algorithm to keep lowering some functional. The existence of such a functional
is likely only if all the substeps of the algorithm are optimized for the same kind of
errors. If p=q,boththecoarsesteporthelocalstepwilleitherremovelowfrequency
errors (small p and q) or high frequency ones (high p and q). An efficient numerical
algorithmshouldhavesubstepsoptimizedforcompletelydifferentkindoferrors. This
iswhyefficientnumericalalgorithmsareusuallytheonesforwhichtheconvergence
proofsarethemoredifficult.
6
6 Conclusion
In this paper, we have introduced a new discontinuous coarse space algorithm, the
DCS-RJMin,thatissuitableforcell-centeredfinitevolumediscretizations. Thecoarse
space greatly improve numerical convergence. It would be of great interest to study
which is the optimal low-dimensional subspace of all piecewise discontinuous piece-
wise harmonic functions. Future work also includes the development of a possible
alternative to coarse space in order to get scalability: “Piecewise Krylov Methods”
where the same minimization problem than the one used in DCS-RJMin is used but
where the coarse space are made of piecewise, per subdomain, differences between
consecutiveone-leveliterates.
References
[1] Bruno Despre´s. Domain decomposition method and the helmholtz problem. In
Gary C. Cohen, Laurence Halpern, and Patrick Joly, editors, Mathematical and
numerical aspects of wave propagation phenomena, volume 50 of Proceedings
inAppliedMathematicsSeries,pages44–52.SocietyforIndustrialandApplied
Mathematics,1991.
[2] VictoriaDolean,Fre´de´ricNataf,RobertScheichl,andNicoleSpillane. Analysis
of a two-level schwarz method with coarse spaces based on local dirichlet to
neumann maps. Computational Methods in Applied Mathematics, 12(4):391–
414,2012.
[3] Maksymilian Dryja and Olof B. Widlund. An additive variant of the Schwarz
alternatingmethodforthecaseofmanysubregions. TechnicalReport339,also
Ultracomputer Note 131, Department of Computer Science, Courant Institute,
1987.
[4] Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-
Neumann type for three-dimensional elliptic finite element problems. Comm.
PureAppl.Math.,48(2):121–155,February1995.
[5] OlivierDubois. OptimizedSchwarzMethodsfortheAdvection-DiffusionEqua-
tionandforProblemswithDiscontinuousCoefficients. PhDthesis,McGillUni-
versity,2007.
[6] Olivier Dubois and Martin J. Gander. Convergence behavior of a two-level op-
timizedSchwarzpreconditioner. InDomainDecompositionMethodsinScience
andEngineeringXXI.SpringerLNCSE,2009.
[7] Olivier Dubois, Martin J. Gander, Sebastien Loisel, Amik St-Cyr, and Daniel
Szyld. TheoptimizedSchwarzmethodwithacoarsegridcorrection. SIAMJ.on
Sci.Comp.,34(1):A421–A458,2012.
7
[8] MartinJ.Gander,LaurenceHalpern,andKe´vinSantugini.Anewcoarsegridcor-
rectionforRAS. InDomainDecompositionMethodsinScienceandEngineering
XXI.SpringerLNCSE,2013.
[9] MartinJ.Gander,LaurenceHalpern,andKe´vinSantugini-Repiquet. Discontin-
uous coarse spaces for dd-methods with discontinuous iterates. In Domain De-
compositionMethodsinScienceandEngineeringXXI.SpringerLNCSE,2013.
[10] MartinJ.Gander,FelixKwok,andKe´vinSantugini. OptimizedSchwarzatcross
points: Finitevolumecase. Inpreparation,2013.
[11] Pierre-LouisLions.OntheSchwarzalternatingmethod.III:avariantfornonover-
lappingsubdomains. InTonyF.Chan, RolandGlowinski, JacquesPe´riaux, and
Olof Widlund, editors, Third International Symposium on Domain Decomposi-
tionMethodsforPartialDifferentialEquations, heldinHouston, Texas, March
20-22,1989,pages202–223,Philadelphia,PA,1990.SIAM.
[12] Jan Mandel. Balancing domain decomposition. Communications in Numerical
MethodsinEngineering,9(3):233–241,mar1993.
[13] JanMandelandMarianBrezina. Balancingdomaindecompositionforproblems
withlargejumpsincoefficients. Math.Comp.,65:1387–1401,1996.
[14] Jan Mandel and Radek Tezaur. Convergence of a Substructuring Method with
LagrangeMultipliers. Numer.Math.,73:473–487,1996.
[15] Fre´de´ric Nataf, Hua Xiang, Victorita Dolean, and Nicole Spillane. A coarse
sparse construction based on local Dirichlet-to-Neumann maps. SIAM J. Sci.
Comput.,33(4):1623–1642,2011.
[16] Roy A. Nicolaides. Deflation conjugate gradients with application to boundary
valueproblems. SIAMJ.Num.An,,24(2):355–365,1987.
[17] Ke´vin Santugini. A discontinuous galerkin like coarse space correction for do-
main decomposition methods with continuous local spaces: the dcs-dglc algo-
rithm. 2014. Inrevision.
8