NASA Technical Reports Server (NTRS) 20110005564: A Critical Study of Agglomerated Multigrid Methods for Diffusion

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A Critical Study of Agglomerated Multigrid Methods for
Diffusion
HiroakiNishikawa,NationalInstituteofAerospace,Hampton,VA23666∗
BorisDiskin,NationalInstituteofAerospace,Hampton,VA23666†
JamesL.Thomas,NASALangleyResearchCenter,HamptonVA23681‡
Agglomeratedmultigridtechniquesusedinunstructured-gridmethodsarestudiedcriticallyforamodel
problem representative of laminar diffusion in the incompressible limit. The studied target-grid discretiza-
tionsanddiscretizationsusedonagglomeratedgridsaretypicalofcurrentnode-centeredformulations. Ag-
glomeratedmultigridconvergenceratesarepresentedusingarangeoftwo-andthree-dimensionalrandomly
perturbed unstructured grids for simple geometries with isotropic and stretched grids. Two agglomeration
techniquesareusedwithinanoveralltopology-preservingagglomerationframework. Theresultsshowthat
multigrid with an inconsistent coarse-grid scheme using only the edge terms (also referred to in the litera-
tureasathin-layerformulation)providesconsiderablespeedupoversingle-gridmethodsbutitsconvergence
deterioratesonfinergrids.MultigridwithaGalerkincoarse-griddiscretizationusingpiecewise-constantpro-
longationandaheuristiccorrectionfactorisslowerandalsogrid-dependent. Incontrast,grid-independent
convergence rates are demonstrated for multigrid with consistent coarse-grid discretizations. Convergence
ratesofmultigridcyclesareverifiedwithquantitativeanalysismethodsinwhichpartsofthetwo-gridcycle
arereplacedbytheiridealizedcounterparts.
Introduction
Multigridtechniques1areusedtoaccelerateconvergenceofcurrentReynolds-AveragedNavier-Stokessolversfor
steadyandunsteadyflowsolutions,especiallyforstructured-gridapplications. Mavriplisetal.2–5 pioneeredagglom-
eratedmultigridmethodsforlarge-scaleunstructured-gridapplications. Impressiveimprovementsinefficiencyover
single-gridcomputationshavebeendemonstrated.Duringarecentdevelopmentofmultigridmethodsforunstructured
grids,6itwasrealizedthatsomeofthecurrentapproachesforcoarse-griddiscretizationofviscousfluxesusedinstate-
of-the-art codes have serious limitations on highly-refined grids. The purpose of this paper is to critically study the
currenttechniquesforasimplePoissonequation(representinglaminardiffusionintheincompressiblelimit), assess
theirperformanceingridrefinement,anddevelopimprovedapproaches.
The paper is organized as follows. The model diffusion equation is presented from a general finite-volume dis-
cretization (FVD) standpoint. Elements of multigrid algorithms are described, including a tabulation of target and
coarse-grid discretizations. Quantitative analysis methods, in which parts of the actual multigrid cycle are replaced
by their idealized counterparts, are described in the next section. The target grids and typical agglomerated grids
developedwithinatopology-preservingframeworkarenextshown,followedbytwodimensional(2D)andthreedi-
mensional (3D) results. Results from applying analysis methods to 3D computations are also reported. The final
sectioncontainsconclusions.
I. ModelDiffusionEquationandBoundaryConditions
TheFVDschemesconsideredarederivedfromtheintegralformofthediffusionequation,
∗ResearchScientist,100ExplorationWay,[email protected]. ThisauthorwassupportedbytheNationalInstituteofAerospaceunderNASA
fundamentalAeronauticsProgramthroughNRAContractNNL07AA23C
†AssociateFellow,100ExplorationWay,[email protected];VisitingAssociateProfessor,MechanicalandAerospaceEngineeringDepart-
ment,UniversityofVirginia,Charlottesville. ThisauthorwassupportedbytheNationalInstituteofAerospaceunderNASAFundamentalAero-
nauticsProgramthroughNRAContractNNL07AA31C
‡SeniorResearchScientist,ComputationalAerosciencesBranch,MailStop128,FellowAIAA,[email protected].
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I ZZ
(∇U ·nˆ) dΓ= fdΩ, (1)
Γ Ω
wheref isaforcingfunctionindependentofthesolutionU,ΩisacontrolvolumewithboundaryΓ,nˆistheoutward
unit normal vector, and ∇U is the solution gradient vector. The boundary conditions are taken as Dirichlet, i.e.,
specifiedfromaknownexactsolutionoverthecomputationalboundary. Testsareperformedforsimplemanufactured
solutions, namely collections of polynomial or sine functions. The corresponding forcing functions are found by
substitutingthesesolutionsintothedifferentialformofthediffusionequation,
∆U =f, (2)
and boundary conditions. The discretization error, E = U −Uh, is defined as the difference between the exact
d
continuoussolution,U,tothedifferentialequation(2)andtheexactdiscretesolution,Uh,ofthediscretizedequation
(1).Thealgebraicerroristhedifferencebetweentheapproximateandexactdiscretesolutions.Aschemeisconsidered
asdesign-orderaccurate,ifitsdiscretizationerrorscomputedonasequenceofconsistently-refinedgrids7,8 converge
withthedesignorderinthenormofinterest.
1
2
0
3 4
Figure 1. Illustration of a node-centered median-dual control volume (shaded).
Dualfacesconnectedgemidpointswithprimalcellcentroids.Numbers0-4denote
gridnodes.
The general FVD approach requires partitioning the domain into a set of non-overlapping control volumes and
numerically implementing equation (1) over each control volume. Node-centered schemes define solution values at
themeshnodes. In2D,theprimalmeshesarecomposedoftriangularandquadrilateralcells;in3D,theprimalcells
aretetrahedral,prismatic,pyramidal,orhexahedral. Themedian-dualpartition9,10 usedtogeneratecontrolvolumes
isillustratedinFigure1for2D.Thesenon-overlappingcontrolvolumescovertheentirecomputationaldomainand
composeameshthatisdualtotheprimalmesh.
The control volumes of each agglomerated grid are found by summing control volumes of a finer grid. Any
agglomeratedgridcanbedefinedintermsofaconservativeagglomerationoperator,R ,as
0
Ωc =R Ωf, (3)
0
where superscripts c and f denote entities on coarser and finer grids, respectively. On the agglomerated grids, the
control volumes become geometrically more complex than their primal counterparts and the details of the control-
volume boundaries are not retained. The directed area of a coarse-grid face separating two agglomerated control
volumes, ifrequired, isfoundbylumpingthedirectedareasofthecorrespondingfiner-gridfacesandisassignedto
thevirtualedgeconnectingthecentersoftheagglomeratedcontrolvolumes.
II. Multigrid
Elements of the multigrid algorithm are presented in this section. A V-cycle,1 denoted as V(ν ,ν ), uses ν
1 2 1
relaxationsperformedateachgridbeforeproceedingtothecoarsergridandν relaxationsaftercoarse-gridcorrection;
2
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thecoarsestgridissolvedexactly(withmanyrelaxations). Residuals,r,correspondingtotheintegralequation(1)are
restrtictedtothecoarsegridusingR ,as
0
rc =R rf. (4)
0
TheprolongationsP andP areexactforpiecewise-constantandlinearfunctions,respectively. Theprolongation
0 1
P is the transpose of R . The operator P is constructed locally using linear interpolation from a triangle (2D) or
0 0 1
tetrahedra (3D) defined on the coarse grid. The geometrical shape is anchored at the coarser-grid location of the
agglomerate that contains the given finer control volume. Other nearby points are found using the adjacency graph.
An enclosing simplex is sought that avoids prolongation with non-convex weights and, in situations where multiple
geometrical shapes are found, the first one encountered is used. Where no enclosing simplex is found, the simplex
withminimalnon-convexweightsisused. Thecoarse-gridsolutionapproximationisrestrictedas
R (UfΩf)
Uc = 0 . (5)
Ωc
ThecorrectionδU tothefinergridisprolongedtypicallythroughP ,as
1
(δU)f =P (δU)c. (6)
1
Target-GridDiscretizations
Green-Gauss
Avg-LSQ
Table1. Summaryoftarget-griddiscretizations.
The available consistent target-grid discretizations are listed in Table 1 where Avg-LSQ stands for average-
least-squares. TheseschemesarerepresentativeofviscousdiscretizationsusedinReynolds-AveragedNavier-Stokes
unstructured-grid codes. The main target discretization of interest is the Green-Gauss scheme, which is the most
widely-usedviscousdiscretizationfornode-centeredschemesandisequivalenttoaGalerkinfinite-elementdiscretiza-
tionfortriangular/tetrahedralgrids.Formixedelements,edge-basedcontributionsareusedtoincreasetheh-ellipticity
oftheoperator. Theavg-LSQschemeusestheaverageofthedual-volumeLSQgradientstodetermineagradientat
theface,whichisaugmentedwiththeedge-baseddirectionalcontributiontodeterminethegradientusedintheflux.
AfulllinearizationhasbeenimplementedfortheGreen-Gaussscheme,enablingarobustmulticolorGauss-Seidel
relaxation.Theavg-LSQschemehasacomparativelylargerstencilanditsfulllinearizationhasnotbeenimplemented;
insteadrelaxationoftheavg-LSQschemereliesonanapproximatelinearization,whichconsistsofedgetermsonly.
Sofar,weobservegoodsmoothingrateswiththisapproach,butpreviousanalysishasshownthatthesmoothingrate
can deteriorate on highly skewed grids.6 The estimates for the smoothing rates obtained with quantitative analysis
methods12 areshowninSectionVI.NotethattheGreen-Gaussschemereliesonanelement-baseddatastructureand
henceisnotconsideredforagglomeratedgrids.
Coarse-GridDiscretizations
Avg-LSQ
DirectDiscretization
Edge-Terms-Only
R AfP∗
GalerkinDiscretization 0 0
R AfP
0 1
Table2. Summaryofcoarse-griddiscretizations.
Theavailablecoarse-griddiscretizationsaresummarizedinTable2,listingtwopossibledirectdiscretizationsand
twopossibleGalerkindiscretizations,inwhichthecoarsegridoperatorsarederivedfromthefinegridoperator. The
edge-terms-only discretization is often cited as a thin-layer discretization in the literature;2–4 it is a positive scheme
butisnotconsistent(i.e.,itsdiscretesolutiondoesnotconvergetotheexactcontinuoussolutionwithconsistentgrid
refinement)unlessthegridisorthogonal.7,8,11 Anorthogonalgridwouldhaveeachedgeconnectingnodesacrossa
facebeingco-linearwiththeface-normaldirectionnˆ.
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TheGalerkincoarse-gridoperator1 isdenotedbyRAP. Becausethegoverningequationisasecond-orderequa-
tion,theGalerkinconstruction,R AfP ,isformallyinconsistent;theheuristiccorrectionfactoradoptedbyMavriplis2
0 0
isused,
1
Ac =R AfP∗ = R AfP . (7)
0 0 2 0 0
Thecorrectionfactorisderivedbyenforcingconsistencyonuniformly-agglomeratedhexahedralmeshes.TheGalerkin
construction, R AfP , is consistent, but was found to be unstable. Dirichlet boundary conditions are enforced
0 1
strongly. Thecoarse-gridoperatorisoverwrittenwiththeboundaryconditionlinearizationatboundarynodes.
III. QuantitativeAnalysisofUnstructuredMultigridSolvers
The quantitative analysis methods for unstructured multigrid solvers considered in this section are idealized re-
laxation(IR)andidealizedcoarse-grid(ICG)iterationsintroducedinRef.[12]. Themethodsanalyzethemaincom-
plementarypartsofamultigridcycle: relaxationandcoarse-gridcorrection. Inmultigrid,relaxationandcoarse-grid
correctionareassignedcertaintasks: relaxationisrequiredtosmooththealgebraicerrorandcoarse-gridcorrectionis
requiredtoreducesmoothalgebraicerrors.
To apply the analysis, we first choose a desired sample fine-grid solution (zero is a natural choice for linear
problems)andsubstituteitintotheequationstogeneratethecorrespondingsourceandboundarydata. Thenweform
aninitialguess(forexample, arandomperturbationofthesolution); thus, thefine-gridalgebraicerrorisknown. In
the analysis, idealized iterations probe the actual two-grid cycle to identify parts limiting the overall efficiency. In
these iterations, one part of the cycle is actual, and its complementary part is replaced with an idealized part. The
idealized parts do not depend on the operators to be solved. They are numerical procedures acting directly on the
knownalgebraicerrortoefficientlyfulfillthetaskassignedtothecorrespondingpartofthetwo-gridcycle.Theresults
oftheanalysisarenotsingle-numberestimates;theyareratherconvergencepatternsoftheiterationsthatmayeither
confirmorrefuteourexpectationsastowhatpartoftheactualcycleisnotefficientincarryingouttheassignedtask.
These IR and ICG analysis methods can be regarded as a numerical extension of the Fourier analysis to problems
wheretheclassicalFourieranalysisisinapplicable,inparticular,tounstructured-gridsolvers.
IR and ICG iterations are analysis methods that test computational efficiency of a two-grid cycle. The two-grid
cycleamplificationmatrix,M,transformstheinitialfine-gridalgebraicerror,eold,intotheafter-cycleerror,enew,
enew =Meold. (8)
Theamplificationmatrixcanbedefinedas
M =Sν2CSν1. (9)
Here, ν andν aresmallnonnegativeintegersrepresentingthenumberofpre-andpost-relaxationsweeps,S isthe
1 2
fine-gridrelaxationamplificationmatrix,andC istheamplificationmatrixofthecoarse-gridcorrection.
C =E−P (Ac)−1R Af, (10)
0 0
whereAc andAf arethecoarseandfinegridoperatormatrices, P andR aretheprolongationandagglomeration
0 0
matrices,andE isthefine-grididentitymatrix.
ForIRiterations,thecoarse-gridcorrectionpartisactualandtherelaxationisidealized. Theidealizedrelaxation
maybedefinedasanexpliciterror-averagingprocedure. Inthispaper,weemploytheIRprocedurethatreplacesthe
algebraicerrorateachdualcellwithanaverageofalgebraicerrorsatedge-adjacentcells. Ateachrelaxationstep,the
knownexactsolution,ifnotzero,issubtractedfromthecurrentapproximationtoobtainthealgebraicerrorfunction.
Theexplicitaveragingprocedureisapplieddirectlytotheerrorfunction. Thenumberofsweepsthroughoutthegrid
istakenasν orν andwedenoteresultsasIR(ν ,ν ). Theexactsolutionisthenaddedback. Slowconvergenceof
1 2 1 2
IRiterationsindicatesinsufficientcoarse-gridcorrection.
InICGiterations,therelaxationschemeisactualandthecoarse-gridcorrectionisidealized.Assumingthattheag-
glomerationandprolongationoperatorsaresuitableforefficientmultigridsolution,theidealizedcoarse-gridcorrection
involvesidealizedfineandcoarseoperatorsAf andAc ,suchthatDc(Ac )−1isanaccurateapproximationto
ideal ideal Ω ideal
Df(Af )−1forsmootherrorcomponents. Here,Dc andDf arediagonalmatriceswithcorrespondingcoarse-and
Ω ideal Ω Ω
fine-gridvolumesonthediagonals. Thesimplestidealizedoperatorsarecorrespondingfineandcoarsegrididentity
matrices. Withthischoice,theidealizedcoarse-gridcorrectionbecomes
C =E−P (Dc)−1R Df. (11)
ideal 0 Ω 0 Ω
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Note that the operator (Dc)−1R Df represents volume-weighted averaging. In ICG analysis, the idealized C
Ω 0 Ω ideal
is applied directly to the known algebraic errors obtained after pre-relaxation sweep(s) of the actual relaxation. In
implementation,thealgebraicerrorisaveragedtothecoarsegrid,changedinsign,andthenprolongedtothefinegrid.
SlowconvergenceobservedintheICGiterationsisasignofpoorsmoothinginrelaxation. WedenoteICGresultsas
ICG(ν ,ν ).
1 2
IV. TargetGridsandAgglomerations
1
0.8
0.6
z
0.4
0.2
0
0 0.5 1
x
Figure2. Atypical2Dtargetgrid.
ThegridsconsideredaregeneratedbysplittingisotropicmappedCartesiangridsintotriangular(2D)ortetrahedral
elements (3D) and then randomly perturbing the grid points by up to one quarter of the local mesh size. A typical
targetgridisshowninFigure2for2Dwith33pointsineachdirection. Anorthographicviewoftheboundarygrids
ofatypicaltarget3DgridisshowninFigure3,againfor33pointsineachdirection.
Thegridsareagglomeratedwithinatopology-preservingframework,inwhichhierarchiesareassignedbasedon
connections to the computational boundaries. Corners are identified as grid points with three or more boundary-
condition-type closures (or three or more boundary slope discontinuities). Ridges are identified as grid points with
two boundary-condition-type closures (or two boundary slope discontinuities). Valleys are identified as grid points
withasingleboundary-condition-typeclosureandinteriorsareidentifiedasgridpointswithnoboundaryclosure.The
agglomerations proceed hierarchically from seeds within the topologies, first corners, then ridges, then valleys, and
finallyinteriors. Rulesareenforcedtomaintaintheboundaryconditiontypesofthefinergridwithintheagglomerated
grid. Candidatevolumestobeagglomeratedarevettedagainstthehierarchyofthecurrentlyagglomeratedvolumes
usingtherulessummarizedinTable3. Theallowedentriesdenotethatinteriorvolumescanbeagglomeratedtoany
existingagglomerate. Thesingledisallowedentryenforcesthattwocornerscannotbeagglomerated. Theconditional
entries denote that further inspection of the connectivity of the topology must be considered before agglomeration
is allowed. For example a ridge can be agglomerated into a corner if the ridge is part of the boundary condition
specification associated with the corner. As another example, a ridge can be agglomerated into an existing ridge
agglomerationifthetwoboundaryconditionsassociatedwitheachridgearethesame.Also,theprolongationoperator
P ismodifiedtoprolongonlyfromhierarchiesequalorabovethehierarchyoftheprolongedpoint. Hierarchieson
1
eachagglomeratedgridareinheritedfromthefinergrid.
There are two agglomeration schemes, referred to as scheme 1 and 2, that have evolved historically within this
development. Theagglomerationscheme1ordersthepossiblepointswithinahierarchyusingthedistancefromthe
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Figure3. Orthographicviewofatypical3Dtargetgrid.
Figure4. Controlvolumeboundaries(non-lumped)for2Dagglomerationsusingscheme1(toprow)andscheme2(bottomrow).
cornersofthegridandclosestpointsaretakenfirst. Givenaseed,atriadisconstructedusingasurroundingcloudof
points,definedfromtheadjacencylist. Thefirstlegofthetriadisdefinedbytheseedandthenearestpoint. Thenext
legofthetriadisdefinedbyincludinganotherpointfromtheentriesinthecloudsuchthatthelegismostorthogonal
tothefirstleg. Thethirdlegisfoundastheonemostparalleltothecross-productofthefirsttwolegs. Pointswithin
thevolumedefinedbythetriads(extendedtoinfinitelength)aretaken,firstfortheedge-adjacenciesinthecloudand
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Figure5. Controlvolumeboundaries(non-lumped)for3Dagglomerationsusingscheme2.
HierarchyofAgglomeration HierarchyofAddedVolume AgglomerationDecision
corner interior allowed(cornertointerior)
corner valley conditional
corner ridge conditional
corner corner disallowed(twocorners)
ridge interior allowed(ridgetointerior)
ridge valley conditional
ridge ridge conditional
valley interior allowed(valleytointerior)
valley valley conditional
interior interior allowed(interiortointerior)
Table3. Admissibleagglomerations.
subsequentlyfortheentireadjacency, tosatisfyaglobalcoarseninggoal(4volumesagglomeratedfor2Dand8for
3D). The agglomeration scheme 2 also starts from corners. After all corners have been agglomerated, a front list is
defined by collecting nodes adjacent to the agglomerated corners. It then proceeds to agglomerate nodes in the list
(whileupdatingthelistasagglomerationproceeds)inthefollowingorder:ridges,valleys,interiors.Anodeisselected
amongthoseinthesamehierarchy,thathastheleastnumberofnon-agglomeratedneighborstoreducetheoccurrences
ofagglomerationswithsmallnumbersofvolumes. Foragivenseed,itcollectsallneighborsandagglomeratesthem
uptoaspecifiedmaximumnumber,e.g.,8in3D.Theagglomerationcontinuesuntilthefrontlistbecomesempty. For
eitheragglomerationscheme,agglomerationscontainingonlyafewvolumesarecombinedwithotheragglomerations,
asistypicalofthemethodsusedintheliterature.
For highly stretched meshes, the agglomerations are first constructed along the boundary of the grid (corners,
ridges,andvalleys)andthenthecellsareagglomeratedfromtheboundarytotheendsoftheimplicitlinesassociated
with the stretched grid. The boundary agglomerate is merged with the volumes corresponding to the next point
in the line. The agglomeration continues to the end of the shortest line in the boundary agglomerate, forming an
agglomeration from the volumes associated with the next two entries in the line. After agglomeration of lines, the
algorithmusesthepointagglomerationmethodfortherestofthedomain.
Figure4showsthreeagglomeratedgridsgeneratedfromtheprimalgridinFigure2usingschemes1and2.Figure5
showsthreeagglomeratedgridsgeneratedfromtheprimalgridinFigure3usingscheme2. Theagglomerationsare
representativeofthoseintheliterature.
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FineGrid DirectDiscretization GalerkinDiscretization
Avg-LSQ Edge-Terms-Only R AfP∗ R AfP
0 0 0 1
33x33 0.15 0.20 0.51 divergent
65x65 0.22 0.26 0.53 divergent
129x129 0.21 0.32 0.60 divergent
257x257 0.17 0.45 — —
Table4. Summaryofmulti-levelasymptoticconvergenceratesperV(2,1)multigridcyclewithagglomerationscheme1forvariouscoarse-
gridoperators.
FineGrid DirectDiscretization GalerkinDiscretization
Avg-LSQ Edge-Terms-Only R AfP∗ R AfP
0 0 0 1
33x33 0.17 0.29 0.49 divergent
65x65 0.19 0.44 0.58 divergent
129x129 0.18 0.54 0.68 divergent
Table5. Summaryofmulti-levelasymptoticconvergenceratesperV(2,1)multigridcyclewithagglomerationscheme2forvariouscoarse-
gridoperators.
V. TwoDimensionalResults
A summary of multigrid convergence rates is compiled in Tables 4 and 5 for the two agglomeration schemes,
respectively,foraV(2,1)multigridcyclewithvariouscoarse-gridoperators. Multigridcyclesemployasmanylevels
aspossible;e.g.,thereare6levelsusedforthe129x129targetgridand4levelsforthe33x33targetgrid. Somewhat
surprisingly, with the Galerkin coarse-grid operator constructed via R AfP , the multigrid algorithm is divergent.
0 1
Thereason,confirmedbyanalysis,isthatthecoarse-gridoperator,althoughaccurate,losesh-ellipticity.1 Thislossof
h-ellipticityfortheGalerkinoperatorwithsimplex-basedP -prolongationhasbeenobservedevenwithquadrilateral
1
grids,forwhichbilinearprolongationisknowntoresultinh-ellipticcoarse-gridoperators.
WiththeGalerkincoarse-gridoperatorconstructedviaR AfP∗,themultigridalgorithmisstable. However,the
0 0
convergenceratesdegradeonfinergridswitheitheragglomerationscheme. Withthecoarse-gridoperatorusingonly
theedgeterms,theconvergencepercycleisbetter,butagainshowsadeteriorationonfinergrids. Thedeteriorationis
noticeablyworsewiththeagglomerationscheme2, althoughitishardtojudgethereasonfromvisualinspectionof
theagglomeratedgrids. Withtheavg-LSQscheme,theconvergencepercycleis0.22orbetterandgrid-independent.
1stOrderVariation
100 2ndOrderVariation
Agglomerates(65x65primal) Edge-Terms-Only
Agglomerates(65x65primal) Avg-LSQ
Agglomerates(129x129primal)Avg-LSQ
10-1
or
Err10-2
n
o
ati
cretiz10-3
s
Di
10-4
10-5
0.05 0.1 0.150.2
EquivalentMeshSize,h
V
Figure6. Spatialconvergenceofdiscretizationerrorforagglomeratefamilies.
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Thespatialconvergenceofdiscretizationerrorforagglomeratefamilieswiththeavg-LSQtarget-griddiscretiza-
tion is shown in Figure 6. Results with the edge-terms-only (thin-layer) discretization are also shown for reference.
The manufactured solution is U = sin(πx+0.8πy)+0.1x+0.2y and the coarser grids were generated using ag-
glomerationscheme2.Eachagglomeratefamilyiscomposedofasingleprimalgridandagglomeratedgridsgenerated
recursively;aparticularagglomeratefamilyisdenotedbythedensityoftheprimalmeshinparentheses. TheL norm
1
ofthediscretizationerrorisshownversusanequivalentmeshsize,takenastheL normofalocalcharacteristicdis-
1
tance,i.e.,h =||Ω1/d||,wheredisthenumberofspatialdimensions. Theedge-terms-onlydiscretizationshowsno
V
orderproperty,asexpected,buttheavg-LSQschemeshowsasecond-orderconvergenceofdiscretizationerror. This
resultisbelievedtobethefirsttoshoworderpropertiesfordiffusiononafamilyofagglomeratedmeshes.
10-3
PrimalGrid
Agglomerate1
10-5 Agglomerate2
Agglomerate3
Error10-7 Agglomerate4
al
u
d
si
Re10-9
10-11
10-13
0 2 4 6 8 10 12 14 16 18 20
MultigridCycle
Figure7. Multigridconvergenceforagglomeratefamilycomposedofthe129x129primalmeshanditscoarsenedagglomerates.
Forthefineragglomeratefamily,multigridconvergenceisshowninFigure7usingtheavg-LSQdiscretizationon
allgrids. MultilevelV(2,2)cyclesareusedwith2levelsonthecoarsestagglomerateand6levelsontheprimalmesh.
Theinitialconditionsaretakenastheexactsolutionwitharandomlyperturbederroroneachgrid. Grid-independent
convergenceisshownwithapproximatelyanorderofmagnitudereductioninresidualpercycle.
0.5 0.5
0.45 0.45
z z
0.4 0.4
0.5 0.55 0.6 0.65 0.5 0.55 0.6 0.65
x x
(a)Agglomerationscheme1. (b)Agglomerationscheme2.
Figure8. Interiorviewofagglomeratedgridsgeneratedfrom257x257element-basedgrid.
Thegrid-dependentconvergenceofmultigridcycleswiththeedge-terms-onlyscheme(Tables4and5)isattributed
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to the poor coarse-grid correction, which is confirmed by quantitative analysis. Both ICG and IR were applied to a
family of element-based grids (33x33, 65x65, 129x129, and 257x257) with coarser grids constructed in turn using
eachofthetwoagglomerationschemes. ConvergenceoftheICG(3,3)schemewaslessthan0.1percycleinallcases,
indicatingthemulticolorrelaxationisnotasourceofthegrid-dependentconvergence. TheresultsofapplyingIR(3,3)
areshowninTable6withthecoarse-gridcorrectionusingtheavg-LSQandtheedge-terms-onlyschemesforeachof
the two agglomeration schemes. The asymptotic convergence per cycle and the number of cycles to reach machine
precisionresidualsfromtherandomperturbationappliedinitiallyaretabulated. Withthecoarse-gridcorrectionusing
theavg-LSQscheme,theconvergenceratespercyclearegrid-independentandbetterthan0.21;thenumberofcycles
toconvergeis12atmost. Withthecoarse-gridcorrectionusingtheedge-terms-onlyscheme, theconvergencerates
andnumberofcyclestoconvergearegrid-dependent.
Asintheactualcycleresultsshownpreviously,IRresultswiththeagglomerationscheme2showmoredependence
onthegrid. Comparingthetwoforthe257x257parentgrid,agglomerationschemes1and2provideagglomeration
ratiosof3.3and3.9,respectively;thus,scheme2isclosertotherequestedratioof4. Aninteriorsectionofthetwo
agglomerationsisshowninFig.8.Scheme2resultsinapiecewisenear-regulargridwithadistinctskewanglenear45
deg;thegridfromscheme1isnoticeablylessregular. Themaximumskewangleishigherwithscheme1(70degas
comparedto60deg);however,withscheme2,47percentofthenodeshavecontrolvolumeswithskewanglesgreater
than40degascomparedtoonly23percentofthenodeswithscheme1. Itishypothesizedtheconsistentskewingof
scheme 2 causes the increased degradation with the edge-terms-only discretization. In any event, the differences in
convergencebetweenthetwoschemesarefarsmallerwithaconsistentdiscretizationonthecoarsegrid.
AgglomerationScheme1 AgglomerationScheme2
Coarse-GridDiscretization Coarse-GridDiscretization
Element-BasedGrid Avg-LSQ Edge-Terms-Only Avg-LSQ Edge-Terms-Only
33x33 0.11(9) 0.32(15) 0.14(10) 0.55(26)
65x65 0.13(10) 0.49(21) 0.15(10) 0.72(44)
129x129 0.20(11) 0.54(26) 0.21(12) >0.99(>200)
257x257 0.17(10) 0.61(28) 0.20(11) >0.99(>500)
Table6. AsymptoticconvergencepercycleusingIR(3,3)analysis;cyclesareinparentheses.
With a consistent coarse-grid discretization, such as the avg-LSQ scheme, we expect good 2-level convergence
rates. With the avg-LSQ scheme, relaxation is implemented within a defect-correction setting in which only the
linerizationoftheedge-terms-onlyschemeisretained. TheviabilityofthisapproachischeckedusingICG(3,3)for
the family of grids agglomerated from the parent 257x257 grid. The convergence per cycle is shown in Table 7 for
differentagglomerationlevels, wheretheelement-basedgridisdenotedaslevel0. Inallcases, theedge-terms-only
schemeprovidesadequaterelaxation,yieldinganorderofmagnitudeconvergenceperICG(3,3)cycle.
AgglomerationScheme
AgglomerationLevel 1 2
4 0.06(8) 0.06(8)
3 0.06(8) 0.05(7)
2 0.07(8) 0.07(8)
1 0.06(8) 0.08(8)
0 0.07(8) 0.08(8)
Table7. AsymptoticconvergencepercycleusingICG(3,3)analysisforfamilyofagglomeratedgrids;cyclesareinparentheses.
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