Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method

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rchive

T

OULOUSE

A

rchive

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uverte (

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To cite this version : Guittet, Arthur and Lepilliez, Mathieu and

Tanguy, Sébastien and Gibou, Frédéric Solving elliptic problems with

discontinuities on irregular domains – the Voronoi Interface Method.

(2015) Journal of Computational Physics, vol. 298. pp. 747-765. ISSN

0021-9991

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Solving

elliptic

problems

with

discontinuities

on

irregular

domains

–

the

Voronoi

Interface

Method

Arthur Guittet

a

,

∗

,

Mathieu Lepilliez

c

,

d

,

e

,

Sebastien Tanguy

c

,

Frédéric Gibou

a

,

b

a_Department_of_Mechanical_Engineering,_University_of_California,_Santa_Barbara,_CA_93106-5070,_United_States b_Department_of_Computer_Science,_University_of_California,_Santa_Barbara,_CA_93106-5110,_United_States c_Institut_de_Mécanique_des_Fluides_de_Toulouse,_2bis_allée_du_Professeur_Camille_Soula,₃₁₄₀₀_Toulouse,_France d_Centre_National_d’Etudes_Spatiales,₁₈_Avenue_Edouard_Belin,₃₁₄₀₁_Toulouse_Cedex_9,_France

e_Airbus_Defence_&_Space,₃₁_Avenue_des_Cosmonautes,₃₁₄₀₂_Toulouse_Cedex_4,_France

a

b

s

t

r

a

c

t

Keywords:

Level-set

Ellipticinterfaceproblems Discontinuouscoeﬃcients Irregulardomains Voronoi Finitevolumes Quad/octrees

Adaptivemeshreﬁnement

We introduce asimple method, dubbedthe Voronoi InterfaceMethod, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method producesalinearsystemthatissymmetricpositivedefinitewithonlyitsright-hand-side affected by the jump conditions.The solution and the solution’s gradients are second-order accurateand first-order accurate, respectively, inthe L∞ norm, even inthe case of large ratios in the diffusion coefficient. Thisapproach is alsoapplicable to arbitrary meshes. Additionaldegrees offreedom are placedclose tothe interface and aVoronoi partitioncentered ateachofthesepointsisused todiscretizethe equations inafinite volumeapproach.BoththelocationsoftheadditionaldegreesoffreedomandtheirVoronoi discretizationsarestraightforwardintwoandthreespatialdimensions.

1. Introduction

WefocusontheclassofEllipticproblemsthatcanbewrittenas:

∇ · (β∇

u

)

+

ku

=

f in

−

∪

+

,

[

u

] =

g on

,

[β∇

u

·

n

] =

h on

,

(1)

wherethe computational domain

iscomposed oftwo subdomains,

− and

+, separatedby a co-dimensionone in-terface

(see Fig. 1), withn the outward normal. Here,

β

= β(

x

),

with x

∈ R

n (n

∈ N

), is boundedfrom below by a positiveconstant and

[

q

]

=

q+

−

q− indicates adiscontinuityinthe quantityq across

,

f is inL2, g,h andk aregiven. Notethatthisgeneralformulationincludespossiblediscontinuitiesinthediffusioncoeﬃcient

β

andinthegradientofthe solution

∇

u. DirichletorNeumann boundaryconditionsare appliedon theboundaryof

,

denoted by

∂.

Thisclassof equations,wheresome orall ofthe jumpconditionsare non-zero,isa cornerstone inthemodeling ofthedynamics of

*

Correspondingauthor.

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Fig. 1. Geometry of the problem.

importantphysicalandbiologicalphenomenaasdiverseasmultiphaseﬂowswithandwithoutphasechange,biomolecules’ electrostatics,electrokinetics(Poisson–Nerntz–Planck)modelswithsourcetermorelectroporationmodels.

Giventheimportanceofthisclassofequations,severalapproacheshavebeenpursuedtocomputationallyapproximate their solutions,each withtheirown prosandcons.The finiteelementmethod(FEM)isone oftheearliest approachesto solvethisproblem[4,10,12,19,32,35]andhastheadvantageofprovidingasimplediscretizationformalismthatguarantees the symmetryanddefinitepositiveness ofthe correspondinglinearsystem, eveninthecaseofunstructuredgrids. Italso provides a framework whereapriori errorestimatescan be usedto bestadaptthe meshinorder tocapturesmallscale details.However,theFEMisbasedonthegenerationofmeshesthatmustconformtotheirregulardomain’sboundaryand must satisfy some restrictive quality criteria, a taskthat is difficult,especially in three spatial dimensions. The difficulty is exacerbatedwhen thedomain’sboundaryevolves during thecourseofacomputation, asit isthecaseformostofthe applicationsmodeledbytheseequations.Meshgenerationisthefocusofintenseresearch[58],asthecreationofunwanted sliverelementscandeterioratetheaccuracyofthesolution.

Methods based on capturing the jump conditions do not depend on the generation of a mesh that conformsto the domain’s boundary, hence avoiding the mesh generation diﬃculty altogether. However, they must impose the boundary conditions implicitly,which isa non-trivialtask.A popularapproach isthe ImmersedInterfaceMethod(IIM) of Leveque and Li[36], andthe morerecent development ofImmersed FiniteElement Method(IFEM) andImmersedFinite Volume Method (IFVM)[40,27,23]. Thebasis of the IIMisto use Taylorexpansionsofthe solution oneach side of theinterface andmodifythestencilslocaltotheinterfaceinordertoimposethejumpconditions.Assuch,solutionscanbeobtainedon simpleCartesiangridsandthesolutionissecond-orderaccurateintheL∞norm.Thecorrespondinglinearsystem,however, isasymmetricunlessthecoeﬃcient

β

hasnojumpacrosstheinterface.Anotherdiﬃcultyistheneedtoapproximatesurface derivativesalong

aswellastheevaluationofhigh-order jumpconditions.Thesedifficultieshavebeenaddressedinthe Piecewise-polynomial InterfaceMethod ofChenand Strain[11] andseveral other approacheshaveimproved theefficacy of theIMM[38,39,64,9,1–3].We notealso thattheearliest approachon Cartesiangrid isthat ofMayo[43],whoderived an integral equation tosolve thePoissonandthebi-harmonic problemswithpiecewise constantcoefficientson irregular domains; thesolution issecond-order accurate in the L∞ norm.We alsorefer the interested researcherto the matched interfaceandboundary(MIB)method[66,65].

Theﬁniteelementcommunityhasalsoproposedembeddedinterface approaches,includingdiscontinuous Galerkinand theeXtendedFiniteElementMethod(XFEM)[37,29,48,17,8,47,33,22,28,62].Thebasicideaistointroduceadditionaldegrees offreedom1 neartheinterfaceandaugmentthestandardbasis functionson theseelements withbasis functionsthatare combinedwithaHeavisidefunctioninordertohelpcapturethejumpconditions.

In[16],theauthorsintroduceasecond-orderaccuratediscretizationinthecaseof,possiblyadaptive,Cartesianmeshes using acut-cell approach.The jumpcondition is imposedby determiningthe ﬂuxesonboth sideof theinterface, which are constructedfromacombinationofleastsquaresandquadraticapproximations.In[51],theauthorsalsouseacutcell approachbutimposethejumpwiththehelpofacompact27-pointstencil.

The GhostFluidMethod(GFM), originallyintroduced toapproximatetwo-phasecompressibleflows [21],hasbeen ap-pliedtothesystemproblem(1)in[41].Thebasicideaistoconsiderfictitiousdomains andghostvaluesthatcapturethe jumpconditionsinthediscretizationatgridnodesneartheinterface.Anadvantageofthisapproachisthatonlythe right-hand-sideofthelinearsystemisaffectedbythejumpconditions.However,inordertoproposeadimension-by-dimension approach,theprojectionofthenormaljumpconditionsmustbeprojectedontotheCartesiandirections.Asaconsequence, the tangential componentof thejump isignored. Nonetheless, themethodhas beenshownto be convergent with first-orderaccuracy[42].TheGFMwasalsoshowntoproducesymmetricpositivedefinitesecond-orderaccuracy[25] andeven fourth-order accuracy[24],butforadifferentclassofproblem, namelyforsolving Ellipticproblemsonirregular domains withDirichletboundaryconditions.Infact,symmetricpositivedefinitesecond-orderaccuratesolutionscanalsobeobtained inthecasewhereNeumannorRobinboundaryconditionsareimposedonirregulardomains[55,50,53,54].Thesemethods canbetriviallyextendedtothecaseofadaptiveCartesiangridsandwerefertheinterestedreaderstothereviewofGibou, MinandFedkiw[26]formoredetails.Inthecaseofjumpconditions,CocoandRusso[14]havealsousedafictitiousdomain

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approach,wherearelaxationschemeisusedtoimposetheboundarycondition;thesolutionissecond-orderaccurate.The sameauthorshavealsointroducedamethodtoconsiderDirichlet,NeumannandRobinboundaryconditionsonanirregular interface [15].Latigeet al.[46] havealso presenteda methodbased on ﬁctitiousdomains usinga piecewise polynomial representationofthesolutiononadualgrid,alsoobtainingsecond-orderaccurate solutions.Finally,[31] haveappliedthe ghostﬂuidideainavariationalframework.

Relatedideasareusedinmethodscombiningﬁctitiousdomainsandvariationalformulations[49,7],dubbedvirtualnodes approaches.SomeoftheseapproachescanbeconsideredsimilartoXFEMmethods[30,60,18],whileothersaredifferentand offer advantages when considering under-resolved, possibly non-smooth, interfaces [49,59,56]. This philosophy has been used in[34],which introduces a virtual node algorithm forsolving Elliptic problems onirregular geometries withjump conditionsas well asDirichletor Neumannboundary conditionsimposed on

.

Thesolutions are second-order accurate inthe L∞ normandthe approachprovides a unifyingtreatment forDirichlet, Neumannandjump boundaryconditions; howeversacriﬁcingsimplicity.

Ratherrecently,CisterninoandWeynans[13]introducedasecond-orderaccuratemethodthatusesadditionaldegreesof freedomonthedomain’sboundaryandusethemtodiscretizethePoissonoperatorwithjumpconditionsina dimension-by-dimension framework. The authors also present how to carefully approximate the gradients. The method produces second-order accuratesolutions andanonsymmetric linearsysteminpartbecauseoftheneed tochangethe sizeofthe stencilfornodesadjacenttothedomain’sboundary.

Weintroduce acapturingcomputational approach,theVoronoiInterfaceMethod,that producessecond-order accurate solutionsinthe L∞ norm.ThisapproachisbasedonbuildingaVoronoidiagramlocaltotheinterface,whichenablesthe directdiscretization ofthejumpconditions inthenormaldirection. The linearsystemissymmetricpositive deﬁniteand the jump conditionsonly inﬂuence the right-hand-side. The construction ofthe local Voronoimesh is a straightforward andparallelizableprocessandcanbebuiltwithexistinglibrariesthatarefreelyavailable.Inthepresentwork,weusethe excellent

Voro++

libraryinthreespatialdimensions[57].Thismethodisdifferentfrombody-fittedmethodsinthatitrelies onthepost-processingofanexisting backgroundmesh,thusavoidingthestandard difficultiesassociatedwithbody-fitted approaches.WenotethatpreviousworkshavedevelopedsolversforthePoissonequationonVoronoidiagrams(see[63,61]

andthereferencestherein);howeverdiscontinuitiesacrossanirregularinterfacewerenotconsidered.

2. Thegeometricaltools 2.1. Thelevel-setmethod

The level-set method[52] isa powerful wayof representingirregular interfaces asthe zero contourof a continuous function.Thisrepresentationisconvenientinthatitcanbeappliedtothecaseofmovingboundariesthatcanchangetheir topology.Italsoprovidesaframeworkthatlendsitselftodesignsharp discretizations.

We use the level-set set framework to capture the irregular interface on which the discontinuities are enforced. We deﬁnealevel-setfunction

φ

onthedomain

suchthattheirregularinterface

isdescribedby

= {

x

∈ R

n

_∈

_|

_{φ (}

_x

₎

₌

₀

_}

and

φ

isnegative onone sideoftheinterfaceandpositiveon theotherside,aspicturedinFig. 1.Eventhough inﬁnitely manyfunctionssatisfythiscriteria,itisconvenienttoworkwithasigneddistancefunctiontotheirregularinterface,i.e.a functionthatisnegativeononesideoftheinterface,positiveontheothersideandsuchthatitsmagnitudeateverypoint isthe distancefromthe pointto theinterface.Constructing a signeddistancefunctionfroman arbitraryfunctioncan be doneforexamplebyfollowingtheprocedureexplainedin[44].Thenormaltotheinterfaceisthenobtainedas

n

=

∇φ

,

andthecurvatureas

κ

= ∇ ·

n

.

Notethatifthelevel-setfunctionisasigneddistancefunction,

∇φ

=

1,andtheprojection onto

ofanygivenpointx

iseasilycomputedas:

x

=

x

− φ(

x

)

∇φ(

x

).

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2.2. Voronoidiagrams

ThesolverwepresentisbasedonVoronoidiagrams,whichcanbegeneratedlocallywithexistingproceduresandfreely availablelibraries.Inthiswork,weusetheexcellent

Voro++

library[57].Forthesakeofclarity,weintroducetheVoronoi diagram:givenasetofpoints,whichwecallseeds,theVoronoicellofagivenseedconsistsofallthepointsofthedomain that are closer to that seed than to anyother seed.Hence, the collection ofall theVoronoi cellsof a set of seeds is a tessellationofthedomain,i.e.atilingthatﬁllsthedomainanddoesnotcontainanyoverlaps.

Givenacomputationalmesh,whichweconsidertobeuniforminthissectionforclarity,weproposetomodifythemesh sothattheirregularinterfacecoincideswiththeedgesofthenewmeshandthedegreesoffreedomclosetotheinterface arealllocatedatthesamedistancefromtheinterface.

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Fig. 2. IllustrationoftheprocedureforgeneratingaVoronoidiagrambasedcomputationalmesh.Theleftfigurepresentstheoriginaluniformmeshandthe rightfigureshowsthefinalcomputationalmesh.Thepurplesquaredegreesoffreedomhavebeenremovedandtheorangedotsdegreesoffreedomhave beenaddedclosetotheinterface.

Fig. 3. ExampleofaVoronoimeshwhereadegreeoffreedomisconnectedtomorethanoneotherdegreeoffreedomlocatedontheothersideofthe irregularinterface.Thesmoothingprocedureisillustratedontheright.

The procedure is illustrated inFig. 2.Starting from auniform grid,we ﬁnd the projection ofthe degreesoffreedom whose control volumeis crossedby the irregularinterface ontothe interface using(2),andwe removethose degreesof freedom from the original list of unknowns. If a projected point is within

diag

/5 of

a previously computed projected point, where

diag

isthelengthofthediagonalofthesmallestgridcell,we skipthispoint.Otherwise,weadd twonew degrees offreedom located ata distance d of theinterface in the normaldirectionon either side ofthe interface. We

repeat this procedure for all projected points. The new set of degrees of freedom is therefore made up of the original degreesoffreedom whosecontrolvolume isnotcrossed bythe interfaceandofthenewdegreesoffreedomaddednext to theinterface. Thisconstitutesthe setofseeds forthe Voronoidiagramcomputational meshonwhich weperformthe computations.EachVoronoicell canthenbe generatedindependentlybasedonthelocalneighborhood ofeachdegree of freedom,makingthegenerationoftheVoronoimeshembarrassinglyparallel.

Notethatallthenewdegreesoffreedomareplacedatthesamedistanced fromtheinterface.Thisisafreeparameter

ofourmethod,andexperimentingwithvariousreasonablevaluesshowslittleimpactonthenumericalresults.Wechoose

d

= diag/

5. This simple procedure will be shown in Sections 4.1 and 4.2 to be suﬃcient to construct second-order

accuratesolutionsintheL∞ norm.

2.3. Smoothingthemesh

Thealgorithmdescribedintheprevioussectioncanleadtoundesirablegeometricalconfigurationsinthecasewhenthe interface isnotsufficientlyresolved.Fig. 3presentsonesuchconfiguration.Thecontrolvolumesofsomeofthedegreesof freedomareconnectedbyafacethatisnotcapturingproperlytheinterface.

It ispossibletoremediatethisissueby modifyingtheVoronoipartition inapost-processing step.Thecontrolvolume of anydegree offreedom that has beenadded next tothe interface should be connectedto exactly one control volume associated toadegreeoffreedom ontheotherside oftheirregularinterface.Consequently,ifmorethanoneneighbor is foundacrosstheinterface,wedisconnecttheundesiredonesbyremovingtheconnectingedgeasshowninFig. 3.Theedge anditstwoassociatedverticesarethenreplacedbyasinglevertexlocatedinthemiddleoftheremovededge.Thecontrol volumeofallthedegreesoffreedomoftheresultingmeshareconnectedtoatmostonecontrolvolumeontheotherside oftheinterfaceandtheinterfaceiscapturedproperly.

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Fig. 4. Nomenclaturefortheﬁnitevolumediscretization forthedegreeoffreedomi.Foreachneighboringdegreeoffreedom j,wecalldi jthedistance betweeni andj,si jthelengthoftheedge(orsurfaceofthepolygoninthreespatialdimensions)connectingi andj,andui jthevalueofu atthemiddle ofthesegment[i,j].Notethatbyconstructionui jcanbeconsideredtobeexactlyontheirregularinterface,inwhichcasewedeﬁneu+i jandu−i j.

However, this procedure alters the mesh which is no longer a Voronoi diagram, and the edge between two degrees of freedom is not guaranteed to be orthogonal to the line connecting the two degrees of freedom. The impact of this post-processing algorithm is analyzed in Sections4.1 and 4.2 anddoes not seem to improvethe method, we therefore recommendnotusingit.

2.4. Interpolatingbacktotheoriginalmesh

Ingeneral,ifsolvingadiffusionequationwithdiscontinuitiesispartofalargersolver,itisnecessarytointerpolatethe solutionfromtheVoronoimeshbacktotheoriginalmesh.Thisisaneasytaskgivensomebasicbookkeepinginformation linking the original degreesoffreedom to the onesgeneratedfor theVoronoi mesh.Withthis information,thesolution on theVoronoi meshcan be accessedforthe samecost than accessingthe dataon theoriginal mesh.The algorithm to interpolatethesolutionatagivenpoint

(

x

,

y

)

fromtheVoronoimeshisthenasfollows:

1. locatethecelloftheoriginalmeshcontaining

(

x

,

y

),

2. usingthebookkeepinginformation,identifytheVoronoidegreeoffreedom v

(

x

,

y

)

closestto

(

x

,

y

),

3. ﬁndthetwoneighborsofv

(

x

,

y

)

closestto

(

x

,

y

)

andonthesamesideoftheinterface,

4. computethemultilinearinterpolationofthesolutionusingthosethreedegreesoffreedomandevaluateitat

(

x

,

y

).

Thissimpleprocedureproducesasecond-orderinterpolationatanygivenpoint

(

x

,

y

).

3. SolvingaPoissonequationonVoronoidiagrams

Wediscretizeequation(1)withaﬁnitevolumeapproachontheVoronoidiagramintroducedinSection2.2.Weusethe notationsfromFig. 4.Weconsiderthedegreeoffreedomi withthesetofVoronoineighbors

{

j

}

.Applyingaﬁnitevolume approachtotheproblemati,wecanwrite

C

∇ · (β∇

u

)

dV

=

∂C

(β

∇

u

)

·

n_Cdl

≈

j si j

β

i ui j

−

ui di j

/

2

,

wheren_C istheouternormaltothe faceof

C

connectingi and j,si j isthelengthofthat face(orarea ofthesurfacein

threespatialdimensions)anddi j isthedistancebetweenthedegreesoffreedomi and j.Forthecasewheni and j areon eithersideoftheinterfacewith

φ

i

>

0,where

φ

i isthevalue ofthelevel-setfunctionatthedegreeoffreedomi,wecan

matchtheﬂuxattheirregularinterfaceasfollows, si j

β

i u+_{i j}

−

ui di j

/

2

=

si j

β

j uj

−

u−_{i j} di j

/

2

−

si j

β

∇

u

·

n

.

Wealsoknowthatu+_{i j}

=

u−_{i j}

+ [

u

]

.Injectingthisintothepreviousexpressiongives si j

β

i u+_{i j}

−

ui di j

/

2

=

si j

β

j uj

−

u+_{i j}

+ [

u

]

di j

/

2

−

si j

β

∇

u

·

n

⇔

u+_{i j}

(β

i

+ β

j

)

= β

juj

+ β

iui

+ β

j

[

u

] −

di j 2

β

∇

u

·

n

⇔

u+_{i j}

=

1

β

i

+ β

j

β

juj

+ β

iui

+ β

j

[

u

] −

di j 2

β

∇

u

·

n

.

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Inthecasewheni and j areonthesamesideoftheinterface,thederivationisthesamebutthecontributionsfromthe discontinuities vanish.The contributionfrom theinteraction betweenthe degreesof freedom i and j,in the casewhen

φ

i

>

0,totheﬁnitevolumediscretizationof(1)canthenbewritten

β

isi j u+_{i j}

−

ui di j

/

2

=

2

β

i

β

i

+ β

j si j di j

β

juj

+ β

iui

− (β

i

+ β

j

)

ui

+ β

j

[

u

] −

di j 2

β

∇

u

·

n

=

2

β

i

β

j

β

i

+ β

j si j uj

−

ui di j

−

2

β

i

β

j

β

i

+ β

j si j di j

−[

u

] +

di j 2

β

j

β

∇

u

·

n

= ˜β

i jsi j uj

−

ui di j

− ˜β

i j si j di j

−[

u

] +

di j 2

β

j

β

∇

u

·

n

,

where

˜β

i j istheharmonicmeanbetween

β

iand

β

j,i.e.

˜β

i j

=

|φ

i

| + |φ

j

|

|φ

i

|/β

i

+ |φ

j

|/β

j

=

2

β

i

β

j

β

i

+ β

j

.

Thecontributionoftheinteractionbetweenthedegreesoffreedom i and j tothelinearsystemistherefore

˜β

i jsi j

uj

−

ui

di j

,

whilethecontributiontotheright-handsideis

˜β

i j si j di j

−[

u

] +

di j 2

β

j

β

∇u

·

n

.

Note that we made useof the fact that

φ

is a distancefunction and ui j ismidway between i and j tosimplify the

expression.Similarly,wecanderivethecontributionoftheinteractionbetweeni and j forthecasewhen

φ

i

<

0 andobtain

thegeneralexpressionfortheinteractionbetweenanydegreesoffreedomi and j

˜β

i jsi j uj

−

ui di j

+ Vol(

C

)

·

ki

·

ui

= ˜β

i j si j di j

−

sign

(φ

i

)

[

u

] +

di j 2

β

j

β

∇

u

·

n

+ Vol(

C

)

·

fi

,

where

Vol

(

C)

isthe volume ofthe Voronoicell associated tothe degree offreedom i andki and fi are the respective values of k and f at the degree of freedom i. This discretization is entirelyimplicit and leads to a symmetric positive definite matrix.Thediscontinuities contributeonlytothe right-handsideofthelinearsystem. Notethatthisformulation is identicaltotheGhostFluidMethodof[41] inthecasewheretheirregularinterface isorthogonalto thefluxbetween the two degreesof freedom and located midway. In fact, the Voronoi Interface Method can be interpreted as a Ghost FluidMethodwherethefluxbetweentwo degreesoffreedomisguaranteedtobeorthogonaltothefaceconnectingtheir respectivecontrol volumes.Wesolve thelinearsystemwiththeConjugateGradientiterativesolverprovidedbythePetsc libraries[5,6]andpreconditionedwiththeHypremultigrid[20].WeenforceDirichletboundaryconditionson

∂.

4. Numericalvalidationonuniformmeshes

Inthissection,wevalidatetheadditionofthedegreesoffreedomalongtheirregularinterfaceandanalyzethe conver-genceofourmethodonvariousexamples.Inordertodemonstratethatoursolvercapturesthediscontinuitiesproperly,all theresultsinthissectionarepresentedonmeshesthatareuniformawayfromtheinterface.Doingso,wemakesurethat the errorontheinterface dominatestheoverall error.Sincethedegreesoffreedom aretheseeds oftheVoronoicells, it isconvenienttocompute thegradientofthesolutionateverypointlocatedinthemiddleoftwodegreesoffreedom,i.e.

∇

ui j

·

n_{i j}

= (

uj

−

ui

)/

di j whereni j isthenormaltotheedgeoftheVoronoicellconnectingthedegreesoffreedomi and j.

Theerrorspresentedarenormalized.

4.1. ValidationoftheconstructionoftheVoronoidiagramsclosetotheinterface

In thisﬁrst example,weare interested intheinﬂuenceofthequality ofthemeshcloseto theinterface.We consider three differentpossibilities, representedinFig. 5 fora circularirregularinterface described by

φ (

x

,

y

)

= −

x2

₊

_y2

₊

_r

0,

withr0

=

0.5,inadomain

= [−

1,1

]

2.

For theﬁrst case, we placethe newdegreesoffreedom atregular intervalson theirregular interface, making useof theexplicitparametricexpressionavailableforacircle.WechoosetoplaceN

=

1.5

2πr0

min(xmin,ymin)

newdegreesoffreedom

on eitherside oftheinterface, ata distance

diag

/5 from

theinterface with

diag

=

x2_min

+

y2_min.Forthe secondcase, we placethe newdegreesof freedom accordingto the procedure described inSection 2.2, ata distance diag

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Fig. 5. Visualization of the three different meshes on a resolution 24_×₂4_{for Section}_4.1_.

Fig. 6. Left:representationofthesolutionforExample 4.1.Right:visualizationofthelocalizationoftheerroronanon-smoothedmesh(case2)ofresolution 27_×₂7_.

Table 1

ConvergenceoftheerroronthesolutionintheL∞normforExample 4.1.Theﬁrstcasecorrespondstothedegreesoffreedomplacedalongtheinterface usingtheexplicitparametrization,thesecondcasecorrespondstothemeshobtainedfollowingthemethoddescribedinSection2.2,andthethirdcaseis thesmoothedversionofcase 2.

Resolution Explicit Non-smoothed Smoothed

Solution Order Solution Order Solution Order

23 ₃ .66·10−3 _– ₁ .27·10−2 _– ₁ .20·10−2 _– 24 ₁ .79·10−3 _1.03 ₂ .34·10−3 _2.44 ₂ .20·10−3 _2.45 25 ₅_.₇₇_·₁₀−4 _1.63 ₆_.₁₇_·₁₀−4 _1.92 ₆_.₀₂_·₁₀−4 _1.87 26 ₁_.₅₆_·₁₀−4 _1.89 ₁_.₆₂_·₁₀−4 _1.93 ₁_.₆₁_·₁₀−4 _1.91 27 ₄_.₃₂_·₁₀−5 _1.86 ₄_.₄₅_·₁₀−5 _1.87 ₄_.₂₃_·₁₀−5 _1.86 28 ₁_.₁₃_·₁₀−5 _1.94 ₁_.₁₄_·₁₀−5 _1.97 ₁_.₁₄_·₁₀−5 _1.96 29 ₂_.₈₃_·₁₀−6 _1.99 ₂_.₉₆_·₁₀−6 _1.95 ₂_.₉₅_·₁₀−6 _1.94 210 ₇ .19·10−7 _1.98 ₇ .46·10−7 _1.99 ₇ .45·10−7 _1.98

interface.Finally,forthethirdcase,we startfromthepartitionobtainedwiththesecond caseandmodifyitaccordingto theproceduredescribedinSection2.3toobtainasmoothedmesh.

Wemonitortheconvergenceofourmethodonthesethreemeshesforthefollowingsolutiontakenfrom[64], u

(

x

,

y

)

=

1

+

log

(

2

x2

₊

_y2

₎

_if

_{φ (}

_x

_,

_y

_{) <}

₀

_,

1 if

φ (

x

,

y

) >

0

,

and

β(

x

,

y

)

=

1.Notethatforthiscasewehave

[

u

]

=

0 and

[∇

u

·

n

]

=

2,withcontinuous

β

andadiscontinuityintheﬂux acrosstheinterface.ArepresentationofthesolutionisgiveninFig. 6togetherwithavisualizationofthelocalizationofthe error.WereporttheconvergenceofthesolveronthisexampleforthethreedifferentmeshesinTables 1 and 2.Weobserve second-order convergence forthe solution andﬁrst-order convergence forthe gradient ofthe solution, andvery similar errorsforall three meshes.We concludethat smoothingthemeshobtainedwiththe procedureexplained inSection 2.3

doesnotseemtoimprovetheaccuracyofthesolver.

4.2. Inﬂuenceofthesmoothingofthemesh

WefurtherconsidertheinﬂuenceofthesmoothingproceduredescribedinSection2.3.Thistime,weconsideran inter-facedescribedby

φ (

x

,

y

)

= −

x2

₊

_y2

₊

_r₀

₊

_r₁_{cos(5θ ),}_with_r₀

₌

_0.5,_r₁

₌

_{0.15 and}

_θ

_the_angle_between

₍

_x

_,

_y

₎

_and

_(1,

_0),

(9)

Table 2

ConvergenceoftheerroronthegradientofthesolutionintheL∞ normforExample 4.1.Theﬁrstcasecorrespondstothedegreesoffreedomplaced alongtheinterfaceusingtheexplicitparametrization,thesecondcasecorrespondstothemeshobtainedfollowingthemethoddescribedinSection2.2, andthethirdcasecorrespondstoitssmoothedversion.

Resolution Explicit Non-smoothed Smoothed

Gradient Order Gradient Order Gradient Order

23 ₄_.₆₀_·₁₀−2 _– ₄_.₇₁_·₁₀−2 _– ₄_.₂₄_·₁₀−2 _– 24 ₃_.₂₈_·₁₀−2 _0.49 ₂_.₄₀_·₁₀−2 _0.98 ₂_.₃₆_·₁₀−2 _0.84 25 ₁_.₇₉_·₁₀−2 _0.87 ₁_.₂₃_·₁₀−2 _0.96 ₁_.₂₃_·₁₀−2 _0.94 26 ₉_.₅₆_·₁₀−3 _0.91 ₈_.₃₄_·₁₀−3 _0.56 ₈_.₃₅_·₁₀−3 _0.56 27 ₅ .15·10−3 _0.89 ₃ .94·10−3 _1.08 ₃ .94·10−3 _1.08 28 ₂ .56·10−3 _1.01 ₂ .12·10−3 _0.90 ₂ .12·10−3 _0.90 29 ₁_.₃₀_·₁₀−3 _0.98 ₁_.₁₂_·₁₀−3 _0.92 ₁_.₁₂_·₁₀−3 _0.92 210 ₆_.₆₁_·₁₀−4 _0.98 ₅_.₆₁_·₁₀−4 _1.00 ₅_.₆₁_·₁₀−4 _1.00

Fig. 7. Visualization of the non-smoothed (left) and smoothed (right) meshes on a resolution 24_×₂4_{for Section}_4.2_.

Fig. 8. Left: representation of the solution forExample 4.2. Right: visualization of the error on a non-smoothed mesh and for a resolution 27_×₂7_.

inadomain

= [−

1,1

]

2_._Since_we_do_not_have_an_explicit_{parametrization}_of_the_interface_that_would_enable_to_place_the

degreesoffreedom atregularintervals, weonlyconsiderthemeshgeneratedformtheproceduredescribedinSection2.2

andits smoothed version obtainedbyapplying theprocedure describedin Section 2.3. Fig. 7givesa visualization ofthe meshesobtained.

Forthissection,wechoosetoworkwiththeexactsolutiontakenfrom[9]

u

(

x

,

y

)

=

0 if

φ (

x

,

y

) <

0

,

excos

(

y

)

if

φ (

x

,

y

) >

0

,

with

β

−

= β

+

=

1.ThesolutionisrepresentedinFig. 8.WemonitortheconvergenceofthesolverinTable 3andobserve second-orderconvergenceforthesolutionandﬁrst-orderconvergenceforthegradientofthesolutioninbothcases.Given thatthesmoothingalgorithmrequiresadditionalprocessinganddoesnotseemtoimprovetheaccuracy(infact,wenotice forthisparticularexamplethatthenon-smoothedresultsaremoreaccurate),wechoosetowork withthenon-smoothed meshconstructedasdescribedinSection2.2fortheremainingofthisarticle.

(10)

Table 3

ConvergenceoftheerroronthesolutionanditsgradientintheL∞normforExample 4.2.

Resolution Non-smoothed Smoothed

Solution Order Gradient Order Solution Order Gradient Order

23 ₂_.₃₉_·₁₀−3 _– ₃_.₀₁_·₁₀−2 _– ₂_.₀₀_·₁₀−2 _– ₁_.₀₆_·₁₀−1 _– 24 ₁_.₀₆_·₁₀−3 _1.17 ₅_.₀₈_·₁₀−1 ₋₄_.₀₇ ₁_.₄₄_·₁₀−2 _0.47 ₅_.₂₇_·₁₀−1 ₋₂_.₃₂ 25 ₃ .43·10−4 _1.63 ₈ .42·10−3 ₅ .91 4.21·10−3 _1.78 ₂ .84·10−2 ₄ .21 26 ₆ .82·10−5 _2.33 ₃ .95·10−3 ₁ .09 1.62·10−3 _1.38 ₁ .66·10−2 ₀ .78 27 ₂_.₇₉_·₁₀−5 _1.29 ₂_.₉₃_·₁₀−3 ₀_.₄₃ ₄_.₅₅_·₁₀−4 _1.83 ₈_.₀₃_·₁₀−3 ₁_.₀₄ 28 ₇_.₃₅_·₁₀−6 _1.92 ₁_.₄₀_·₁₀−3 ₁_.₀₆ ₁_.₁₁_·₁₀−4 _2.03 ₂_.₆₆_·₁₀−3 ₁_.₅₉ 29 ₁_.₈₉_·₁₀−6 _1.96 ₆_.₃₁_·₁₀−4 ₁_.₁₅ ₂_.₇₆_·₁₀−5 _2.01 ₉_.₆₂_·₁₀−4 ₁_.₄₇ 210 ₄_.₇₅_·₁₀−7 _1.99 ₃_.₂₈_·₁₀−4 ₀_.₉₄ ₆_.₈₆_·₁₀−6 _2.01 ₃_.₃₄_·₁₀−4 ₁_.₅₃

Fig. 9. Two examples of solutions forExample 4.3. Left:β−=1 andβ+=10. Right:β−=10 andβ+=1.

4.3. Examplewithadiscontinuityinthediffusioncoeﬃcient

Wenowconsidertheexactsolution

u

(

x

,

y

)

=

⎧

⎨

⎩

x(ρ+1)−x(ρ−1)r2₀/r2 ρ+1+r2 0(ρ−1) if

φ (

x

,

y

) <

0

,

2x ρ+1+r2₀(ρ−1) if

φ (

x

,

y

) >

0

,

withr0

= .

5,r

=

x2

₊

_y2_,

_{φ (}

_x

_,

_y

₎

_{= −}

_r2

₊

_r2

0 and

ρ

= β

+

/β

−inthedomain

= [−

1,1

]

2.ThiscorrespondstoExample 7.3

from[64].Inthiscase, u iscontinuous,butthediffusioncoeﬃcient

β

experiencesalargejumpacrosstheirregular inter-face

.

We alsohave

[∇

u

·

n

]

=

0.Avisualization ofthe solutionisgivenin Fig. 9.The gradientof thesolutionis given by

∇

u

(

x

,

y

)

=

⎧

⎪

⎨

⎪

⎩

1 ρ+1+r2 0(ρ−1)

ρ

+

1

−

r2 0

(

ρ

−

1

)

y2₋_x2 (x2₊_y2₎2 r2₀

(

ρ

−

1

)

2xy (x2₊_y2₎2

if

φ (

x

,

y

) <

0

,

1 ρ+1+r2₀(ρ−1)

2 0

if

φ (

x

,

y

) >

0

.

The errors on the solution andits gradient are monitored inTable 4 which shows second-order convergenceon the solutionandfirst-orderconvergenceonitsgradient.Fig. 10providesavisualizationofthelocalizationoftheerror.Wealso monitorthe evolutionofthe 1-normcondition numberofthematrix ofthelinear systemasthemeshis refinedin and presenttheresultsinTable 5.Theconditionnumberdependsonthemeshresolutionandonthediffusioncoefficient.When thediffusioncoefficientislarge,the conditionnumbergetslargerapidly.However, thediscontinuitiesattheinterface are entirelycapturedbytherighthandsideandthereforedonotaffecttheconditioningofthematrix.

4.4. Acompleteexample

Thisexampleismeanttotestourmethodtoitsfullcapacity,withdiscontinuitiesinallfourquantities(thesolution,its gradient,thediffusioncoeﬃcientandtheﬂuxacrosstheinterface),andwithacomplexirregularinterface.Wechoosethe exactsolution

u

(

x

,

y

)

=

ex if

φ (

x

,

y

) <

0

,

(11)

Table 4

ConvergenceonthesolutionanditsgradientforExample 4.3,fortwodifferentcombinationsofdiffusioncoeﬃcients.

Resolution β−=1,β+=105 _β−₌₁₀5_,_β+₌₁

Solution Order Gradient Order Solution Order Gradient Order

23 ₁ .10·10−2 _– ₃ .84·10−2 _– ₁ .74·10−2 _– ₁ .14·10−1 _– 24 ₃ .42·10−3 _1.68 ₁ .93·10−2 _1.00 ₄ .70·10−3 _1.89 ₅ .65·10−2 _1.01 25 ₁_.₃₉_·₁₀−3 _1.30 ₁_.₁₇_·₁₀−2 _0.72 ₁_.₂₆_·₁₀−3 _1.90 ₂_.₅₄_·₁₀−2 _1.15 26 ₃_.₈₂_·₁₀−4 _1.86 ₆_.₁₉_·₁₀−3 _0.92 ₃_.₃₉_·₁₀−4 _1.90 ₁_.₇₇_·₁₀−2 _0.53 27 ₁_.₃₄_·₁₀−4 _1.51 ₃_.₅₉_·₁₀−3 _0.79 ₁_.₀₅_·₁₀−4 _1.69 ₈_.₀₆_·₁₀−3 _1.13 28 ₃_.₄₃_·₁₀−5 _1.97 ₁_.₈₄_·₁₀−3 _0.96 ₂_.₆₇_·₁₀−5 _1.98 ₄_.₈₇_·₁₀−3 _0.73 29 ₉_.₇₆_·₁₀−6 _1.81 ₁_.₀₅_·₁₀−3 _0.82 ₇_.₅₅_·₁₀−6 _1.82 ₂_.₄₀_·₁₀−3 _1.02 210 ₂ .58·10−6 _1.92 ₅ .41·10−4 _0.95 ₁ .96·10−6 _1.95 ₁ .25·10−3 _0.94 Table 5

EvolutionoftheconditionnumberasthemeshisreﬁnedforExample 4.3.Theconditionnumberdependssolelyonthediffusioncoeﬃcientβandonthe resolutiononthemeshsincethediscontinuitiesarecapturedbytherighthandsideofthelinearsystem.

Resolution β−=1,β+=105 _β−₌₁₀5_,_β+₌₁ _β−_{= β}+₌₁₀5 _β−_{= β}+₌₁ 23 ₂_.₃₈_·₁₀6 ₂_.₀₂_·₁₀6 ₅_.₅₉_·₁₀6 ₁_.₇₂_·₁₀2 24 ₁_.₁₇_·₁₀7 ₅_.₀₉_·₁₀6 ₁_.₂₃_·₁₀7 ₈_.₁₁_·₁₀2 25 ₄_.₈₆_·₁₀7 ₁_.₅₇_·₁₀7 ₂_.₆₉_·₁₀7 ₃_.₆₆_·₁₀3 26 ₂ .09·108 ₇ .18·107 ₅ .72·107 ₁ .58·104 27 ₈ .52·108 ₃ .14·108 ₁ .19·108 ₆ .59·104 28 ₃_.₈₆_·₁₀9 ₁_.₃₂_·₁₀9 ₂_.₄₂_·₁₀8 ₂_.₇₀_·₁₀5 29 ₁_.₄₈_·₁₀10 ₅_.₄₃_·₁₀9 ₄_.₈₉_·₁₀8 ₁_.₀₉_·₁₀6 210 ₆_.₅₉_·₁₀10 ₂_.₂₀_·₁₀10 ₉_.₈₄_·₁₀8 ₄_.₃₉_·₁₀6

Fig. 10. VisualizationofthelocalizationoftheerrorintheL∞normonagridofresolution27_×₂7_for_{Example 4.3}_._Left:_β−₌_{1 and}_β+₌₁₀5_._Right:

β−=105_and_β+₌_1.

inadomain

= [−

1,1

]

2 with

φ (

x

,

y

)

= −

x2

₊

_y2

₊

_r

0

+

r1cos(n

θ ),

wherer0

=

0.5,r1

=

0.15 andn

=

5,andwedeﬁne

thediffusioncoeﬃcientas

β(

x

,

y

)

=

y2_ln

₍

_x

₊

₂

₎

₊

_{4 if}

_{φ (}

_x

_,

_y

_{) <}

₀

_,

e−y if

φ (

x

,

y

) >

0

.

Note that with our method,each degree of freedom hasa control volume that is entirely on one side ofthe irregular interface,andthereforewecaneasilydefineaforcingtermforanyanalyticalsolution.Theexactsolutionandthediffusion coefficient are represented inFig. 11. The convergenceis summarized in Table 6and once againindicates second-order convergenceforthesolutionandfirst-orderconvergenceforthegradientofthesolution.Avisualizationofthelocalization oftheerroronthesolutionisgiveninFig. 12.

Fig. 13 presentsthe percentageofthe runtimeconsumedby thefour principalcomponents ofthealgorithm, i.e con-structingtheVoronoimesh,assemblingthematrix,computingtherighthandsideandsolvingthelinearsystem.Forcoarse grids, thebottleneckofthecomputationistheconstructionoftheVoronoimesh,butforhighresolutionweobservethat invertingthelinearsystemisthecostliest.Theseresultscorrespondtoourimplementationintheabsenceofparallelization. Theﬁnestresolutionof1024

×

1024 takes18 sonasinglecoreofanInteli7-26003.40 GHzcpu.

4.5. Addingsubdomains

We nowpropose an examplewithmultiplesubdomains. Notethat themethodwe proposeleads naturally toa linear system with N rows, the number of degrees of freedom. This number increases slightly as the number of subdomains increasesandadditionaldegreesoffreedomareaddednexttotheirregularinterfaces.

(12)

Fig. 11. Left: visualization of the solution u forExample 4.4. Right: visualization of the diffusion coeﬃcientβ.

Fig. 12. Visualization of the localization of the error forExample 4.4on a resolution of 27_×₂7_.

Table 6

ConvergenceonthesolutionanditsgradientforExample 4.4.

Resolution Solution Order Gradient Order

23 ₃_.₉₇_·₁₀−3 _– ₄_.₃₇_·₁₀−1 _– 24 ₉_.₉₈_·₁₀−4 _1.99 ₅_.₀₁_·₁₀−1 ₋₀_.₂₀ 25 ₂_.₉₀_·₁₀−4 _1.78 ₃_.₆₇_·₁₀−3 ₇_.₀₉ 26 ₈ .84·10−5 _1.71 ₁ .73·10−3 ₁ .09 27 ₂ .06·10−5 _2.10 ₉ .84·10−4 ₀ .81 28 ₅_.₂₂_·₁₀−6 _1.98 ₄_.₇₇_·₁₀−4 ₁_.₀₅ 29 ₁_.₃₃_·₁₀−6 _1.97 ₂_.₄₅_·₁₀−4 ₀_.₉₆ 210 ₃_.₃₉_·₁₀−7 _1.97 ₁_.₂₃_·₁₀−4 ₀_.₉₉

Forsimplicity,we choose towork withnon-intersectingirregularinterfaces,butourmethod issuitedfor anygeneral conﬁguration.Wedividethecomputationaldomain

= [−

1,1

]

2 in4subdomainsrepresentedinFig. 14andseparatedby thethreecontoursdeﬁnedby

0

=

(

x

,

y

), φ

0

(

x

,

y

)

=

x2

₊

_y2

₋

₀

_.

₂

,

1

=

(

x

,

y

), φ

1

(

x

,

y

)

=

x2

₊

_y2

₋

₀

_.

₅

₊

₀

_.

_{1 cos}

₍

₅

_{θ )}

,

2

=

(

x

,

y

), φ

2

(

x

,

y

)

=

x2

₊

_y2

₋

₀

_.

₈

,

where

θ

istheanglebetween

(

x

,

y

)

andthex-axis.Wechoosetheexactsolution

u

(

x

,

y

)

=

⎧

⎪

⎨

⎪

⎩

ex

+

1

.

3 if

(

x

,

y

)

∈

0

,

cos

(

y

)

+

1

.

8 if

(

x

,

y

)

∈

1

,

sin

(

x

)

+

0

.

5 if

(

x

,

y

)

∈

2

,

−

x

+

ln

(

y

+

2

)

if

(

x

,

y

)

∈

3

,

(13)

Fig. 13. Representationofthecomputationtimeconsumedbythefourmainsectionsofourimplementation,constructingthemesh,assemblingthematrix, computing therighthandsideandsolvingthelinearsystemforExample 4.4.Forcoarsegrids,buildingtheVoronoipartitiontakesthemosttime,butas theresolutionofthegridincreases,theinversionofthelinearsystembecomesthecostliest.

Fig. 14. Illustration of the division of the computational domain into four subdomains, together with the Voronoi mesh generated, forExample 4.5.

andthediffusioncoeﬃcient

β(

x

,

y

)

=

⎧

⎪

⎨

⎪

⎩

y2

+

1 if

(

x

,

y

)

∈

0

,

ex if

(

x

,

y

)

∈

1

,

y

+

1 if

(

x

,

y

)

∈

2

,

x2

+

1 if

(

x

,

y

)

∈

3

.

ThesolutionandthediffusioncoeﬃcientarerepresentedinFig. 15.

The convergenceofthesolverispresentedinTable 7.Weobservesecond orderconvergenceforthesolutionandﬁrst orderconvergenceforitsgradient.

4.6. Applicationtothreespatialdimensions

We now present an example in three spatial dimensions and with a spherical interface of radius 0.5 in a domain

= [−

1,1

]

3.Weworkwiththeexactsolution

u

(

x

,

y

,

z

)

=

ez if

φ (

x

,

y

,

z

) <

0

,

cos

(

x

)

sin

(

y

)

if

φ (

x

,

y

,

z

) >

0

,

β(

x

,

y

,

z

)

=

y2_ln

₍

_x

₊

₂

₎

₊

_{4 if}

_{φ (}

_x

_,

_y

_,

_z

_{) <}

₀

_,

e−z if

φ (

x

,

y

,

z

) >

0

.

Thegeometry,togetherwithasliceofthesolutionandofthediffusioncoeﬃcient,isrepresentedinFig. 16.Table 8presents thenumericalresultsandindicatessecond-orderconvergenceforthesolutionandﬁrst-orderconvergenceforitsgradient.

(14)

Fig. 15. Visualization of the solution (left) and the diffusion coeﬃcient (right) forExample 4.5.

Table 7

ConvergenceonthesolutionanditsgradientintheL∞normforExample 4.5.

24 ₁_.₀₀_·₁₀−3 _– ₃_.₃₃_·₁₀−3 _– 25 ₂_.₃₃_·₁₀−4 _2.11 ₁_.₀₁_·₁₀−3 _1.72 26 ₆_.₂₃_·₁₀−5 _1.90 ₃_.₄₆_·₁₀−4 _1.54 27 ₁_.₅₆_·₁₀−5 _2.00 ₁_.₅₉_·₁₀−4 _1.12 28 ₄ .00·10−6 _1.96 ₆ .82·10−5 _1.22 29 ₁ .01·10−6 _1.99 ₄ .00·10−5 _0.77 210 ₂_.₅₅_·₁₀−7 _1.99 ₂_.₂₄_·₁₀−5 _0.84

Fig. 16. Left:representationoftheirregularinterfaceandtheassociatedVoronoimeshonaresolution24_×₂4_×₂4_for_{Example 4.6}_._Center:_{visualization} ofthesolutionontheslicex=0.Right:visualizationofthediffusioncoeﬃcientontheslicex=0.Notethatthesurfaceshavebeentranslatedtofacilitate thevisualization.

Table 8

ConvergenceonthesolutionanditsgradientintheL∞normonasphere(Example 4.6).

23 ₃ .61·10−3 _– ₁ .13·10−2 _– 24 ₁_.₂₁_·₁₀−3 _1.58 ₇_.₆₉_·₁₀−3 _0.56 25 ₃_.₀₄_·₁₀−4 _1.99 ₃_.₈₃_·₁₀−3 _1.01 26 ₇_.₇₄_·₁₀−5 _1.98 ₂_.₄₃_·₁₀−3 _0.66 27 ₁_.₉₇_·₁₀−5 _1.98 ₁_.₂₄_·₁₀−3 _0.98

(15)

Fig. 17. Representation of the irregular interface and the associated Voronoi mesh on a resolution 24_×₂4_×₂4_for_{Example 4.7}_. 4.7. Acomplexgeometryinthreespatialdimensions

Foramorecomplicatedgeometry,weselecttheintricatecontourborrowedfrom[34]andparametrizedby

trefoil

=

R 3

₍

₂

₊

_cos

₍

_3t

₎₎

_cos

₍

_2t

₎

(

2

+

cos

(

3t

))

sin

(

2t

)

sin

(

3t

)

,

t

∈ [

0

,

2π

]

,

where R

=

0.7 isthemajorradiusofthetrefoil.Wethendeﬁne

+

=

x

∈ R,

min y∈trefoil

x

−

y

2

<

r

,

withr

=

0.15 theminorradiusofthetrefoil.ThecontourisrepresentedinFig. 17. Forthisexample,weusetheexactsolution

u

(

x

,

y

,

z

)

=

yz sin

(

x

)

if

φ (

x

,

y

,

z

) <

0

,

xy2

₊

_z3 _if

_{φ (}

_x

_,

_y

_,

_z

_{) >}

₀

_,

β(

x

,

y

,

z

)

=

y2

+

1 if

φ (

x

,

y

,

z

) <

0

,

ex+z if

φ (

x

,

y

,

z

) >

0

.

Slices of the solution and of the diffusion coeﬃcient are displayed in Fig. 18. The convergence of our method on this complexirregular interfaceisreportedinTable 9andonce moreindicatessecond orderconvergenceforthesolutionand ﬁrstorderconvergenceforitsgradient.

4.8. ThescreenedPoissonequation

Thisexampleandthefollowingoneaddanon-zerok tothepreviousExample 4.7.Inthisexample,wechoosek

<

0 as k

(

x

,

y

,

z

)

=

−

ex _if

_{φ (}

_x

_,

_y

_,

_z

_{) <}

₀

_,

−

cos

(

y

)

sin

(

z

)

−

2 if

φ (

x

,

y

,

z

) >

0

.

TheconvergenceresultsarepresentedinTable 10andshowsecondorderconvergencewitherrorsverysimilartotheones obtainedwhenk

=

0.

4.9. TheHelmholtzequationcase

Our last example on a uniform base mesh is exactly the same than the one from the previous section but for the Helmholtzequationcase,i.e.k

(

x

,

y

,

z

)

>

0.Weset

k

(

x

,

y

,

z

)

=

ey if

φ (

x

,

y

,

z

) <

0

,

cos

(

x

)

sin

(

z

)

+

2 if

φ (

x

,

y

,

z

) >

0

.

Inthiscase, thelinearsystemobtainedismorecomplicatedtosolvebecausethematrixisnolongerdiagonallydominant, meaningthattheproblemisnotconvexanditerativesolverssuchastheConjugateGradientusedsofararenotguaranteed toconverge.Instead,wesolvethelineardirectlywithanLUdecomposition.ThenumericalresultsarepresentedinTable 11

andarealmostidenticaltotheresultsfromtheprevioussection,illustrating thesecond orderconvergenceofourmethod fortheHelmholtzequation.

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Fig. 18. Visualizationofthesolution(toprow)andthediffusioncoeﬃcient(bottomrow)onthreeslicesforExample 4.7.Theslicesaretaken,fromleftto right,atx=0.3,x= −0.3 andx= −0.5.Notethatthesurfacesofthediffusioncoeﬃcienthavebeentranslatedtofacilitatethevisualization.

Table 9

ConvergenceonthesolutionanditsgradientintheL∞normonacomplexthree-dimensionalcontour(Example 4.7).

24 ₃_.₈₉_·₁₀−3 _– ₁_.₄₄_·₁₀−1 _– 25 ₁_.₃₄_·₁₀−3 _1.54 ₂_.₅₆_·₁₀−2 _2.67 26 ₃ .45·10−4 _1.96 ₉ .75·10−3 _1.21 27 ₈ .25·10−5 _2.07 ₅ .04·10−3 _0.95 Table 10

Convergenceonthesolutionanditsgradientinthe L∞normonacomplexthree-dimensionalcontourforthescreenedPoisson equation(Example 4.8).

24 ₃ .87·10−3 _– ₁ .44·10−1 _– 25 ₁ .34·10−3 _1.53 ₂ .25·10−2 _2.67 26 ₃_.₄₄_·₁₀−4 _1.96 ₉_.₇₄_·₁₀−3 _1.21 27 ₈_.₂₂_·₁₀−5 _2.06 ₅_.₀₄_·₁₀−3 _0.95 Table 11

ConvergenceonthesolutionanditsgradientintheL∞normonacomplexthree-dimensionalcontourfortheHelmholtzequation (Example 4.9).

24 ₃_.₉₁_·₁₀−3 _– ₁_.₄₄_·₁₀−1 _–

25 ₁_.₃₅_·₁₀−3 _1.54 ₂_.₂₆_·₁₀−2 _2.67

26 ₃_.₄₆_·₁₀−4 _1.96 ₉_.₇₅_·₁₀−3 _1.21

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Fig. 19. Example of a Quadtree grid.

Fig. 20. Left:visualizationofthesolutionforExample 5.2withα=10.Highervaluesofαnarrowthepeaks.Right:representationoftheerrorinterpolated onthebaseQuadtreemeshoflevel8/10 forα=50.

5. Extensiontoadaptivemeshes

Intheprevioussectionwedemonstratedtheeﬃciencyofourmethodbasedonuniformmeshesinbothtwoandthree spatialdimensions.However,itcanbeappliedstraightforwardlytoanymeshandinthissectionweproposean implemen-tationonQuad/Oc-trees.

5.1. IntroductiontotheQuad/Oc-treedatastructure

A Quad/Oc-tree grid refers to a Cartesian grid that uses the Quad/Oc-tree datastructure forits storagein two/three spatial dimensions. Starting froma rootcellcorresponding to theentiredomain, four(eight inthree spatial dimensions) children are created ifthecell satisfies a givensplittingcriterion. The process isiterated recursively untilthe maximum allowed levelisreached.Therootcellhaslevel0andthefinestcells havethemaximumlevelallowed.We denotea tree withcoarsestleveln andfinestlevelm byleveln

/

m.Theprocess isillustratedinFig. 19.Thisdatastructureprovidesan

O

(ln(

n

))

accesstothe datastoredattheleaves. Werefer thereaderto[45] forfurtherdetails ontheQuad/Oc-tree data structureandtheassociateddiscretizationtechniques.

Since thisarticleis aproof ofconcept andwe knowtheexact solutioninall the numericalexamplewe propose,we makeuseofthisknowledgeintheconstructionofthetree.Wewillusesolutionsoftheform

u

(

x

)

=

e−αx−x022

_,

andanygivenleaf

L

ofthetreewithcentercoordinatesxc issplitif

x0

−

xc

2

< λ

· diag

,where

diag

isthelengthof

thediagonalof

L

and

λ

controlsthespreadofthemesharoundthepeaksofthesolution.

5.2. SolutiononaQuadtreemesh

Forthisexample,weconsidertheexactsolution,representedinFig. 20, u

(

x

,

y

)

=

e−50((x+0.7)2+(y−0.7)2) if

φ (

x

,

y

) <

0

,

e−50((x−0.1)2+(y+0.1)2) _if

_{φ (}

_x

_,

_y

_{) >}

₀

_,

with

β

−

= β

+

=

1 andonthedomain

= [−

1,1

]

2_._We_use_the_same_geometry_as_in_Section_4.4_,_i.e.

_{φ (}

_x

_,

_y

₎

_{= −}

_x2

₊

_y2

₊