A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods

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three-dimensional time-harmonic Maxwell equations

discretized by discontinuous Galerkin methods

Victorita Dolean, Stephane Lanteri, Ronan Perrussel

To cite this version:

Victorita Dolean, Stephane Lanteri, Ronan Perrussel. A domain decomposition method for solving the

three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods.

Journal of Computational Physics, Elsevier, 2008, 227 (3), pp.2044-2072. �10.1016/j.jcp.2007.10.004�.

�inria-00155231v3�

a p p o r t

d e r e c h e r c h e

9

-6

3

9

9

IS

R

N

IN

R

IA

/R

R

--6

2

2

0

--F

R

+

E

N

G

Thème NUM

Une méthode de décomposition de domaine

pour la résolution numérique des équations

de Maxwell tridimensionnelles

en domaine harmonique

discrétisées par des méthodes de type

Galerkin discontinu

Victorita Dolean — Stéphane Lanteri — Ronan Perrussel

N° 6220

Juin 2007

de Maxwell tridimensionnelles

en domaine harmonique

dis rétisées par des méthodes de type

Galerkin dis ontinu

Vi torita Dolean

∗

,Stéphane Lanteri

†

, Ronan Perrussel

‡

ThèmeNUMSystèmesnumériques

ProjetCaiman

Rapport dere her he n°6220Juin200739pages

Résumé: Nous présentonsi iune méthodede dé ompositionde domainepourla

résolu-tion deséquations de Maxwelltridimensionnelles en domaineharmonique dis rétisées par

uneméthodeGalerkin dis ontinu. Pourpermettreletraitementde géométriesirrégulières,

laméthode Galerkindis ontinuest formulée enmaillages tétraédriques non-stru turés.La

stratégie de résolution par dé omposition de domaine onsiste en un algorithme de type

S hwarzoùdes onditionsabsorbantesdupremierordresontimposéesauxinterfa esentre

sous-domaines voisins. Un solveur dire t reux multifrontal est utilisé pour la résolution

des problèmes lo aux posés dans haque sous-domaine. La méthode de dé omposition de

domainerésultantepeuts'interpréter ommeunsolveurhybrideitératif/dire tpourles

sys-tèmesalgébriques reux à oe ients omplexesrésultantdeladis rétisationdeséquations

deMaxwellendomaineharmoniquepardesméthodesdetypeGalerkin dis ontinu.

Mots- lés : éle tromagnétisme numérique, équations de Maxwell en domaine

harmo-nique, méthode Galerkin dis ontinu, maillages non-stru turés, méthode de dé omposition

dedomaine,algorithmedeS hwarz.

∗

UniversitédeNi e-SophiaAntipolisLaboratoireJ.A.Dieudonné,CNRSUMR662106108Ni eCedex,

Fran e

†

INRIA,2004RoutedesLu ioles,BP9306902SophiaAntipolisCedex,Fran e

‡

dis retized by

dis ontinuous Galerkin methods

Abstra t: Wepresentherea domain de omposition method for solvingthe

three-dimen-sionaltime-harmoni Maxwellequationsdis retizedbyadis ontinuousGalerkinmethod.In

order to allow the treatment of irregularly shaped geometries, the dis ontinuous Galerkin

methodisformulatedonunstru turedtetrahedralmeshes.Thedomainde omposition

strat-egy takes theform of aS hwarz-typealgorithm where arst-order absorbing onditionis

imposed at the interfa es between neighboring subdomains. A multifrontal sparse dire t

solver is used at the subdomain level. The resulting domain de omposition strategy an

beviewed as a hybrid iterative/dire t solutionmethod for the large, sparse and omplex

oe ientsalgebrai systemresultingfromthedis retizationofthetime-harmoni Maxwell

equationsbyadis ontinuousGalerkinmethod.

Key-words: omputational ele tromagnetism, time-harmoni Maxwell's equations,

dis- ontinuousGalerkinmethod,unstru turedmeshes,domainde ompositionmethod,S hwarz

Table des matières

1 Introdu tion 4

2 Formulation of the ontinuous problem 5

3 A lassi al domainde ompositionmethod 6

4 Dis retization 7

4.1 Dis retizationofthemono-domainproblem . . . 8

4.2 Commentsonthedis retization . . . 10

4.3 Dis retizationofthedomainde ompositionalgorithm . . . 12

4.3.1 Dis ontinuousGalerkinformulationofthemulti-domainproblem . . . 12

4.3.2 Formulationofaninterfa esystem . . . 13

5 Numeri al and performan eresults 14 5.1 Implementedformulationsandexperimentaltestbed . . . 14

5.2 Solutionstrategies . . . 15

5.3 Dira tionofaplanewavebyaPECsphere. . . 16

5.4 Dira tionofaplanewavebyaPEC ube . . . 21

5.4.1 Propagationin va uum . . . 22

5.4.2 CoatedPEC ube . . . 22

5.5 Dis ussionofthenumeri alandparallelperforman es . . . 25

5.6 Abioele tromagneti sappli ation. . . 28

1 Introdu tion

Thisworkaimsatdevelopingahigh-performan enumeri almethodologyforthe

ompu-ter simulationof time-harmoni ele tromagneti wavepropagationproblems in irregularly

shapeddomainsandheterogeneousmedia.Inthis ontext,wearenaturallyledto onsider

volumedis retizationmethods(i.e.nitedieren e,nitevolumeorniteelementmethods)

asopposed to surfa edis retization methods (i.e.boundaryelement method).Most ofthe

relatedexistingworksdealwiththese ond-orderformofthetime-harmoni Maxwell

equa-tionsdis retized bya onforming nite elementmethod [30℄. Morere ently, dis ontinuous

Galerkinmethods [24℄havealsobeen onsideredforthispurpose.Here,we on entrateon

the rst-order form of the time-harmoni Maxwell equations dis retized by dis ontinuous

Galerkinmethodsonunstru turedtetrahedralmeshes.

Theoreti al results on erning dis ontinuous Galerkin methods applied to the

time-harmoni Maxwell equations have been obtained by several authors. Most of these works

useamixedformulation[31,25℄butdis ontinuousGalerkinmethodsonthenon-mixed

for-mulation havealso been proved to onverge(interior penalty te hniques[24, 7℄ and lo al

dis ontinuousGalerkinmethods[7℄).However,toourknowledge,adire t onvergen e

ana-lysis of dis ontinuous Galerkin methods applied to the rst-order time-harmoni Maxwell

systemhasnotbeen ondu tedsofar.Our ontribution in[12℄isanumeri alstudyofthe

onvergen e of dis ontinuous Galerkin methods based on entered and upwind uxes and

nodalpolynomialinterpolationapplied totherst-ordertime-harmoni Maxwellsystemin

the two-dimensional ase. These methods have previouslybeen shown to onverge in the

time-domain ase[23,17℄.

Inthispaper,weare on ernedwiththeappli ationofsu hdis ontinuousGalerkin

me-thodstothedis retizationofthethree-dimensionaltime-harmoni Maxwellequationstaken

in theformof arst-order systemofpartial dierentialequations.Oureortsaretowards

thedesignofaparallel solutionstrategy fortheresultinglarge,sparseand omplex

oe- ientsalgebrai systems.Indeed,as farasnon-trivialpropagationproblemsare onsidered,

theasso iatedmatrixoperatorsareinmost asessolvedwithdi ultyby lassi aliterative

methods.Thepre onditioningissuesforhighlyindeniteandnon-symmetri matri esisfor

instan edis ussedbyBenzietal.in[3℄inthe ontextofin ompletefa torizationandsparse

approximateinversepre onditioners.Ifarobustande ientsolverissoughtthenasparse

dire t method is the most pra ti al hoi e. Over the last de ade, signi ant progress has

beenmade in developingparallel dire t methods for solvingsparse linear systems,due in

parti ulartoadvan esmadeinboththe ombinatorialanalysisofGaussianelimination

pro- ess,and onthedesignofparallel blo k solversoptimizedfor high-performan e omputers

[2,22℄.However,dire tmethodswillstillfailtosolveverylargethree-dimensionalproblems,

duetothepotentiallyhugememoryrequirementsforthese ases.Iterativemethods anbe

usedtoover omethismemoryproblem.However,abettersolution anbefound, ombining

advantagesofbothiterativeanddire tmethods.Forexample,apopularapproa hisdomain

usesadire tsolverinsideea hsubdomain oupledwithaniterativesolverontheinterfa es

(arti ialboundaries) betweensubdomains.Thisapproa hisadoptedinthis work.

Domainde ompositionmethodsareexibleandpowerfulte hniquesfortheparallel

nu-meri alsolutionofsystemsofpartialdierentialequations.Con erningtheirappli ationto

time-harmoni wave propagation problems, the simplest algorithm was proposed by

Des-prés [10℄ for solving the Helmholtz equation and then extended and generalized for the

time-harmoni Maxwell equations in [11, 9, 1℄. The analysis of a larger lass of S hwarz

algorithms has been performed re ently in [13℄. Our ultimate obje tive is the design and

appli ation of optimized S hwarz algorithms in onjun tion with dis ontinuous Galerkin

methods. The rststep in this dire tion is understanding and analyzing lassi al

overlap-pingandnon-overlappingS hwarzalgorithmsinthedis reteframeworkofthesemethods.To

ourknowledge,ex eptinHelluy [20℄,wheresu hanalgorithmisapplied toadis retization

oftherst-ordertime-harmoni Maxwellequationsbyanupwindnite volumemethod,no

otherattempts forhigher orderdis ontinuous Galerkinmethods ordierentkind ofuxes

an be found in the literature. A lassi al domain de omposition strategy is adopted in

thisstudywhi htakestheformof aS hwarz-typealgorithmwhere Després onditions[11℄

areimposedattheinterfa esbetweenneighboringsubdomains.Amultifrontalsparsedire t

solver is used at the subdomain level. The resulting domain de omposition strategy an

beviewed as a hybrid iterative/dire t solutionmethod for the large, sparse and omplex

oe ientsalgebrai systemresultingfromthedis retizationofthetime-harmoni Maxwell

equationsbyadis ontinuousGalerkinmethod.

Therestofthis paperisorganizedasfollows.Inse tion2,weformulate the ontinuous

boundaryvalueproblemtobesolved.Then,inse tion3,theadoptedS hwarz-typedomain

de ompositionmethodisintrodu ed.Se tion4isdevotedtothedis retizationoftheglobal

anddomainde omposedboundaryvalueproblems.Awell-posednessresultforaperturbed

dis rete problem whi h generalizes the idea of [6℄ to higher order dis ontinuous Galerkin

methodsisestablished.Finally,inse tion5,numeri alstrategiesforsolvinglo alproblems

aswellasparallel omputingaspe tsaredis ussedand experimentalresultsarepresented.

Beside lassi als atteringtestproblems,wealso onsideramore hallengingsituationwhi h

onsists in the propagation ofa planewavein arealisti geometri model of humanhead

tissues.

2 Formulation of the ontinuous problem

Thesystemofnon-dimensionedtime-harmoni Maxwell'sequations anbewrittenunder

thefollowingform:

(

iωε

r

E

− curl H = −J,

iωµ

r

H

+ curl E = 0,

(1)

where

E

and

H

are the unknown ele tri and magneti elds and

J

is a known urrent sour e. The parameters

ε

r

and

µ

r

are respe tively the omplex-valued relative diele tri permittivity(integratingtheele tri ondu tivity)andtherelativemagneti permeability;

we onsiderherethe aseoflinearisotropi media.Theangularfrequen yoftheproblemis

givenby

ω

.Equation(1)issolvedinaboundeddomain

Ω

.Ontheboundary

∂Ω = Γ

a

∪ Γ

m

, thefollowingboundary onditionsareimposed:

-aperfe tele tri ondu tor(PEC) onditionon

Γ

m

: n × E = 0,

-aSilver-Müller(rst-orderabsorbingboundary) ondition

on

Γ

a

: n × E + n × (n × H) = n × E

inc

_{+ n × (n × H}

inc

_).

(2) Theve tors

E

inc

and

H

inc

representthe omponentsof anin ident ele tromagneti wave

and

n

denotestheunitaryoutwardnormal.Equations(1)and(2) anbefurtherrewritten, assuming

J

equalsto0,underthefollowingform:











iωG

0

W

+ G

x

∂

x

W

+ G

y

∂

y

W

+ G

z

∂

z

W

= 0

in

Ω,

(M

Γ

m

− Gn)W = 0

on

Γ

m

,

(M

Γ

a

− Gn)(W − W

inc

) = 0

on

Γ

a

.

(3) where

W

=

 E

H



is the newunknown ve torand

G

0

=

 ε

r

I

3

0

3×3

0

3×3

µ

r

I

3



. Theterms

I

3

and

0

3×3

denote respe tively the identity matrix and a null matrix, of dimensions

3 × 3

. Therealpartof

G

0

issymmetri positivedeniteanditsimaginarypart,whi happearsfor instan einthe aseof ondu tivematerials,issymmetri negative.Denotingby

(e

x

_{, e}

y

_{, e}

z

₎

the anoni albasisof

R

3

,thematri es

G

l

with

l ∈ {x, y, z}

aregivenby:

G

l

=

0

3×3

Ne

l

N

t

e

l

0

3×3



whereforave tor

v

=





v

x

v

y

v

z





, Nv =





0

v

z

−v

y

−v

z

0

v

x

v

y

−v

x

0





.

Inthe following wedenote by

Gn

thesum

G

x

n

x

+ G

y

n

y

+ G

z

n

z

and by

G

+

n

and

G

−

n

its positiveandnegativeparts

1

.Wealsodene

|Gn|= G

+

n − G

−

n

.Inordertotakeintoa ount theboundary onditions,thematri es

M

Γ

m

and

M

Γ

a

aregivenby:

M

Γ

m

=

 0

3×3

Nn

−N

_n

t

0

3×3



and

M

Γ

a

= |Gn|.

3 A lassi al domain de omposition method

We onsidernowthe problem(3). In orderto ease thepresentation wede omposethe

domain

Ω

intotwooverlappingornon-overlappingsubdomains

Ω

1

and

Ω

2

buttheextension oftheformulationofthemethodtoanynumberofsubdomainsisstraightforward.Wedene

Γ

12

= ∂Ω

1

∩ Ω

2

and

Γ

21

= ∂Ω

2

∩ Ω

1

.Inthefollowingwedenoteby

n

ij

theoutwardnormal 1

If

T

ΛT

−1

istheeigenfa torization of

Gn

then

G

±

n

= T Λ

±

T

−1

where

Λ

+

(resp.

Λ

−

)onlygathersthe

ve tor to the interfa e

Γ

ij

with

i, j

in

{1, 2}

. We solve system (3) in both subdomains and weenfor e onthe subdomain interfa esthe ontinuity of the in oming hara teristi

variables whi h provides aso- alled lassi al S hwarz algorithm (see [13℄ for details). The

lassi alS hwarz algorithm allowsto omputethe

(n + 1)

-th iterate of the solution from the

n

-thiterate,startingfromanarbitraryinitialguess,bysolvinglo alproblemsandthen ex hanging information between arti ial boundaries, alled interfa es. This algorithm is

givenby:















iω W

1,n+1

+

X

l∈{x,y,z}

G

l

∂

l

W

1,n+1

= 0

in

Ω

1

,

G

−

_n

12

W

1,n+1

_{= G}

−

n

12

W

2,n

on

Γ

12

, +Boundary onditionson

∂Ω

1

∩ ∂Ω,















iω W

2,n+1

+

X

l∈{x,y,z}

G

l

∂

l

W

2,n+1

= 0

in

Ω

2

,

G

−

_n

21

W

2,n+1

_{= G}

−

n

21

W

1,n

on

Γ

21

, +Boundary onditionson

∂Ω

2

∩ ∂Ω,

(4)

where subs ripts denote omponents,and supers ripts denote thesubdomain numberand

theiteration ount.

Thisalgorithmhasbeenanalyzedin[13℄andits onvergen eratehasbeen omputedin

the aseofaninnitedomain

Ω = R

3

.

4 Dis retization

The subproblems of the S hwarz algorithm (4) are dis retized using a dis ontinuous

Galerkin formulation. In this se tion, we rst introdu e this dis retization method in the

one-domain ase.Then, westate awell-posednessresult foraperturbeddis rete problem.

Finally,weestablishthedis retizationoftheinterfa e onditionofalgorithm(4)withrespe t

totheadopteddis ontinuousGalerkinformulation.

Let

Ω

h

denote adis retization ofthedomain

Ω

intoaunionof onformingtetrahedral elements

Ω

h

=

[

K∈T

h

K

.Welookfortheapproximatesolution

W

h

=

 E

h

H

h



of(3)in

V

h

×V

h

wherethefun tionalspa e

V

h

isdened by:

V

h

=

U ∈ [L

2

(Ω)]

3

/ ∀K ∈ T

h

, U

|K

∈ P

p

(K) .

(5) where

P

p

(K)

denotes aspa e of ve torswith polynomial omponents of degreeat most

p

overtheelement

K

.

4.1 Dis retization of the mono-domain problem

Following the presentation of Ern and Guermond [15, 16℄, the dis ontinuous Galerkin

dis retizationofsystem(3)yieldstheformulationofthedis reteproblem:



















































































Find

W

h

in

V

h

× V

h

su h that:

Z

Ω

h

(iωG

0

W

h

)

t

V

dv +

X

K∈T

h

Z

K





X

l∈{x,y,z}

G

l

∂

l

(W

h

)





t

V

dv

+

X

F ∈Γ

m

_∪Γ

a

Z

F

 1

2

(M

F,K

− I

F K

Gn

F

)W

h



t

V

ds

−

X

F ∈Γ

0

Z

F

(Gn

F

JW

h

K)

t

{V }ds +

X

F ∈Γ

0

Z

F

(S

F

JW

h

K)

t

JV Kds

=

X

F ∈Γ

a

Z

F

 1

2

(M

F,K

− I

F K

Gn

F

)W

inc



t

V

ds,

∀V ∈ V

h

× V

h

,

(6) where

Γ

0

,

Γ

a

and

Γ

m

respe tively denote the set of interior (triangular) fa es, the set of

fa eson

Γ

a

andthesetoffa eson

Γ

m

.Theunitarynormalasso iatedtotheorientedfa e

F

is

n

F

and

I

F K

standsforthein iden ematrixbetweenorientedfa esand elementswhose entries aregivenby:

I

F K

=











0

ifthefa e

F

doesnotbelongtoelement

K

,

1

if

F ∈ K

andtheirorientationsmat h,

−1

if

F ∈ K

andtheirorientationsdonotmat h.

Wealsodenerespe tivelythejump andtheaverageofave tor

V

of

V

h

× V

h

onafa e

F

sharedbytwoelements

K

and

K

˜

:

JV K = I

F K

V

|K

+ I

F ˜

K

V

| ˜

K

and

{V } =

1

2



V

|K

+ V

| ˜

K



.

Finally,thematrix

S

F

,whi hishermitianpositive,allowstopenalizethejumpofaeldor of some omponents ofthis eld on thefa e

F

and thematrix

M

F,K

, to be dened later, insurestheasymptoti onsisten ywiththeboundary onditionsofthe ontinuousproblem.

Problem (6) is often interpreted in terms of lo al problems in ea h element

K

of

T

h

oupled by the introdu tion of an element boundary term alled numeri al ux (see also

Inthisstudy,we onsidertwo lassi alnumeri aluxes,whi hleadtodistin tdenitions

formatri es

S

F

and

M

F,K

:

- a entered ux(see[17℄forthetime-domainequivalent).Inthis ase

S

F

= 0

forall thefa es

F

and,fortheboundaryfa es,weuse:

M

F,K

=











I

F K



0

3×3

Nn

F

−N

t

n

F

0

3×3



if

F ∈ Γ

m

_,

|Gn

F

|

if

F ∈ Γ

a

_.

(7)

- an upwind ux(see [32,15℄).Inthis ase:

S

F

=

α

E

F

Nn

F

N

t

n

F

0

3×3

0

3×3

α

H

F

N

n

t

F

Nn

F



,

M

F,K

=











η

F

Nn

F

N

t

n

F

I

F K

Nn

F

−I

F K

N

n

t

F

0

3×3



if

∈ Γ

m

_,

|Gn

F

|

if

F ∈ Γ

a

_,

(8) with

α

E

F

,

α

H

F

and

η

F

equalsto

1/2

forhomogeneousmedia.

Remark 1 Theformulationofthedis ontinuousGalerkins heme above(inparti ular, the

enteredand upwind uxes) a tually applies tohomegeneous materials. For des ribing the

uxinthe inhomogeneous ase,letus dene:

Z

K

=

1

Y

K

=

r µ

r

ε

r

, Z

F

=

Z

K

_{+ Z}

K

˜

2

and

Y

F

₌

Y

K

+ Y

˜

K

2

,

(9)

where

F = K ∩ ˜

K

.With these denitions, the dis ontinuousGalerkin s heme inthe inho-mogeneous ase anbewrittenformally as (6) butbymodifying

S

F

as:

S

F

=

1

2







1

Z

F

Nn

F

N

t

n

F

0

3×3

0

3×3

1

Y

F

N

t

n

F

N

n

F







,

(10)

andbyusing forthe average, aweighted average

{·}

F

forea h fa e

F

:

{V}

F

=

1

2

















Z

K

˜

Z

F

0

3×3

0

3×3

Y

˜

K

Y

F









V

_|K

₊







Z

K

Z

F

0

3×3

0

3×3

Y

K

Y

F







V

| ˜

K









.

(11)

Inordertosimplify thepresentationinthefollowingse tions,weonlyretainthe

4.2 Comments on the dis retization

Afewworkshave onsideredthedis retizationofthetime-harmoni Maxwellequations

by a dis ontinuous Galerkin formulation ombined to the numeri al uxes (7) and (8).

Con erning the onvergen e properties of su h dis ontinuous Galerkin formulations, the

state-of-artisthefollowing:

 whenthedis ontinuousGalerkin methodis ombinedto theupwindux (8) ,

onver-gen eresultshavebeenobtainedbyHelluyandDaymain[21℄foraperturbedproblem,

i.e.repla ing

iω

by

iω + ν

with

ν

astri tlypositiveparameter.Theirresultstatesthat, if thesolutionis su iently regularand if apolynomial approximationof order

p

is used in ea h element

K

, the

L

2

-norm errorof the ele tromagneti elds behaves as

h

p+1/2

where

h

isthemesh parameter.

 thedis ontinuousGalerkinmethod ombinedtothe enteredux(7)hasbeenstudied

by Fezoui et al.in [17℄ for thetime-domain Maxwell equations.In this ase,the

L

2

-norm error ofthe ele tromagneti elds behavesas

h

p

. This resultshould extend to

thetime-harmoni asehoweverno onvergen eproofsareavailablesofar.

The onvergen eofthedis ontinuousGalerkinmethods onsideredhereisstudied

nume-ri allyin the ontext ofthetwo-dimensionaltime-harmoni Maxwell equationsdis retized

ontriangularmeshesin [12℄.

Beside, we anstudy thesolvability ofthe dis reteproblem in the ase ofaperturbed

problem(werepla e

iω

by

iω + ν

with

ν > 0

).Werea llheretheproalreadypresentedin [12℄.Inthis settingand assuminghomogeneousboundary onditions,theproblemat hand

anbesimplywritten as:

(

Find

W

h

in

V

h

× V

h

su h that:

a(W

h

, V ) + b(W

h

, V ) = 0, ∀V ∈ V

h

× V

h

,

(12) with,

∀U , V ∈ V

h

× V

h

:

a(U , V ) =

Z

Ω

h

((iω + ν)G

0

U

)

t

V

dv +

X

F ∈Γ

a

Z

F

 1

2

|Gn

F

|U



t

V

ds

+

X

F ∈Γ

m

Z

F

 1

2

M

F,K

U



t

V

ds +

X

F ∈Γ

0

Z

F

(S

F

JU K)

t

JV K

F

ds,

(13) and:

b(U , V ) =

X

K∈T

h

Z

K





X

l∈{x,y,z}

G

l

∂

l

(U )





t

V

dv

−

X

F ∈Γ

a

_∪Γ

m

Z

F

 1

2

I

F K

Gn

F

U



t

V

ds

−

X

F ∈Γ

0

Z

F

(Gn

F

JU K)

t

{V }ds.

(14)

Then,wehavethefollowingresult.

Proposition1 Thesolution of problem (12)is equaltozero.

Proof Let

ℜ(G

0

)

and

ℑ(G

0

)

respe tively denote the real and imaginary parts of

G

0

. First, onsidering the fa t that the matri es

|Gn

F

|

,

S

F

,

ℜ(G

0

)

and

−ℑ(G

0

)

are hermi-tiananddenotingby

H(M

F,K

)

thehermitian partof

M

F,K

for

F

in

Γ

m

,whi hisequalto

η

F

Nn

F

N

t

n

F

0

3×3

0

3×3

0

3×3



, onehas:

ℜ(a(W

h

, W

h

)) =

Z

Ω

h

((νℜ(G

0

) − ωℑ(G

0

))W

h

)

t

W

h

dv

+

X

F ∈Γ

0

Z

F

(S

F

JW

h

K)

t

JW

h

K

F

ds

+

X

F ∈Γ

a

Z

F

 1

2

|Gn

F

|W

h



t

W

h

ds

+

X

F ∈Γ

m

Z

F

 1

2

H(M

F,K

)W

h



t

W

h

ds.

(15)

Then, we rewrite using the orresponding Green identity an equivalent expression of the

sesquilinearform

b

:

b(U , V ) = −

X

K∈T

h





Z

K

U

t





X

l∈{x,y,z}

G

l

∂

l

(V )





dv

−

X

F ∈∂K

Z

F

(I

F K

Gn

F

U

|K

)

t

V

|K

ds

#

−

X

F ∈Γ

a

_∪Γ

m

Z

F

 1

2

I

F K

Gn

F

U



t

V

ds

−

X

F ∈Γ

0

Z

F

(Gn

F

JU K)

t

{V }ds, ∀U , V ∈ V

h

× V

h

.

(16)

Bynoti ingthat onafa e

F ∈ Γ

0

separatingtwoelements

K

and

K

˜

:

(Gn

F

{U })

t

JV K + (Gn

F

JU K)

t

_{{V } = (I}

F K

Gn

F

U

|K

)

t

V

|K

+ (I

_{F ˜}

_K

Gn

F

U

| ˜

K

)

t

V

_{| ˜}

_K

,

whi h isinpartdue tothefa tthat

Gn

F

ishermitian,onededu es:

b(U , V ) = −

X

K∈T

h

Z

K

U

t





X

l∈{x,y,z}

G

l

∂

l

(V )





dv

+

X

F ∈Γ

a

_∪Γ

m

Z

F

 1

2

I

F K

Gn

F

U



t

V

ds

+

X

F ∈Γ

0

Z

F

(Gn

F

{U })

t

JV Kds, ∀U , V ∈ V

h

× V

h

.

(17)

Thus,itisnowstraightforwardtosee that

b

isanti-hermitianand onsequently:

ℜ(a(W

h

, W

h

) + b(W

h

, W

h

)) =

Z

Ω

h

((νℜ(G

0

) − ωℑ(G

0

))W

h

)

t

W

h

dv

+

X

F ∈Γ

0

Z

F

(S

F

JW

h

K)

t

JW

h

K

F

ds

+

X

F ∈Γ

a

Z

F

 1

2

|Gn

F

|W

h



t

W

h

ds

+

X

F ∈Γ

m

Z

F

 1

2

H(M

F,K

)W

h



t

W

h

ds,

From (12),

ℜ(a(W

h

, W

h

) + b(W

h

, W

h

))

is also equal to zero. As

νℜ(G

0

) − ωℑ(G

0

)

is positivedeniteand

|Gn

F

|

,

S

F

and

H(M

F,K

)

arepositive,theve toreld

W

h

iszero. 4.3 Dis retization of the domain de omposition algorithm

4.3.1 Dis ontinuousGalerkin formulation ofthe multi-domainproblem

Let us nowassume that thedomain

Ω

is de omposed into

N

s

subdomains

Ω =

N

s

[

i=1

Ω

i

. Asupers ript

i

indi ates that somenotationsarerelativeto thesubdomain

Ω

i

and notto thewhole domain

Ω

. Thus, wewill referto

T

i

h

and

V

i

h

withobviousdenitionsfrom those of

T

h

and

V

h

andwealsodene

Γ

i

m

= Γ

m

∩ ∂Ω

i

,

Γ

i

a

= Γ

a

∩ ∂Ω

i

and

Γ

i

0

= Γ

0

∩ ∂Ω

i

with their orrespondingsetsof fa es

Γ

m,i

,

Γ

a,i

and

Γ

0,i

. Finally

Γ

ij

willdenotetheset offa es

whi h belongsto

Γ

ij

= ∂Ω

i

∩ Ω

j

.

A ordingtoalgorithm(4),theinterfa e onditionon

Γ

ij

writesas:

G

−

_n

_F

(W

i,n+1

_h

− W

j,n

_h

) = 0

forall

F

belongingto

Γ

ij

,

where

W

i,n+1

h

denotes the approximationof

W

i,n+1

for

i = 1, 2

. Thus, the dis ontinuous Galerkindis retization ofalo alproblemof algorithm(4) anbewritten using(6) ,asthe

solutionofthefollowingproblem:



















































































































































Find

W

i,n+1

h

in

V

i

h

× V

h

i

su hthat:

Z

Ω

i

h



iωG

0

W

i,n+1

_h



t

V

dv +

X

K∈T

i

h

Z

K





X

l∈{x,y,z}

G

l

∂

l

(W

i,n+1

_h

)





t

V

dv

+

X

F ∈Γ

m,i

Z

F

 1

2

(M

F,K

− I

F K

Gn

F

)W

i,n+1

h



t

V

ds

+

X

F ∈(Γ

a,i

_∪Γ

ij

₎

Z

F



I

F K

G

−

_n

_F

W

i,n+1

_h



t

V

ds

−

X

F ∈Γ

0

,i

Z

F



Gn

F

JW

i,n+1

h

K



t

{V }ds

+

X

F ∈Γ

0

,i

Z

F



S

F

JW

i,n+1

h

K



t

JV Kds

=

X

F ∈Γ

a,i

Z

F

I

F K

G

−

_n

_F

W

inc

t

V

ds

+

X

F ∈Γ

ij

Z

F

I

F K

G

−

_n

_F

W

j,n

t

V

ds,

∀V ∈ V

h

i

× V

h

i

.

(18)

4.3.2 Formulation of aninterfa e system

Inthetwo-domain asetheS hwarzalgorithm anbewrittenformallyasfollows:







LW

1,n+1

= f

1

,

in

Ω

1

,

B

1

(W

1,n+1

) = λ

1,n

,

on

Γ

12

,

+Boundary onditionson

∂Ω

1

∩ ∂Ω,







LW

2,n+1

= f

2

,

in

Ω

2

,

B

2

(W

2,n+1

) = λ

2,n

,

on

Γ

21

,

+Boundary onditionson

∂Ω

2

∩ ∂Ω,

(19) andthen:



λ

1,n+1

_{= B}

1

(W

2,n+1

)

on

Γ

12

,

λ

2,n+1

_{= B}

2

(W

1,n+1

)

on

Γ

21

,

(20)

where

L

isalineardierentialoperator,

f

1,2

denotesrighthandsidesasso iatedto

Ω

1,2

and,

B

1

and

B

2

are the interfa e operators. The S hwarz algorithm (19)-(20) an be rewritten as:







λ

1,n+1

₌

_B

1

(W

2

(λ

2,n

, f

2

)),

λ

2,n+1

₌

_B

2

(W

1

(λ

1,n

, f

1

)),

where

W

j

_{= W}

j

_(λ

j

_{, f}

j

)

arethesolutionofthelo alproblems.Bylinearityoftheoperators involved,aniterationoftheS hwarzalgorithm isequivalentto :

λ

n+1

= (

Id

− T )λ

n

_{+ d,}

whi h isaxedpointiterationtosolvetheinterfa esystem:

T λ = g,

(21)

where

λ

= (λ

1

_{, λ}

2

₎

. From thedis retepointof view,theglobalproblem ondomain

Ω

an bewrittenin thematrixform:









A

1

0

R

1

0

0

A

2

0

R

2

0

−B

2

I

0

−B

1

0

0

I

















W

1

h

W

2

h

λ

1

h

λ

2

_h









=









f

1

h

f

2

h

0

0









,

where

A

1,2

are lo al matri es oupling onlyinternal unknowns,

R

1,2

express the oupling betweeninternalunknownsandinterfa eunknownsandthesubs ript

h

denotesthedis rete ounterpartofagivenquantity(e.g.

λ

1,2

h

arethedis retizedunknownve tors orresponding to

λ

1,2

).The eliminationof theinternal unknowns

W

1,2

h

leadstothe dis rete ounterpart oftheinterfa eproblem(21),

T

h

λ

h

= g

h

,with:

T

h

=





I

B

2

A

−1

2

R

2

B

1

A

−1

1

R

1

I





and

g

h

=





B

2

A

−1

2

F

2

B

1

A

−1

1

F

1





,

where

T

h

and

g

h

arethedis retizationof

T

and

d

.ThissystemisfurthersolvedbyaKrylov subspa emethod asdis ussedinthefollowingse tion.

5 Numeri al and performan e results

5.1 Implemented formulations and experimental testbed

Forthisstudy,theimplementationofthedis ontinuousGalerkinformulationsdes ribed

inse tion(4.1)hasbeenlimitedtoa

P

0

approximationwiththe enteredux(7)(whi his equivalentto anitevolumemethod whi hwill bereferredasDG-

P

0

- in thesequel) and a

P

1

approximation(i.e.alineardis ontinuousGalerkin method) witheither the entered ux (7)ortheupwindux (8) andnodal polynomialbasisfun tions (respe tivelyreferred

asDG-

P

1

- andDG-

P

1

-uinthesequel).

Unlessotherwiseindi ated, omputationshavebeenperformedin64bitarithmeti .The

experimental testbedis a luster of AMD Opteron 2GHzdual nodes with2GB of RAM

memory,inter onne tedbyaGigabit Ethernet swit h. The omputer odesfortheDG-

P

0

andDG-

P

1

methodshavebeenprogrammedin Fortranandtheparallelizationreliesonthe MPI(MessagePassingInterfa e).Theimplementation ofthedomainde omposition solver

requiresapartitioningoftheunderlyingtetrahedralmeshwhi hisobtainedusingtheMeTiS

5.2 Solution strategies

An unpre onditionedBiCGstab

(ℓ)

Krylovsubspa emethod[35℄isusedforthesolution oftheinterfa esystem(21).Afterdierenttestsforassessingthe onvergen eofthemethod

and the asso iated omputation time, the parameter

ℓ

has been set to 6. This method is adapted to linear systemsinvolvingnon-symmetri matri eswith omplexspe trum.The

onvergen e of the iterative solution of the interfa e system is evaluated in terms of the

eu lidian norm of the residual normalizedto the norm of the right-hand side ve tor.The

orresponding linear threshold has been set to

ε

i

= 10

−6

. Ea h iteration of this Krylov

subspa e method requires a ertain number of matrix-ve tor produ ts with the interfa e

matrixof system(21).Withinthedomainde ompositionframeworkofalgorithm(4),su h

a matrix-ve tor produ t translates into the solution of the subdomain dis rete problems

(18).Forthispurpose,severalstrategieshavebeen onsidered:

 apre onditionedrestartedGMRES(

m

)[34℄(with

m = 10

)orapre onditioned BiCGstab

(ℓ)

(with

ℓ = 1

) method where the pre onditioner is taken to be a LU fa torization omputed and stored in singlepre ision arithmeti using the MUMPS

multifrontalsparsedire tsolver[2℄,whiletheKrylovsubspa emethodworksondouble

pre isionarithmeti ve tors.In both ases,thelinearthresholdhas been set to

ε

i

=

10

−6

.Thesesolutionstrategieswillbereferredrespe tivelyasDD-gmresandDD-bi gl.

 aLUfa torizationwheretheLand Ufa torsare omputedand storedinsingle

pre- ision(32bits)arithmeti andaniterativerenementpro edureisapplied tore over

doublepre isionarithmeti (64bits).Morepre isely,assumingthat thelinearsystem

is

Ax = b

,theiterativerenementpro edureisasfollows:

x ← 0

REPEAT

r ← b − Ax

%

residualevaluationstep. Solve

Ly = r

Solve

U z = y

x ← x + z

%

updatingstep. UNTIL

k r k< ε

l

where thetriangular solves

Ly = r

and

U z = y

are performed using singlepre ision arithmeti while the residual evaluation and updating step are omputed in double

pre isionarithmeti .Inpra ti e,weset

ε

l

= 10

−10

andamaximumofveiterationsof

theabovepro edure.Inthesequel,thissolutionstrategywillbereferredasDD-itref.

Thesestrategieshavebeensele tedwiththeaimtoredu ethememoryrequirementsfor

storingtheL andUfa torsand thus allowingtota klelargeproblems.Wenote thatsu h

mixed-pre ision strategies have re ently been onsidered in the linear algebra ommunity

essentiallyforperforman eissues[27, 28℄onmodernhigh-performan epro essors.Inthese

works,themixing ofsingleanddouble pre ision omputations isperformedin the ontext

a urate pre onditioner and onsequently, a few iterations of the pre onditioned Krylov

subspa emethodsaresu ientforsolvingthesubdomainproblems.Inpra ti eweuseone

iterationofBiCGstabandtwoiterationsofGMRES.

Inthefollowingtablesandgures:

 L min ,L max andL avg

respe tivelydenotetheminimum,maximumandaveragelength

ofanedgeinagiventetrahedralmesh,



N

s

is thenumberofsubdomains whi h isalso thenumberof pro essesinvolvedin a parallelsimulation,

 'CPU' is the CPU time whi h is evaluated on ea h pro ess of a parallel simulation

and,forthisreason,wegiveboththeminimumandmaximumvaluesofthisquantity,

 'REAL'isthereal(orelapsed)timeofaparallelsimulation,

 'RAM'is thememoryrequirementforstoringtheLandU fa torswhi hisevaluated

onea hpro ess ofaparallelsimulationand, asfor the'CPU'quantity,wegiveboth

theminimumandmaximumvaluesofthisquantity.

5.3 Dira tion of a plane wave by a PEC sphere

Thersttestproblem thatwe onsider isthedira tionofaplanewaveby aperfe tly

ondu ting sphere with radius

R = 1

m entered at the origin. The arti ial boundary onwhi hthe rst-orderabsorbing ondition(2) appliesis dened byasphere withradius

R

a

= 1.5

m enteredat theorigin.Themediumis onsideredhomogeneouswith

ε

r

and

µ

r

equalto one.Thefrequen y ofthe in ident planewaveis

F = 600

MHzand

ω = 2πF/F

0

with

F

0

= 300

MHz.Itspolarizationissu h that:

k

= (0, 0, −k

z

)

t

, E = (E

x

, 0, 0)

t

and

H

= (0, H

y

, 0)

t

_.

Four tetrahedral meshes of in reased resolution have been used and their hara teristi s

aresummarized inTable1.Viewsofthetriangulationsin theplaneZ=0.0m aregivenon

Figure1.Note that themeshwith thenestresolution issu h that L avg

=

λ/11

whilethis ratioisequalto6forthemeshwiththe oarsestresolution. Numeri alsolutionsareshown

Mesh #verti es #tetrahedra L min (m) L max (m) L avg (m) M1 32,418 172,800 0.051990 0.152832 0.086657 M2 70,422 384,000 0.039267 0.118029 0.066279 M3 151,452 843,648 0.030206 0.091805 0.051038 M4 244,834 1,382,400 0.025665 0.078819 0.043431

Tab.1Dira tionofaplanewavebyaPECsphere,F=600MHz.

Chara teristi softhetetrahedralmeshes('#'refersto'thenumberof').

onFigures2and3in theform ofthe ontourlinesintheplaneZ=0.0m ofthe

E

x

and

E

y

omponents.Figures 2(a)and 2(b) orrespond to theanalyti alsolutionfor this problem,

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

(a)meshM1.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

(b)meshM3.

Fig.1Dira tionofaplanewavebyaPECsphere,F=600MHz.

TriangulationintheplaneZ=0.0m.

DG-

P

0

- method, andonmesh M1using theDG-

P

1

- /DG-

P

1

-u methodsarein verygood agreementwiththereferen eresult.

Mesh Method Strategy

N

s

# it CPU(min/max) REAL M1 DG-

P

1

- DD-bi gl 32 10 441se /772se 929se - - DD-gmres - 10 227se /271se 442se

- - DD-itref - 9 197se /300se 480se

M1 DG-

P

1

-u DD-bi gl 32 10 544se /616se 842se - - DD-gmres - 9 259se /284se 464se

- - DD-itref - 9 170se /200se 344se

M2 DG-

P

0

- DD-bi gl 16 8 215se /379se 390se

- - - 32 9 98se /132se 143se

- - DD-gmres 16 8 110se /139se 143se

- - - 32 9 46se /58se 68se

- - DD-itref 16 8 215se /379se 390se

- - - 32 9 101se /159se 172se

M3 DG-

P

0

- DD-bi gl 32 8 244se /352se 456se

- - - 64 9 116se /178se 184se

- - DD-gmres 32 8 121se /164se 249se

- - - 64 9 56se /87se 98se

- - DD-itref 32 8 116se /197se 256se

- - - 64 9 53se /98se 111se

M4 DG-

P

0

- DD-bi gl 64 9 197se /432se 460se - - DD-gmres - 10 109se /173se 211se

- - DD-itref - 9 101se /193se 233se

Tab.2Dira tionofaplanewavebyaPECsphere,F=600MHz.

Computationtimes(solutionphase).

Mesh Method

N

s

CPU(min/max) RAM(min/max) M1 DG-

P

1

- 32 198se /301se 1217MB/1457MB M1 DG-

P

1

-u 32 211se /329se 1257MB/1512MB M2 DG-

P

0

- 8 220se /359se 1365MB/1679MB - - 16 56se /121se 492MB/733MB - - 32 11se /26se 156MB/249MB M3 - 32 69se /185se 586MB/959MB - - 64 17se /52se 210MB/370MB M4 - 64 43se /135se 425MB/737MB

Tab.3Dira tionofaplanewaveinva uumbyaPEC sphere,F=600MHz.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.45

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

(a)Analyti alsolution,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35

-0.4

-0.45

-0.5

-0.55

-0.6

-0.65

-0.7

(b)Analyti alsolution,

E

y

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

( ) MethodDG-

P

0

- ,meshM2,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.63

0.58

0.53

0.48

0.43

0.38

0.33

0.28

0.23

0.18

0.13

0.08

0.03

-0.02

-0.07

-0.12

-0.17

-0.22

-0.27

-0.32

-0.37

-0.42

-0.47

-0.52

-0.57

-0.62

(d)MethodDG-

P

0

- ,meshM2,

E

y

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

(e) MethodDG-

P

0

- ,meshM3,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.64

0.59

0.54

0.49

0.44

0.39

0.34

0.29

0.24

0.19

0.14

0.09

0.04

-0.01

-0.06

-0.11

-0.16

-0.21

-0.26

-0.31

-0.36

-0.41

-0.46

-0.51

-0.56

-0.61

-0.66

(f)MethodDG-

P

0

- ,meshM3,

E

y

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.4

1.35

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

(a)MethodDG-

P

0

- ,meshM4,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.63

0.58

0.53

0.48

0.43

0.38

0.33

0.28

0.23

0.18

0.13

0.08

0.03

-0.02

-0.07

-0.12

-0.17

-0.22

-0.27

-0.32

-0.37

-0.42

-0.47

-0.52

-0.57

-0.62

-0.67

(b)MethodDG-

P

0

- ,meshM4,

E

y

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

( ) MethodDG-

P

1

- ,meshM1,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.57

0.52

0.47

0.42

0.37

0.32

0.27

0.22

0.17

0.12

0.07

0.02

-0.03

-0.08

-0.13

-0.18

-0.23

-0.28

-0.33

-0.38

-0.43

-0.48

-0.53

-0.58

(d)MethodDG-

P

1

- ,meshM1,

E

y

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REX

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

(e)MethodDG-

P

1

-u,meshM1,

E

x

.

-1.5

-1

-0.5

0

0.5

1

1.5

Y

0

Z

-1.5

-1

-0.5

0

0.5

1

1.5

X

X

Y

Z

REY

0.54

0.49

0.44

0.39

0.34

0.29

0.24

0.19

0.14

0.09

0.04

-0.01

-0.06

-0.11

-0.16

-0.21

-0.26

-0.31

-0.36

-0.41

-0.46

-0.51

-0.56

(f)MethodDG-

P

1

-u,meshM1,

E

5.4 Dira tion of a plane wave by a PEC ube

Thetestproblem onsideredhere onsistsinthedira tionofaplanewavebyaperfe tly

ondu ting ubeofsidelength

C = 1/3

m enteredattheorigin.Thearti ialboundaryon whi h therst-order absorbing ondition(2)applies isdened by aunitary ube entered

attheorigin.Thefrequen yofthein identplanewaveis F=900MHzanditspolarization

issu hthat :

k

= (k

x

, 0, 0)

t

, E = (0, E

y

, 0)

t

and

H

= (0, 0, H

z

)

t

_.

Five tetrahedral meshes havebeen used whose hara teristi s are summarized in Table 4

(seealso Figure 4(a)).The exteriordomain is alwaysthe va uum but twosituations have

been onsidered for the ube: either it is stri tly aperfe t ondu tororit is oated by a

diele tri materialwith

ε

r

= 4.0

(seealsoFigure4(b)foraviewofthe orrespondingzone).

Mesh #verti es #tetrahedra L min (m) L max (m) L avg (m) M1 9,136 46,704 0.05000 0.08660 0.06343 M2 29,062 156,000 0.03333 0.05773 0.04242 M3 67,590 373,632 0.02500 0.04330 0.03187 M4 129,276 725,424 0.02000 0.03464 0.02552 M5 220,122 1,248,000 0.01666 0.02886 0.02128

Tab.4Dira tionofaplanewavebyaPEC ube,F=900MHz.

Chara teristi softhetetrahedralmeshes.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

(a)Meshexampleofthedomain.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

(b)Cube oatedbyadiele tri material.

Fig.4Dira tionofaplanewavebyaPEC ube,F=900MHz.

5.4.1 Propagation in va uum

Numeri al solutions are shown on Figures 5 and 6 in the form of the ontourlines in

theplaneZ=0.5mofthe

E

x

and

E

y

omponents.One annotethatthesolutionresulting fromtheDG-

P

1

- methodappliedonmeshM1isverysimilartotheoneobtainedusingthe DG-

P

0

- method withmeshM4.Moreover,theformersolutionexhibitsabettersymmetry withregardstothedistributionofthe

E

y

omponent.TimingmeasuresaregiveninTable5 (solutionphase)and 6(fa torizationphase).

Mesh Method Strategy

N

s

# it CPU(min/max) REAL M1 DG-

P

1

- DD-bi gl 8 6 202se /352se 355se - - DD-gmres - 6 106se /118se 124se

- - DD-itref - 6 102se /130se 136se

M2 DG-

P

1

- DD-bi gl 32 9 253se /440se 506se

- - - 64 10 105se /202se 236se

- - DD-gmres 32 9 115se /151se 207se

- - - 64 10 60se /82se 117se

- - DD-itref 32 9 108se /152se 168se

- - - 64 10 47se /72se 91se

M2 DG-

P

1

-u DD-bi gl 32 9 343se /389se 430se

- - - 64 10 161se /207se 234se

- - DD-gmres 32 9 170se /204se 258se

- - - 64 11 90se /116se 137se

- - DD-itref 32 9 114se /131se 174se

- - - 64 10 51se /69se 94se

M2 DG-

P

0

- DD-bi gl 16 6 48se /61se 64se - - DD-gmres - 6 26se /32se 35se

- - DD-itref - 6 20se /27se 31se

M3 DG-

P

0

- DD-bi gl 16 7 150se /184se 199se

- - - 32 8 81se /101se 122se

M4 DG-

P

0

- DD-bi gl 16 7 345se /395se 452se

- - - 32 8 161se /224se 238se

- - - 64 9 87se /108se 120se

Tab.5Dira tionofaplanewavein va uumbyaPEC ube,F=900MHz.

Computationtimes(solutionphase).

5.4.2 Coated PEC ube

Numeri alsolutionsareshownonFigure7in theform ofthe ontourlinesintheplane

Z=0.5mofthe

E

x

and

E

y

omponents.Thistime,weonlyreportonresultsobtainedusing the DG-

P

0

- method applied to mesh M5 and the DG-

P

1

- /DG-

P

1

-u methods applied to

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REX

0.51

0.46

0.41

0.36

0.31

0.26

0.21

0.16

0.11

0.06

0.01

-0.04

-0.09

-0.14

-0.19

-0.24

-0.29

-0.34

-0.39

-0.44

-0.49

-0.54

(a)MethodDG-

P

0

- ,meshM2,

E

x

.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REY

0.92

0.845

0.77

0.695

0.62

0.545

0.47

0.395

0.32

0.245

0.17

0.095

0.02

-0.055

-0.13

-0.205

-0.28

-0.355

-0.43

-0.505

-0.58

-0.655

-0.73

-0.805

-0.88

(b)MethodDG-

P

0

- ,meshM2,

E

y

.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REX

0.58

0.53

0.48

0.43

0.38

0.33

0.28

0.23

0.18

0.13

0.08

0.03

-0.02

-0.07

-0.12

-0.17

-0.22

-0.27

-0.32

-0.37

-0.42

-0.47

-0.52

-0.57

-0.62

( ) MethodDG-

P

0

- ,meshM3,

E

x

.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REY

0.955

0.88

0.805

0.73

0.655

0.58

0.505

0.43

0.355

0.28

0.205

0.13

0.055

-0.02

-0.095

-0.17

-0.245

-0.32

-0.395

-0.47

-0.545

-0.62

-0.695

-0.77

-0.845

-0.92

(d)MethodDG-

P

0

- ,meshM3,

E

y

.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REX

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35

-0.4

-0.45

-0.5

-0.55

-0.6

-0.65

(e) MethodDG-

P

0

- ,meshM4,

E

x

.

0

0.2

0.4

0.6

0.8

1

Y

0

0.5

1

Z

0

0.2

0.4

0.6

0.8

1

X

X

Y

Z

REY

0.965

0.89

0.815

0.74

0.665

0.59

0.515

0.44

0.365

0.29

0.215

0.14

0.065

-0.01

-0.085

-0.16

-0.235

-0.31

-0.385

-0.46

-0.535

-0.61

-0.685

-0.76

-0.835

-0.91

(f)MethodDG-

P

0

- ,meshM4,

E

y

.