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Time-ha monic elec omagne ics wi h exac con ollabili y and disc e e ex e io calculus
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Published e sion
Mönkölä, Sanna; Räbinä, Jukka; Rossi, Tuomo
Mönkölä, S., Räbinä, J., & Rossi, T. (2023). Time-ha monic elec omagne ics wi h exac
con ollabili y and disc e e ex e io calculus. Comp es Rendus. Mecanique, Online i s .
h ps://doi.o g/10.5802/c meca.234
2023
Comp es Rendus
Mécanique
Sanna Mönkölä, Jukka Räbinä and Tuomo Rossi
Time-ha monic elec omagne ics wi h exac con ollabili y and disc e e ex e io
calculus
Published online: 13 Decembe 2023
h ps://doi.o g/10.5802/c meca.234
Pa o Special Issue: The scien i ic legacy o Roland Glowinski
Gues edi o s: G egoi e Allai e (CMAP, Ecole Poly echnique, Ins i u Poly echnique de Pa is,
Palaiseau, F ance), Jean-Michel Co on (Labo a oi e Jacques-Louis Lions, So bonne
Uni e si é) and Vi e e Gi aul (Labo a oi e Jacques-Louis Lions, So bonne Uni e si é)
This a icle is licensed unde he
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e-ISSN : 1873-7234
Comp es Rendus
Mécanique
Published online: 13 Decembe 2023
h ps://doi.o g/10.5802/c meca.234
The scien i ic legacy o Roland Glowinski / L’hé i age scien i ique de Roland
Glowinski
Time-ha monic elec omagne ics wi h exac
con ollabili y and disc e e ex e io calculus
Élec omagné ique ha monique empo elle a ec
con ôlabili é exac e e calcul ex é ieu disc e
Sanna Mönkölä ,a, Jukka Räbinä aand Tuomo Rossi ,∗,a
aFacul y o In o ma ion Technology, Uni e si y o Jy äskylä, P.O. Box 35, FI-40014
Uni e si y o Jy äskylä, Finland
E-mail: uomo.j[email p o ec ed] (T. Rossi)
Abs ac . In his pape , we apply he exac con ollabili y concep o ime-ha monic elec omagne ic sca e -
ing. The p oblem is p esen ed in e ms o he di e en ial o ms, and he disc e e ex e io calculus is u ilized
o spa ial disc e iza ion. Acco dingly, he physical p ope ies o he p oblem a e main ained. Despi e we con-
side ime-ha monic p oblems, we concen a e on ansien wa e equa ions ea ed by he exac con ollabil-
i y app oach. Essen ially, we use a con olled a ia ion o he asymp o ic app oach wi h pe iodic cons ain s,
in which he ime-dependen equa ion is simula ed in ime, un il he ime-ha monic solu ion is eached.
Résumé. Dans ce a icle, nous appliquons le concep de con ôlabili é exac e à la dispe sion élec omagné-
ique empo elle. Le p oblème es p ésen é en e mes de o mes di é en ielles e le calcul ex é ieu disc e
es u ilisé pou la disc é isa ion spa iale. En conséquence, les p op ié és physiques du p oblème son main-
enues. Bien que nous considé ions des p oblèmes ha moniques empo els, nous nous concen ons su les
équa ions d’ondes ansi oi es ai ées pa l’app oche de con ôlabili é exac e. Essen iellemen , nous u ili-
sons une a ia ion con ôlée de l’app oche asymp o ique a ec des con ain es pé iodiques, dans laquelle
l’équa ion dépendan du emps es simulée dans le emps, jusqu’à ce que la solu ion ha monique empo elle
soi a ein e.
Keywo ds. Maxwell equa ions, Elec omagne ic sca e ing, Di e en ial o ms, Disc e e ex e io calculus,
Exac con ollabili y.
Mo s-clés. Équa ions de Maxwell, Di usion élec omagné ique, Fo mes di é en ielles, Calcul ex é ieu dis-
c e , Con ôlabili é exac e.
Funding. Academy o Finland (G an ag eemen nos. 259925 and 260076).
Published online: 13 Decembe 2023
∗Co esponding au ho .
ISSN (elec onic) : 1873-7234 h ps://comp es- endus.academie-sciences. /mecanique/
2Sanna Mönkölä e al.
1. In oduc ion
Wa e equa ions a e impo an o modeling acous ic, elas ic, elec omagne ic, and quan um
mechanical sys ems in di e en ields o science, enginee ing, and echnology. The e has ecen ly
been g owing in e es in di e en ial o m -based app oaches o conside ing wa e equa ions
(see, e.g., [1–3]). The e a e se e al easons why he amewo k is a good al e na i e o he
adi ional ec o ield p esen a ion. The me ic- ee na u e o he di e en ial o ms allows he
cons uc ion o di e en ial ope a o s ha a e independen o he coo dina e sys em [4]. Fu he ,
he disc e e spaces and exac di e en ial ope a o s can be cons uc ed o mimic hei con inuous
coun e pa s (see, e.g., [5, 6]). This p ope y also implies ha ce ain p ope ies o he sys em,
such as, ene gy, a e conse ed, p o ided ha he scheme is s able.
In his s udy, we conside ime-ha monic elec omagne ic wa es. We p esen he Maxwell
equa ions as a i s -o de sys em in e ms o he di e en ial o ms (see, e.g., [7]) o elec omag-
ne ic sca e ing p oblems. Tha is, ins ead o he ec o p esen a ion E=(E1,E2,E3) o he elec-
ic ield, we p esen a 1- o m e
E=E1dx1+E2dx2+E3dx3, whe e dis he ex e io de i a i e. Fu -
he , he elec ic lux densi y is conside ed, ins ead o he ec o D=(D1,D2,D3), as a 2- o m
e
D=D1dx2∧dx3+D2dx3∧dx1+D3dx1∧dx2, whe e ∧is a gene aliza ion o he c oss p oduc ,
known as he wedge p oduc o ex e io p oduc . We also p esen a 1- o m e
H o he magne ic
ield and a 2- o m e
B o he magne ic lux densi y.
Fo he spa ial disc e iza ion, he disc e e coun e pa s o he a iables p esen ed as di e en-
ial o ms a e appoin ed a geome ic objec s such as poin s, lines, su aces, and olumes. We
apply, in pa icula , he disc e e ex e io calculus (DEC) ollowing he g oundwo k p esen ed by
Ma sden and his g oup [8] and pionee ed o elec omagne ics simula ions by Bossa i and Ke -
unen [9]. This cons uc ion p o ides, e.g., he conse a ion o ene gy, he elimina ion o non-
physical modes, and exac di e en ial ope a o s a he disc e e le el [10]. The o he ela ed ap-
p oaches include he ini e in eg a ion echnique (FIT), he disc e e geome ic app oach, and
mixed ini e elemen echniques wi h Nédélec and Ra ia –Thomas elemen s (see [11] and he
e e ences he ein).
F om a ma hema ical poin o iew, i is con enien o use a ou -dimensional ( h ee spa ial
dimensions and one empo al dimension) space– ime p esen a ion. The amewo k also allows
he applica ion o he gene alized heo ies p oposed in [12,13]. S ill, he ou -dimensional sys em
leads o la ge nume ical sys ems ha also equi e sophis ica ed compu a ional g ids. To demon-
s a e how he con en ional h ee-dimensional p oblem is inhe i ed om he gene alized wa e
model, we i s p esen he ou -dimensional backg ound a he modeling phase and go on o
p oceed in ime wi h h ee-dimensional spa ial disc e iza ion. Ea lie we ha e p esen ed a nu-
me ical scheme o ansien elec omagne ics wi h he DEC and high-quali y g ids [14, 15]. In
his pape , we sys ema ically apply and es hese echniques o simula ing ime-ha monic elec-
omagne ic sca e ing.
We ollow he exac con ollabili y concep (see, e.g., [16]) and sol e he ime-ha monic p ob-
lem by using he model p esen ed in he ime domain. This concep has a his o y da ing back o
he ea ly 1990s, when Glowinski e al. (see [17–19]) i s conside ed a nume ical me hod based on
i . I s i s applica ion in h ee-dimensional elec omagne ics was, o ou knowledge, p esen ed
in he INRIA epo [20]. The d awback o he app oach was ha i used H1-con o ming ini e
elemen s, ende ing i applicable only o limi ed classes o p oblems due o spu ious modes ha
can a ise in he nume ical solu ion. Addi ional compu a ional cos s a e due o he second-o de
o mula ion o he Maxwell’s equa ions ha was used, leading o slowe con e gence and he
need o p econdi ioning o he minimiza ion p ocess, since he ene gy space is no o L2- ype.
To imp o e he me hodology, Glowinski and Rossi [21] conside ed he wa e equa ion as a i s -
o de sys em, and hey de i ed he associa ed exac con ollabili y scheme. The main bene i o
Sanna Mönkölä e al. 3
hei app oach is ha he sys em’s ene gy space, whe e he minimiza ion is ealized, is o L2- ype,
and he minimiza ion con e ges wi hou p econdi ioning. The d awback, howe e , is he need
o use mixed me hods (e.g. Ra ia –Thomas ini e elemen s [22]) in disc e izing he p oblem.
Recen ly, a i s -o de o m based on a discon inuous Gale kin me hod was de eloped o he
h ee-dimensional Maxwell case [23].
Ins ead o using mixed ini e elemen s, ano he app oach, based on DEC, s a ed o p og ess
wi h he PhD hesis o Räbinä [24]. The idea o his hesis a ose om he heo e ical a icle [25].
Räbinä also conside ed exac con ollabili y echniques o he compu a ion o ime-ha monic
wa es. Se e al imp o emen s o he ime in eg a ion o he Maxwell’s sys em, such as adap i e
local ime-s epping, ha monic Hodge s a app oxima ion, and spa ial ilings inspi ed by c ys al-
log aphy, we e hen in oduced in [14]. In [15], he DEC app oach was u he gene alized o a
class o wa e p oblems co e ing acous ics, elas odynamics, elec omagne ism, and e en quan-
um mechanics ( he Weyl equa ion). Fu he applica ions in quan um mechanics we e hen con-
side ed in [26–28]. The sys ema iza ion o scien i ic so wa e de elopmen co e ing undamen-
al conse a ion laws was ske ched in [29], and a gene ic ca ego y- heo e ic amewo k o space–
ime linea wa e phenomena was inally p oposed by [30] in 2022. Such o malisms allow, e.g.,
mo ing and de o ming compu a ional domains [29]. Ano he esea ch ack conside s highe -
o de disc e e ex e io calculus in ol ing highe o de Whi ney o ms and no el Hodge s a o -
mula ions [31–33].
The exac con ollabili y app oach was ex ensi ely s udied in he la e 2000s and ea ly 2010s
by Mönkölä e al. In pa icula , highe -o de spec al elemen disc e iza ions and luid-s uc u e
in e ac ion p oblems we e conside ed in [34–36]. The app oach was ecen ly ex ended o h ee-
dimensional isco-elas ic equa ions [37]. Essen ially, he app oach is a con olled a ia ion o
he asymp o ic app oach wi h pe iodic cons ain s, in which he ime-dependen equa ion is
simula ed in ime un il he ime-ha monic solu ion is eached. The ime disc e iza ion is ealized
wi h a s agge ed leap og- ype scheme wi h asynch onous ime s eps (see, [14,38,39]). In p ac ice,
he esidual o he exac con ollabili y algo i hm de ines a each i e a ion how a he solu ion is
om a pe iodic one. I also gi es an an impulse o he sys em o accele a e he con e gence a e.
Roughly speaking, he numbe o compu a ional ope a ions g ows linea ly wi h he numbe o
deg ees o eedom in ol ed in he spa ial disc e iza ion.
The es o his a icle is o ganized as ollows. In Sec ion 2, we p esen he ou -dimensional
di e en ial o m o mula ion. The model is simpli ied in o sepa a e ime and space dimensions
in Sec ion 3, whe e we also ca y ou disc e iza ion. To elimina e disc e iza ion e o s, we ad-
jus he scheme o ime-ha monic p oblems wi h spa ial and empo al co ec ions. This ype o
s a egy has been used o imp o e he accu acy o ime disc e iza ion in elec omagne ics, e.g., by
Ma and Chen [40] and Appelö e al. [41]. The spa ial co ec ion e m is de i ed wi hin he disc e e
ex e io calculus ha we use o space disc e iza ion, and he empo al co ec ion e m a ises
om adap ing he leap og ime disc e iza ion o ime-ha monic equa ions. Ne e heless, bo h
co ec ion e ms can be encapsula ed in o disc e e Hodge ope a o s. In Sec ion 4, we ocus on a
ime-ha monic p oblem and conside how i can be sol ed e icien ly in he ime domain wi h he
exac con ollabili y me hod. The p oblem is e o mula ed as a leas -squa es op imiza ion p ob-
lem ha is sol ed by he conjuga e g adien algo i hm. In Sec ion 5, we apply he app oach o
h ee-dimensional elec omagne ic sca e ing p oblems and es he pe o mance o he me hod
wi h se e al ypes o compu a ional meshes. The concluding ema ks a e p esen ed in Sec ion 6.
2. Model
We conside a linea combina ion o he di e en ial o ms in he ou -dimensional Euclidean
space (x0,x1,x2,x3)Tas ˜
F=˜
u1+˜
u2dx0+˜
u3dx1+˜
u4dx2+˜
u5dx3+˜
u6(dx0∧dx1)+˜
u7(dx0∧dx2)+

4Sanna Mönkölä e al.
˜
u8(dx0∧dx3)+˜
u11(dx2∧dx3)+˜
u10(dx3∧dx1)+˜
u9(dx1∧dx2)+˜
u14(dx0∧dx2∧dx3)+˜
u13(dx0∧
dx3∧dx1)+˜
u12(dx0∧dx1∧dx2)+˜
u15(dx1∧dx2∧dx3)+˜
u16(dx0∧dx1∧dx2∧dx3), whe e he
coe icien s ˜
ui,i=1,...,16 a e scala - alued. The basis 1- o ms a e he coo dina e di e en ials
dx0,dx1,dx2, and dx3, and highe o de basis k- o ms, k=2,3,4, a e cons uc ed as ex e io
p oduc s o kbasis 1- o ms, such ha , dxi∧dxi=0 and dxi∧dxj=−dxj∧dxi. This o mula ion
allows us o p esen a se o linea models wi h he i s -o de di e en ial ope a o in a sho o m,
(d+δ)˜
F=˜
J, (1)
whe e d=P3
i=0(dxi∂i∧) is he ex e io de i a i e, δis he code i a i e o d, and ˜
Jis a sou ce
e m p esen ed as a linea combina ion o he di e en ial o ms. Acco dingly, Equa ion (1) is
equi alen o he sys em















−∂0∇·
−∂0∇·
−∂0−∇× ∇
−∂0−∇× ∇
∇ ∇× −∂0
∇ ∇× −∂0
∇· −∂0
∇· −∂0






























˜
u15
˜
u16
˜
u3
˜
u6
˜
u11
˜
u14
˜
u1
˜
u2















=















−˜
b16
˜
b15
−˜
b6
˜
b3
−˜
b14
˜
b11
−˜
b2
˜
b1















, (2)
whe e
˜
u3:=

˜
u3
˜
u4
˜
u5
,˜
u6:=

˜
u6
˜
u7
˜
u8
,˜
u11 :=

˜
u11
˜
u10
˜
u9
, and ˜
u14 :=

˜
u14
˜
u13
˜
u12
,
and
˜
b3:=

˜
b3
˜
b4
˜
b5
,˜
b6:=

˜
b6
˜
b7
˜
b8
,˜
b11 :=

˜
b11
˜
b10
˜
b9
, and ˜
b14 :=

˜
b14
˜
b13
˜
b12
.
We in e p e ∂0as a ime de i a i e and ∇,∇·, and ∇× as h ee-dimensional g adien , di-
e gence, and cu l ope a o s, espec i ely. By he choice ˜
u1=˜
u2=˜
u15 =˜
u16 =0, ows 1, 2, 5,
and 6 o Equa ion (2) gi e he comple e se o he Maxwell equa ions wi h he magne ic ield
s eng h ˜
u3=(H1,H2,H3)T, he elec ic ield s eng h ˜
u6=(E1,E2,E3)T, he magne ic lux densi y
˜
u11 =−(B1,B2,B3)T, and he elec ic lux densi y ˜
u14 =(D1,D2,D3)T, espec i ely. The magne ic
cu en densi y is −˜
b14, he elec ic cu en densi y is −˜
b11, and magne ic and elec ic cha ges
a e modeled by −˜
u15 and −˜
u16. A Maxwell-like sys em is also ob ained om ows 3, 4, 7, and 8 o
Equa ion (2).
We e u n o Equa ion (1) and conside
˜
F=e
H+dx0∧e
E−e
B+dx0∧e
D, (3)
˜
J= −e
JE−dx0∧e
JH−e
JρH−dx0∧e
JρE−e
JB−dx0∧e
JD−e
JρD−dx0∧e
JρB, (4)
whe e espec i ely, elec ic and magne ic ield s eng hs a e ep esen ed by spa ial 1- o ms e
E=
E1dx1+E2dx2+E3dx3and e
H=H1dx1+H2dx2+H3dx3, elec ic and magne ic lux densi ies
and elec ic and magne ic cu en densi ies by spa ial 2- o ms e
D=D1dx2∧dx3+D2dx3∧dx1+
D3dx1∧dx2,e
B=B1dx2∧dx3+B2dx3∧dx1+B3dx1∧dx2,e
JE=JE1(dx2∧dx3)+JE2(dx3∧
dx1)+JE3(dx1∧dx2), and e
JH=JH1(dx2∧dx3)+JH2(dx3∧dx1)+JH3(dx1∧dx2), and elec-
ic and magne ic cha ges, ρEand ρH, by spa ial 3- o ms e
JρE=ρE(dx1∧dx2∧dx3) and
Sanna Mönkölä e al. 5
e
JρH=ρH(dx1∧dx2∧dx3). Fu he , in Equa ion (3), he e a e also sou ce o ms o cu en den-
si ies and cha ges co esponding o he dual componen s o he elec ic and magne ic ones, e-
e ed o wi h subsc ip s Band D, espec i ely. F om he choices made abo e, we ge om Equa-
ion (1) o he di e en ial o m o mula ions







de
H−∂0e
D=−e
JE,
dx0∧(de
E−de
B)=−dx0∧e
JH,
−dx0∧de
B=−dx0∧e
JρH,
−dx0∧de
D=−dx0∧e
JρE,
(5)







−⋆(dx0∧de
H)=−⋆(dx0∧e
JρB),
−⋆(dx0∧de
E)=−⋆(dx0∧e
JρD),
⋆(dx0∧(de
B−de
E)) =−⋆(dx0∧e
JD),
⋆(de
D−∂0e
H)=−⋆e
JB,
(6)
whe e ⋆is he Hodge s a ope a o . The sys ems (5) and (6) a e dual o each o he , and in
wha ollows, we p oceed wi h Equa ion (5) in sepa a e space and ime dimensions. Fo u he
in o ma ion on di e en ial o ms in elec omagne ics, he eade is e e ed o, e.g., [42–44] and
he e e ences he ein.
3. Disc e iza ion
The quali y o he disc e e model, based on he s uc u e o he compu a ional g id, has a signi -
ican impac on he e iciency o he me hod. As p esen ed in [14], we gene a e pa ly-s uc u ed
non-uni o m polygonal g ids imi a ing he close packing in c ys al la ices and disc e ize he op-
e a o s in he spa ial domain wi h he DEC.
We disc e ize he h ee-dimensional spa ial domain by a pai o p imal and dual g ids. The
elemen s o a g id a e called k-cells, whe e k=0,1,2,3 is he dimension o he cell. Basically, a
k-cell is an o ien ed, con ex, polygon s yle objec , and i is de ined in a ecu si e manne by a
lis o k−1-cells. A 0-cell (node) is a e ex. A 1-cell (edge) is de ined as a line segmen be ween
wo 0-cells and o ien ed om he i s o he second node o he lis . A 2-cell ( ace) is a con ex
polygon su ounded by a ini e se o edges. The bounda y edges a e p esen ed wi h coun e -
clockwise o ien a ion, also p o iding, he o ien a ion o he ace elemen . A 3-cell (body) is a
con ex polyhed on su ounded by a ini e se o aces. The g id cons uc ion has a c ucial ole in
he accu acy o he me hod.
In p ac ice, we gene a e a pa ly s uc u ed g id ha consis s o s uc u ed a eas sepa a ed
by uns uc u ed ones. The app oach allows us o model bounda ies as accu a ely as wi h ully
uns uc u ed g ids, and i is also possible o modi y elemen sizes inside he domain. The g id
gene a ion is based on he Delaunay iangula ion (see [45]). Wi h he help o a Vo onoi diag am,
i is possible o c ea e sophis ica ed polyhed al g ids o ill he s uc u ed pa s o he g id. By
main aining he numbe o uns uc u ed elemen s a a ela i ely low le el, he g id gene a ion
p ocess is conside ably sped up. By cons uc ing he dual elemen s ha a e o hogonal o he
co esponding p imal elemen s, we ge diagonal disc e e Hodge ope a o s, p o iding a signi ican
sa ing in compu ing ime.
3.1. Disc e e di e en ial o ms
The disc e e di e en ial k- o ms (k-cochains) a e a iables associa ed by he de Rham map wi h
he k-cells in he g id (see, e.g., [24, 46]) ha , in he h ee-dimensional cons uc ion, a e 1-cells
(edges) and 2-cells ( aces). We deno e he i h edge o he g id by ei,i=1,...,ne, and he j h ace
by j,j=1,...,m , and he co esponding dual elemen s by ∗
iand e∗
j, espec i ely. Acco dingly,
6Sanna Mönkölä e al.
he componen s o he ec o - alued disc e e di e en ial 1- o ms Eand H, p esen ing he
disc e ized elec ic and magne ic ields and associa ed wi h he p imal and dual edges, a e
Ei=Zeie
E,Hj=Ze∗
je
H,i=1,...,ne,j=1,...,m . (7)
Respec i ely, Dand Bp esen he disc e ized elec ic and magne ic luxes, such ha ,
Di=Z ∗
ie
D,Bj=Z je
B,i=1,...,ne,j=1,...,m , (8)
a e he disc e e di e en ial 2- o ms associa ed wi h he dual and p imal aces o he g id ele-
men s. Also, he ec o s o sou ce o ces, JEand JH, a e conside ed as disc e e di e en ial 2-
o ms
JEi=Z ∗
ie
JE,JHj=Z je
JH,i=1,...,ne,j=1,...,m , (9)
associa ed wi h he dual and p imal aces.
The ela ion be ween he disc e e 1- o ms Eand Hand disc e e 2- o ms Dand Bis p esen ed
by he cons i u i e equa ions
D=⋆ϵE,B=⋆µH, (10)
whe e he disc e e Hodge s a ope a o s ⋆ϵand ⋆µco e he ma e ial p ope ies, pe mi i i y ϵ
and pe meabili y µ, espec i ely, and he me ic p ope ies o he space o coo dina e sys em. In
his pape , he disc e e Hodge s a ope a o s a e diagonal ma ices based on he o hogonali y o
he p imal and dual elemen s and de ined as
⋆α=| |
|e|| ⋄e|Z ⋄eαne·ned , (11)
whe e neis he uni o ien a ion ec o o edge e, and ⋄eis a con ex hull including bo h e
and . The i h diagonal componen o ⋆ϵis composed by se ing α=ϵ,e=ei, and = ∗
iin
Equa ion (11). The j h diagonal componen o ⋆µis composed by se ing α=µ,e=e∗
j, and
= jin Equa ion (11).
The key componen o he disc e e ex e io calculus is he disc e e ex e io de i a i e, o
incidence ma ix, d ep esen ing he neighbo ing ela ions and ela i e o ien a ions o he p imal
edges and aces. We use he no a ion d1 o emphasize ha in his case, he disc e e ex e io
de i a i e ope a es disc e e 1- o ms. The ma ix en y, d1ji , is non-ze o i and only i he edge
eiis included in he bounda y o he ace j. Fu he , he non-ze o en ies ha e he alue ±1,
indica ing he ela i e o ien a ion de ined by coun e -clockwise ci cula ion.
Wi h he no a ions p esen ed abo e, he elec omagne ic sys em a e space disc e iza ion in
a h ee-dimensional domain Ωis
⋆ϵ
∂E
∂ −dT
1H=−JE, in Ω×[0,τ], (12)
⋆µ
∂H
∂ +d1E=−JH, in Ω×[0,τ], (13)
wi h he ini ial condi ions, E(0) and H(0), se a he ini ial ime 0. The inal s a e is conside ed a
he inal ime τ. By deno ing
M=µ⋆ϵ0
0⋆µ¶,K=µ0−dT
1
d10¶, (14)
we can w i e Equa ions (12)–(13) wi h he ini ial condi ions as
(M∂0+K)µE
H¶=−µJE
JH¶, in Ω×[0,τ], (15)
µE(0)
H(0)¶=µE0
H0¶, in Ω, (16)
Sanna Mönkölä e al. 7
he solu ion o which, a ime , is
µE( )
H( )¶=e− M−1KµE0
H0¶−Z
0e−( −s)M−1KM−1µJE(s)
JH(s)¶ds, (17)
whe e
e− M−1K=I+³0⋆−1
ϵdT
1
−⋆−1
µd1 0´+³0⋆−1
ϵdT
1
−⋆−1
µd1 0´2
2! +³0⋆−1
ϵdT
1
−⋆−1
µd1 0´3
3! +³0⋆−1
ϵdT
1
−⋆−1
µd1 0´4
4! +...
=Ãcosµ³⋆−1
ϵdT
1⋆−1
µd1´1
2 ¶⋆−1
ϵdT
1³⋆−1
µd1⋆−1
ϵdT
1´−1
2sinµ³⋆−1
µd1⋆−1
ϵdT
1´1
2 ¶
−⋆−1
µd1³⋆−1
ϵdT
1⋆−1
µd1´−1
2sinµ³⋆−1
ϵdT
1⋆−1
µd1´1
2 ¶cosµ³⋆−1
µd1⋆−1
ϵdT
1´1
2 ¶!. (18)
The de i a i e o he o al ene gy
d
d E(E,H)=ET⋆ϵ∂0E+HT⋆µ∂0H=−(ETJE+HTJH) (19)
is ob ained by mul iplying Equa ion (15) om he le by (ETHT). The igh -hand side o Equa-
ion (19) includes he sou ce unc ions se on he abso bing bounda y o laye , such as he Sil e –
Mülle bounda y condi ion [47] o pe ec ly ma ched laye (PML) [48]. By assuming hem o be o
he o ms JE=⋆σEEand JH=⋆σHH, whe e σEand σHp esen he elec ic and magne ic con-
duc i i ies, espec i ely, we ge (d/d )E(E,H)= −(ET⋆σEE+HT⋆σHH)≤0, implying ha he
ene gy is conse ed o dissipa ed as he wa es a e abso bed om he domain. The o al ene gy
o he elec omagne ic sys em is
E(E,H)=1
2(ET⋆ϵE+HT⋆µH). (20)
3.2. Spa ial ha monic co ec ion
To imp o e he accu acy o he simula ions in ol ing ime-ha monic wa es wi h he ime-
dependence o he o m eiω , whe e ωis he angula equency and i is he imagina y uni , we
o mula e spa ially ha monic co ec ions o he disc e e Hodge ope a o s. Since he elec omag-
ne ic wa e speed is c=1/pµε, we can p esen ime as dis ance mul iplied by pµε. F om his
basis, he concep o spa ial co ec ion is based on ob aining he componen s o Eand Hin Equa-
ion (7) by in eg a ing o e ˆ
Eeiω and ˆ
Heiω , wi h complex- alued ˆ
Eand ˆ
H, ins ead o in eg a ing
o e e
Eand e
H. Respec i ely, in Equa ions (8) we in eg a e o e ϵiˆ
Eeiω and µjˆ
Heiω , ins ead o e
D
and e
B, o ob ain he componen s o Dand B. Acco dingly, by minimizing he squa ed no m o
e o s based on he cons i u i e equa ions, we ind spa ially ha monic disc e e Hodge s a ope -
a o s ⋆spa
ϵand ⋆spa
µ o sa is y he cons i u i e equa ions. These a e diagonal ma ices wi h he
diagonal componen s
⋆spa
ϵi=⋆ϵiκi,i=1,...,ne,⋆spa
µj=⋆µjκ∗
j,j=1,...,m , (21)
whe e he spa ial co ec ion e ms, κiand κ∗
j, a e
κi=

1−κ ∗
5+κ2
∗
56
1−κ ∗
10 −κe
120 +κ2
∗
280 +κ ∗κe
1680 +κ2
e
22,400


,κ∗
j=

1−κ
5+κ2
56
1−κ
10 −κe∗
120 +κ2
280 +κ κe∗
1680 +κ2
e∗
22,400


,
wi h κe=ω2ϵµ|ei|2,κ ∗=ω2ϵµ 2
∗
i,κe∗=ω2ϵµ|e∗
j|2, and κ =ω2ϵµ 2
jand he squa ed adii o
he aces ∗
iand j, 2
∗
iand 2
j, espec i ely (see [14,24]). Respec i ely,
⋆spa
σEi=⋆σEiκi,i=1,...,ne,⋆spa
σHj=⋆σHjκ∗
j,j=1,...,m . (22)
The la ge he g id elemen s, he mo e ad an ages a e gained by eplacing ⋆ϵby ⋆spa
ϵ,⋆µby
⋆spa
µ,⋆σEby ⋆spa
σE, and ⋆σHby ⋆spa
σHin ime-ha monic simula ions.
14 Sanna Mönkölä e al.
Figu e 2. Rela i e e o o he Muelle ma ix, in eg a ed o e all sca e ing di ec ions.
bounda y o he PML a he adius 2.7. To epo he accu acy o he me hod, he nea - ield so-
lu ion is ans e ed o a - ield sca e ing esul s by applying a nea - ield o a - ield ans o ma-
ion [51,52]. The a - ield sca e ing da a a e applied o p oduce he Muelle ma ix [53] cha ac-
e izing how a ma e ial in e ac s wi h elec omagne ic wa es and p o iding in o ma ion abou
sca e ing in ensi ies and pola iza ion in all sca e ing di ec ions. The accu acy is e alua ed by
compa ing he Muelle ma ix solu ion wi h he analy ical Mie sca e ing solu ion [54]. Figu e 2
illus a es he ela i e e o o he Muelle ma ices compa ed o he exac Mie solu ion o each
g id ype wi h i e di e en mesh esolu ions (numbe s o elemen s pe wa eleng h). We see ha
he ha monic co ec ion imp o es he accu acy o he me hod wi h all he g id ypes. The mos
accu a e esul s, ob ained wi h he C15- and Z-g ids and ha monic co ec ion wi h espec o
space and ime, a e almos an o de o magni ude mo e accu a e han he leas accu a e esul s
ob ained wi h he cubical g id and con en ional Yee’s disc e e Hodge ope a o .
5.2. Non-con ex obs acle
In he second es , we use he S an o d bunny [55] as a non-con ex obs acle (see Figu e 3). Inside
he bunny, we se ϵ=4 and µ=1, while he media su ounding he bunny is modeled wi h
ϵ=1 and µ=1. The bunny, wi h heigh 6.173 ( om ea ip o he g ound), leng h 5.265, and
dep h 4.782, is cen e ed in a ec angula p ism o heigh 9.453, leng h 9.508, and dep h 8.107.
The ou e mos laye , o hickness 1.5, o he ec angula p ism ea u es he PML condi ion o
Equa ion (57). In he in e io o he obs acle, we cons uc ed o each g id ype a se o g ids wi h
a ying esolu ion. Ou side he obs acle, we used a egula g id wi h an edge leng h o one en h
o he wa eleng h. As an inciden wa e, we used a ci cula ly pola ized plane wa e p opaga ing in
he di ec ion o he posi i e x1-axis.
The simula ion esul is illus a ed in Figu e 3. The ed, g een, and blue componen s in he
illus a ion p esen he x1,x2, and x3componen s o he elec ic ield, espec i ely. The e can
be seen a li le o e i e sca e ing wa es in he ho izon al di ec ion inside he bunny, which
is in good ag eemen wi h he wa eleng h and he bunny’s wid h. To assess he accu acy, he
e e ence solu ion is compu ed wi h he BCC g id wi h 40 elemen s pe wa eleng h, and he
esul s o he elec ic and magne ic wa e componen s compa ed wi h he e e ence solu ion a e
shown in Figu e 4. Again, we obse e a signi ican di e ence be ween he e o s compu ed wi h
he con en ional Yee’s disc e e Hodge ope a o and wi h he ha monic Hodge ope a o . When he
ha monic Hodge ope a o is used, he cubical g id is he mos inaccu a e, bu he e a e only sligh
di e ences in accu acy be ween he o he g id ypes.
In he las examples, we used he BCC ype bunny mesh wi h 50,190 nodes, 269,621 edges,
and 403,215 aces o compa e he con ol me hod wi h he asymp o ic app oach and o conside

Sanna Mönkölä e al. 15
Figu e 3. The S an o d bunny objec is illus a ed on he le -hand side. The x1–x3plane
c oss-sec ion o he in e io sca e ed ield is illus a ed on he igh -hand side.
Figu e 4. Rela i e e o o he in e io sca e ed ields.
he numbe o i e a ions equi ed by he con ol me hod wi h inc easing angula equency. In
hese es s, we unca ed he bunny by applying he Sil e –Mülle bounda y condi ion wi hou
addi ional abso bing laye s. We ound ha despi e each i e a ion o he con ol me hod in ol es
sol ing bo h he s a e and adjoin s a e equa ions, i emains mo e e icien han he asymp o ic
app oach (see Figu e 5). This is because he numbe o i e a ions equi ed by he con ol me hod
is less han hal o he numbe o ime pe iods needed o he asymp o ic app oach o achie e he
accu acy le el o he s opping c i e ion o he con ol me hod. Figu e 6 shows he con e gence o
he con ol me hod a ou di e en angula equencies. The numbe o i e a ions inc eases wi h
he angula equency.
6. Conclusion
We p esen ed a ou -dimensional o igin o he elec omagne ic wa e p oblem and de i ed om
i a di e en ial o m o mula ion whe e he spa ial and empo al a iables a e sepa a ed. We
comple ed he disc e iza ion in space by he disc e e ex e io calculus. Since ou main ocus was
on ime-ha monic p oblems, o imp o e he accu acy, we applied an exac ime-s epping scheme
and a ha monic co ec ion on he Hodge ope a o ma ices used a he disc e e le el. In p inciple,
we used a ansien wa e sol e wi h space and ime disc e iza ion adap ed o ime-pe iodic
16 Sanna Mönkölä e al.
Figu e 5. Compa ison o CPU usage be ween he exac con ollabili y me hod and he
asymp o ic app oach a a ious angula equencies.
Figu e 6. Numbe o i e a ions o he exac con ollabili y me hod a a ious angula
equencies.
p oblems. We u he accele a ed he con e gence o he ime-ha monic solu ion by using he
exac con ollabili y app oach ealized by he conjuga e g adien me hod. The ene gy no m is
a weigh ed L2-no m, and we minimize he disc e e quad a ic unc ional spanned by a diagonal
mass ma ix. Thus, he conjuga e g adien algo i hm ope a es in an L2- ype Hilbe space, and
no p econdi ioning is needed. The nume ical esul s demons a e he accu acy imp o emen s o
he ha monic co ec ions. They also show he capabili y o he exac con ollabili y me hod wi h
he chosen disc e iza ion s a egies in non-con ex domains.
Decla a ion o in e es s
The au ho s do no wo k o , ad ise, own sha es in, o ecei e unds om any o ganiza ion
ha could bene i om his a icle, and ha e decla ed no a ilia ions o he han hei esea ch
o ganiza ions.
Sanna Mönkölä e al. 17
Dedica ion
The manusc ip was w i en h ough con ibu ions o all au ho s. All au ho s ha e gi en app o al
o he inal e sion o he manusc ip .
Acknowledgmen s
This p ojec has been pa ially unded by he Academy o Finland, g an s 259925 and 260076. The
au ho s app ecia e wo anonymous e iewe s o hei insigh ul and cons uc i e ema ks.
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