darwin and higher order approximations to maxwell's equations in r3

Darwin and higher order approximationsto Maxwell’s equations in R3

Sebastian BauerUniversitat Duisburg-Essen

in close collaboration with the Maxwell group around

Dirk PaulyUniversitat Duisburg-Essen

Special Semester on Computational Methods in Science and EngineeringRICAM, October 20, 2016

Sebastian Bauer Darwin approximation Universitat Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equationsElectro-and magnetostatics

div E =ρ

ε0rotB = µ0j

rotE = 0 divB = 0

Faraday’s law of induction, no charge conservation, Eddy current model

div E =ρ

ε0rotB = µ0j

∂tB + rotE = 0 divB = 0

Maxwell’s displacement current, charge conservation, Lorentz invariance

div E =ρ

ε0− 1

c2∂tE + rotB = µ0j



Another system with charge conservation but ellipticequations

Maxwell’s equations

div E =ρ

ε0− 1

c2∂tE + rotB = µ0j


Darwin equations E = EL + ET with rotEL = 0 and div ET = 0

div EL =ρ

ε0− 1

c2∂tE

L + rotB = µ0j ∂tB + rotET = 0

rotEL = 0 divB = 0 div ET = 0

charge conservation, three elliptic equations which can be solved successively


Problems/Questions and Outline of the talk

Questions

Dimensional analysis: In which situations is the Darwin system a reasonableapproximation? What are lower order and what are higher orderapproximations?

solution theory for all occuring problems

rigorous estimates for the error between solutions of approximate equationsand solutions of Maxwell’s equations

Outline of the talk

dimensional analysis and asymptotic expansion

bounded domains

exterior domains


In which situations is the approximation reasonable? –dimensionless equations

x characteristic length-scale of the charge and current distributions

t characteristic time-scale, in which a charge moves over a distant x , slowtime-scale

ρ characteristic charge density

v = xt characteristic velocity of the problem

x = xx ′, t = tt ′, E = EE ′, B = BB ′, ρ = ρρ′, j = j j ′, E ′(t ′) = E(tt′)

E...

Maxwell’s dimensionless equations

ε0E

x %div′ E ′ = %′

v E

c2B∂t′ E

′ − rot′ B ′ = −µ0j x

Bj ′

v B

E∂t′ B

′ + rot′ E ′ = 0 div′ B ′ = 0

charge conservation%v

j∂t′ %

′ + div′ j ′ = 0


units and dimensionless equations

Degond, Raviart (’92):

E = x ρε0, j = c ρ, B = x ρ

cε0and η = v

c leads to

div E = ρ −η ∂tE + rotB = j

η ∂tB + rotE = 0 divB = 0

together with charge conservation η ∂tρ+ div j = 0.

Schaeffer (’86), plasma physics with Vlasov matter

E = x ρε0, j = v ρ, B = x ρ

cε0and η = v

c leads to

div E = ρ −η ∂tE + rotB = ηj

η ∂tB + rotE = 0 divB = 0

together with charge conservation 1 ∂tρ+ div j = 0.

Assumption: η 1


Formal expansion in powers of η and equations in theorders of η

div Eη = ρη −η ∂tEη + rotBη = ηjη

η ∂tBη + rotEη = 0 divBη = 0

Ansatz: Eη = E 0 + ηE 1 + η2E 2 + . . . , Bη = B0 + ηB1 + η2B2 + . . .

For simplicity: ρη = ρ0, jη = j0 with ∂tρ0 + div j0 = 0

resulting equations (for the plasma scaling)

O(η0)

div E 0 = ρ0, rotB0 = 0rotE 0 = 0, divB0 = 0

O(η1)

div E 1 = 0, rotB1 = j0 + ∂tE0

rotE 1 = − ∂tB0, divB1 = 0,

O(η2)

div E 2 = 0, rotB2 = ∂tE1,

rotE 2 = − ∂tB1, divB2 = 0,

O(ηk)

div E k = 0, rotBk = ∂tEk−1,

rotE k = − ∂tBk−1, divBk = 0,


Comparsion with eddy current and Darwin, plasma case

We can consistently set : E 1 = E 2k−1 = 0 and B0 = B2k = 0

first order : Set E = E 0 + ηE 1 = E 0 and B = B0 + ηB1 = ηB1

div E = ρ0 rotB = j0

η∂tB + rotE = 0 divB = 0

second order: Set EL = E 0, ET = η2E 2, and B = ηB1, then

div EL = ρ0 rotB = j0 + η ∂tEL rotET = −η ∂tB



Formal expansion in powers of η and equations in theorders of η, Degond Raviart scaling

div Eη = ρη −η ∂tEη + rotBη = jη

η ∂tBη + rotEη = 0 divBη = 0

Ansatz: Eη = E 0 + ηE 1 + η2E 2 + . . . , Bη = B0 + ηB1 + η2B2 + . . .

For simplicity: ρη = ρ0, jη = j0 + ηj1 .

resulting equations

O(η0)

div E 0 = ρ0, rotB0 = j0

rotE 0 = 0, divB0 = 0

O(η1)

div E 1 = 0, rotB1 = j1 + ∂tE0

rotE 1 = − ∂tB0, divB1 = 0,

O(η2)

div E 2 = 0, rotB2 = ∂tE1,

rotE 2 = − ∂tB1, divB2 = 0,

O(ηk)

div E k = 0, rotBk = ∂tEk−1,

rotE k = − ∂tBk−1, divBk = 0,


Comparsion with eddy current and Darwin, Degond Raviartscaling

zeroth order: quasielectrostatic and quasimagnetostatic, div j0 = 0

div E 0 = ρ0 rotB0 = j0

rotE 0 = 0 divB0 = 0

second order:E = E 0 + ηE 1 + η2E 2, EL = E 0, ET = ηE 1 + η2E 2

B = B0 + ηB1 and j = j0 + ηj1

div EL = ρ0 rotB = j0 + η ∂tEL rotET = −η ∂tB



Solution theory

Maxwell’s time-dependent equations: L2 setting, selfadjoint operator, spectralcalculus or halfgroup theory or Picard’s theorem, independently of thedomain, very flexible.

Iterated rot-div systems. Solution of the previous step enters as source term.


the general L2-setting for rot, div and grad

rot :C∞(Ω) ⊂ L2 → L2 , R(Ω) = D(rot∗) = H(curl,Ω)

rot = rot∗ : R(Ω) ⊂ L2 → L2 ,R (Ω) = D(rot∗) = E ∈ R(Ω) |E ∧ ν = 0

rot= rot∗ :

R (Ω) ⊂ L2 → L2 ,

rot = rot∗∗ = rot and

rot∗ = rot

L2-decomposition

L2 = rotR⊕R0 = rotR⊕

R0

In the same manner

D = H(div, Ω) = D(grad∗)D = D(grad∗) = E ∈ D |E · ν = 0

H1 = D(div∗) H1 = D(

div ∗)

L2 decompositions

L2 = gradH1 ⊕D0 = grad H1⊕

D0

L2 = divD = divD ⊕ Lin 1


L2-decompositions in bounded domainsLet Ω ⊂ R3 be a bounded domain. The following embeddings are compact, if theboundary is suffenciently regular (weakly Lipschitz is enough).

R ∩D → L2 , R∩

D→ L2

If these embeddings are compact we can skip the bars:

L2 =

D0︷︸︸︷

rotR ⊕HN ⊕ grad H1 = rotR⊕

R0︷︸︸︷

HD ⊕ gradH1

L2 = divD = divD ⊕H1

0

Dirichlet fields HD =R0 ∩D0 and Neumann fields HN = R0∩

D0

refinement of the decomposition

L2 = rot( R ∩D0

)⊕HN ⊕ grad H1 = rot

(R∩

D0

)⊕HD ⊕ grad

H1

L2 = div(D∩

R0

)= div

( D ∩R0

)⊕ Lin 1


rot-div-problems in bounded domains

L2(Ω)3 = rot( R ∩D0

)⊕HN ⊕ grad H1 = rot

(R∩

D0

)⊕HD ⊕ grad

H1

L2(Ω) = div(D∩

R0

)= div

( D ∩R0

)⊕ Lin 1

The problems rotE = Fdiv E = fE ∧ ν = 0

E ⊥ HD

and

rotB = GdivB = gB · ν = 0

B ⊥ HN

are uniquely solvable iff F ∈D0, F ⊥ HN , G ∈ D0, G ⊥ HD and

∫g dx = 0.


Comparison of the asymptotic expansion with the fullsolution

η ∂te − rot b = j η ∂tb + rot e = k

||e(t)||2L2 + ||b(t)||2L2 =: w2(t) ||j ||2L2 + ||k ||2L2 =: m2(t)

η

2

d

dt

∫Ω

|e|2 + |b|2 dx +

∫Ω

(− rot b · e + rot e · b) dx =

=

∫Ω

(j · e + k · e) dx

w2(t) ≤ w2(0) +2

η

∫ t

0

w(s)m(s) ds

w(t) ≤ w(0) +2

η

∫ t

0

m(s) ds


e := Eη −k+1∑j=0

ηkE k and b := Bη −k+1∑j=0

Bk , then

η ∂te − rot b = −ηk+2 ∂tEk+1 and η ∂tb + rot e = −ηk+2 ∂tB

k+1 .

w(t) ≤ w(0) + 2ηk+1

∫ t

0

(|| ∂tE k+1(s)||2 + || ∂tBk+1(s)||2

)1/2ds


Theorem (Degond, Raviart 1992)

∣∣∣∣∣∣∣∣∣∣∣∣Eη(t)−

∑j=0

ηkE k(t)

∣∣∣∣∣∣∣∣∣∣∣∣L2

≤ ηk+1∣∣∣∣E k+1(s)

∣∣∣∣L2 + ηk+1

∫ t

0

mk+1(s) ds

∣∣∣∣∣∣∣∣∣∣∣∣Bη(t)−

∑j=0

ηkBk(t)

∣∣∣∣∣∣∣∣∣∣∣∣L2

≤ ηk+1∣∣∣∣Bk+1(s)

∣∣∣∣L2 + ηk+1

∫ t

0

mk+1(s) ds

if the initial data Eη0 and Bη0 are suitable matched and j0 and j1 fullfill certaininitial conditions.


Comparison of the asymptotic expansion with the Darwinmodell

Theorem (Degond, Raviart, 1992)

Let ED = EL + ET and BD be the solution of the dimensionless Darwin modell,then

EL = E0,ET = ηE 1 + η2E 2,BD = B0 + ηB1

and ∣∣∣∣Eη(t)− ED(t)∣∣∣∣L2 ≤ 2η3

∣∣∣∣E 3(s)∣∣∣∣L2 + η3

∫ t

0

m3(s) ds

∣∣∣∣Bη(t)− BD∣∣∣∣L2 ≤ 2η2

∣∣∣∣B2(s)∣∣∣∣L2 + η2

∫ t

0

m2(s) ds

if the initial data Eη0 and Bη0 is suitable matched.

different boundary conditions are studied in Raviart, Sonnendrucker 94 and 96

finite element convergence by Ciarlet and Zou, 97


Asymptotic expansions in an exterior domains

Li and Ying 2003, Liao and Ying 2008, Fang and Ying 2008, givingL2-bounds of the error

Approximations to solutions of Vlasov-Maxwell system: phase-spacedistribution f (t, x , v), (x , v) ∈ R3 × R3

∂t f + v · ∇x f ± (E + 1/cv ∧ B) · f = 0

ρ(t, x) = ±∫

f (t, x , v) dv j(t, x) = ±∫

v f (t, x , v) dv

Schaeffer ’86, B and Kunze ’05, B and Kunze and Rein and Rendell ’06, Band Kunze ’06

low-frequency asysmptotics for exterior domains in accustics: Weck andWitsch, series of papers ’90 - ’93

low-frequency asymptotics in linear elasticity: Weck and Witsch ’94, ’97 and’97

low-frequency asymptotics for Maxwell: Pauly, ’06, ’07, ’08


Asymptotic expansions in an exterior domains,

rotR 6= rotR, divD 6= divD, grad H1 6= grad H1

concept of polynomially weighted L2-Sobolev spaces: w(x) = (1 + |x |2)1/2

u ∈ Hls ⇔ u ∈ Hl

loc and w s+|α| ∂α u ∈ L2 for all 0 ≤ |α| ≤ l

E ∈ Rs ⇔ E ∈ Rloc and E ∈ L2s , rotE ∈ L2

s+1

Poincare type estimates, Picard ’82

||u||L2−1≤ C ||grad u||L2 and ||E ||L2

−1≤ C (||rotE ||L2 + ||div E ||L2 )

rotR−1 = rotR, divD−1 = divD and grad H1−1 = grad H1

decomposition L2 = rotR−1 ⊕ grad H1−1 and L2 = divD−1, but

new problem: the potentials are not L2 and we can’t iterate


weighted L2-decompositions

McOwen, 1979: ∆s−2 : H2s−2 −→ L2

s is a Fredholm-operator iff s ∈ R \ Iwith I = 1

2 + Z. In this case ∆s−2 is injective if s > −3/2 and

Im (∆s−2) =

u ∈ L2s | 〈u, p〉 = 0 for all p ∈

<s−3/2⋃n=0

Hn

=: Xs

where Hn is the 2n + 1 dimensional space of harmonic polynoms which arehomogenous of degree n.

Generalizing to vector fields with −∆E = (rot rot− grad div)E

Xs ⊂ rotRs−1 + divDs−1

Goal: exact characterization of rotRs−1 + divDs−1

calculus for homogenous potential vectorfields in spherical co-ordinates,(developed by Weck, Witsch 1994 for differential forms of rank q)


spherical harmonics expansion of harmonic functions andpotential vector fields

spherical harmonics Y nm = Ym

n (θ, ϕ) give an complete L2 ONB ofEigenfunctions of the Beltrami operator Div Grad on the sphere S2:

(Div Grad +n(n + 1))Ymn = 0 for all n = 0, 1, 2, . . . , −n ≤ m ≤ n

pn,m := rnYmn , −n ≤ m ≤ n basis of homogenous harmonic polynoms of

degree n.

Umn = GradYm

n , Vmn = ν ∧ Um

n , n = 1, 2, . . . ,−n ≤ m ≤ n gives a completeL2-ONB of tangential vector fields on the sphere S2, see e.g. Colton, Kress

homogenous potential vector fields in spherical co-ordinates

Hn = P1n ⊕ P2

n ⊕ P3n ⊕ P4

n

e.g. P3n+1 = Lin

P3n+1,m = −

√n+1n rn+1Ym

n er + rn+1Umn − n ≤ m ≤ n

and P1

1 = LinP1

1,0 = rY 00

and void if n 6= 1.


Fine structure and decomposition of L2s

P3n+1

rot−→ P2n

rot−→ P4n−1

rot−→ 0 bijectiv for all n = 1, 2, . . .

P3n+1

div−→ Lin pn,mgrad−→ P4

n−1div−→ 0 bijectiv for all n = 1, 2, . . .

Theorem (Weck, Witsch 1994 for q-forms in RN , formulation forvector-fields)

L2s decomposition: Let s > −3/2 and s 6∈ 1

2 + Z, then

L2s (R3)3 = D0,s ⊕R0,s ⊕ Ss = rotRs−1 ⊕ grad H1

s−1 ⊕ Ss ,L2s (R3) = divDs−1 ⊕ Ts

where Ss is dual to P4<s−3/2 and Ts is dual to Lin pn,m<s−3/2 w.r.t the L2

s − L2−s

duality given by 〈·, ·〉L2 .

Pauly 2008 L2 decomposition of q-forms in exterior domains withinhomogenous and anisotropic media


Mapping properties of rot and div in weighted L2-spaces

Theorem (Weck, Witsch 1994)

Let s > −3/2 and s 6∈ 12 + Z

rots−1 : Rs−1 ∩ D0,s−1 → D0,s and divs−1 : Ds−1 ∩R0,s−1 → L2s

are injectiv Fredholm-operator with

Im (rots−1) =F ∈ D0,s | 〈F ,P〉 = 0 for all P ∈ P2

<s−3/2

Im (divs−1) =

f ∈ L2

2 | 〈f , p〉 = 0 for all p ∈ Lin pn,m , n < s − 3/2

Pauly 2007 for q-forms in exterior domains with inhomogenous andanisotropic media


Iteration scheme, zeroth order


<s−3/2

Im (divs−1) =

f ∈ L2


Assumptions on the data: ρη = ρ0, jη = ηj1 with ∂tρ0 + div j1 = 0

O(η0)

rotE 0 = 0, div E 0 = ρ0,rotB0 = 0, divB0 = 0.

E 0 ∈ L2 iff ρ0 ∈ L21, B0 = 0.


Iteration scheme, first order


<s−3/2

Im (divs−1) =

f ∈ L2


O(η1)

rotE 1 = 0, div E 1 = 0,rotB1 = j1 + ∂tE

0, divB1 = 0 .

E 1 = 0

B1 ∈ L2 iff j1 + ∂tE0 ∈ D0,1 (and

⟨∂t j

1 + ∂tE0,P2

n,m

⟩= 0 for all

n < 2− 3/2)I ∂tE

0 ∈ L21 iff ∂tρ ∈ L2

2 and⟨∂tρ

0, pn,m⟩

= 0 for all n < 2− 3/2, that meanscharge conservation.


Iteration scheme second order


<s−3/2

Im (divs−1) =

f ∈ L2


O(η2)

div E 2 = 0 rotB2 = 0rotE 2 = − ∂tB1 divB2 = 0 .

E 2 ∈ L2 iff ∂tB1 ∈ D0,1 (and

⟨∂tB

1,P2n,m

⟩= 0 for all n < 1− 3/2)

I ∂tB1 ∈ D0,1 iff ∂t j

1 + ∂2t E

0 ∈ D0,2

I ∂2t E

0 ∈ D2 iff ∂2t ρ

0 ∈ L23 and

⟨∂2t ρ

0, pn,m⟩

= 0 for all n < 3− 3/2

B2 = 0


space of regular convergence

In order to estimate the error of the approximation in third order we need theapproximation in third order:

B3 ∈ L2 iff ∂3t ρ

0 ∈ L24 and ∂3

t

⟨ρ0, pn,m

⟩= 0 for all n < 4− 3/2

Theorem (Space of Regular Convergence, B. 2016?, prepreprint)

The Darwin order approximation is well defined in L2 iff the the second timederivative of the dipole contribution vanishes:∫

x ∂2t ρ

0 dx = 0

It is an approximation of order O(η3) if in addition the third time derivative of thequadrupole moment vanishes.


Outlook

What happens if sources are not in the space of regular convergence?I decomposition of the sources in a regular part and a radiating partI solve Maxwell’s equations for the radiating part and expand for the regular

part orI use correction operators in the asymptotic expansion

general initial conditions, asymptotic matching

non-trivial topologies

(linear) media

different boundary conditions

Thank you for your attention


darwin and higher order approximations to maxwell's equations in r3

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