Darwin and higher order approximations to Maxwells equations in R 3 - - PowerPoint PPT Presentation

Darwin and higher order approximations to Maxwell’s equations in R3 Sebastian Bauer

Universit¨ at Duisburg-Essen in close collaboration with the Maxwell group around

Dirk Pauly

Universit¨ at Duisburg-Essen Special Semester on Computational Methods in Science and Engineering RICAM, October 20, 2016

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations

Electro-and magnetostatics div E = ρ ε0 rot B = µ0j rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ ε0 rot B = µ0j ∂tB + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ ε0 − 1 c2 ∂tE + rot B = µ0j ∂tB + rot E = 0 div B = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations

Electro-and magnetostatics div E = ρ ε0 rot B = µ0j rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ ε0 rot B = µ0j ∂tB + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ ε0 − 1 c2 ∂tE + rot B = µ0j ∂tB + rot E = 0 div B = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Historical development of Maxwell’s equations

Electro-and magnetostatics div E = ρ ε0 rot B = µ0j rot E = 0 div B = 0 Faraday’s law of induction, no charge conservation, Eddy current model div E = ρ ε0 rot B = µ0j ∂tB + rot E = 0 div B = 0 Maxwell’s displacement current, charge conservation, Lorentz invariance div E = ρ ε0 − 1 c2 ∂tE + rot B = µ0j ∂tB + rot E = 0 div B = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Another system with charge conservation but elliptic equations

Maxwell’s equations div E = ρ ε0 − 1 c2 ∂tE + rot B = µ0j ∂tB + rot E = 0 div B = 0 Darwin equations E = E L + E T with rot E L = 0 and div E T = 0 div E L = ρ ε0 − 1 c2 ∂tE L + rot B = µ0j ∂tB + rot E T = 0 rot E L = 0 div B = 0 div E T = 0 charge conservation, three elliptic equations which can be solved successively

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Problems/Questions and Outline of the talk

Questions Dimensional analysis: In which situations is the Darwin system a reasonable approximation? What are lower order and what are higher order approximations? solution theory for all occuring problems rigorous estimates for the error between solutions of approximate equations and solutions of Maxwell’s equations Outline of the talk dimensional analysis and asymptotic expansion bounded domains exterior domains

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

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, slow time-scale ¯ ρ characteristic charge density ¯ v = ¯

x ¯ t characteristic velocity of the problem

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

tt′) ¯ E

... Maxwell’s dimensionless equations ε0 ¯ E ¯ x ¯ ̺ div′ E ′ = ̺′ ¯ v ¯ E c2 ¯ B ∂t′ E ′ − rot′ B′ = −µ0 ¯ j ¯ x ¯ B j′ ¯ v ¯ B ¯ E ∂t′ B′ + rot′ E ′ = 0 div′ B′ = 0 charge conservation ¯ ̺¯ v ¯ j ∂t′ ̺′ + div′ j′ = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

units and dimensionless equations

Degond, Raviart (’92): ¯ E = ¯

x ¯ ρ ε0 ,

¯ j = c ¯ ρ, ¯ B =

¯ x ¯ ρ cε0 and η = ¯ v c leads to

div E = ρ −η ∂tE + rot B = j η ∂tB + rot E = 0 div B = 0 together with charge conservation η ∂tρ + div j = 0. Schaeffer (’86), plasma physics with Vlasov matter ¯ E = ¯

x ¯ ρ ε0 ,

¯ j = ¯ v ¯ ρ, ¯ B =

¯ x ¯ ρ cε0 and η = ¯ v c leads to

div E = ρ −η ∂tE + rot B = ηj η ∂tB + rot E = 0 div B = 0 together with charge conservation 1 ∂tρ + div j = 0. Assumption: η ≪ 1

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Formal expansion in powers of η and equations in the

• rders of η

div E η = ρη −η ∂tE η + rot Bη = ηjη η ∂tBη + rot E η = 0 div Bη = 0 Ansatz: E η = 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, rot B0 = 0 rot E 0 = 0, div B0 = 0 O

• η1

div E 1 = 0, rot B1 = j0 + ∂tE 0 rot E 1 = − ∂tB0, div B1 = 0, O

• η2

div E 2 = 0, rot B2 = ∂tE 1, rot E 2 = − ∂tB1, div B2 = 0, O

• ηk

div E k = 0, rot Bk = ∂tE k−1, rot E k = − ∂tBk−1, div Bk = 0,

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

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 rot B = j0 η∂tB + rot E = 0 div B = 0 second order: Set E L = E 0, E T = η2E 2, and B = ηB1, then div E L = ρ0 rot B = j0 + η ∂tE L rot E T = −η ∂tB rot E L = 0 div B = 0 div E T = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Formal expansion in powers of η and equations in the

• rders of η, Degond Raviart scaling

div E η = ρη −η ∂tE η + rot Bη = jη η ∂tBη + rot E η = 0 div Bη = 0 Ansatz: E η = 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, rot B0 = j0 rot E 0 = 0, div B0 = 0 O

• η1

div E 1 = 0, rot B1 = j1 + ∂tE 0 rot E 1 = − ∂tB0, div B1 = 0, O

• η2

div E 2 = 0, rot B2 = ∂tE 1, rot E 2 = − ∂tB1, div B2 = 0, O

• ηk

div E k = 0, rot Bk = ∂tE k−1, rot E k = − ∂tBk−1, div Bk = 0,

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Comparsion with eddy current and Darwin, Degond Raviart scaling

zeroth order: quasielectrostatic and quasimagnetostatic, div j0 = 0 div E 0 = ρ0 rot B0 = j0 rot E 0 = 0 div B0 = 0 second order: E = E 0 + ηE 1 + η2E 2, E L = E 0, E T = ηE 1 + η2E 2 B = B0 + ηB1 and j = j0 + ηj1 div E L = ρ0 rot B = j0 + η ∂tE L rot E T = −η ∂tB rot E L = 0 div B = 0 div E T = 0

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Solution theory

Maxwell’s time-dependent equations: L2 setting, selfadjoint operator, spectral calculus or halfgroup theory or Picard’s theorem, independently of the domain, very flexible. Iterated rot-div systems. Solution of the previous step enters as source term.

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

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 = rot

• R ⊕ R0 = rot R⊕
• 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 = grad

• H1 ⊕ D0 = grad H1⊕
• D0

L2 = div D = div

• D ⊕ Lin {1}

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

L2-decompositions in bounded domains

Let Ω ⊂ R3 be a bounded domain. The following embeddings are compact, if the boundary is suffenciently regular (weakly Lipschitz is enough).

• R ∩D ֒

→ L2 , R∩

• D֒

→ L2 If these embeddings are compact we can skip the bars: L2 =

• D0
• rot
• R ⊕HN ⊕ grad H1 = rot R ⊕
• R0
• HD ⊕ grad
• H1

L2 = div D = div

• D ⊕H1

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}

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

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        rot E = F div E = f E ∧ ν = E ⊥ HD and        rot B = G div B = g B · ν = B ⊥ HN are uniquely solvable iff F ∈

• D0, F ⊥ HN, G ∈ D0, G ⊥ HD and
• g dx = 0.

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Comparison of the asymptotic expansion with the full solution

η ∂te − rot b = j η ∂tb + rot e = k | |e(t)| |2

L2 + |

|b(t)| |2

L2 =: w 2(t)

| |j| |2

L2 + |

|k| |2

L2 =: 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 w 2(t) ≤ w 2(0) + 2 η t w(s)m(s) ds w(t) ≤ w(0) + 2 η t m(s) ds

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

e := E η −

k+1

• j=0

ηkE k and b := Bη −

k+1

• j=0

Bk, then η ∂te − rot b = −ηk+2 ∂tE k+1 and η ∂tb + rot e = −ηk+2 ∂tBk+1 . w(t) ≤ w(0) + 2ηk+1 t

• |

| ∂tE k+1(s)| |2 + | | ∂tBk+1(s)| |21/2 ds

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Theorem (Degond, Raviart 1992)

• E η(t) −
• j=0

ηkE k(t)

• L2

≤ ηk+1

• E k+1(s)
• L2 + ηk+1

t mk+1(s) ds

• Bη(t) −
• j=0

ηkBk(t)

• L2

≤ ηk+1

• Bk+1(s)
• L2 + ηk+1

t mk+1(s) ds if the initial data E η

0 and Bη 0 are suitable matched and j0 and j1 fullfill certain

initial conditions.

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Comparison of the asymptotic expansion with the Darwin modell

Theorem (Degond, Raviart, 1992)

Let E D = E L + E T and BD be the solution of the dimensionless Darwin modell, then E L = E0, E T = ηE 1 + η2E 2, BD = B0 + ηB1 and

• E η(t) − E D(t)
• L2

≤ 2η3

• E 3(s)
• L2 + η3

t m3(s) ds

• Bη(t) − BD
• L2

≤ 2η2

• B2(s)
• L2 + η2

t m2(s) ds if the initial data E η

0 and Bη 0 is suitable matched.

different boundary conditions are studied in Raviart, Sonnendr¨ ucker 94 and 96 finite element convergence by Ciarlet and Zou, 97

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Asymptotic expansions in an exterior domains

Li and Ying 2003, Liao and Ying 2008, Fang and Ying 2008, giving L2-bounds of the error Approximations to solutions of Vlasov-Maxwell system: phase-space distribution f (t, x, v), (x, v) ∈ R3 × R3 ∂tf + ˆ v · ∇xf ± (E + 1/c ˆ v ∧ B) · f = 0 ρ(t, x) = ±

• f (t, x, v) dv

j(t, x) = ±

• ˆ

vf (t, x, v) dv Schaeffer ’86, B and Kunze ’05, B and Kunze and Rein and Rendell ’06, B and Kunze ’06 low-frequency asysmptotics for exterior domains in accustics: Weck and Witsch, 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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Asymptotic expansions in an exterior domains

Li and Ying 2003, Liao and Ying 2008, Fang and Ying 2008, giving L2-bounds of the error Approximations to solutions of Vlasov-Maxwell system: phase-space distribution f (t, x, v), (x, v) ∈ R3 × R3 ∂tf + ˆ v · ∇xf ± (E + 1/c ˆ v ∧ B) · f = 0 ρ(t, x) = ±

• f (t, x, v) dv

j(t, x) = ±

• ˆ

vf (t, x, v) dv Schaeffer ’86, B and Kunze ’05, B and Kunze and Rein and Rendell ’06, B and Kunze ’06 low-frequency asysmptotics for exterior domains in accustics: Weck and Witsch, 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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Asymptotic expansions in an exterior domains

Li and Ying 2003, Liao and Ying 2008, Fang and Ying 2008, giving L2-bounds of the error Approximations to solutions of Vlasov-Maxwell system: phase-space distribution f (t, x, v), (x, v) ∈ R3 × R3 ∂tf + ˆ v · ∇xf ± (E + 1/c ˆ v ∧ B) · f = 0 ρ(t, x) = ±

• f (t, x, v) dv

j(t, x) = ±

• ˆ

vf (t, x, v) dv Schaeffer ’86, B and Kunze ’05, B and Kunze and Rein and Rendell ’06, B and Kunze ’06 low-frequency asysmptotics for exterior domains in accustics: Weck and Witsch, 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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Asymptotic expansions in an exterior domains,

rot R = rot R, div D = div D, grad H1 = grad H1 concept of polynomially weighted L2-Sobolev spaces: w(x) = (1 + |x|2)1/2 u ∈ Hl

s

⇔ u ∈ Hl

loc

and w s+|α| ∂α u ∈ L2 for all 0 ≤ |α| ≤ l E ∈ Rs ⇔ E ∈ Rloc and E ∈ L2

s,

rot E ∈ L2

s+1

Poincare type estimates, Picard ’82 | |u| |L2

−1 ≤ C |

|grad u| |L2 and | |E| |L2

−1 ≤ C (|

|rot E| |L2 + | |div E| |L2) rot R−1 = rot R, div D−1 = div D and grad H1

−1 = grad H1

decomposition L2 = rot R−1 ⊕ grad H1

−1 and L2 = div D−1, but

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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

weighted L2-decompositions

McOwen, 1979: ∆s−2 : H2

s−2 −

→ L2

s is a Fredholm-operator iff s ∈ R \ I

with I = 1

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

Im (∆s−2) =   u ∈ L2

s | 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 are homogenous of degree n. Generalizing to vector fields with −∆E = (rot rot − grad div)E Xs ⊂ rot Rs−1 + div Ds−1 Goal: exact characterization of rot Rs−1 + div Ds−1 calculus for homogenous potential vectorfields in spherical co-ordinates, (developed by Weck, Witsch 1994 for differential forms of rank q)

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

spherical harmonics expansion of harmonic functions and potential vector fields

spherical harmonics Y n

m = Y m n (θ, ϕ) give an complete L2 ONB of

Eigenfunctions of the Beltrami operator Div Grad on the sphere S2: (Div Grad +n(n + 1))Y m

n = 0

for all n = 0, 1, 2, . . . , −n ≤ m ≤ n pn,m := r nY m

n , −n ≤ m ≤ n basis of homogenous harmonic polynoms of

degree n. Um

n = Grad Y m n , V m n = ν ∧ Um n , n = 1, 2, . . . , −n ≤ m ≤ n gives a complete

L2-ONB of tangential vector fields on the sphere S2, see e.g. Colton, Kress homogenous potential vector fields in spherical co-ordinates Hn = P1

n ⊕ P2 n ⊕ P3 n ⊕ P4 n

e.g. P3

n+1 = Lin

• P3

n+1,m = −

• n+1

n r n+1Y m n er + r n+1Um n

− n ≤ m ≤ n

• and P1

1 = Lin

• P1

1,0 = rY 0

• and void if n = 1.

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Fine structure and decomposition of L2

s P3

n+1 rot

− → P2

n rot

− → P4

n−1 rot

− → bijectiv for all n = 1, 2, . . . P3

n+1 div

− → Lin {pn,m}

grad

− → P4

n−1 div

− → bijectiv for all n = 1, 2, . . .

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

L2

s decomposition: Let s > −3/2 and s ∈ 1 2 + Z, then

L2

s(R3)3

= D0,s ⊕ R0,s ⊕ Ss = rot Rs−1 ⊕ grad H1

s−1 ⊕ Ss,

L2

s(R3)

= div Ds−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 with inhomogenous and anisotropic media

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Mapping properties of rot and div in weighted L2-spaces

Theorem (Weck, Witsch 1994)

Let s > −3/2 and s ∈ 1

2 + Z

rots−1 : Rs−1 ∩ D0,s−1 → D0,s and divs−1 : Ds−1 ∩ R0,s−1 → L2

s

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 and

anisotropic media

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Iteration scheme, zeroth order

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}

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

O

• η0

rot E 0 = 0, div E 0 = ρ0, rot B0 = 0, div B0 = 0. E 0 ∈ L2 iff ρ0 ∈ L2

1, B0 = 0.

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Iteration scheme, first order

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}

• O
• η1

rot E 1 = 0, div E 1 = 0, rot B1 = j1 + ∂tE 0, div B1 = 0 . E 1 = 0 B1 ∈ L2 iff j1 + ∂tE 0 ∈ D0,1 (and

• ∂tj1 + ∂tE 0, P2

n,m

• = 0 for all

n < 2 − 3/2)

◮ ∂tE 0 ∈ L2

1 iff ∂tρ ∈ L2 2 and

• ∂tρ0, pn,m
• = 0 for all n < 2 − 3/2, that means

charge conservation.

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Iteration scheme second order

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}

• O
• η2

div E 2 = 0 rot B2 = 0 rot E 2 = − ∂tB1 div B2 = 0 . E 2 ∈ L2 iff ∂tB1 ∈ D0,1 (and

• ∂tB1, P2

n,m

• = 0 for all n < 1 − 3/2)

◮ ∂tB1 ∈ D0,1 iff ∂tj1 + ∂2

t E 0 ∈ D0,2

◮ ∂2

t E 0 ∈ D2 iff ∂2 t ρ0 ∈ L2 3 and

• ∂2

t ρ0, pn,m

• = 0 for all n < 3 − 3/2

B2 = 0

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Iteration scheme second order

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}

• O
• η2

div E 2 = 0 rot B2 = 0 rot E 2 = − ∂tB1 div B2 = 0 . E 2 ∈ L2 iff ∂tB1 ∈ D0,1 (and

• ∂tB1, P2

n,m

• = 0 for all n < 1 − 3/2)

◮ ∂tB1 ∈ D0,1 iff ∂tj1 + ∂2

t E 0 ∈ D0,2

◮ ∂2

t E 0 ∈ D2 iff ∂2 t ρ0 ∈ L2 3 and

• ∂2

t ρ0, pn,m

• = 0 for all n < 3 − 3/2

B2 = 0

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Iteration scheme second order

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}

• O
• η2

div E 2 = 0 rot B2 = 0 rot E 2 = − ∂tB1 div B2 = 0 . E 2 ∈ L2 iff ∂tB1 ∈ D0,1 (and

• ∂tB1, P2

n,m

• = 0 for all n < 1 − 3/2)

◮ ∂tB1 ∈ D0,1 iff ∂tj1 + ∂2

t E 0 ∈ D0,2

◮ ∂2

t E 0 ∈ D2 iff ∂2 t ρ0 ∈ L2 3 and

• ∂2

t ρ0, pn,m

• = 0 for all n < 3 − 3/2

B2 = 0

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space of regular convergence

In order to estimate the error of the approximation in third order we need the approximation in third order: B3 ∈ L2 iff ∂3

t ρ0 ∈ L2 4 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 time derivative of the dipole contribution vanishes:

• x ∂2

t ρ0 dx = 0

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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

space of regular convergence

In order to estimate the error of the approximation in third order we need the approximation in third order: B3 ∈ L2 iff ∂3

t ρ0 ∈ L2 4 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 time derivative of the dipole contribution vanishes:

• x ∂2

t ρ0 dx = 0

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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen

Outlook

What happens if sources are not in the space of regular convergence?

◮ decomposition of the sources in a regular part and a radiating part ◮ solve Maxwell’s equations for the radiating part and expand for the regular

part or

◮ 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

Sebastian Bauer Darwin approximation Universit¨ at Duisburg-Essen, Campus Essen