Darwin and higher order approximations to Maxwell s equations in R 3. Sebastian Bauer Universität Duisburg-Essen

Darwin and higher order approximations to Maxwell s equations in R 3 Sebastian Bauer Universität Duisburg-Essen in close collaboration with the Maxwell group around Dirk Pauly Universität Duisburg-Essen Special Semester on Computational Methods in Science and Engineering RICAM, October 20, 2016

Historical development of Maxwell s equations Electro-and magnetostatics div E = ρ ε 0 rot B = µ 0 j rot E = 0 div B = 0 Faraday s law of induction, no charge conservation, Eddy current model div E = ρ ε 0 rot B = µ 0 j t B + rot E = 0 div B = 0 Maxwell s displacement current, charge conservation, Lorentz invariance div E = ρ ε 0 1 c 2 te + rot B = µ 0 j t B + rot E = 0 div B = 0

Another system with charge conservation but elliptic equations Maxwell s equations div E = ρ ε 0 1 c 2 te + rot B = µ 0 j t B + 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 ε 0 1 c 2 te L + rot B = µ 0 j t B + 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

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

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 = ĒE, B = BB, ρ = ρρ, j = jj, E (t ) = E( tt )... Ē Maxwell s dimensionless equations ε 0 Ē x ϱ div E = ϱ vē c 2 B t E rot B = µ 0 j x B j v B Ē t B + rot E = 0 div B = 0 charge conservation ϱ v j t ϱ + div j = 0

units and dimensionless equations Degond, Raviart ( 92): Ē = x ρ ε 0, j = c ρ, B = x ρ cε 0 and η = v c div E = ρ leads to η t E + rot B = j η t B + rot E = 0 div B = 0 together with charge conservation η t ρ + div j = 0. Schaeffer ( 86), plasma physics with Vlasov matter Ē = x ρ x ρ ε 0, j = v ρ, B = cε 0 and η = v c div E = ρ leads to η t E + rot B = ηj η t B + rot E = 0 div B = 0 together with charge conservation 1 t ρ + div j = 0. Assumption: η 1

Formal expansion in powers of η and equations in the orders of η div E η = ρ η η t E η + rot B η = ηj η η t B η + rot E η = 0 div B η = 0 Ansatz: E η = E 0 + ηe 1 + η 2 E 2 +..., B η = B 0 + ηb 1 + η 2 B 2 +... For simplicity: ρ η = ρ 0, j η = j 0 with t ρ 0 + div j 0 = 0 resulting equations (for the plasma scaling) O ( η 0) div E 0 = ρ 0, rot B 0 = 0 rot E 0 = 0, div B 0 = 0 O ( η 1) div E 1 = 0, rot B 1 = j 0 + t E 0 rot E 1 = t B 0, div B 1 = 0, O ( η 2) div E 2 = 0, rot B 2 = t E 1, rot E 2 = t B 1, div B 2 = 0, O ( η k) div E k = 0, rot B k = t E k 1, rot E k = t B k 1, div B k = 0,

Comparsion with eddy current and Darwin, plasma case We can consistently set : E 1 = E 2k 1 = 0 and B 0 = B 2k = 0 first order : Set E = E 0 + ηe 1 = E 0 and B = B 0 + ηb 1 = ηb 1 div E = ρ 0 rot B = j 0 η t B + rot E = 0 div B = 0 second order: Set E L = E 0, E T = η 2 E 2, and B = ηb 1, then div E L = ρ 0 rot B = j 0 + η t E L rot E T = η t B rot E L = 0 div B = 0 div E T = 0

Formal expansion in powers of η and equations in the orders of η, Degond Raviart scaling div E η = ρ η η t E η + rot B η = j η η t B η + rot E η = 0 div B η = 0 Ansatz: E η = E 0 + ηe 1 + η 2 E 2 +..., B η = B 0 + ηb 1 + η 2 B 2 +... For simplicity: ρ η = ρ 0, j η = j 0 + ηj 1. resulting equations O ( η 0) div E 0 = ρ 0, rot B 0 = j 0 rot E 0 = 0, div B 0 = 0 O ( η 1) div E 1 = 0, rot B 1 = j 1 + t E 0 rot E 1 = t B 0, div B 1 = 0, O ( η 2) div E 2 = 0, rot B 2 = t E 1, rot E 2 = t B 1, div B 2 = 0, O ( η k) div E k = 0, rot B k = t E k 1, rot E k = t B k 1, div B k = 0,

Comparsion with eddy current and Darwin, Degond Raviart scaling zeroth order: quasielectrostatic and quasimagnetostatic, div j 0 = 0 div E 0 = ρ 0 rot B 0 = j 0 rot E 0 = 0 div B 0 = 0 second order: E = E 0 + ηe 1 + η 2 E 2, E L = E 0, E T = ηe 1 + η 2 E 2 B = B 0 + ηb 1 and j = j 0 + ηj 1 div E L = ρ 0 rot B = j 0 + η t E L rot E T = η t B rot E L = 0 div B = 0 div E T = 0

Solution theory Maxwell s time-dependent equations: L 2 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.

the general L 2 -setting for rot, div and grad rot : C (Ω) L 2 L 2, R(Ω) = D(rot ) = H(curl, Ω) rot = rot : R(Ω) L 2 L 2, rot= rot : R (Ω) L 2 L 2, L 2 -decomposition In the same manner D = H(div, Ω) = D(grad ) L 2 decompositions H 1 = D(div ) R (Ω) = D(rot ) = {E R(Ω) E ν = 0} rot = rot = rot and L 2 = rot R R 0 = rot R R0 rot = rot D = D(grad ) = {E D E ν = 0} H 1 = D( div ) L 2 = grad H 1 D 0 = grad H 1 D0 L 2 = div D = div D Lin {1}

L 2 -decompositions in bounded domains Let Ω R 3 be a bounded domain. The following embeddings are compact, if the boundary is suffenciently regular (weakly Lipschitz is enough). R D L 2, R D L 2 If these embeddings are compact we can skip the bars: { D0 }} { { R0 }} { L 2 = rot R H N grad H 1 = rot R H D grad H 1 L 2 = div D = div D H 1 0 Dirichlet fields H D = R0 D 0 and Neumann fields H N = R 0 D0 refinement of the decomposition ( R ) L 2 = rot D0 ) L 2 = div ( D R0 ( H N grad H 1 = rot = div ( D R0 ) Lin {1} R D0 ) H D grad H 1

rot-div-problems in bounded domains ( R ) L 2 (Ω) 3 = rot D0 ) L 2 (Ω) = div ( D R0 ( H N grad H 1 = rot = div ( D R0 ) Lin {1} R D0 ) H D grad H 1 The problems rot E = F div E = f E ν = 0 E H D and rot B = G div B = g B ν = 0 B H N are uniquely solvable iff F D0, F H N, G D 0, G H D and g dx = 0.

Comparison of the asymptotic expansion with the full solution η t e rot b = j η t b + rot e = k e(t) 2 L 2 + b(t) 2 L 2 =: w 2 (t) j 2 L 2 + k 2 L 2 =: m2 (t) η d e 2 + b 2 dx + ( rot b e + rot e b) dx = 2 dt Ω Ω = (j e + k e) dx Ω w 2 (t) w 2 (0) + 2 η w(t) w(0) + 2 η t 0 t 0 w(s)m(s) ds m(s) ds

k+1 k+1 e := E η η k E k and b := B η B k, j=0 j=0 then η t e rot b = η k+2 t E k+1 and η t b + rot e = η k+2 t B k+1. t w(t) w(0) + 2η k+1 ( t E k+1 (s) 2 + t B k+1 (s) 2) 1/2 ds 0

Theorem (Degond, Raviart 1992) E η (t) η k E k (t) j=0 Bη (t) η k B k (t) j=0 L 2 L 2 η k+1 E k+1 (s) t L 2 + η k+1 m k+1 (s) ds η k+1 B k+1 (s) t L 2 + η k+1 m k+1 (s) ds 0 0 if the initial data E η 0 and Bη 0 are suitable matched and j 0 and j 1 fullfill certain initial conditions.

Comparison of the asymptotic expansion with the Darwin modell Theorem (Degond, Raviart, 1992) Let E D = E L + E T and B D be the solution of the dimensionless Darwin modell, then E L = E 0, E T = ηe 1 + η 2 E 2, B D = B 0 + ηb 1 and E η (t) E D (t) L 2 2η 3 E 3 (s) L 2 + η 3 t B η (t) B D L 2 2η 2 B 2 (s) L 2 + η 2 t 0 0 m 3 (s) ds m 2 (s) ds if the initial data E η 0 and Bη 0 is suitable matched. different boundary conditions are studied in Raviart, Sonnendrücker 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, giving L 2 -bounds of the error Approximations to solutions of Vlasov-Maxwell system: phase-space distribution f (t, x, v), (x, v) R 3 R 3 t f + ˆv x f ± (E + 1/c ˆv B) f = 0 ρ(t, x) = ± f (t, x, v) dv j(t, x) = ± ˆvf (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

Asymptotic expansions in an exterior domains, rot R rot R, div D div D, grad H 1 grad H 1 concept of polynomially weighted L 2 -Sobolev spaces: w(x) = (1 + x 2 ) 1/2 u H l s u H l loc and w s+ α α u L 2 for all 0 α l E R s E R loc and E L 2 s, rot E L 2 s+1 Poincare type estimates, Picard 82 u L 2 C grad u 1 L 2 and E L 2 C ( rot E 1 L 2 + div E L 2) rot R 1 = rot R, div D 1 = div D and grad H 1 1 = grad H1 decomposition L 2 = rot R 1 grad H 1 1 and L2 = div D 1, but new problem: the potentials are not L 2 and we can t iterate

weighted L 2 -decompositions McOwen, 1979: s 2 : H 2 s 2 L 2 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 H n =: X s where H n 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 X s rot R s 1 + div D s 1 Goal: exact characterization of rot R s 1 + div D s 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 and potential vector fields spherical harmonics Y n m = Y m n (θ, ϕ) give an complete L 2 ONB of Eigenfunctions of the Beltrami operator Div Grad on the sphere S 2 : (Div Grad +n(n + 1))Y m n = 0 for all n = 0, 1, 2,..., n m n p n,m := r n Yn m, n m n basis of homogenous harmonic polynoms of degree n. U m n = Grad Y m n, V m n = ν U m n, n = 1, 2,..., n m n gives a complete L 2 -ONB of tangential vector fields on the sphere S 2, see e.g. Colton, Kress homogenous potential vector fields in spherical co-ordinates H n = Pn 1 Pn 2 Pn 3 Pn 4 { } e.g. Pn+1 3 = Lin Pn+1,m 3 = n+1 n r n+1 Yn m e r + r n+1 Un m n m n and P1 1 = Lin { P1,0 1 = ry } 0 0 and void if n 1.

Fine structure and decomposition of L 2 s P 3 n+1 P 3 n+1 rot Pn 2 div Lin {p n,m } rot grad P 4 n 1 P 4 n 1 rot 0 bijectiv for all n = 1, 2,... div 0 bijectiv for all n = 1, 2,... Theorem (Weck, Witsch 1994 for q-forms in R N, formulation for vector-fields) L 2 s decomposition: Let s > 3/2 and s 1 2 + Z, then L 2 s (R 3 ) 3 = D 0,s R 0,s S s = rot R s 1 grad H 1 s 1 S s, L 2 s (R 3 ) = div D s 1 T s where S s is dual to P 4 <s 3/2 and T s is dual to Lin {p n,m } <s 3/2 w.r.t the L 2 s L 2 s duality given by, L 2. Pauly 2008 L 2 decomposition of q-forms in exterior domains with inhomogenous and anisotropic media

Mapping properties of rot and div in weighted L 2 -spaces Theorem (Weck, Witsch 1994) Let s > 3/2 and s 1 2 + Z rot s 1 : R s 1 D 0,s 1 D 0,s and div s 1 : D s 1 R 0,s 1 L 2 s are injectiv Fredholm-operator with { } Im (rot s 1 ) = F D 0,s F, P = 0 for all P P<s 3/2 2 Im (div s 1 ) = { f L 2 2 f, p = 0 for all p Lin {p n,m, n < s 3/2} } Pauly 2007 for q-forms in exterior domains with inhomogenous and anisotropic media

Iteration scheme, zeroth order { } Im (rot s 1 ) = F D 0,s F, P = 0 for all P P<s 3/2 2 Im (div s 1 ) = { f L 2 2 f, p = 0 for all p Lin {p n,m, n < s 3/2} } Assumptions on the data: ρ η = ρ 0, j η = ηj 1 with t ρ 0 + div j 1 = 0 E 0 L 2 iff ρ 0 L 2 1, B0 = 0. O ( η 0) rot E 0 = 0, div E 0 = ρ 0, rot B 0 = 0, div B 0 = 0.

Iteration scheme, first order { } Im (rot s 1 ) = F D 0,s F, P = 0 for all P P<s 3/2 2 Im (div s 1 ) = { f L 2 2 f, p = 0 for all p Lin {p n,m, n < s 3/2} } O ( η 1) rot E 1 = 0, div E 1 = 0, rot B 1 = j 1 + t E 0, div B 1 = 0. E 1 = 0 B 1 L 2 iff j 1 + t E 0 D 0,1 (and t j 1 + t E 0, Pn,m 2 = 0 for all n < 2 3/2) te 0 L 2 1 iff tρ L 2 2 and tρ 0, p n,m = 0 for all n < 2 3/2, that means charge conservation.

Iteration scheme second order { } Im (rot s 1 ) = F D 0,s F, P = 0 for all P P<s 3/2 2 Im (div s 1 ) = { f L 2 2 f, p = 0 for all p Lin {p n,m, n < s 3/2} } O ( η 2) div E 2 = 0 rot B 2 = 0 rot E 2 = t B 1 div B 2 = 0. E 2 L 2 iff t B 1 D 0,1 (and t B 1, P 2 n,m = 0 for all n < 1 3/2) tb 1 D 0,1 iff tj 1 + 2 t E 0 D 0,2 2 t E 0 D 2 iff 2 t ρ 0 L 2 3 and 2 t ρ 0, p n,m = 0 for all n < 3 3/2 B 2 = 0

space of regular convergence In order to estimate the error of the approximation in third order we need the approximation in third order: B 3 L 2 iff 3 t ρ 0 L 2 4 and 3 t ρ 0, p n,m = 0 for all n < 4 3/2 Theorem (Space of Regular Convergence, B. 2016?, prepreprint) The Darwin order approximation is well defined in L 2 iff the the second time derivative of the dipole contribution vanishes: x 2 t ρ 0 dx = 0 It is an approximation of order O(η 3 ) if in addition the third time derivative of the quadrupole moment vanishes.

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