Generalized formulations of Maxwell's equations for numerical Vlasov-Maxwell simulations

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Generalized formulations of Maxwell’s equations for numerical Vlasov-Maxwell simulations

Régine Barthelmé, Patrick Ciarlet, Eric Sonnendrücker

To cite this version:

Régine Barthelmé, Patrick Ciarlet, Eric Sonnendrücker. Generalized formulations of Maxwell’s equa- tions for numerical Vlasov-Maxwell simulations. [Research Report] RR-5850, INRIA. 2006, pp.26.

�inria-00070176�

ISRN INRIA/RR--5850--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

Generalized formulations of Maxwell’s equations for numerical Vlasov-Maxwell simulations

Régine Barthelmé — Patrick Ciarlet — Eric Sonnendrücker

N° 5850

Février 2006

Unité de recherche INRIA Lorraine

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 

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(·, ·) 0

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Q ⁰ × Q

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h., .i Q

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H(div , Ω) = { f ∈ L ² (Ω) ³ | div f ∈ L ² (Ω) }, H (curl , Ω) = { f ∈ L ² (Ω) ³ | curl f ∈ L ² (Ω) ³ },

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k f k 0,div = Z

Ω

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,

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4

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γ n : H (div , Ω) → H ^−1/2 (Γ), f 7→ f · n _|

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Γ .

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H 0 (div , Ω) = { f ∈ H(div , Ω) | f · n _|

Γ = 0 }, H 0 (curl , Ω) = { f ∈ H(curl , Ω) | f × n _|

Γ = 0 }.

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X = H (curl , Ω) ∩ H(div , Ω)

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V ⊂ H (curl , Ω)

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b : H × M → R,

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χ ∈ M ⁰

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B ∈ L(H ; M ⁰ )

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hBv, µi = b(v, µ) ∀v ∈ H, ∀µ ∈ M.

£ hŒuh]xf‚„vwh§µsr

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B

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()Y

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M ⁰ .

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V =

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(B)

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V ^◦ = {g ∈ H ⁰ ; < g, v >= 0 ∀v ∈ V }

^¶

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kB ⁰ µk H ⁰ ≥ βkµk M ∀µ ∈ M ;

)) -

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dt + Au = f

[0, T ], u(0) = u 0 .

4— 6

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f - 2

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dt + Au + λu = f

[0, T ], u(0) = u 0 .

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e ^λt f (t)

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ρ,

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div E ₀ = ρ(., 0)

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∂t ² + c ² curl curl E + c ² ∇ ∂p

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∂ t B

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t = 0

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( E , B , p)

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— 6

¶±…±‚[x4t}|Œuh7{

U = ∂ E

∂t − c ² curl B + c ² ∇p + 1 0

J .

cd4h7x

∂ U

∂t = ∂ ² E

∂t ² − c ² curl ∂ B

∂t + c ² ∇ ∂p

∂t + 1 0

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∂t = −c ² curl (curl E + ∂ B

∂t ) = 0.

3 t

U (., 0) = 0

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46

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U = 0

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QFE c% ]!c R-]!P(_Qc=K aPMK!c R-]!P(_ ]+_

E

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J ∈ H (div , Ω)

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div J ∈ L ² (Ω)

^°§†„x4Œ

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t > 0

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ψ ∈ H ₀ ¹ (Ω)

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∆ψ = div J

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)

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P = −∂ t ψ/ 0 − c ² ∂ t p

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( E , P ) ∈ X 0 × L ² (Ω)

-

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∂t ² , F

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0 + P, div F

0 = − 1 0

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0 = 1 0 ρ, q

0 , ∀q ∈ L ² (Ω),

4’ 6

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∂ t B + curl E = 0

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4

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∂t , F

X 0 = − 1 0

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∂t , F

X 0 + 1 0

∂∇ψ

∂t , F

0

= ∂ ² E

∂t ² , F

X 0 + c ²

curl curl E , F

X 0 + c ²

∇ ∂p

∂t , F

X 0 + 1 0

∇ ∂ψ

∂t , F

0

= ∂ ² E

∂t ² , F

X 0 + c ²

curl curl E , F

H 0 (curl,Ω) + c ²

∇ ∂p

∂t , F

H(div ,Ω)

+ 1 0

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∂t , F

0

= ∂ ² E

∂t ² , F

X 0 + c ² curl E , curl F

0 − c ² ∂p

∂t + 1 0

∂ψ

∂t , div F

0 .

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P = −∂ t ψ/ 0 − c ² ∂ t p

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5

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4˜ 6

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p(·, t) = − 1 c ²

Z t 0

P(·, s)ds + 1 0

(ψ(·, t) − ψ(·, 0))

...

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L ² (Ω)

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0 +

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p |Γ

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( E , P ) ∈ X 0 × L ² (Ω)

t}y4Ždv}d4†„v

 



 

∂ ² E

∂t ² + A E + B ⁰ P = − 1 0

∂ ˜ J

∂t

|‰x

X ₀ ⁰ , B E = 1

ρ

^|–x

L ² (Ω) ⁰ .

4'6

£ h<t¦dZ†„‡‰‡†[Œf†[‹uvv}dfh$‹4{}‚s‚„ª‚„ª |–{Ž†„yf‡–v†„x4Œ Š1†Kˆs|†„{}v8

;

vw‚Et¦d4‚¯ v}dZ†v"v}df|t"vw|–g+h<Œuh]‹Zh]x4Œuh7x>v

g+|–¨uh]Œ‹f{}‚'µf‡‰h7g)d4†'t1†Xyfxf|~>yfht¦‚'‡–yuvw|–‚'xÀ¶¬psh7v

V =

^"h7{

B = {v ∈ X 0 | div v = 0}, V ρ = {v ∈ X 0 | div v = ρ/ 0 }, V ^◦ = {g ∈ X ₀ ⁰ | hg, vi = 0 ∀v ∈ V }, V ^⊥ = {u ∈ X 0 |(u, v) X = 0 ∀v ∈ V }.

£ hxf‚[v}h[°fv}d4†„vk™[|‰ˆ[h]x†[x>rh7‡‰h7g+h7x>v

ϕ

^‚„ª

H ₀ ¹ (Ω)

t¦yZŽdvwd4†v

∆ϕ ∈ L ² (Ω)

°4‚[x4hd4†[t

∇ϕ ∈ X 0

¶

®x¹†'ŒfŒu|–v}|‰‚[xÀ°svwdfh7{wh"df‚[‡Œftµsr<|‰x>v}h7™'{w†„v}|‰‚[xµ>rœ‹4†„{}vwt

(∇ϕ, v) X = (∇ϕ, v) 0 = 0, ∀ϕ ∈ H ₀ ¹ (Ω)

^t]¶v]¶

∆ϕ ∈ L ² (Ω), v ∈ V.

^{4!5 “ 6}

®xº‚[v}dfh]{¯‚[{ŽŒft]°

∇ϕ

µHh7‡‰‚[xf™>tv}‚

V ^⊥

¶cd4|‰t‹4{}‚'‹Zh]{¦v(r$¯|‰‡–‡6µZhy4t}h]Œdfh]{}hS†ªèv}h]{]¶

±Â   - ) )(

b

^{[ )7 ) )}

X 0 × L ² (Ω)

λ∈L inf ² (Ω) sup

v∈X 0

b(v, λ) kvk X kλk 0

≥ β.

+f £ hxfh7hSŒœvw‚+‹f{}‚ˆ'hkvwd4†vvwdfh7{whh¨u|t(vŽt

β > 0

t¦yZŽdœvwd4†v

∀λ ∈ L ² (Ω), sup

v∈X 0

b(v, λ) kvk X

≥ βkλk 0

hv

λ ∈ L ² (Ω)

¶cdfh7{whh7¨s|t¦vwt

ξ ∈ H ₀ ¹ (Ω)

t¦y4ŽdEv}d4†„v

∆ξ = λ

^|–x

Ω

¶l¹‚'{}h]‚ˆ[h]{]°

kξk 1 ≤ Ckλk 0

¯|–v}d

C > 0

|‰x4Œuh7‹Hh7xZŒuh7x>v1‚„ª

λ

^¶cdfh7x

u = ∇ξ

ˆ[h]{}|4h]t

u ∈ L ² (Ω) ³ , div u = λ ∈ L ² (Ω), curl u = 0 ∈ L ² (Ω) ³ , u × n _|

Γ = ∇ξ × n _|

Γ = 0

^†[t

ξ ∈ H ₀ ¹ (Ω).

·kh7x47h

u

µZh]‡–‚'xf™'tvw‚

X 0

¶ <mxœvwdfh‚„vwdfh7{dZ†„x4Œ©°

b(u, λ) = (div u, λ) 0 = kλk ² ₀

^°f†„x4Œ

kuk ² _X = kuk ² ₀ + kdiv uk ² ₀

= k∇ξk ² ₀ + kλk ² ₀

≤ kξ k ² ₁ + kλk ² ₀

≤ (1 + C ² )kλk ² ₀ .

cd4h7{whª³‚[{wh[°

b(u, λ) kuk X

= kλk 0

kλk 0

kuk X

≥ kλk 0 (1 + C ² ) ^−1/2 ,

5:

†[x4Œt¦‚

sup

v∈X 0

b(v, λ) kvk X

≥ b(u, λ) kuk X

≥ βkλk 0 ,

¯|–v}d

β = (1 + C ² ) ^−1/2

|‰x4Œuh]‹Zh]x4Œuh]x'v1‚[ª

λ

^¶

ÂsÆ Ÿ  > [- - )) = S

( E ₀ , E ₁ ) ∈ X 0 (curl ) × X 0 ,

^{4+5 576}

[>

ρ

J

J ∈ C ² ([0, T ]; H (div , Ω)), ρ ∈ C ² ([0, T ]; L ² (Ω)), ( ∂ρ

∂t + div J )(., 0) = 0.

^{4!5 — 6}

%^> Y ^ - ,

( E , P )

⁾

E ∈ C ⁰ ([0, T ]; X 0 (curl )) ∩ C ¹ ([0, T ]; X 0 ) ∩ C ² ([0, T ]; H (div , Ω)),

P ∈ C ⁰ ([0, T ]; L ² (Ω)).

+f <mxºv}dfh3‚[xfhdZ†„x4Œ©°Zv}d4hµf|‰‡–|‰xfh]†[{1ª³‚'{}g

b

ˆ[h]{}|4h]t1v}dfh¥|–xuªè­®t}yf‹_‚'x4Œu|–v}|‰‚[x¹|‰x¹v}d4h3t}‹4†[7h

X 0 × L ² (Ω)

¶5cdfh]xÀ°±†'7‚'{wŒf|–xf™_v}‚ h]gXg$†

5 °

B

|‰t+†„x5|‰t}‚[g+‚'{}‹fd4|‰t}g ª³{}‚'g

V ^⊥

^‚[x>v}‚

L ² (Ω) ⁰

^¶

·kh7x47h§v}dfh]{}hh7¨s|t¦vwt†Xyfxf|~>yfh

E ^⊥ ∈ V ^⊥

t}y4Ždœvwd4†v

B E ^⊥ = ρ/ 0 .

lE‚[{wh7‚ˆ[h]{]°sŒuyfh§vw‚$|–xfhS~>y4†„‡‰|´v(r

4

|‰|–|

6

|–x h7g+g$†

5

†[x4Œ†[t

ρ ∈ C ² ([0, T ]; L ² (Ω))

^°

E ^⊥ ∈ C ² ([0, T ]; V ^⊥ ).

<mx¥v}dfh‚[v}dfh]{ d4†„x4ŒÀ°„‡–h7v

ϕ ∈ H ₀ ¹ (Ω)

µZh†"t¦‚'‡–yfv}|‰‚[xvw‚

∆ϕ = ρ/ 0

¶ 3

]‚'{wŒu|‰xf™kv}‚

4!5

“ 6 °

∇ϕ ∈ V ^⊥

^¶

r<y4xf|‰~>yfh]xfh]twt

4

B

|t1†[x|t¦‚'g+‚[{w‹fdf|t¦g ‚'x

V ^⊥

^{6 °}

E ^⊥ = ∇ϕ

¶cdf|t|–g+‹f‡‰|‰h]tvwd4†v

A E ^⊥ = 0,

^†„x4Œ

E ^⊥ , F

X = E ^⊥ , F

0 = 0

ª³‚[{1†[‡–‡

F ∈ V.

»mh7x4‚„v}hµsr

A V

v}dfh‡‰|‰xfh]†[{‚[‹Hh7{Ž†v}‚'{Œuh 4x4h]Œµ>r

A V : X 0 → V ⁰

^°

F 7→ A V F

°4t}y4Ždv}d4†„v

hA V F , G i = c ² (curl F , curl G ) 0 , ∀( F , G ) ∈ X 0 × V.

Yª

( E , P )

|t†mt¦‚'‡–yfv}|‰‚[x"v}‚kv}dfhg+|´¨uhSŒ‹f{w‚[µf‡‰h7g 4 6

°„|–v|‰g+‹f‡‰|–hSt6vwd4†v

E

|t9†kt}‚[‡‰yuv}|‰‚[x"vw‚kvwdfhª³‚'‡–‡‰‚¯|–x4™

‹4{}‚'µf‡–h]g){wh]t¦v}{w|‰v}h]Œœv}‚

V

^•|‰x4Œ

E ∈ V ρ

t¦yZŽdœvwd4†v

∂ ² E

∂t ² + A V E = − 1 0

∂ J ˜

∂t

|–x

V ⁰ .

3 ]‚'{wŒu|‰xf™3v}‚

4!5

“ 6

°uv}df|t‹f{w‚[µ4‡–h]g)|‰thS~>yf|–ˆ†[‡–h]x'vvw‚

•9|–xZŒ

W = E − E ^⊥ ∈ V

t}y4Ždv}d4†„v

∂ ² W

∂t ² + A V W = − 1 0

∂ ˜ J

∂t

|‰x

V ⁰ .

^{4+5 . 6}

puhv¦vw|–x4™

u = W

∂ t W

, A =

0 −I A V 0

, f =

0

−∂ t J / 0

, u 0 =

E ₀ − E ^⊥ (., 0) E ₁ − ∂ t E ^⊥ (., 0)

,

vwdf|t‹f{w‚[µf‡‰h7g)µHh]7‚[g+h]t

d

dt u + Au = f, u(0) = u 0 .

^{4+5: 6}

®x>vw{}‚uŒuy47h

H = V × H (div 0, Ω)

D(A) = {u ∈ H | Au ∈ H } = V (curl ) × V.

hvy4t1†'Œug+|´vª³‚'{v}dfhg+‚'gXh]x>vvwdfh"ª³‚[‡‰‡‰‚¯|–xf™ h7g+g$†f¶

±Â   - >S

I + A : D(A) → H

^{[ %') L YS}

H

^f

3 ]‚[{ŽŒu|‰xf™v}‚_Š1h7g$†„{>1”—u°9¯h<7†„x/†[‹f‹f‡‰rEv}d4hœ·1|‰‡–‡‰h­‚'t}|Œf†¹cdfh7‚'{}h]g

†[twt¦y4gX|‰xf™v}dZ†v

u 0 ∈ D(A)

^†[x4Œ

f ∈ C ¹ ([0, T ], H)

°‹f{}‚'µf‡‰h7g

4+5: 6

d4†[tœ† yfxf|~'y4hEt¦‚'‡–yuvw|–‚'x

u ∈ C ⁰ ([0, T ], D(A)) ∩ C ¹ ([0, T ], H)

^¶

cdfhdsrs‹H‚„v}d4h]t}h]t‚„ªv}dfh·k|–‡‰‡‰h­‚'t}|‰Œf†+cdfh]‚[{wh7g7‚[{w{}hSt¦‹H‚[xZŒ$v}‚

f ∈ C ¹ ([0, T ], H) ⇐⇒ ∂ t J ˜ ∈ C ¹ ([0, T ], H(div 0, Ω))

^¶

cdfh{w|–™'d'v1t}|‰Œfh§d4‚[‡Œft¯dfh7x

J ∈ C ² ([0, T ]; H(div , Ω))

^†„xZŒ

ρ ∈ C ² ([0, T ]; L ² (Ω))

^¶

u 0 ∈ D(A) ⇐⇒ E ₀ − E ^⊥ (., 0) ∈ V (curl )

^†„xZŒ

E ₁ − ∂ t E ^⊥ (., 0) ∈ V

^¶

hS7†„yZt¦h‚[ª©v}dfhmd>rs‹H‚„vwdfh]t}h]t±†„x4Œ$†'t

curl E ^⊥ = 0

°'‚'xfh1d4†'t

E ₀ − E ^⊥ (., 0) ∈ X 0 (curl )

¶l¹‚[{wh7‚ˆ'h7{S°

div ( E ₀ − E ^⊥ (., 0)) = div E ₀ − ρ(., 0)/ 0 = 0,

d4h7x47h

E ₀ − E ^⊥ (., 0)

µZh]‡–‚'xf™'tRv}‚

V (curl )

^¶ ®xXv}d4hkt}†[g+h¯†Kr'°

E ₁ −∂ t E ^⊥ (., 0) ∈ X 0

°'†[x4ŒXµZhS7†[y4t¦h

‚[ª

4 6

div ( E ₁ − ∂ E ^⊥

∂t (., 0)) = div E ₁ − 1 0

∂ρ

∂t (., 0)

= div (c ² curl B ₀ − J (., 0)/ 0 ) − 1 0

∂ρ

∂t (., 0)

= − 1 0

(div J + ∂ρ

∂t )(., 0) = 0.

·kh7x47h

E ₁ − ∂ t E ^⊥ (., 0)

µHh7‡‰‚[xf™>tv}‚

V

^¶

cdfh]{}h7ª³‚[{wh[°uvwdfhdsrs‹Z‚[v}dfhSt¦hSt‚„ª v}d4h·1|‰‡–‡‰h­‚'t}|‰Œ4†$cdfh7‚'{}h]g †„{wh†„‡‰‡Àˆ[h]{}|4hSŒ©°4†„xZŒ¯h™'hv1h¨u|t(­

vwh7x47h†„x4Œyfxf|~>yfh7x4h]twt‚„ª

u ∈ C ⁰ ([0, T ], D(A)) ∩ C ¹ ([0, T ], H),

5

¯d4|‰Žd|‰t1h]~>yf|‰ˆK†[‡–h]x>vvw‚

W ∈ C ⁰ ([0, T ], V (curl )) ∩ C ¹ ([0, T ], V ) ∩ C ² ([0, T ], H (div 0, Ω)),

y4xf|‰~>yfh¹t}‚[‡‰yuv}|‰‚[x5v}‚

4+5

. 6 ¶ pu|–x47h

X 0 = V ⊕ V ^⊥

°¯hdZ†Kˆ[hh¨u|‰t¦v}h]x4hº†„xZŒOyfx4|‰~>yfh]xfh]twt+‚„ª§v}dfh t}‚[‡‰yuvw|–‚'x

E

^vw‚ 4 6

°4†[]‚[{ŽŒu|‰xf™3v}‚+v}d4ht¦‹f‡‰|–v¦v}|‰xf™

E = W + E ^⊥ ∈ C ⁰ ([0, T ], X 0 (curl )) ∩ C ¹ ([0, T ], X 0 ) ∩ C ² ([0, T ], H (div , Ω)).

Yv {}h]g+†[|–xZt©v}‚ 4x4Œ

P

^¶ h]]†„y4t}h‚[ªfv}dfh|‰xuªè­Yt}yf‹¥7‚[x4Œf|´vw|–‚'x

4

‡‰h7g+g$†

5 6 °

B ⁰

|t9†„x|t¦‚'g+‚[{w‹fdf|t¦g ª³{w‚[g

L ² (Ω)

^‚[x

V ^◦

^¶yuv

− 1 0

∂ ˜ J

∂t − ∂ ² E

∂t ² − A E = − 1 0

∂ ˜ J

∂t − ∂ ² W

∂t ² − A W

! +

− ∂ ² E ^⊥

∂t ² − A E ^⊥

∈ V ^◦ .

·kh7x47h§v}dfh]{}hh7¨s|t¦vwt†Xyfxf|~>yfh

P

^|‰x

L ² (Ω)

t¦y4Ždv}dZ†v

B ⁰ P = − 1 0

∂ J ˜

∂t − ∂ ² E

∂t ² − A E ∈ V ^◦ .

jmt¦|‰xf™+xf‚¯ |–xfhS~>y4†„‡‰|´v(r

4

|‰|

6

‚[ª h7g+g$†

5

°f†„x4Œvwdfh"ªÑ†[vvwd4†v

− ∂ ˜ J

∂t − ∂ ² E

∂t ² − A E ∈ C ⁰ ([0, T ], L ² (Ω) ³ ),

¯h"‚[µuvŽ†„|‰xœvwd4†v

P ∈ C ⁰ ([0, T ]; L ² (Ω)).

c‚<‚'gX‹4‡–h7v}h§v}dfh‹4{}‚s‚„ª(°u¯h"xfh7hSŒv}‚+‹f{w‚ˆ[h

h7g+g$†

. ¶

+f

8€¬{}‚s‚„ª9‚„ª

h7g+g$†

. ¶;

h7v1y4tt¦d4‚¯Wv}dZ†v

I + A

|‰tg$†„¨s|‰g$†„‡©g+‚[x4‚„v}‚'xfh§‚[x

H

^¶

…±‚'x4t}|‰Œuh]{

u ∈ D(A)

^¶

((I + A)u, u) _H = (u 1 − u 2 , u 1 ) V + (u 2 + A V u 1 , u 2 ) 0,div 0

= Z

Ω

|u 1 | ² + c ² |curl u 1 | ² − u 1 · u 2 − c ² curl u 1 · curl u 2

+ |u 2 | ² + c ² curl u 1 · curl u 2

d x

= Z

Ω

c ² |curl u 1 | ² + |u 1 | ² + |u 2 | ² − u 2 · u 1

d x ≥ 0.

·kh7x47h§v}dfh‚'‹Zh]{w†„v}‚[{

I + A

|tgX‚'xf‚„vw‚[xfh'¶

…±‚'x4t}|‰Œuh]{1xf‚¯

f ∈ H

^{¶ £ h¥‡‰‚s‚ 1œª³‚[{}

u ∈ D(A)

t¦yZŽdºv}d4†„v

(2I + A)u = f

°4v}d4†„vm|t7°H¯±h¥‡–‚s‚ 1œª³‚[{

u 1 ∈ V (

^7yf{}‡

)

^†„xZŒ

u 2 ∈ V

t¦y4Ždv}dZ†v

2u 1 − u 2 = f 1 ∈ V,

2u 2 + A V u 1 = f 2 ∈ H (

^Œu|‰ˆ

0, Ω).

cd4|‰tt}rut(vwh7g|th]~>yf|‰ˆ†„‡‰h7x>vv}‚

u 2 = 2u 1 − f 1

4u 1 + A V u 1 = f 2 + 2f 1 .

cd4h¥µf|‰‡–|‰xfhS†„{mª³‚[{wg

(u 1 , v 1 ) 7→ (4u 1 , v 1 ) 0 + c ² (curl u 1 , curl v 1 ) 0

|‰t"7‚[x>v}|‰xsyf‚[y4tm†„x4Œ_‚sh7{Ž|‰ˆ[h‚'x

vwdfh‡‰‚'t}h]Œt}yfµ4t}‹4†[7h

V

^‚„ª

X 0

¶

f 2 + 2f 1 ∈ H (

^Œu|‰ˆ

0, Ω)

^Œuh 4xfh]tk†3‡‰|‰xfh]†[{k‚'x'vw|–xsyf‚'y4tª³‚[{wg)‚'x

V

^¶

»myfh"vw‚3vwdfh †„¨s­Yl¹|‰‡‰™[{Ž†„g)cdfh]‚[{wh7gº°>v}dfh]{}hh7¨s|t¦vwt†Xyfxf|~>yfh

u 1 ∈ V

t}y4Ždœvwd4†v

h4u 1 + A V u 1 , v 1 i V = (f 2 + 2f 1 , v 1 )

ª³‚'{1†„‡‰‡

v 1 ∈ V.

£ h‹f{w‚ˆ[h§xf‚¯Wvwd4†v

c ² curl curl u 1 = f 2 + 2f 1 − 4u 1

|–xvwdfht¦h]x4t}h§‚[ª9Œu|t¦v}{w|–µfyfv}|‰‚[x4t]¶

hv

ϕ

µHhm†[x>r3ª³yfx4v}|‰‚[x<|‰x

D(Ω)

¶Rcdfh7{wh1h7¨s|t¦vwt¬†yfxf|~'y4h

λ ∈ H ₀ ¹ (Ω)

t}y4Žd+vwd4†v

∆λ = div ϕ

^¶¬puhv

ψ = ϕ − ∇λ

¯±h"ˆ[h]{}|–ª³r$h]†'t¦|‰‡‰r3vwd4†v

(∇λ, ψ) ∈ H 0 (curl , Ω) × V

^¶ <mxfhmdZ†[t]°uµ>r<Œfh 4xf|–v}|‰‚[x‚„ª

u 1

°

hc ² curl curl u 1 , ψi V = hA V u 1 , ψi V = (f 2 + 2f 1 − 4u 1 , ψ) 0 .

lE‚[{wh7‚ˆ[h]{

hc ² curl curl u 1 , ∇λi H 0 (curl ,Ω) = 0.

cd4h7{whª³‚[{wh[°

c ² curl curl u 1 = f 2 + 2f 1 − 4u 1

ª³‚[‡‰‡–‚¯1t|‰x

D(Ω) ⁰

^†„xZŒt}‚

u 1 ∈ V (curl )

^¶

£ hŒfh]Œuy47h<vwdfhh7¨u|‰t¦v}h]x4h<‚[ª

u 2 = 2u 1 − f 1 ∈ V

^¶ cdfhœ‚'‹Zh]{w†„v}‚[{

2I + A

|t¥‚'x'vw‚E‚[x

H

^°

¯d4h7x47h

I + A

|tg$†¨u|–g$†[‡©gX‚'xf‚„vw‚[xfh"‚'x

H

^¶

cdfh|‰xf|–v}|†„‡À‹4{}‚'µf‡–h]g7†[x Zx4†„‡‰‡–rœµZht}‚[‡‰ˆ[hSŒ©¶

ÂsÆ Ÿ Â

-

B ₀ ∈ H 0 (div 0, Ω) ∩ H (curl , Ω),

Y - ,[

( E , B , p)

⁾

E ∈ C ⁰ ([0, T ], X 0 (curl )) ∩ C ¹ ([0, T ], X 0 ) ∩ C ² ([0, T ], L ² (Ω) ³ ), B ∈ C ¹ ([0, T ], H 0 (div 0, Ω) ∩ H (curl , Ω)) ∩ C ² ([0, T ], H 0 (div 0, Ω)), p ∈ C ¹ ([0, T ], H ₀ ¹ (Ω)).

+f p>vw†[{¦vw|–x4™$ª³{}‚'gvwdfh¯h]† 1t}‚[‡‰yuv}|‰‚[x

( E , P )

‚[µuvŽ†„|‰xfh]Œª³{w‚[g cdfh7‚'{}h]g¿—u°Z¯hµfyf|‰‡‰Œ

B

^†„xZŒ

p

°sv}dfh]xº¯±h‹f{w‚ˆ[hkv}d4†„v

( E , B , p)

|t†+t(vw{}‚'xf™$t¦‚'‡–yuvw|–‚'xœvw‚+‹f{}‚'µf‡‰h7g 4

— 6 ¶

»myfh§v}‚

4’ 6 °

div E = ρ/ 0

|–x

L ² (Ω)

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