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Applicazioni ed esercizi di modellistica numerica per problemi differenziali (UNITEXT La Matematica per il 3+2) [1 ed.]
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Zitiervorschau

A Francesca, Paola, Laura

L. Formaggia F. Saleri A. Veneziani

Applicazioni ed esercizi di modellistica numerica per problemi differenziali

12 3

LUCA FORMAGGIA FAUSTO SALERI ALESSANDRO VENEZIANI MOX - Dipartimento di Matematica “F. Brioschi” Politecnico di Milano

L’immagine di sfondo della copertina rappresenta una simulazione numerica del campo di moto attorno a una imbarcazione da canottaggio da competizione (per gentile concessione di CD ADAPCO Ltd. e Filippi Lido s.r.l.). Nei riquadri: in basso, geometria semplificata e griglia di un disco freno per automobili; in alto, griglia di un modello di carotide fornito da D. Liepsch e dalla F.H. di Monaco di Baviera (gentile concessione di K. Perktold e M. Prosi). Entrambe le griglie sono state generate con il codice Netgen di J. Schöberl (http://nathan.numa.uni-linz.ac.at/netgen/usenetgen.html).

Springer-Verlag fa parte di Springer Science+Business Media springer.it © Springer-Verlag Italia, Milano 2005 ISBN 10 88-470-0257-5 ISBN 13 978-88-470-0257-9 Quest’opera è protetta dalla legge sul diritto d’autore. Tutti i diritti, in particolare quelli relativi alla traduzione, alla ristampa, all’uso di figure e tabelle, alla citazione orale, alla trasmissione radiofonica o televisiva, alla riproduzione su microfilm o in database, alla diversa riproduzione in qualsiasi altra forma (stampa o elettronica) rimangono riservati anche nel caso di utilizzo parziale. Una riproduzione di quest’opera, oppure di parte di questa, è anche nel caso specifico solo ammessa nei limiti stabiliti dalla legge sul diritto d’autore, ed è soggetta all’autorizzazione dell’Editore. La violazione delle norme comporta le sanzioni previste dalla legge. L’utilizzo di denominazioni generiche, nomi commerciali, marchi registrati, ecc, in quest’opera, anche in assenza di particolare indicazione, non consente di considerare tali denominazioni o marchi liberamente utilizzabili da chiunque ai sensi della legge sul marchio. Riprodotto da copia camera-ready fornita dagli Autori Progetto grafico della copertina: Simona Colombo, Milano Stampato in Italia: Signum Srl, Bollate (Milano)

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%

# (

{φ1 , . . . , φnl }" % (& nl p ( x) = i ( p)φi ( x)" i=1 σ ( & 1 ( p ∈ P(K) p ∈ P (K) K $

#

( TK

$ ( (

σ i (φj ) = δij

−1 p(x) = p(TK (x)).

B"BC

( ( %

−1 φK,i = φi ◦ TK "

% 1 " 5 I048℄"

- & % & 0 C & ( " 1 r i ∈ K i = 1, . . . , nl (K) ⊂ P (K) N r P

# ( "

&

i ) ( σ i ( p) = p(N i ◦ T −1 ( # K φK,i = φ K i ) $ i& σK,i (p) = p(NK,i ) NK,i = TK (N K" ( %

0 1 - $ P(K) = Pr (K) TK $ '" K r 1

Xh (Ωh ) - & r

Xh " % ' (

Pr

5 % 1

Xhr = {vh ∈ C 0 (Ωh ) :

vh |K ∈ Pr (K),

7 B"C

( (

Pr

∀K ∈ Th }.

B"34

( (

%"

% /

i j

k

n

( -

!

- " %

7 B"C

% 1 d

$ nl = d! j=1 (r + j)" % ##

(

. 1 " % (

r=1

2

%

3

P1

" %

φn = ξi ,

r=2

P2

1≤i≤d+1

n = 1, . . . , d + 1.

%

φn = ξi (2ξi − 1), 1 ≤ i ≤ d + 1 n = 1, . . . , d + 1, φn = 4ξi ξj , 1 ≤ i < j ≤ d + 1 n = d + 1, . . . , nl .

r=3 φn φn φn φn

P3

1 2 ξi (3ξi − 1)(3ξi 9 2 ξi (3ξi − 1)ξj , 9 2 ξj (3ξj − 1)ξi ,

= = = = 27ξi ξj ξk ,

%

− 2), 1 ≤ i ≤ d + 1 n = 1, . . . , d + 1, 1 ≤ i < j ≤ d + 1 n = d + 1, . . . , 3d + 1, 1 ≤ i < j ≤ d + 1 n = 3d + 2, . . . , 5d + 1, 1 ≤ i < j < k ≤ d + 1 n = 5d + 2, . . . , nl .

P1 x b2

P2 x b2

3

B"3B

P3 x b2

3

3

9

8

6 5

10

5 1

2

1

x b1

x b3 4

3

x b2

4

x b1

x b3 4

10

3 9

7 6 2

1

2

x b1

1

x b2

4

13

16 15 19

10

2

x b1

1

6

3

20 9 17

5

x b2 14

18 12 7

2

7

x b1

x b3 4

8 5

1

6

8 11

2

x b1

Pr

P3

#

!

$

& C 0

% %

#

(

&

(

"

M ( ## (

# ( # ( (

# "

% $ ( # " $ ( ( ( %

$

(

"

//

1 % % ( 1 B"3B ( P2 "

(

& //

## 1

j ) = δij φi (N

j N

7 B"C" 6 # "

nl = 6

" A

%

#

#

% #

%

i 1 2 3

ξ1 1 0 0

ξ2 0 1 0

ξ3 0 0 1

( x1 , x 2 ) B"B9

i ξ1 ξ2 ξ3 4 1/2 1/2 0 5 1/2 0 1/2 6 0 1/2 1/2

3 ( ( φi = ξi (2ξi − 1) i = 1, 2, 3" φ0 = 0 $

1

( ξi

= 0 ξi = 1/2" # $ %

5 $ i " ξi = 1 φi (N i ) = 1(2 − 1) = 1" N ( ( φ4 = ξ1 ξ2 φ5 = ξ1 ξ3 φ6 = ξ2 ξ3 " %

#

$

3

" F 1 # %

1"

$ / % #

% ( ( 7 B"C"

i ξ1 ξ2 ξ3 1 1 0 0 2 0 1 0 3 0 0 1 4 0 0 0 5 1/2 1/2 0

ξ4 0 0 0 1 0

i ξ1 ξ2 ξ3 ξ4 6 1/2 0 1/2 0 7 1/2 0 0 1/2 8 0 1/2 1/2 0 9 0 1/2 0 1/2 10 0 0 1/2 1/2

*

( # " *

K

♦

# 5

1 ( (

r > 0" = Qr (K)

# x s % P (K) 2 2 3 2 r" x2 x ∈ Q ( K) x ∈ / Q ( K) "

% 1 2 1 ⊂ Qr (K) ⊂ Pdr (K) " ' Pr (K) K $ K ## " >

TK - "

( ( 1 ( - " (

Ni

φi

( ( ( 1 (

TK ( x) =

nl s=1

x), Ni φi (

B"33

K " & 1 # #

K 1 ( ( 1 P ( TK 1 B"33

$ 'N

*

TK $

%L ( & # ) " 1

( ( $ B"33

$ φi ∈ Pm (K) % m K m d # φi ∈ Qm (K) TK ∈ [P (K)] K d " 5 TK ∈ [Qm (K)]

x=

nl i=1

x), Ni φi (

∈ K. x

# 1 5

% d ( K

B"9"8" TK ∈ [Q1 (K)]

d = 2 d = 3"

m = r & m < r &

1 ' m = 1" m > r $ "

# " r 1 Q r > 0 $ 1 &

. ( # (

$ "

,

(K) = Qr (K) $ #

$ Qr

$ P 1 d TK ∈ [Q (K)] " 1 $ −1 vh |K = vh ◦ TK ; vh ∈ Qr (K),

Qh r = {vh ∈ C 0 (Ω h ) :

( ( (

∀K ∈ Th }.

K

( & φ1i i = 0, . . . , r" ξi i = 0, . . . , r [0, 1] ( φ1 (x)φ1 (y) d = 2

( # Qr (K) i j φ1i (x)φ1j (y)φ1k (z) d = 3 i, j, k 0 r" −1

% vh |K $ TK $ '"

//

# 1

d

r+d d

r+d r 1 Qr $ nl = (r + 1)d " d = 2 % d = 3" ' (

% ni=1 i = n(n + 1)/2 % ni=1 i2 = n(n + 1)(2n + 1)/6"

P

r

$

& //

nl =

1 d!

j=1 (r

+ j) =

Pr

=

P (K) = Pr (K)" * P (K) = Pr (K) % ( r" 6 d = 2" r $ ( i j r

$ ( x1 x2 i ≥ 0 j ≥ 0 i + j = s s = 0, . . . , r" # s $ s + 1" ( i % 0 s ## # j j = s − i" * r $

## %

# $

r

(s + 1) =

s=0

r+1

s = (r + 1)(r + 2)/2,

s=1

( "

% s ( xi1 xj2 xk3 i+j+k = s" * i 0 s j k ( j + k = s − i" # % i % s − i + 1 # j k # " * s $ s i=0

(s + 1 − i) =

s i=0

(s + 1) −

s i=0

i = (s + 1)2 − s(s + 1)/2 =

1 2 (s + 3s + 2). 2

/

r $ 1 r 2 s=0 s + 3s + 2"

( ( 2

#

%

# "

r 1 Q & r 1 Q (K)" ( r 1 $ r + 1 ( Q

♦

( "

//

Xhr

% 1

1 B"34

d

r

$

r=1 r=2 r=3 2 Nv Nv + Nl Nv + 2Nl + Ne " 3 Nv Nv + Nl Nv + 2Nl + Nf & // −1 p ◦ TK

K"

r ## % p ∈ Xh ( % p|K = r p ∈ P " TK # p

% p $

>

" 6 &

K1

% % %

r " V

% % p ∈ Xh

r

% p|K1 (V) = p|K2 (V) p ∈ Xh

5 $ # V $

# " (

K2

p V" K1 K2 ## Γ12 " & p

% p|K1 (x) = p|K2 (x) x ∈ Γ12 " 6 5 $ # p Γ12 $ 1 Γ12 " r

% r + 1 " % ## # %

% # r − 1 " * ( 3 + 3(r − 1) = 3r

"

B"9"B % Pr (K) (r + 1)(r + 2)/2 # (r + 1)(r + 2)/2 − 3r " r = 1

1 Xh Nv H r = 2 nl = 6 # & %

Nh = Nv + Nl " r = 3

1 #

2 $

$ # $ %

1

3

Nh = Nv + 2Nl + Ne "

$ "

0

"

% (

" 6

B"9". C 0 " >

% #

'

( #

//

" K

♦

K1 K2 e = K1 ∩ K2 $ ( ' ( K TK1 TK2 % e = " e K e) e) = TK2 ( TK1 ( 1 , x 2 ∈ e

e x x2 ) = x ∈ e" % C 0 x1 ) = TK2 ( TK1 ( r

( P

%

e K

"

Th

%

$ %

& //

7

( 7 B"B4" TK1 TK2 & = ∂K1 TK2 (∂ K) = ∂K2

K1 K2 TK1 (∂ K) " A B

e" 1 %

K j j = 1, 2, 3

( K V Ki j = 1, 2, 3

V j

Ki i = 1, 2" 6%

1 ) = VK1 = A" % K1 TK1 (V 1 2 ) = VK1 = B" (

VK1 1 TK1 (V 2 j j = 1, 2, 3 ( & " K2 TK2 (V 2 ) = V K2 = A K 2 K2 TK2 (V1 ) = V1 = B" TK2 (V1 ) = A TK2 (V2 ) = B ## ( &

#

K2 "

V 1 V 2 5 % e

K e) TK2 ( e)

e % " TK1 (

1 ' B"C ( ( %

1 V

$ (

(0, 0)

1 x

%

1) = A TK1 (V

, x) = A + FK1 x TK1 (

2 ) = B TK1 (V

1) = B TK2 (V

%

. TK2 ( x) = B + FK2 x

2) = A TK2 (V

2 = B − A, FK1 V

( x1 , x 2 )

$

1 V

%

2 = A − B. FK2 V

2 " 6 e x 2 = −FK V 1

FK1 V 2

" 2 % 0 ≤ α ≤ 1 %

1

2 − V 1) 1 = α(V x

(

2 − V 1 )" 0 2 = (1 − α)(V x ( % V1 % (0, 0)

2 = 2 = B + (A − B) − αFK V 2 = A − αFK V x1 ) = A + αFK1 V TK1 ( 2 2 2 = TK2 [(1 − α)V 2 ] = TK2 ( x2 ). B + (1 − α)FK2 V

Xh $ C 0 ( vh e

$ vh |K1 (x) = vh |K2 (x) x ∈ e" F %

5 $ # i ∈ e $ # "

Ni e N * Ni ∈ e TK1 TK2 ( Ni " 6 $ % 1

e " 7 B"BB

" ♦

A

V1K1 V1K2

TK1

K1 e

K2

V2K1 V2K2 B

TK2 b3 V

TK2 TK1

b K

eb

b1 V

b2 V

Kb 3

TK1 TK2

//

TK &

% K % &

7 B"B3

Q1

# "

" K

TK

$ #

$ "

e

e

TK1

TK1

TK2

eb

TK2

eb

be

3

e - . 4

-.

V3 V3

V4

V2

V1

V4

x2 V1

K

x2

K

θ4

TK V2 x b2 b3 V

b1 V

TK

b3 V

b4 V b K

x1

x b2

x1

b2 V x b1

b1 V

b4 V b K

TK &

& 5

& //

Q1

b2 V x b1

A

% ( ( (

φ1 ( x1 , x 2 ) = (1 − x 1 )(1 − x 2 ), x1 , x 2 ) = (1 − x 1 ) x2 , φ3 (

lij

( %

= Vj − Vi ## nl φi = 1 i=1

φ2 ( x1 , x 2 ) = x 1 (1 − x 2 ), φ4 ( x1 , x 2 ) = x 1 x 2 .

% 1 #

TK ( x1 , x 2 ) =

4 i=1

φi ( x1 , x 2 )Vi = [1 −

4 i=2

φi ( x1 , x 2 )]V1 +

4 i=2

φi ( x1 , x 2 )Vi

= V1 + φ2 ( x1 , x 2 )l12 + φ3 ( x1 , x 2 )l13 + φ4 ( x1 , x 2 )l14 .

"

6 TK K 12 TK 1 x 2 = 0 0 ≤ x 1 ≤ 1 " 5 12 TK ( x1 ) = TK ( x1 , 0) = V1 + φ2 ( x1 , 0)l12 + φ3 ( x1 , 0)l13 + φ4 ( x1 , 0)l14 .

i ## % φ3 ( x1 , 0) = φ4 ( x1 , 0) = 0 1 ( # φ 12 TK ( x1 ) = TK ( x1 , 0) = V1 + φ2 ( x1 , 0)l12 = V1 + x 1 l12 % $ 12 V1 l12 " TK (0) = V1 12 12 TK (1) = V2 " * x = TK ( x1 ) ( x1 , 0)

K

6%

TK

1 V

2 V

l12 " $ - '

x 1 x 2

x 1 x 2 x 3

K

## (

.

J(TK ) & " #

" *

G #

% %

TK

|J(TK )| > 0

% 1

" K

TK,1

w

R2

∂TK,1 ∂TK,1

∂ x1 ∂ x2

|J(TK )| =

.

∂TK,2 ∂TK,2

∂ x1 ∂ x2

A

1

v

v × w = |v||w| sin(θ), θ

v

w

∂TK = (1 − x 2 )l12 + x 2 l34 , ∂x1

B"3.

5

1 %

TK,2

|J(TK )| =

∂TK ∂TK " × ∂ x1 ∂ x2

∂TK = (1 − x 1 )l13 + x 1 l24 . ∂x2

13 24 13 24 = lij × lkl |J(TK )| = φ1 l12 + φ2 l12 + φ3 l34 + φ4 l34 " ( ( φj K ( 0 1 ( % $ Q1 " $ B" |J(TK )| ( K kl lij

kl kl min(lij ) = min(lij )

i

kl φi ≤ |J(TK )| ≤ max(lij )

i

kl φi = max(lij ),

$ "

##

&

" '

J(TK ) $ kl

% lij " A

( B"3. $

%

%

K

##

#

π"

1 %

$ % " ( l12 13 l13 ( π ## l12 ≤ 0 |J(TK )| ≤ 0 V1

"

7 B"B3

Q1

$ - "

♦

/

( Q1

%℄&!!% %℄&!!% " !!% ' & ( ) * % ( ' + * % ! !% ! ! %

//

x 1 x 2

.

[0, 1]

0.02

K1 K2

% " K1 / (0, 0) (1.5, 0) (0.1, 1.4) (1.6, 1.8) K2 / (1.3, 1.1) (1.5, 0) (0.1, 1.4) (1.6, 1.8)"

"

& //

(

/0( #

"

" ## 1

%

1

2+3 3 4 3 3 5 6℄ 2&+$ 4 3 3 5 6℄

"

(

1

!

+*℄(#333&*333&%

+*&℄1 #2**%

1

%

0#*&*"##%%% #+3*3*℄%

" 1

+*&℄1 #2&**% 0#*&*"##%%% #+3*3*℄% 1

7 B"B. # & "

x 2 → K2 K

%

1 /

% (

$ #

$

( x1 , x 2 )"

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0

0.2

0

0.2

0.4

0.6

x b2 b V3

.

♦

b1 V

0.8

1

1.2

1.4

1.6

b4 V b K

b2 V x1

0

0

0.2

0.4

0.6

x b2 b V3

b1 V

Q1

0.8

1

1.2

1.4

1.6

b4 V b K

b2 V x1

- . -

* 5

$

! Xh ⊂ V (Ωh ) ( V (Ωh ) $ Πh

1

Th

Πh : V (Ωh ) → Xh ,

Πh v(x) =

K ∈ Th ΠK v(x) =

σi (v)φi (x).

i=0

$

Nh

b N

σi (v)φi (x),

i=0

ΠK : V (K) → PK

x ∈ K.

Ωh " Ωh ⊆ Ω " -

$ 1 ( (

Ω

$

5 I6 ;D℄"

1 $ " 6

r r > 0 TK '" * r r P Q (

" r Πh % ( v ∈ C 0 (Ωh )

Πhr v(x) =

Nh

x ∈ Ωh ,

v(Ni )φ(x),

i=1

φi i& ( ( Ni " (

$ /

K

1

TK q > d2 − 1 l = min(r, q)" q+1 0 ≤ m ≤ l + 1 ∀v ∈ H (K)

'

%

r ≥ 1 1 C > 0 hK

m |v − Πhr (v)|H m (K) ≤ Chl+1−m γK |v|H l+1 (K) . K

γK

m>0

B"38

% ( (

"

r 1 # % Xh $ H & ( m ( H m > 1" %

Hm

%

|v − Πhr (v)|H m (Ωh ) ≤ C

K∈Th

m > 1" m = 0, 1

2(l+1−m) 2m γK |v|2H l+1 (K) hK

1/2

≤

Cγhm hl+1−m |v|H l+1 (Ωh ) ,

B"39

*

$ γh ≤ γ K∈Th

γh = max γK "

h

|v − Πhr (v)|H m (Ωh ) ≤ Chl+1−m |v|H l+1 (Ωh ) .

5

B"3:

5 1

5 I*2C8 048℄"

Qr

r≥1

# " 6%

TK # 1

K

Th 4 K Vj

Kj j = 1, . . . , 4"

% B"9"8"

ρK = min ρKj 1≤j≤4

γK =

hK . ρK

'

γK % (

1 ' " Qr

ΠK K " C > 0 % 0 ≤ m ≤ r + 1 ∀v ∈ H r+1 (K) r

max(4m−1,1) k+1−1

Q |v − ΠK |H m (K) ≤ CγK

h

|v|H r+1 (K) .

B"3;

'

γK ## ' % L2

m = 0"

5 % ) " h * " Pr + # % Qr , -

. Pr /

Parte I

Problemi stazionari

"#$ %

#

# /

u ∈ V : a(u, v) = F (v) ∀v ∈ V.

3"B

# $ O& I* 4.℄ %

/

' V 0 + a(·, ·) : V × V → R #

"

"$ α+ F : V → R # / 0 / 1 023 4

1/2 / 3

||u||V ≤

||F ||V ′ =

1 ||F ||V ′ , α

|F (v)| . v∈V,||v||V =0 ||v||V sup

#

Ω ⊂ Rd d = 1, 2" V ⊂ H 1 (Ω)"

%

( "

# )

% #

Ω

%

%

V =

ΓD = ∅

HΓ1D

ΓD "

1

≡ {v ∈ H (Ω) : v|ΓD = 0}.

V

3"3

4

uL2 (Ω) ≤ CP ∇uL2 (Ω) ,

3".

1

67 %

CP Ω " 0 3". 5 2 1 ( L (Ω) H (Ω)

∇u2L2(Ω) ≤ uH 1 (Ω) ≤ (1 + CP2 )∇u2L2 (Ω) . ΓD

$

V = H 1 (Ω)

3"8

$

$

# " 1 # ) &

u = g ΓD g = 0" '

5 #

% #

Ω

"

g

R

g

$ '

% # 1/2 g ∈ H (ΓD )"

Ω

C1

Ω

Ω

% &

$

$

"

( "B. & G ∈ H 1 (Ω) G|ΓD = g

I* 4.℄"

◦

u≡ u − G, ◦

u∈ V ≡ HΓ1D (Ω)

# / ◦

%

a(u, v) = Fg (v) ≡ F (v) − a(G, v)

∀v ∈ V.

# 5

O&

Fg $ F (·) a(·, ·)" * % # 1

%

(

O&

a(·, ·)

F (·)

(

# ( "

%

Wg ≡ {v ∈ H 1 (Ω) : vΓD = g}

3"9

Wg ) $ % %

$ ( $ % " '"

#

% 5 ( ( % $ O& "

67 %

Gh

Gh

x0

ΓD

x1

xn

+

' ( " u ∈ Wg : a(u, v) = F (v) ∀v ∈ V

. /

≡ HΓ1D ,

3":

" Wg ≡ {v ∈ H 1 (Ω) : vΓ = g}+ g ∈ H 1/2 (ΓD ) Ω ! / ' # a(·, ·) * +

" V × V # F (·) * V + $ / , D

||u||V ≤ C(||F ||V ′ + ||g||H 1/2 (ΓD ) ),

" C

"$ α cγ /2 / 6

( ( $

G

Wg

G+V"

$

5 #

" # 1

"

(

i∈BΓD g(xi )φi |ΓD BΓD $ xi %

# ΓD φi ( # " 6 $ Gh = i∈BΓD g(xi )φi " ( 1 Vh

# 5

gh =

% 1 % % % #

% 1

7 3"B" * $

% #

% #

!" * ( ( # 3": 0 < & 1 r Gh ∈ Xh 1 B"34 # 1 / uh ∈ Gh + Vh ≡ Wg,h

67 %

Gh + Vh ≡ {vh ∈ Xhr : vh|ΓD = Gh } %

a(uh , vh ) = F (vh )

vh ∈ Vh

3";

Vh = {vh ∈ Xhr : vh|ΓD = 0}.

% (

#

u % du du d + σu = f, ν +β − dx dx dx

) ( /

x ∈ (a, b),

# '

σ

(

x"

3"D

ν β f

) 1

(

"

%

#

# 6 . )

"

( ( % # 3"D H m+2 (a, b) f ∈ H m (a, b) m ≥ 0"

'

/ /

# #

B"

−u′′ + u = 0 x ∈ (0, 1), u(0) = 1, u(1) = e.

3"C

( # # # $

"

3" 0 < & 1

(

1

#

( h" 1 % H 1 (0, 1) L2 (0, 1) h % 1/10 1/320 %

$

u(x) = ex "

5

& / /

# %

% " % ( #

3"C1 ( 1 v ∈ V ≡ H0 (0, 1)H (0, 1) ##

1

′′

− u v dx + 0

1

∀v ∈ V.

uv dx = 0

0

%

v(0) = v(1) = 0

1

′ ′

u v dx +

1

∀v ∈ V.

uv dx = 0

0

0

3"B4

ΓD = {0, 1} ( # / u ∈ Wg 1 3"9 % a(u, v) = 0 ∀v ∈ V a : V × V → R $ (

#

a(u, v) ≡

1

′ ′

u v dx +

0

1

uv dx.

3"BB

0

# # 3"BB 6 3"B/ ## 1 % ( #

V ×V" &

&

a

a

$

$

## %/

a(·, ·)

1

1

|a(u, v)| ≤

u′ v ′ dx

+

uv dx

0 0 ≤ u′ L2 (0,1) v ′ L2 (0,1) + uL2(0,1) vL2 (0,1) ≤ 2uV vV ;

"

a(u, u) = u′ 2L2 (0,1) + u2L2 (0,1) = u2V . * 6 3"B 3"BB

$ # "

Vh V Vh → V (

∞

dim(Vh ) → uh ∈ Gh + Vh

1 % 0 < 3"BB $/

!

67 %

%

Vh

a(uh , vh ) = 0 ∀vh ∈ Vh " 1 Pr Vh ≡ {vh ∈ Xhr : vh = 0

Xhr

r = 1, 2, 3

ΓD } ,

3"B3

$ B". 6 B"

(xj−1 , xj )

Th

(0, 1)

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3"B.

( # # "

3" 1

(

&

( "

& / /

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u ∈ V ≡ H 1 (0, 1)

%

5

1 0

u′ v ′ dx − u′ (1)v(1) + u′ (0)v(0) +

1

uv dx = 0

0

,

∀v ∈ V.

3"B.

( #/

1

u∈V

%

2 u v dx + u(0)v(0) + 5 ′ ′

0

1 0

3 uv dx = ev(1) − v(0) ∀v ∈ V, 5

3"B8

( #

a(u, v) ≡ (

1

′ ′

u v dx +

0

1

2 uv dx + u(0)v(0) 5

0

F (v) ≡ ev(1) − 3/5v(0)"

( # $

2 a(u, u) = u′ 2L2 (0,1) + u2L2(0,1) + u2 (0) ≥ u2V . 5 F

2 |a(u, v)| ≤ u′ L2 (0,1) v ′ L2 (0,1) + uL2 (0,1) vL2 (0,1) + |u(0)| |v(0)| 5 ≤ (2 + 2C/5)uH 1(0,1) vH 1 (0,1) , 6 %@& %P

" (

F

$ "

# 0 < 1 / uh ∈ Vh % 1 vh ∈ Vh Vh Xh "

aij

a(uh , vh ) = F (vh )

aij ≡ a(ϕj , ϕi ) fi ≡ F (ϕi )"

u % %

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ui # I* 4.℄

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C $ h" h = h1 h = h2 q q h1 e(h1 ) ≃ ⇒ q ≃ log(e(h1 )/e(h2 ))/ log(h1 /h2 ). e(h2 ) h2

5

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p

T % L x ∈ (0, L) # ) /

A

−kAT ′′ + σpT = 0, x ∈ (0, L), T (0) = T0 ,

$

A k

T ′ (L) = 0,

3"B:

'

'

(

T0

σ

$ "

B"

"

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( # 3"B:

0 < & 1 " H01 (0, L) k σ p T0 % " ." 1 % 1 ##

1 # 1 T0 = 10 σ = 2 k = 200"

(

!1

67 %

& / /

) 3"B: (

T (x) = C0 eλ0 x + C1 eλ1 x ,

3"B;

C0 C1 # # λ0 λ1

m =

σp/(kA)

−kAλ2 + σp = 0. λ0 = −m λ1 = m & T (x) = C0 e−mx + C1 emx " C0

C1 #

% % C0 + C1 = T0 −C0 e−mL + C1 emL = 0"

C0 =

T0 emL −mL e + emL

=

T0 emL T0 e−mL T0 e−mL , C1 = −mL , = mL 2 cosh(mL) e +e 2 cosh(mL)

T (x) =

cosh(m(x − L)) T0 . em(L−x) + e−m(L−x) = T0 2 cosh(mL) cosh(mL)

( # 3"B:" ΓD v ∈ V ≡ HΓ1D

3"B: (

= {0}

&

L L ′′ −kA T (x)v(x) dx + σp T (x)v(x) dx = 0 ∀v ∈ V. 0

0

L L ′ ′ kA T (x)v (x) dx + σp T (x)v(x) dx − kA[T ′ v]L 0 = 0, 0

[f ]L 0

0

f (L) − f (0)" 6 T ∈ Wg

( #/

%

a(T, v) = 0

3"BD

a(·, ·) : V × V → R

∀v ∈ V,

$ ( #

3"BC

5

!

L L ′ ′ a(u, v) ≡ kA u (x)v (x) dx + σp u(x)v(x) dx. 0

0

2 1 %

" ( # $

|a(u, v)| ≤ M uV vV

M = kA + σp

$ %

V

%L

a(u, u) = kAu′ 2L2 (0,L) + σpu2L2 (0,L) ≥ kAu′ 2L2 (0,L) ≥ αu2V ,

α = kA/(1 + L2 /2)"

( %

v(0) = 0

3"34

%

2

x

x x

2 ′ 2

|v(x)| = v (x)dx ≤ 1 dx (v ′ (x))2 dx ≤ xv ′ 2L2 (0,L)

0

0

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L vL2 ≤ √ |v ′ 2L2 (0,L) , 2

% $ % L"

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=

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+

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% % %

a(u, u) ≥ min(kA, σp)u2V . * % ( # %

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% %

"

Vh V 1 Nh " & #

Gh

0 < 3"BC $ /

Th ∈ Gh + Vh

%

a(Th , vh ) = 0

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(0, L)

!

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u ∈ H q+1 (0, L)

q>0

CM s h |u|H s+1 (0,L) , α

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%

⎧ ⎨

d2 T = α(H − y)2 (0, 2H), dy 2 ⎩ T (0) = Tinf , T (2H) = Tsup , −

α = 4U 2 µ/(H 4 κ) κ = 0.60 $ '

µ = 0.14

∇ · (µ∇u)v dω +

Ω

b · ∇uv dω =

Ω

v ∈ V ≡

f v dω.

Ω

( 0

µ∇u · ∇v dω +

Ω

b · ∇uv dω =

Ω

Ω

f v dω +

gN v dγ.

ΓN

( # 3".: $ / Wg = {w ∈ H 1 (Ω) : w|ΓD = gD } %

u∈

a(u, v) = F (v) ∀v ∈ V, 3".; a(u, v) ≡ Ω µ∇u · ∇v dω + Ω b · ∇uv dω F (v) ≡ Ω f v dω + ΓN gN v dγ "

2 1 % 6 3"B" #

a(·, ·)

F (·)

"

γT

|F (v)| ≤ ||f ||L2 (Ω) ||v||L2 (Ω) + ||gN ||L2 (ΓN ) ||v||L2 (ΓN ) # $ ≤ ||f ||L2 (Ω) + γT ||gN ||L2 (ΓN ) ||v||H 1 (Ω) ,

%

"B3"

%

a

$

% (

vb · ∇v dω =

1 2

Ω

b · ∇v 2 dω =

Ω

∇ · b = 0"

( ( %

a(u, u) =

b·n ≥0

ΓN

%

b · nv 2 ,

ΓN

µ|∇u|2 dω +

Ω

1 2

v∈V

*

1 2

b · nu2

ΓN

%

a(u, u) ≥ µ0 ∇u2L2 (Ω) .

L % $

#

u

# % ( # $

2

α = µ0 /(1 + CP ) CP L" % a(·, ·) $ % " ) ##

|a(u, v)| ≤ µ∇u · ∇v dω + b · ∇uv dω

Ω

Ω

≤ µ1 ∇uL2 (Ω) ∇vL2 (Ω) + b1 ∇uL2 (Ω) vL2 (Ω)

≤ M uV vV ,

M = max{µ1 , b1 }"

## 1 % $

"

0 < # 3".; $/

%

a(uh , vh ) = F (vh )

Vh

uh ∈ Wg,h

∀vh ∈ Vh ,

1 B 3

V"

)

"

' %L ( f (x, y) = 4π 2 (sin(2π(x +

# )

y)) + sin(2π(x − y))) + 2π cos(2π(x + y)) gD = sin(2πx) cos(2πy) gN = −2π cos(2πx) cos(2πy)" 1 #

*

67 % 1

10

10 10

1

0

0

1

−1

10

1

10

2

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- . & - .

0" "#& % #& % 0" V ≡ a(v, u) =V < u, Lv >V ′ ,

%

L = LS + LSS ,

con

⎧ 1 ∗ ⎪ ⎪ ⎨ LS = 2 (L + L ) , ⎪ ⎪ ⎩ L = 1 (L − L∗ ) . SS 2

) H01 (0, 1)

( #

1

′′

(−µu + σu) vdx = µ

0

0

1

′′

′ ′

u v dx +

1 0

σuvdx ≡ aDR (u, v).

Lu = −µu + σu < Lu, v >V "

( 1 V ′

2 6 ( (

v

% %

aDR (·, ·)

,$

u′ v ′

u

$

"

6 "

%

b " aT (u, v) ≡ b

1 0

( #

(

′

u vdx = −b

1 0

."B

b $ $

LS u = −µu′′ + σu,

6

aT (u, v) =

uv ′ dx = −aT (v, u).

1

b

1 bu vdx = 2 ′

0

LSS u = bu′ .

."3

" %

1

1 bu vdx + 2 ′

0

1 0

1 (bu) vdx − 2 ′

1

b′ uvdx.

0

$ #

"

(

1 2

1 0

# ′ $ 1 bu + (bu)′ vdx = − 2

1 ′ LS = −µu + σ − b u, 2 ′′

/ /

1 0

$ # (bv)′ + bv ′ udx.

LSS =

1 ′ 1 ′ bu + (bu) . 2 2

♦

) & & (

Lu = −µ△u + b · ∇u + σu

µ

σ

& / /

.".

b

u ∈ H01 (Ω)"

> %

b · ∇u =

1 1 1 b · ∇u + ∇ · (bu) − (∇ · b) u. 2 2 2

,*

; V

< Lu, v >V ≡

′

Ω

1 2

Ω

µ∇u · ∇vdω +

Ω

1 2

(∇ · b) uvdω +

(−µ△u + b · ∇u + σu) vdω =

1 2

Ω

(b · ∇u) vdω +

Ω

σuvdω ≡ a(u, v).

Ω

∇ · (bu)vdω−

µ∇u · ∇vdω + Ω (σ − 12 ∇ · b)uvdω $

Ω ( "

( #

aS (u, v) ≡

( 0

aSS (u, v) ≡ 1 − 2

1 2

Ω

(b · ∇u) vdω +

1 ∇ · (bv)udω − 2 Ω

Ω

1 2

Ω

(b · ∇v) udω = −aSS (v, u).

$ LS u 1 1

$ LSS u = 2 (b · ∇u) + 2 ∇

/ /

∇ · (bu)vdω =

= −µ△u− 12 (∇·b)u+σu · (bu). ♦

# /

⎧ 2 ⎪ ⎨− ⎪ ⎩

∂2u ∂2u ∂2u ∂u ∂2u +β 2 +γ +δ 2 +η = f in Ω, ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 i,j=1 ∂xi ∂xj

u=0

su ∂Ω,

β γ δ η '

f x = (x1 , x2 ) ∈ Ω "

B"

."8

$ (

'

%

#"

3"

% 0 < & 1 "

."

# $

& ( #

K

%

#

"

& / /

# 5 (

2 −∇ · (K∇u) + B · ∇u = f

Ω K B

,, ."9

-

M 1

⎤ 1 (1 − γ) ⎥ ⎢ 2 ⎥, K≡⎢ ⎣ ⎦ 1 (1 − γ) 1 − δ 2 ⎡

⎡ ⎤ η B ≡ ⎣ ⎦. 0

1−β

."9

#

u=0

∂Ω "

)

# $

1 I*2C8℄ 6 " : 2 µ0 > 0 % χ R ## 2

i,j=1

µKij χi χj ≥ µ0 ||χ||2 .

.":

1

K

@ I*

44 ℄ #

$ 1

" 1

%

1 (1 − β)(1 − δ) − (1 − γ)2 > 0. 4

1 − β > 0,

.";

% - ."; 5

δ V "

% # $

##

% ( # $

"

( #

1 2 (∇u)u = ∇|u| L 2 a(u, u) = (K∇u) · ∇udω + B · (∇u)udω ≥ Ω

2 µ0 ||∇u||L2

1 + 2

Ω

Ω

$ # 2 ∇ · Bu2 dω ≥ α||u||V +

α = µ0 /(1 + CP2 ) CP

%

2

B · nu2 dγ = α||u||V

∂Ω

$ L" &

$

O&

"

V

Vh ⊂ Xhr Xhr

$ B"34 6 B"

Vh r ( Xh #" ( # / uh ∈ Vh % vh ∈ Vh ## a(uh , vh ) = F (vh )" # $ # (

# "

u ∈ H s+1 (Ω)

I* 4.℄ 6 " .

γ Chq |u|H q+1 , α

||u − uh ||H 1 ≤

M

$ ( #

q = min(r, s)"

|η| ≫ µ0

γ = µ1 + |η| 1 "

C

$

% %

γ/α ≫ 1

#

+ # ," 1 %

η=0

( #

."B"3" (

a(u, v) =

(K∇u) · ∇vdω =

Ω

K

a(u, v)

$

∇u · (K∇v) dω = a(v, u).

Ω

#

%

) '

Aij = a(ϕj , ϕi )

%

η =0

$

2

,0

% 1 -

%

( # " I*

44 ℄/

B" ( #

# 6%H

%

( & % ( & "

A

(

$ ( 6% 0. 1 + CP2 %

min ψ ′′ < 0

( "

6 ( # ##

u′ = ρ′ eψ/µ + ρeψ/µ

ρ"

u = ρeψ/µ

ψ′ ⇒ µu′ = µρ′ eψ/µ + ψ ′ u. µ

- #

u

%

u(0) = u(1) = 0 ⇒ ρ(0) = ρ(1) = 0. #

ρ

# )

⎧ # $′ ⎪ ⎨ − µeψ/µ ρ′ = 1 x ∈ (0, 1), ⎪ ⎩ ρ(0) = 0,

ρ(1) = 0,

( #

# $/

µ

1

eψ/µ ρ′ v ′ dx =

0

( # $ %L

0

ρ∈V

%

1

vdx

∀v ∈ V.

ψ ∈ L∞ (0, 1)

% ## ( "

$ ( %

µ > 0

%

#

% 5

L"

0*

; 6 1

ψ = αx"

6

( ( #

C1

′ −µ eαx/µ ρ′ = 1

x −αx/µ , e ρ = C1 − µ ′

da cui

$ "

µ x µ ρ = −C1 e−αx/µ + e−αx/µ + 2 e−αx/µ + C2 , α α α C2 $ " C1 C2 #" 6M (

ρ=

! ! 1 eαx/µ − 1 1 1 − e−αx/µ −αx/µ ⇒ u = + xe x − . α 1 − eα/µ α eα/µ − 1

E # ) 1

# % ( ."./

||ψ ′ ||L∞ (0,1) h 2µ

e

||ψ ′′ ||L∞ (0,1) h2 . 6µ

1

2µ , h< ′ ||ψ ||L∞ (0,1)

h
1

ρ

#

ρ

u

/

1

1

; h = 0.1 h = 0.01 h = 0.001 K(A) 3.917901e + 34 3.526591e + 42 1.282097e + 44 eL2 0.00049809 0.00010876 1.2482e − 06 eH 1 0.049641 0.02894 0.0032238

8 ρ µ = 0.01 α = 1

h = 0.1

h = 0.01

h = 0.001

0,01F ! */,** F! ! 1/ ,0F!* /0$F!1 0 $, F 0

0 !F !!0F! ,,,0!$!F! $1/1F , 8 ρ µ = 0.01 α = −1 K(A) eL2 eH 1

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 u ρ µ = 0.01 h = 0.001 α = 1 - . α = −1 -. )

&

)

ρ u" 2

α = −1 ρ - u # " # ) &

( M % u

# 7 ."BC"

♦ 5%"

1

#

# &

"

ρ

# #

" '

$ I ;.℄"

"

#% I D8℄ 6 " 9"3 / +E ( - #

%

) ," ( #

# $ &

- %

1

"

//

# ) & &

⎧ −△u + ∇ · (βu) + u = 0 in Ω ⊂ R2 , ⎪ ⎪ ⎪ ⎪ ⎨ u=ϕ su ΓD , ⎪ ⎪ ⎪ ⎪ ⎩ ∇u · n = β · nu su ΓN ,

."B9

Ω $ ∂Ω = ΓD ∪ΓN ΓD ∩ΓN = ∅ ΓD = ∅"

# ( # T ( β = (β1 (x, y), β2 (x, y))

ϕ = ϕ(x, y)" ||β||L∞ (Ω) ≫ 1 # & 1

& 1 0& 1 0 < & 1 " 1

Ω = (0, 1) × (0, 1) β = (103 , 103 )T ΓD = ∂Ω

⎧ 1 per x = 0 0 < y < 1, ⎪ ⎪ ⎪ ⎪ ⎨ ϕ = 1 per y = 0 0 < x < 1, ⎪ ⎪ ⎪ ⎪ ⎩ 0 altrove.

1 % % % # # (

% # "

& //

v ∈ V ≡ HΓ1D (Ω). ( 0 %

(

Ω

∇u · ∇vdω −

∂Ω

∇u · nvdγ +

∂Ω

β · nuvdγ −

β · ∇vudω +

Ω

6

#

uvdω = 0.

Ω

ΓD

ΓN

1!

; −

∇u · nvdγ +

∂Ω

−

β · nuvdγ =

∂Ω

(∇u · n + β · nu) vdγ −

(∇u · n + β · nu) vdγ.

ΓN

ΓD

ΓN ∇u · n − β · nu = 0 ΓN " * 1 % + &

ΓN " ΓD # G(x, y) ( H 1 (Ω) % G(x, y) = ϕ(x, y) G $ % ϕ ∈ H 1/2 (ΓD ) %

,

v = 0"

ΓD "

Ω

'

"

' %L

1 " # ( # 5 /

u∈G+V

%

v∈V

a(u, v) = 0,

a(u, v) ≡

∇u · ∇vdω −

Ω

β · ∇vudω +

Ω

uvdω.

Ω

# (

( 3"B"

% β ( L∞ (Ω)" 6

% ( # $

V

(

a(w, w) = ||∇w||2L2 (Ω) −

1 2

H 1 (Ω)"

w ∈V

β · ∇w2 dω + ||w||2L2 (Ω) .

Ω

||w||2V "

"

( 0 #

1 − 2

1 β · ∇w dω = − 2 2

Ω

1 β · nw dγ + 2 2

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