Applicazioni ed esercizi di modellistica numerica per problemi differenziali (UNITEXT La Matematica per il 3+2) [1 ed.] 9788847002579, 8847002575 [PDF]

Questo testo contiene una raccolta di esercizi riferiti agli argomenti tipici di un corso di metodi analitici e numerici

152 32 6MB

Italian Pages 399 Year 2005

Report DMCA / Copyright

DOWNLOAD PDF FILE

Papiere empfehlen

Modellistica Numerica per Problemi Differenziali, 3a edizione (UNITEXT La Matematica per il 3+2) [3a ed. 2006. Corr. 2a stampa] 8847004934, 9788847004931

111 105 7MB Read more

Matematica numerica (UNITEXT La Matematica per il 3+2) [2 ed.] 8847000777, 9788847000773

104 49 10MB Read more

Matematica: si parte!: Nozioni di base ed esercizi per il primo anno di Ingegneria (UNITEXT La Matematica per il 3+2) [1 ed.] 8847006759, 9788847006751

113 5 1MB Read more

Modellistica numerica per problemi differenziali 978-88-470-0493-1, 978-88-470-0494-8

110 65 9MB Read more

Esercizi di finanza matematica (UNITEXT La Matematica per il 3+2) Italian [1 ed.] 9788847006102, 8847006104

102 22 1MB Read more

Esercizi per il Corso di ebraico biblico

114 27 26MB Read more

Esercizi per il corso di Ebraico biblico

108 94 26MB Read more

Modellistica Numerica per Problemi Differenziali [5 ed.] 978-88-470-2747-3, 978-88-470-2748-0

117 81 17MB Read more

Problemi Di Scacchi Per Principianti

0 0 38KB Read more

Un invito all'Algebra (UNITEXT La Matematica per il 3+2) Italian [1 ed.] 884700313X, 9788847003132

117 41 2MB Read more

Applicazioni ed esercizi di modellistica numerica per problemi differenziali (UNITEXT La Matematica per il 3+2) [1 ed.]
9788847002579, 8847002575 [PDF]

Author / Uploaded
Luca Formaggia
Fausto Saleri
Alessandro Veneziani

0 0 0
Gefällt Ihnen dieses papier und der download? Sie können Ihre eigene PDF-Datei in wenigen Minuten kostenlos online veröffentlichen! Anmelden

Datei wird geladen, bitte warten...

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)

! " # $

%

& #

" ' % # $ # (

% )& # " *

# ( %

( +

, -

- "

+

,

#

# $ # ( %

" ## % &

" 0

% (

" $ %

. /

(

-

&

% (

1

" ##

" *

&2 344. 5 & "

#

%

6

%

( ( "

$

#

# " ( 6 ( % # 7 " $

6 3 . 8" 6 3 #

)

6 . # 6 8

) 1 "

6 9 : # #

#

) 1 1 6 ; & " 7"? % " ?@

/ & ( # (

" 2 % % # &

(" 1 * % % % ( "

& # 1 % " A 1 " 7 !

&

%

% ( "

3449

0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "

#

$#

. . . . . . . . . . . . . . . . . . . . % & % $# ' $# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % (

! %

# ) $#

" #

! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B:9

9"B

) 1 " " " " " " " " " " " " " " B:;

9"3

1 " " " " " " " " " " " " " " " 3BD

!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3..

:"B

# " " " " " " " " " " " " " " " " " " " " " " " " 3..

:"3

#

% " " " " " " " " " " " 3::

! $%& ' (

. . . . . . . 3D9

;"B

# " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 3D:

;"3

# & " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .BC

) ) * + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 "B

6 1 " " " " " " " " " " " " " " " .89

"3

# " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .8:

".

? # " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .8D

"8

# " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .8C "9 Lp H s " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .94 p ":

l " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .9. ";

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .98

, -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

!"B

!

% #

" " " " " " " " " " " " " " " " " " " " " " " " " " " .99

!"3

E

%

" " " " " " " " " " " " " " " " " " " .9;

!".

!"3"B

( 6>> " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .:3

!"3"3

(

!"3".

( 6 A " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .:;

!"3"8

( A " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .:C

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .:.

" " " " " " " " " " " " " " " " " " " " " " " " .;3 !"."B

# " " " " " " " " " " " " " " " " .;.

!"."3

E

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " .;8

!".".

E

+ , " " " " " " " " " " " " " " " " " " " " " " " " " " .;9

!"."8

6 # " " " " " " " " " " " " .;:

. .+ ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .;C * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CB

( 1 " * (

1

"

1 # # (

1 Vh

( # "

Vh

( "

%

[a, b] n % % &

Kj = [Vj−1 , Vj ] hj = Vj −Vj−1 j = 1, . . . , n n+1

Vi i = 0, . . . , n V0 = a Vn = b" E %

% Th (a, b) -

Th $ " # V

x

%

"

c)

0

6

7

1

8

9

2 10

11 3 12

13 4 14 15 5

K2 b)

0

a)

2,0

2,1

2,2

2,3

1

2,1

2,2

2

1,1 1,2

K1 0

K2 1

3,1 3,2 3

K3 2

4,1

4,2 4 5,1 5,2 5

K4 3

K5 4

5

r > 1

a)

Kj

•

K2

K

(

" #

r−1

r≥1

%

r − 1 xj,s = Vj−1 + shj /r j = 1, . . . , n s = 1, . . . , r − 1 7 B"B" xj,0 ≡ Vj−1 xj,r ≡ Vj "

Kj r ( f xj,s s = 0, . . . , r "

Kj

1

%

" *

"

s = 0, . . . , r

" F (

% " $ ( # $

xk = xj,s k = r(j − 1) + s s = 0, . . . r − 1 j = 1, . . . , n

1 xrn+1 = xn,r " ## N = nr + 1

% ( x0 < x1 < . . . < xN " % # # 1

"

% % $ 7 B"B %

( #

j = 1, . . . , n j − 1 s = 0, (j, s) ⇒ n + (j − 1)(r − 1) + s,

(n, r) ⇒ n"

(j, s)

s = 1, . . . , r − 1,

% $ %

n+1

(

xi = Vi

i = 0, . . . , n"

F

%

% #

% "

(

f ∈ C 0 ([a, b]) [a, b] ⊂ R r Th ( Πh f % ( Πhr f |Kj ∈ Pr (Kj ),

x0 , . . . xN

Πhr f (xi ) = f (xi ),

j = 1, . . . , n,

1

i = 0, . . . N,

Pr (Kj ) $ r Kj h = max1≤j≤n hj " h $ " Πhr f 5

#

r φi i = 0, 1, . . . , N

Πhr f (x) =

N

f (xi )φi (x).

B"B

i=0

(

φi (xj ) = δij ,

i, j = 0, . . . , N,

B"3

δij " φi (x) = 0 x Kj xi

%

"

$ N φi (x) = 1, i=0

∀x ∈ [a, b].

$

# B"B ( f = 1" 5 % & r

Πh : C 0 ([a, b]) → Xhr (a, b) 1 r B"B Xh (a, b) ( r Th

Xhr (a, b) ≡ {vh ∈ C 0 ([a, b]) : vh |Kj ∈ Pr (Kj ), j = 1, . . . , n}.

B".

r

( # Xh (a, b) % nr + 1" Πhr $ ( h

f C 0 ([a,b]) ≡ max |f (x)|, a≤x≤b

∀f ∈ C 0 (a, b),

( E #

5 ( H 1 (a, b)" ( % H 1 1 "BB/ -

Cr > 0

%

∀v ∈ H 1 (a, b) Πhr vH 1 (a,b) ≤ Cr vH 1 (a,b) .

B"8

!

φi

φ0

φn

a

1

b xi

x0

xn

" % v ∈ H p+1 (a, b) p > 0 s = min(r, p)

|v − Πh1 v|H 1 (a,b) v − Πh1 vL2 (a,b)

n 2s 2 ≤Cs,r,1 hj |v|H 2 (Kj ) ≤ Cs,r,1 hs |v|H s+1 (a,b) ,

n 2(s+1) 2 |v|H 2 (Kj ) ≤ Cs,r,0 hs+1 |v|H s+1 (a,b) . ≤Cs,r,0 hj

B":

j=1

√ v ∈ H 2 (a, b) % C1,1,0 = 5/24 C1,1,1 = 2/2" 1 / H 2 (a, b) 5 ( v ∈ H (a, b) v ∈ 2

L ( % n 2 2 r v − Πh vL2 (a,b) ≤ Cr hj |v|H 1 (Kj ) ≤ Cr h|v|H 1 (a,b) ,

B"9

j=1

j=1

√ C1 = 2

%

lim |v − Πhr v|H 1 (a,b) = 0.

B";

h→0

//

2 1 B"9 &

(

f1 (x) = sin2 (3x)

f2 (x) = | sin(3x)| sin(3x),

(1, 3) r = 1, 2, 3, 4" & s 2

( % f1 ∈ H (1, 3) s ≥ 0 f2 ∈ H (1, 3) 3 f2 ∈ / H (1, 3)" (

# ( 4

8 16 32

64"

& //

( 1 # 1

(

"

&

'

#

$

&

" L2 H s s ½ "

1

#

"

" A - 1

!"#$ % "#$ %! &!'#"#$ %% "#$ %! (&!$ #$ % #'#"#$ %%"#$ % "#"#$ %%%! )" " #*$*% #)%&

#**(%

& #*&*(% #)% # *+*$℄***"% &#)% # &*+*$℄*-&*(&.**"% & "( 0 (

h

7 B"."

( ## "

#

- (

f2

1

♦

h"

1

K

$ %

(

#

#

TK

K

Rd

'

7 B"8

K = TK (K).

B"D

K ¾ "

%

( "

TK 1

/

%

% & % ' (

2

h

Rd

r

)

d+1

*

0

10 0.2

10

0.1

10

0

h

||f−Π1 f||

1

||f−Π2h f||1

10

−0.1

10

−0.2

−1

10

10

−0.3

10

−0.4

10

−0.5

10

−1

−1

10

10

h

h

−1

10

−1

10

−2

||f−Π4h f||1

3

||f−Πh f||1

10

−2

10

−3

10

−4

10

−3

10

−1

−1

10

10

h

h

" H 1 (1, 3) %

f2

h f1 &

+

r = 1, 2, 3

4

C 1 (K) % " C 1 (K)

G # J(TK )

$ % *

%

" K

TK

!

$ (

, TK ( x) = aK + FK x

B"C

aK ∈ Rd FK ∈ Rd×d $

& |FK | > 0 # | · |

"

% hK K hK = maxx1 ,x2 ∈K x1 − x2 ρK

% ( K "

% x d x ∈ R " Th (Ω) Ω $ K = TK (K) %

Ωh 1 ⎞ ⎛

K⎠ , Ωh = int ⎝

K∈Th (Ω)

#

K1

K2

y

TK1

,

x

TK2

yb b K

x b

+ Kb - &

TK1 b K2 = TK2 (K)

& .

(0,1)

yb

(0,0,1)

(1,0)

(0,1)

yb

(1,1)

(1,0)

x b

x b

zb

(0,1,0)

(0,0,1)

R2

R3

yb

(1,0,0)

zb

(0,1,1)

x b

(1,1,1)

(1,0,1)

yb (1,0,0)

b K1 = TK1 (K)

&

b K

x b

&

Ω % limh→0 d(∂Ω, ∂Ωh ) = 0 h = maxK∈Th (Ω) hK d(A, B) A B Rd " & int(A)

AH ( % Ωh Rd " Ωh = Ω 5

## ( Ω $ TK $ '"

% Th Th (Ω)

% " TK $ ' % !" ' TK ( r r / PK b = P (K) PK = P (K)" K $ $ % K " 1 TK (

"

/

Th 1 Vh " Eh . (

Fh (

"

# K K1 , K2 ∈ Th K1 = K2 ## 7 B":

(

◦

◦

K 1 ∩ K 2 = ∅,

K1 ∩ K2 ∈ Vh ∪ Eh ∪ Fh ∪ ∅.

( "

" - .

- .

nv

&

Nl,b Nv,b

#

%

K Ne Nv Nl

#"

m $

m = 0

$ +( ,

( %

I7044℄

(a) Ne − Nl + Nv = 1 − m, (c) Nv,b = Nl,b .

Nf

cb

(b) 2Nl − Nl,b = nv Ne , m

B"B4

(

(

Nf,b

(a) Ne − Nf + Nl − Nv = m − cb − 1, (c) Nv,b + Nf,b = Nl,b + 2(cb − m). # $ %

Ω

Ω

# %

(b) 2Nf − Nf,b = nv Ne ,

B"BB

$ (

" A

1 % (

∃γ > 0 % ∀h > 0 γK =

hK ≥ γ, ρK

{Th (Ω)}h

∀K ∈ Th (Ω).

h

B"B3

γ

#

$ %

$

0

# $ "

) % (

"

//

. ( (

Ne

Nv (

$ (

Nf,b

#

"

% (

"" * #K

& //

Nf,b 2Ne + " 2

nv = 4

B"BB&b %

Nf =

#

$ &

1

%

$ # ( " B"B4&b 5 ( ( "

(

#

Nl B"B4&b $ Nl,b 1

# # Nl,b B"B4&b $

Th

Ne B"B4&b $ &

1

1

$ % " 1

Nf,b

(

# Th " > 2Nl,b = 3Nf,b Nl,b = 23 Nf,b "

% & % cb = 1 m = 0

" Nv,b = 2 + Nl,b − Nf,b = 2 + 12 Nf,b Nl = Nv + Nf − Ne − 2" ♦

#"

K

$

% ξ = (ξi , . . . , ξd+1 )"

Vi

%

K

x

Rd "

{Vi , i = 1, . . . , d + 1} Vi,j j

%

1

1 ... 1 1

V1,1 . . . Vd,1 Vd+1,1

K ≡ "

. " "

"" "

V1,d . . . Vd,d Vd+1,d

K = ±d!|K|

|·|

B"B.

Rn

$ "

$

K

+

1

V3

V4

z y

V1

V2

V1

x

x

V3

y

V2

2

V4

V3

K2 K1 K3 x

K2 K1 x K3 V1

V3

K4 V1

V2

V2

2

x

K

K $ j = 1, . . . , d + 1

% Kj (x) K

Vj

x" 1 j & x

K ,"

$

%

7 B";"

7 B"D

ξj (x) ≡

Kj (x) , K

j = 1, . . . , d + 1.

B"B8

#

% ( % ( '"

TK

$ '

K

K % K

#

% ( (

ξ1 = 1 −

d s=1

x s ,

ξj+1 = x j , j = 1, . . . , d.

B"B9

B 3 5

#

%

"

!

/ 6 #

%

x

K

℄

! " #℄# #℄# " #

/ 6

#

%

x

K

℄

$ #

1 # ( 1

Xh (Ωh ) ≡ {vh : vh ∈ V (Ωh ),

B"B:

V (Ωh ) $ ( P (K) $ 1

K " * Xh (Ωh ) %

nl

vh K ∈ P (K), ∀K ∈ Th },

Xh $ + # , (& (

%

1

K

( # - " ##

% % Xh V (Ωh )"

% Xh $ ( V V & (" $ ( C 0 %

( H 1 %L P (K) ⊂ H 1 (K) K

Xh (Ωh ) $

Nh " Xh V Xh → V h → 0 v ∈ V ǫ > 0 h0

I*2C8℄"

%

% (

inf v − vh V ≤ ǫ

vh ∈Xh

h ≤ h0 .

$ Σ = {σi : Xh → R, i = 1, . . . , Nh } (

Xh

vh ∈ Xh → (σ1 (vh ), . . . , σNh (vh )) ∈ RNh vh ∈ Xh $ σi (vh ) i = 1, . . . , Nh "

$ 1" ( #

* 1 $

&

#

%

% 1 -

vh

Ni

$ σi (vh ) = vh (Ni )" ( /

A &E% # = 1

K ? % C & (

# % vh " - 1 5 I048 *2C8℄ #

%

φi ∈ Xh

%

σi (φj ) = δij "

Xh

{φi ,

$ (

(

φi

i = 1, . . . , Nh }

# #

# &" 1

%

φi (Nj ) = δij ,

i, j = 0, . . . , Nh

B"B;

vh ∈ Xh

" > (

vh (x) =

Nh

5

σi (vh )φi (x).

B"BD

i=1

Xh & 1 &

K P (K) nl $

ΣK = {σK,i : P (K) → R, i = 1, . . . , Nl }

p ∈ P (K) → (σK,1 (p), σK,2 (p), . . . , σK,nl (p)) ∈ Rnl

P (K)

ΣK {φK,i , i = 1, . . . nl } ( σK,i (φK,j ) = δij " # p ∈ P (K) 5 (

$ 1" % 1 % #

p(x) =

Nl

σK,i (p)φK,i (x)

i=1

(

φK,i

%

&

x ∈ K.

# #

K"

#

% %

( 1 #

φi |K = φK,νK (i) , ∀K ∈ TK ,

!

≤ νK (i) ≤ nl $ ( ( φi & K " % σi (p) = σK,νK (i) (p|K ) ∀p ∈ Xh "

1

1 ( #

K

"

Xh % V & ( $ % P (K) ⊂ V (K) $ '

/ # #

"

1 $ %

( K r 1 K / P (K) ⊂ P (K) % r ≥ 0" P (K)

* $

%/ $ - (

1

% K P (K)

P (K)" & = { Σ σ1 , σ 2 , . . . , σ nl }

% (

%

# (

{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)

h"

A

% #

Nh

Kj ≡

(" (

1 &

$

#

% 7 3"3"

+

1 %

7 3". " %

( % # $ '

( ( "

'

# #

% # 7 3". "

7

%

"

1 % % 1 ##

P

r = 1, 2, 3

"

8 3 7 7

3"8 " 6 %

1 "

/ #

#

#%"

7

"

5

$

8 9

- /.

& 5

(

% 7 3"9" 4"44B9.C. 2 1

L 4"49B9C8 H " 92 1 1

7 3"9 %

1 # "

: 1 : 1

" 1 &

" > & P2 P3

E # 3"B"

% 7 3"9 E # 3"B"

r

% h $

/

H 1 r + 1 L2 1 r" ♦

*

67 %

5 %

&

P1 P2 P3 u − uh L2 u − uh H 1 u − uh L2 u − uh H 1 u − uh L2 u − uh H 1

h

11$$0!

1 ,%1$ 1111**$,* $0 0 %1/

$* *%1*

11$

1

1111 /!/!

111$ 0

11 $,0/

/!$%1*

1111**$

,1 $/%1,

11 $

0* %1$

11 /00

*1$0%1,

!* %1$

,0%1

/,0$/%1/

11 $

!1$ %1$

111*!!0!

11,!%1/

1!1/%1$

!!/0%

100$%1/

111* $

*1 %1*

111

!,

$10 %10

*10%1*

!$ %

,! %10

$1

111* !

*,%1

*$1!/%1,

,!%

0 1/%1

111 $

%1*

,1/%10

5 h

L2 (0, 1)

/ /

H 1 (0, 1)

# )

3"C1

2u(0) − 5u′ (0) = −3, B"

u′ (1) = e.

3"B.

( # # "

3" 1

(

&

( "

& / /

A# " 6 %

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 % %

'

ui # I* 4.℄

Au = f .

3"B9

( 0 & <

%

"

/

67 % A #

(" A #

1 # A# 7 3":"

%

(

(

α = 2/5 β = −3/5

−u′ (0) + αu(0) = β

γ = e = #%

u′ (1) = γ "

"

- 0.1 %- .

1

h = 0.1"

h

> 1

7 3"; %

>> +3 334 33&4 33&4 3335&4 333$&4℄ >> ;& +33336&4$6 3333&354& 45> ? +3344>> 33&4>=5 33&6== 3335=

333$&&> 3335&℄

%

q e h '

$

h

e(h) ≃ Chq

(

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

5

0

−1

10

−2

10

−3

1

10

1 −4

10

−5

10

2

−6

10

1

−7

10

−3

−2

10

−1

10

10

5 h L2 - .

H 1 - . )

>> /;& #;&#&5% ;&#4%% ##&5% #4%%

==66

===>

====

&3333

&3333

3===5

3====

3333

3333

3333

>> /? #?#&5% ?#4%% ##&5% #4%% ♦

( "

/ /

(

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"

"

3"

( # 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

0

L vL2 ≤ √ |v ′ 2L2 (0,L) , 2

% $ % L"

v2H 1

=

v2L2 (0,L)

+

v ′ 2L2 (0,L)

L2 ≤ 1+ v ′ 2L2 (0,L) 2

% $ 3"34" 7

% % %

a(u, u) ≥ min(kA, σp)u2V . * % ( # %

%" # L $

σp

% %

"

Vh V 1 Nh " & #

Gh

0 < 3"BC $ /

Th ∈ Gh + Vh

%

a(Th , vh ) = 0

∀vh ∈ Vh , Th

(

0 < & 1

Nh & {Kj } % % h # Vh $

(0, L)

!

67 %

Vh ≡ {vh ∈ Xhr (0, L) : vh (0) = 0} ,

Xhr

$ B"."

u − uh H 1 (0,L) ≤

C

u ∈ H q+1 (0, L)

q>0

CM s h |u|H s+1 (0,L) , α

s = min{r, q} |·|H s+1 (0,L) H s+1 (0, L)" k σ p

σp

M/α

M/α"

%

5 / #

" * ( M %

0 <

" #

% $

6 ."

A #

(

'

kA = 0.02π σp = 0.04π " 6 h % 1/10 1/320 7 3"D

( h"

% B 3 H 1 L2 " ♦

−2

10

−3

10

1 −4

10

1 −5

10

2 −6

10

1 −7

10

−3

10

−2

10

−1

10

5 L2 (0, 1) - .

H 1 (0, 1)

- .

h

5

/ /

6 = &

2H

7 3"C"

Tsup

%

(

!

Tinf "

U

&

T : (0, 2H) → R = ( #

%

⎧ ⎨

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

−2

−1

10

10

−3

1

1 10

−2

10

2

10

−4

−5

3

−3

10

10 −4

10

−3

10

10

−2

10

−1

10

−6

−7 −3

−2

10

10

10

−1

5 L2 (Ω) H 1 (Ω)

- . & - .

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

ΓN

∇ · βw2 dω.

Ω

%

∇·β ≥0

in Ω

e

β·n≤0

su ΓN ,

."B:

% ( # $

# $ # " 4

β 9 G

% ∇ · β = 0 - *.

8

G H I - G B.

1$

1 & Vh ⊂ Xhr B"34 ( ΓD

uh % vh ∈ Vh

1 & ##

a(uh , vh ) = F (vh ). L

$

Pe ≡

||β||L∞ h ||β||L∞ h = " 2µ 2

%

Pe < 1"

5

+ # ,

7 7 "

# $

" ( # 1 I* 4.℄ 6 " 9/

ah (uh , vh ) = a(uh , vh ) +

h (∇uh , ∇vh ) . ||β||∞

* 1 % ( # ( # &

"

5

## # )

#

%

B 1 M

."3"B" %

% & 1

- ( 0 < $

%

L

' 85 " 7 '8 7 "

)

#

Lu = −△u + ∇ · (βu) + u, &

."B"3

LSS u =

1 1 (β · ∇u) + ∇ · (βu). 2 2

( ( # # 0

ah ( uh , vh ) = a( uh , vh ) +

K∈Th 5

Pe

hK LSS vh , δ L uh , |β|

) 9 %

! ) 9

max i

||β||L∞ (Ki ) hi 2µ

Ki

) i%

hi

Peloc =

1*

;

K

$

1 "

Th

δ

$

% % )&

( (

( " A + # , % & # " ( & # # ( (" 5 % $ '

hr+1/2 r

%

1 "

#

% $ # %

1 "

∇·β = 0

meas(ΓN ) = 0

8 1

."B:

# # &

1

" 1 # #

0 $ 9"

/ # ) 3

1

! .!( /0/0# -!.!1# -! 0# # 0# 2000# 2000# # (3%3%# 000# 4 %5 6789: 202; %.! $ $ $