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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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   (   (  



$    

         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)"

     

      % 

  (   "    

 # )       

%       # 



% 

% 

V = 

ΓD = ∅

  

HΓ1D

ΓD "

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1

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β · ∇vudω +



6     

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β · nuvdγ =

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