An Inverse Problem for the Stokes Equations

An Inverse Problem for the Stokes Equations

G.M. Dairbaeva

Al-Farabi Kazakh Nalional lJniversity, Almaty, Kazakhstan lazat dairbayeYa@roail . ru

Abstract. Wc consider the inilial boundary problem to the Stokcs cquations, which is ill-posed in this paper. The dircct and arljoint problcms are formulatcd to the original cquatiorrs. Sobolev's spacer space of traces of functions and their norms arc considered. The definiliorts of generalizcd solutions to the dircct and ad.ioint problcms are given for the original equations We consider lhc following approach: ill poscd problem is reduced to the invclse problem, which can be written in operator form /q = l, then minimization of the objcctive functional J (q) :< A,l L A,l - f >

is researchcd. Formujation of tle ilverse problern [o the direct problen r's rnade for thc Stokes equations. It is shown that thc initial boundary value problem is reduced lo 1,he inverse problern I with respect to well-poscd problem for the original equations. It is shown that the inverse problem I can be written a.s an operator equation. For numcrical solving lhc dirccl, and adjoirrt problems wc I use a finile element method, its application to ill-posed problems is new. Iror application of the finite I element mcthod a triangulalion of the domain is made. The inverse problem is solved numerically I using a combinat;on of optirnization method and llnii,e element method. 'fhc

estimatio[ of the I convergence rate of the algorithm wi1,h respect to the functional is found.

I(eywords: Slokes ectuations, invcrsc problem, finite eleinelt rrrethod, optimization method. I

1 The problem formulation

I

I Lct's consider the <lornain t : {(",g) € R2:r € (0, 1),9 € (0, 1)}. 0Q - lou fr is the I

boundary of J7, where fr - {(l,g) : y € [0, 1]]. Let's iuragine thc boundary l-o as follos-s I r o : l - o r U - | o z U f t : , w h e r e 1 6 1 : {(r,O):r € [0, 1]], l-0r: {(0,1/) : s e 1 0 , 1 l } , J . 6 3 : { ( 2 , 1 1 : z e [0, f]]. ln thc domain .f/ we considcr thc initial boundary problen for the Stokes equation-.

A u - V p : 0 ,

d i u u : 0 ,

(", s) e l-or U j-os, i f ( r , s ) e fip,

t "- u4,) l r o - J ( r ' . 3 7 1 , ( r , v ) e l o ,

wlrere rr, - (u1,22) is velocity and p is prcssure, q: (.pt,Vz), f : n - (n1,n2) is the outward unit nonnal to the boundary dfl.

Itle re;write thc problcm (2)- (4) n the coordinatc lorm A u t - p , : 0 , ( x , ' y ) e A

A u z - p a - 0 , ( r , y ) e C

" '"- t),1{,

( i

( 3

(1 (.fr,lz) are giverr functions-

( 6

Technologies, Vol 20, 2015 The Bulletin of KazNU, X, 3(86), 2015

problem (2)-{a) is ill-posed [1J. we can see berow that it can be formu]ated a6 an inverse to some well-posed problem for the initial equations.

' h: {|,t),@; f',f ,if !i::' :''''

, , . { o l 1 , U ) - u 1 ' ( I . y ) ' = q t l a \ . A € r c . 1 ) 1 - u 2 , ( 1 . s \ : q z ( a l . s e ( 0 , 1 ) .

(7)

(8)

(e)

( 1 0 )

( 1 1 )

direct problem formulation

the Stokes system equations in the given domain J-) A u 1 - p r : 9 ,

A u 2 - p r = Q , u t r * u z y : 0 , hllowing boundary conditions

( 1 t \

( 1 3 )

( 1 4 \

(15) (16)

poblem (12) - (16), where it is required to determine u(r,,3), p(a,g) from the given ffu), *" will be called direct.

consider the Sobolev space Lr(e), W;@) with the appropriate norms

),ruet:

U.-r)"' , ,,,,,-,,n,: (y,"o,on * 1 ,,^,o,l'''

rider the space of traces of firnctions from W](A) and traces functions on dO and pace by w]/2pal with the norm

tr,ly;,"6i1: (1,*" .

JJry-#*)* ,

^{( 1 7 )}

90 Bu.rr-rc.;rtrtc,tt rrue texHo,lorsu! T ,20, 2011-r

w h e r c z - @,g), h - (.u,u).

Rcstriction furrctiorrs tro W)/2pA1 on the part of the bordcr' ft (-l-o:, fr) we denore : wll'Qo), (wll'1r-1,w)/'z(4)) -ith norm (17) wherc df,l is rcplaced wiih i-6 (?.s2,l'1) .

We givc the definition of a gencralized solution to the direct problcm (12) - (16).

Definition 1. Let 9 e wll21ro"1urra q e (I4lrtl'{-l-r t)-. T}," set of firnctions (z,p) 'u € W}(9), p € L2(9), is callcd the generalized solution to the direct problen (12) (lft ' t h c f . , l L w i n g c o n d i t i o n s h o l d : d l u u : 0 . { 8 ) o t r d f o r a n v r ' : { r ' , . , 2 ) ' r u c h t h a l , ' W } t Q t a rll; : Q, satisfies the equalitY

- t .

llrtsr,r, I u2"tt2, + lrlaury + u2aa2u - p(ut, + u2u)) d'rd71i

I

The direct problcnr (12) - (16) in the scnsc of the delinition of the generalized solutior x wcll posed [2,3].

3 The inverse problem to the direct problem

Suppose that acltlitional information is givcn about thc sohrtion to the clirect problcrn (12) - ri:

<.rrr the boundary l-rl

I u1,ft,0) - "fr(",0), r € 0, 1), lor :

t r,(r,, o) -F zz,(2,0) : !21r,0), r € (0, 1), {1!

- ( - p ( 0 , y ) + 2 1 , ( 0 , s ) -

" f r ( 0 , y ) , y e ( 0 , 1 ) , 1 ( ) 2 i

\ z r " ( o , s ) : fz(o,a), s € ( 0 , 1 ) , ( x

, . ( - r r o k , l ) - l L ( r , 1 ) , . r e ( 0 , 1 ) , t . , - 1 0 : r :

1 p ( r . ' r t u r r ( r , 1 ) - f z l t , l ) , u € ( 0 , 1 ) . \ - - The inverse problem. The inverse problem to the direct problem (12) (16) is conclude:

in finding unknowtt function q: (qr(y), qz(A)) from the additional information (19) - (21).

We consider the opcrator

A :

q : : | p n

o u . - 4 ) 1 , , . .

6 , , l l r , + . l : | P t i -

i t t I a .

where (2,p) is a solution to the probtem (12) (16). Then the invcrse problern (12) (16), (lg - (21) can bc written in the operator forrn

A , t - f , ( 2 2

t , , , t ) - \ ' 1 , , ' "

w h r - r o A : ( 1 4 2 , r f r t ) -

\ W ) ' ' r f o t ) . | { . / r . J ; t i s g i v e t r f u n c t i o n . q : k | ' . q , \ - r r n k n , ' $ n function.

Lct's write in the coordinatc forrl

Becrru< KacIlV, N. 3(86)- 1.'

l I .

t l l q l t g l o ' t ) . u ) | q 2 t g ) 0 2 l | . y ) l d g - - O .

0

q t ( a ) '- p ( \ , ' s ) - z r " ( 1 , s ) , s € ( 0 , 1 ) , ( f 1 ) ( 2 3 )

Technologies, Vol 20, 2015 T h e B u l l c r i n o f K a z N U . N " 3 ( 8 6 r . 2 0 t 5 9 1

124)

( 2 5 )

(26)

i )

I

l x 4

a z \ Y ) :: - u 2 " ( 1 , 3 / ) , g € ( 0 , 1 ) , ( 4 ) ( " t u ( t - 0 ) . r e ( 0 . I ) . ( l o r )

( A q r ) G , a ) - {-a(o,s; *u,(0,y), s€(0, 1), (1br) [ - u ' r ( r . | ). r e (0. l). (fo3)

[ -p(x'o) * uzo@,o), r e (0, i), (rs1) (Aqz\(r. !J\: \ ,2,t0. s), y c (0. r). r,|oz)

( p ( , , 1 ) u 2 o @ , 7 ) , c e ( 0 , 1 ) , ( 1 6 ) is a solution to the direct problem (12) - (16).

the equation (22) we apply the optimization method.

the functional

t(q): ))Aq fll2rxra: llAq, - frll'r,tra + l)Aqz - fzll2;"soy.

t 1

. l ^ t

J ( s r , q z ) -

l ( u 1 , , ( r . O ) I t ( x . O ) ) 2 d - c r l t n , 0 . u ) + u r x ( 0 . s ) I t \ 0 . 0 ) 2 d s t

J . l

1 1

t ^ f

I l ( u z l } . s ) - I z Q , ! t ) t 2 d l 1 + l t p l . r . t \ - u z u ( r . 1 ) - I2l-r.1))2 ( L x .

J . t '

0 0

k sill solve the the problem (22) by minimizing the functional J(g).

The adjoint problem

1 1

f ^ f

+ l ( u t u ( r , l ) + f t k . l ) ) 2 d x | / t - p l r . O t I u 2 u ( t . 0 ) - 7 " 1 r . 0 1 5 2 d . c 1

J . t '

0 0

trt Tt zrt

l€d

the optimization method for solving the oquation (22) let'g consider the adjoint problem tle problem (12) - (16)

A $ t - r " : O , ( r , g ) € 9 ,

A42 - ru - 0, (r,y) e 9, th" -t l;2n :0, (r,il € A

)

n rith the following boundary conditions

o 1 ( r . s ) : p t ( x . s ) . ( r . e ) e r u . , b z Q . E ) : p 2 ( x . s ) . ( r . s ) e f , J .

(28) (2e)

(30)

( 3 1 )

) , 2 0 1 5

(32) ,z) € rblem

(33)

(3rl

"ed )iil

C,omputational Technologies, Vol 20, 2015

end converges with respect to the functional, and the estimate

The Bulletin of KazNU, Ns 3(86), 2015 93

lolds.

From the algorithm of the Landweber method it can be seen that for the calculation of the qrproximate solution q", for each n is necessary to solve the direct problem (12) - (16) a.nd the rdjoint problem (27) - (32). To solve the last problems we apply the finite element method [4].

The method of the finite elements (MFE). We illustrate a finite element method as example of the direct problem. Consider MFE in the case piecewise-linear functions in the tiangles. We perform triangulation of the given domain J-l, triangular grid cells are triangles.

Let's consider any triangle with given tops P6 (zs,96) , h(rt,at), P2(r2,y2) : , - { @ , A ) : P o , \ o - f P t \ r I P z ) z : ) ; ) 0 , ) o i ) r i ^ z : 1 } ,

) i : \ t . ( r , i l , i . : 0 , 1 , 2 .

If we write down .\6 : 1 - )r .\2, then

, - - { @ , a ) : P o * ) r ( & - P o ) r \ z ( P z - P 6 ) : . \ ; > 0 , ) 1 * } : < 1 } ,

\ : \ r ( r ' a ) ' i - r ' 2 .

From (35) we find

(35)

trqs<#4m

. (-c - xo)(a2- yo) - @ - ao)(rz - ro)

/ r l \ r t y l -

a

\ 2 ( r . s ) : (v - ao)(q - ro) - (r - ro)(u - ao) A

A : (rr - ro)(yz - Ao) - (At - Ao)(rz - *o).

On the triangle r we consider linear function

u ( r , y ) : u n * \ ( r , s ) ( q - u o ) + \ z ( r , a ) ( r " - u o ) , f o r o ' n y ( r ' y ) e r , ( 3 6 ) r h e r e u ( z , y ) : ( u 1 ( r , s ) , u 2 ( r , i l ) , u o : ( r { o ) , r l o ) ) , r r : 1 r f t ) , u j t ) 1 , u , : 1 r \ 2 ) , r f ) 1 .

The function u(r,g) is determined uniquely by its values uo : u (Po) ) ut : u (P1) , u2 : t (Pz)

Further, the domain of integration in equation (18) is a union of triangles, subintegral fuctions are replaced on the type formulas (36). We note that'for the scalar function p in (36) we must take the scalar function.

6 Numerical results

For the numerical solution of direct and adjoint problems the finite element method was used, Ite number of the partition sides of the domain n - 30.

As a zero approximation of the inverse problem (22) is taken g(1)0 : 0,q(2)0 : 0. In the Iandweber method parameter a : 0.01. Fig. l-Fig. 3 show the components of the exact solution to the direct problem (12) - (16). Fig. 4-Fig. 6 shows the components of the zero approximation to the direct problem (12) - (16). Fig. 7-Fig. 9 shows the components of an approximate solution to the direct problem at 1200 iterations. Fig. 10, Fig. 11 illustrate the error rate and the pressure p in the grid analog norm of the norm in L2(\) , where the horizontal axis - iteration, vertical exis - error. Fig. 12, Fig. 13 show the components of the exact solution of the inverse problem ) i s

Brr.rac,nrrergsue rexrroJrori.rn. r.20. 2015 Becrnnr KasHV, lYe 3(86).

(22). Fig. 14, Fig. 15 show the components of the approximate solution at 1200 iterations inverse problem (22), and Fig. 16 error q approximate solution of the inverse problem, the horizontal axis - iteration, vertical axis- error. Fig. 17 shows the residual (the functionJ of the inverse problem (22), ar'd, the horizontal axis - iteration, vertical axis - residual.

7 Conclusion

It is shown that ill-posed problem to the Stokes equations is reduced to the inverse with respect to well-posed problem to the original equations. The inverse problem is numerically using a combination of optimization method and the finite element method.

estimate of convergence rate of the algorithm with respect to the functional was obtained.

Fig. 1. The exact solution ofthe direct problern. Velocity Fig. 2. The exact solution ofthe direct problem.

1 t ( t) . u ( 2 ) .

Fig.3. The exact solution ofthe direct problem.Pressure Fig. 4. The zero approximation to the solution ol the

p. direct problem. Velocity tlr1o.

: t , l ' 3 i r s . \' . 1 2 U . 2 0 1 5 'lhc lJrrlletin of K:rzNU N" 3(86)' 2015 95

f

t

to thc soluliol of lhcFig.6. Thr: zcro :rpproxirnill iolr to thc solution of l'hc dircct Ploblem l'rcssure 716;

{ .:.:::.

: . . ' .:tl . 4 . .

Fig.9. 'l'hc alrproxillllLic solutiou of the 'lircct pr-oblcnl l'rcsstltc p

96 Brr.rnc,rzre,rurgle texnoJrorlrrr, r.20, 2Ol5

Fig. 10. The error ratc on the mrmber iter?r.tions of z.

Becrnrx Ka-rHY, N' 3(86 -

of Fig.ll. Tbe crror of lhe pressure orr

number of iterations.

F i g . 1 3 . T h . t s x a c l s o l u l i o n o [ l h e problem. fhe second component qz.

Fig. 12, The exact solution oI the rnverse problem. 'l'he first conponent qr.

Fig. 14. The approximate solution of the inverse problem. The {irst component, q1.

Fig,15. The approximate solrrtion of tb€

inversc problem. Thc second component q2.

Vol 20, 2015

I

error of the apProximate th€ number of iterations

The Bulletin of KazNU' N' 3(86)' 2015

Fie.17' The residual on the number of iteiations (functionar J(q-) - ll\q" - t1\')'

S.I.: Inverse and Ill-Posed Problems. Theory and Applications De Gruyter' Germany (2011) ia. O.A.: Matematicheskie voprosy dinamiki vjazkoj neszhima€moj zhidkosti Gosudarstvennoe trc drko-matematicheskoj literatury Moscow (1961)

il il;;;;;, i , Lesnik", D , xo'to", x ' An ilternating Method for the Stationarv stokes Svstem'

f*f:r ilil ltl*] "11;?1? i;1X,lllf6J.w york,e84

the q2.