Uzawa algorithms for coupled Stokes equations from ... - Springer Link

CALCOLO 46, 37 – 47 (2009) DOI: 10.1007/s10092-009-0158-7

Sang Dong Kim

Uzawa algorithms for coupled Stokes equations from the optimal control problem

Received: May 2008 / Accepted: July 2008 – © Springer-Verlag 2009

Abstract Coupled Stokes equations obtained from the optimal control problem subject to Stokes equations can be approximated by Uzawa finite element methods. The linear convergence is provided with an optimal relaxation parameter for the Richardson pressure update. Keywords Uzawa algorithm, coupled Stokes equations, optimal control problem Mathematics Subject Classification (2000) 65F10, 65M30

1 Introduction

It is known that Uzawa algorithms are quite efficient for solving the stationary Stokes equations discretized with stable finite elements(see [2, 5]). The general convergence proofs for the Uzawa algorithm for Stokes equations can be found in [6] (also see [12]). There is a report on a unified approach for Uzawa algorithms for linear saddle problems (see [1]) such as second-order elliptic partial differential equations, Stokes equations and elasticity problems. Recently, by proving the inf-sup constant β ≤ 1, an optimal relaxation parameter for this algorithm applied to Stokes equations is obtained in [11] to accelerate convergence. Unfortunately, this approach is not focused on a coupled linear saddle This work was supported by KRF-2005-070-C00017. S.D. Kim (B) Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea. E-mail: [email protected]

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S.D. Kim

problem like the coupled Stokes equations: ⎧ −νu + ∇ p − f = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ νv + ∇q + σ1 u = σ1 uˆ ⎪ ⎪ ⎨ ∇·u=0 ⎪ ⎪ ⎪ ⎪ ∇·v=0 ⎪ ⎪ ⎪ ⎪ ⎩ σ2 f + v = 0

in  in  in 

(1.1)

in  in 

with u = v = 0 on the boundary ∂ of an open, connected and bounded   d polygonal domain  ⊂ R (d = 2 or 3) and with  pdx =  qdx = 0, where σ1 and σ2 are positive constants. The coupled Stokes equations (1.1) arose from optimal control problems, which are subjects of interest to experimentalist, involving partial differential equations. For these problems, sophisticated optimization strategies such as Lagrange multiplier methods, adjoint-based gradient methods, quasi-Newton methods have received theoretical and computational attention (see [9]). In particular, the Lagrange multiplier rule is a standard approach for solving control problems constrained by PDEs (see, i.e., [3, 7, 9, 10]) such as the Stokes equations. The optimal control problem subject to Stokes equations leads to the coupled Stokes equations (1.1) (see [9, 10] for example) by minimizing the quadratic functional   σ2 σ1 ˆ 2 dx + |u − u| |f|2 dx (1.2) J (u, f) = 2  2  subject to the Stokes equations ⎧ −νu + ∇ p = f in  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∇ · u = 0 in  ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(1.3)

u = 0 on ∂

 

p dx = 0,

where σ1 and σ2 are positive acceleration parameters. With the Lagrangian functional L(u, v, f, p, q) = J (u, f) +



 

ν(u) · v − ∇ p · v + f · v dx −



∇ · u q dx , (1.4)

the optimality system (1.1) can be derived by setting the first variations of the Lagrangian functional (1.4) to zero. Then the control problem minimizing (1.2) subject to constraint equations (1.3) leads to solutions of the coupled Stokes equations (1.1). For a given velocity

Uzawa algorithms for coupled Stokes equations

39

ˆ we must find optimal state variables (u, p) and a controller f satisfying (1.1). u, The block Gauss-Seidel method was suggested for implementing linear systems arising from optimal control problems in [4]. On the other hand, the Uzawa algorithm using finite element methods, which is known as an efficient method (see [5]) for solving Stokes equations, can be applied to solve (1.1) numerically. The successful results in [11] lead to usage of the Uzawa algorithm with an optimal relaxation parameter on the Richardson pressure update for the coupled Stokes equations by adopting the ideas in [11]. The rest of this paper is organized as follows. In Sect. 2, we review Uzawa algorithms for the coupled Stokes systems. The convergence of Uzawa algorithms is established in Sect. 3 with an optimal relaxation parameter in the Richardson pressure update.

2 Uzawa method for the coupled Stokes system We rewrite the coupled Stokes equations (1.1) by replacing f by − σv2 and q by −q as ⎧ v ⎪ =0 in  −νu + ∇ p + ⎪ ⎪ ⎪ σ2 ⎪ ⎪ ⎨ in  −νv + ∇q − σ1 u = −σ1 uˆ (2.5) ⎪ ⎪ ∇·u=0 in  ⎪ ⎪ ⎪ ⎪ ⎩ ∇·v=0 in  . These coupled Stokes equations (2.5) can be written, by replacing q by −q, as 

−νU + ∇P + AU = F

(2.6)

∇ ·U = 0, where

u U= , v



p P= , q

A=

0 σ12 −σ1 0



,

F=

0 . −σ1 uˆ

(2.7)

In order to use finite element discretizations, shape -quasi regular triangulations T = {K } of  with mesh size h are employed. For the convergence analysis, we use the standard Sobolev spaces L 2 () and H01(), and write   pi dx = 0, i = 1, . . . , d , L 20 ()d := p = ( p1, . . . , pd ) : pi ∈ L 2 (), 

H01()2d

:= {v = (v 1, . . . , v d ) : v i ∈

H01(), i

= 1, . . . , 2d}.

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S.D. Kim

The finite element spaces to be used for approximating the velocity space H01()2d and pressure space L 20 ()d are given as Vh := {Wh ∈ H10()2d : Wh | K ∈ P(K ) for all Ph := {Qh ∈ L 20 ()d : Qh | K ∈ Q(K )

for all

K ∈ T }, K ∈ T },

where P(K ) and Q(K ) are spaces of polynomials with degrees bounded uniformly with respect to K ∈ T (see [5, 8]). On these finite element spaces, the following discrete LBB condition holds. There exists a positive constant β such that (∇ · Wh , Qh ) ≥β, (2.8) inf sup Qh ∈Ph Wh ∈Vh ∇Wh  Qh  where  ·  denotes the usual L 2 -norm and the L 2 inner product (∇ · Wh , Qh ) is interpreted as (∇ · Wh , Qh ) = (∇ · vh , qh ) + (∇ · uh , ph ), Wh = (vh , uh )t , Q h = (qh , ph )t . Recalling the relation between divergence and gradient operators in [11], one may suppose that ∇ · W ≤ ∇W for all W ∈ H01()2d ,

(2.9)

so that, using (2.9), (see [11]) the inf-sup constant β in (2.8) satisfies β ≤ 1.

(2.10)

We also recall the Poincaré inequality, u ≤ m()∇u for all u ∈ H01()2 , where m() is the Poincaré constant. The discrete coupled Stokes problem corresponding to (2.6) is then 

ν(∇Uh , ∇Wh ) − (Ph , ∇ · Wh ) + (AUh , Wh ) = (F, Wh ) for all Wh ∈ Vh , (∇ · Uh , Qh ) = 0

for all Qh ∈ Ph ,

(2.11) which has a unique solution (Uh , Ph ) ∈ Vh × Ph if ν − m()2 |σ2−1 − σ1 | > 0. This can be verified (see [5]) by checking the coercivity of ν(∇Uh , ∇Uh ) + (AUh , Uh ) ≥ (ν − m()2 σ12 )∇Uh 2 , where

1

σ12 := − σ1 . σ2

(2.12)

(2.13)

Uzawa algorithms for coupled Stokes equations

41

Its proof can be obtained from the estimate:





(AUh , Uh ) = σ12 uh · vh d ≤ σ12 uh  vh  

≤ m()2 σ12 ∇uh  ∇vh  ≤ m()2 σ12 ∇Uh 2 . Throughout this paper, we assume that ν − m()2 σ12 > 0 .

(2.14)

The Uzawa algorithm for solving (2.6) can be stated as follows. Algorithm 2.1 (Uzawa method for coupled Stokes) Given a suitable relaxation parameter α > 0 and initial guess P0 : Step 1. find Un+1 ∈ H01()2d satisfying −νUn+1 − ρ∇∇ · Un+1 + ∇Pn + AUn+1 = F in  ; Step 2. find Pn+1 ∈ L 20 ()d from the Richardson update Pn+1 = Pn − αν∇ · Un+1 . The discrete Uzawa method can be stated as follows. Algorithm 2.2 (Discrete Uzawa method for coupled Stokes) Given a suitable relaxation parameter α > 0 and initial guess P0h ∈ Ph : Step 1. find Uhn+1 ∈ Vh satisfying ν(∇Uhn+1 , ∇Wh ) + ρ(∇ · Uhn+1 , ∇ · Wh ) − (Pnh , ∇ · Wh ) + (AUhn+1 , Wh ) = (F, Wh ) ∀ Wh ∈ Vh ;

(2.15)

Step 2. find Phn+1 ∈ Ph from the Richardson update (Phn+1 , Qh ) = (Pnh , Qh ) − αν(∇ · Uhn+1 , Qh )

∀ Qh ∈ Qh .

(2.16)

3 Convergence of Uzawa algorithms For the case where the matrix A is zero, problems (2.5) or (2.6) reduce to two Stokes equations which are not coupled. In this case, linear convergence with an optimal relaxation parameter in the Richardson pressure update is established in [11]. Denote the error functions by Ehn+1 = Uh − Uhn+1 ,

ehn+1 = Ph − Phn+1 .

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S.D. Kim

Then the variational error equations for the velocity Uh and the pressure Ph for the discrete Uzawa algorithm can be obtained immediately. That is to say, subtracting (2.15) from (2.11) leads to ν(∇Ehn+1 , ∇Wh ) + ρ(∇ · Ehn+1 , ∇ · Wh )

(3.17)

− (enh , ∇ · Wh ) + (AEhn+1 , Wh ) = 0 for all Wh ∈ Vh and, from (2.16), by considering ∇ · Uh = 0 we see that (ehn+1 , Qh ) = (enh , Qh ) − αν(∇ · Ehn+1 , Qh )

for all Qh ∈ Ph .

(3.18)

Taking Wh = Ehn+1 in (3.17) and Qh = enh in (3.18) yields ν(∇Ehn+1 , ∇Ehn+1 ) + ρ(∇ · Ehn+1 , ∇ · Ehn+1 ) + (AEhn+1 , Ehn+1 ) = (enh , ∇ · Ehn+1 ) = − =−

(3.19)

1 n+1 (e − enh , enh ) , αν h

 1  n+1 2 eh  − enh 2 − ehn+1 − enh 2 , 2αν

which becomes 2αν 2 ∇Ehn+1 2 + 2ανρ∇ · Ehn+1 2 + 2αν(AEhn+1 , Ehn+1 ) + ehn+1 2

(3.20)

= enh 2 + ehn+1 − enh 2 . For convenience, write σm := max{σ1 ,

1 } σ2

(3.21)

and γ = γ (α) := 1−

  αν ανβ 2 ν − m()2 σ12 2 − . (3.22) ν + m()2 σm ν + m()2 σm + ρ ν + m()2 σm + ρ

Theorem 3.1 Assume that ρ > 0. Then ehn+1  ≤ γ and

n+1 2

e0h 

(3.23)

n

∇Ehn+1  ≤

γ2 e0  . ν − m()2 σ12 h

(3.24)

Uzawa algorithms for coupled Stokes equations

43

Proof By taking Qh = ehn+1 − enh in (3.18) and using the Schwarz inequality, we conclude that ehn+1 − enh 2 = − αν (∇ · Ehn+1 , ehn+1 − enh ) ≤ αν ∇ · Ehn+1  ehn+1 − enh  , (3.25) which leads to ehn+1 − enh 2 ≤ α 2 ν 2 ∇ · Ehn+1 2 .

(3.26)

Now consider the third term of the left-hand side in (3.20). Using the Poincaré inequality, we see that





n+1 n+1 n+1 n+1 , E ) = σ (u − u ) · (v − v ) d (3.27) (AE



12 h h h h 

≤ σ12 u − uhn+1  v − vhn+1  ≤ m()2 σ12 ∇Ehn+1 2 . Substituting (3.26) and (3.27) in (3.20), we obtain   ρ α m()2 σ12 )∇Ehn+1 2 + ∇ · Ehn+1 2 − ∇ · Ehn+1 2 2αν 2 (1 − ν ν 2 + ehn+1 2 ≤ enh 2 .

(3.28)

Replacing  m()2 σ12 αν  ρ α (1 − )∇Ehn+1 2 + ∇ · Ehn+1 2 − ∇ · Ehn+1 2 = − 2 2ρ ν ν +

m()2 σ12 αν (1 − )∇Ehn+1 2 , 2ρ ν

in (3.28), we arrive at  αν  m()2 σ12  (1 − ) ∇Ehn+1 2 + 2αν 2 1 − 2ρ

+

ν

 ρ n+1 2 ∇ · E  h ν − m()2 σ12

α 2ν 3 m()2 σ12 (1 − )∇Ehn+1 2 + ehn+1 2 ≤ enh 2 . ρ ν

Since ρ ρ ≥ , 2 ν − m() σ12 ν + m()2 σm

where σm := max{σ1 ,

1 }, σ2

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S.D. Kim

we have   m()2 σ12  ρ αν  n+1 2 (1 − ) ∇Ehn+1 2 + ∇ · E  2αν 2 1 − h 2ρ ν ν + m()2 σm +

m()2 σ12 α2ν3 (1 − )∇Ehn+1 2 + ehn+1 2 ≤ enh 2 . ρ ν

(3.29)

Since enh ∈ Ph , it follows from (2.8) that there exists a function Vh ∈ Vh such that 1 (3.30) (∇ · Vh , enh ) = enh 2 and ∇Vh  ≤ enh  . β Now with the help of (3.30), (3.17), (2.9) and the Poincaré inequality, it follows that enh 2 = (∇ · Vh , enh )

= ν(∇Ehn+1 , ∇Vh ) + ρ(∇ · Ehn+1 , ∇ · Vh ) + (AEhn+1 , Vh ) ≤ ν∇Ehn+1  ∇Vh  + ρ∇ · Ehn+1  ∇ · Vh  + σm Ehn+1  Vh  

≤ ν∇Ehn+1 2 + ρ∇ · Ehn+1 2 + σm Ehn+1 2

(3.31)

 12

  12 ν∇Vh 2 + ρ∇ · Vh 2 + σm Vh 2 1

(ν + m()2 σm + ρ) 2 n ≤ eh  β   12 (ν + m()2 σm )∇Ehn+1 2 + ρ∇ · Ehn+1 2  ρ 1 + ν+m() 2σ m 2 = (ν + m() σm ) enh  β  1 ρ n+1 2 2 ∇Ehn+1 2 + ∇ · E  , h ν + m()2 σm

which leads to enh 2 ≤

(ν + m()2 σm )(ν + m()2 σm + ρ) × β2   ρ n+1 2 ∇ · E  , ∇Ehn+1 2 + h ν + m()2 σm

(3.32)

and, using (2.9), we find that enh 2 ≤

 ν + m()2 σ + ρ 2 m ∇Ehn+1 2 . β

(3.33)

Uzawa algorithms for coupled Stokes equations

45

Now substituting (3.32) and (3.33) in (3.29) leads to  ν − m()2 σ12 ανβ 2 ehn+1  ≤ 1 − ν + m()2 σm ν + m()2 σm + ρ   12 αν 2− enh  . ν + m()2 σm + ρ These arguments complete the proof of (3.23). From (2.12), (3.17) with Wh = Ehn+1 , we obtain   ν − m()2 σ12 ∇Ehn+1 2 + ρ∇ · Ehn+1 2 ≤ ν(∇Ehn+1 , ∇Ehn+1 ) + ρ(∇ · Ehn+1 , ∇ · Ehn+1 ) + (AEhn+1 , Ehn+1 ) = (enh , ∇ · Ehn+1 ) ≤ enh  ∇ · Ehn+1  , which, using (2.9), leads to (3.24). These arguments complete the proof.



Remark 3.1 Consider the function convergence factor γ (α) defined in (3.22). Note that ν − m()2 σ12 > 0 by assumption so that 0 < c :=

ν − m 2 ()σ12 ≤ 1. ν + m 2 ()σ12

With this constant c, the convergence factor becomes γ (α) = 1 − c

α αβ 2 (2 − ), 1+σ 1+σ

where σ =

m()2 σm + ρ . ν

Note that this convergence factor γ (α) ranges between 0 and 1 if 0 < α < 2(1 + σ ) and it has the minimum 0 < 1 − cβ 2 < 1 at α = 1 + σ . For the zero matrix case A = 0 (that is, for uncoupled Stokes equations), the analysis of the relaxation parameter was given for ρ = 0 in [2] and for ρ ≥ 0 in [11]. Theorem 3.2 Assume that ρ = 0. Then ehn+1  ≤0 (α) and

e0h 

(3.34)

n

∇Ehn+1  where

n+1 2

γ0 (α) 2 ≤ e0  , ν − m()2 σ12 h

  β 2 αν 2ν − 2m()2 σ12 − αν . γ0 (α) := 1 − [ν + m()2 σm ]2

(3.35)

(3.36)

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S.D. Kim

Proof Let ρ = 0. Then it follows from (3.20) that 2αν 2 ∇Ehn+1 2 + 2αν(AEhn+1 , Ehn+1 ) + ehn+1 2

(3.37)

= enh 2 + ehn+1 − enh 2 . Substituting (3.27) and the modified (3.26) with (2.9) in (3.37) leads to   αν 2ν − 2m()2 σ12 − αν ∇Ehn+1 2 + ehn+1 2 ≤ enh 2 .

(3.38)

From (3.33), we have enh  ≤

 1 ν + m()2 σm ∇Ehn+1  . β

(3.39)

Substituting (3.39) in (3.38) and simplifying leads to ehn+1 

   β 2αν 2ν − 2m()2 σ12 − αν  12 n ≤ 1− eh  . [ν + m()2 σm ]2

(3.40)

Hence we obtain (3.34). Next we estimate Ehn+1 . From (2.12), (2.9), (3.17) with ρ = 0 and Wh = Ehn+1 , we have   ν − m()2 σ12 ∇Ehn+1 2 ≤ ν(∇Ehn+1 , ∇Ehn+1 ) + (AEhn+1 , Ehn+1 ) = (enh , ∇ · Ehn+1 ) ≤ enh  ∇Ehn+1  , which leads to ∇Ehn+1  ≤

1 en  . ν − m()2 σ12 h

Then combining (3.40) with (3.41) completes the proof of (3.35).

(3.41)



Remark 3.2 Note that the convergence factor γ0 (α) has the minimum 1 −  2 2σ 2 12 at α = ν−m ν()σ12 which is between 0 and 1. β 2 ν−m() ν+m()2σm Acknowledgements. The author would like to thank Profs. F. Brezzi and R.H. Nochetto for comments and suggestions.

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