a robust all-at-once multigrid method for the stokes...

A robust all-at-once multigrid method for the Stokescontrol problem

Stefan Takacs

Abstract In this paper we present an all-at-once multigrid method for adistributed Stokes control problem (velocity tracking problem). For solvingsuch a problem, we use the fact that the solution is characterized by theoptimality system (Karush-Kuhn-Tucker-system). The discretized optimalitysystem is a large-scale linear system whose condition number depends on thegrid size and on the choice of the regularization parameter forming a part of theproblem. Recently, block-diagonal preconditioners have been proposed, whichallow to solve the problem using a Krylov space method with convergence ratesthat are robust in both, the grid size and the regularization parameter or costparameter. In the present paper, we develop an all-at-once multigrid methodfor a Stokes control problem and show robust convergence, more precisely,we show that the method converges with rates which are bounded away fromone by a constant which is independent of the grid size and the choice of theregularization or cost parameter.

Keywords PDE-constrained optimization · all-at-once multigrid methods ·Stokes control

1 Introduction

In the present paper, we consider the following model problem (distributedvelocity tracking problem). Let Ω ⊆ Rd be a bounded domain with d ∈ 2, 3.Find a velocity field u ∈ [H1(Ω)]d, a pressure distribution p ∈ L2(Ω) and a

The research was funded by the Austrian Science Fund (FWF): J3362-N25.

Stefan TakacsVisiting Postdoc, Mathematical Institute, University of Oxford, United KingdomTel.: +44-1865-(6)15312Fax: +44-1865-273583E-mail: [email protected]: [email protected]

2 Stefan Takacs

control (force field) f ∈ [L2(Ω)]d such that the tracking functional

J(u, f) =1

2‖u− uD‖2L2(Ω) +

α

2‖f‖2L2(Ω)

is minimized subject to the Stokes equations

−∆u+∇p = f in Ω,

∇ · u = 0 in Ω,

u = 0 on ∂Ω.

The cost parameter or regularization parameter α > 0 and the desired state(desired velocity field) uD ∈ [L2(Ω)]d are assumed to be given. To enforceuniqueness of the solution, we additionally require

∫Ωp dx = 0.

Here and in what follows, L2(Ω) and H1(Ω) denote the standard Lebesgueand Sobolev spaces with associated standard norms ‖ · ‖L2(Ω) and ‖ · ‖H1(Ω),respectively.

The main goal of this work is to construct and to analyze numerical meth-ods that produce an approximate solution to the optimization problem, wherethe computational complexity can be bounded by the number of unknownstimes a constant which is independent of the grid level and the choice of theparameter α, in particular for small values of α.

The solution of the optimization problem is characterized by the Karush-Kuhn-Tucker-system (KKT-system). As we are interested in good approxima-tions of the solution, the discretization of the KKT-system leads to a large-scalelinear system. This linear system will be solved with multigrid methods be-cause they are one of the fastest known methods for such problems. Originally,multigrid methods have been designed and analyzed for elliptic problems. Theyalso work well for saddle point problems (like KKT-systems) and have gainedgrowing interest in this area, see, e.g., [5] and the references cited there.

The unknowns of the discretized KKT-system for a PDE-constrained opti-mization problem can be partitioned into primal variables and dual variables.In our case, the primal variables consist of state variables (the velocity field uand the pressure distribution p) and control variables (the force field f). Thedual variables are the Lagrange multipliers which are introduced to incorpo-rate the constraints. One possibility to solve such a system, is to set up aSchur complement formulation consisting only of the control variable, cf. [2].One approach to solve the KKT-system itself is to apply multigrid methodsin every step of an overall block-structured iterative method to equations injust one of these blocks of variables, cf., [21,11,14].

Another approach, which we will follow here, is to apply the multigrididea directly to the (reduced or not reduced) KKT-system, which is called anall-at-once approach. Such methods have been proposed and discussed for theelliptic optimal control problem, e.g., in [15,19].

In this paper we present a convergence proof for multigrid methods basedon the classical splitting of the analysis into smoothing property and approx-imation property, see [9].

A robust all-at-once multigrid method for the Stokes control problem 3

This paper is organized as follows. In Section 2 we will introduce the op-timality system and discuss its discretization. In Section 3 we will introducean all-at-once multigrid approach. Its convergence will be proven in Section 4.Numerical results which illustrate the convergence result will be presented inSection 5. In Section 6 we will close with conclusions.

2 Optimality system and discretization

For setting up the optimality system we need the space H10 (Ω), the space of

functions in H1(Ω) vanishing on the boundary. Moreover, we need the spaceL20(Ω), which is the space of functions in L2(Ω) with mean value 0, i.e.,

L20(Ω) :=

v ∈ L2(Ω) :

∫Ω

v dξ = 0

.

Both spaces are equipped with standard norms, i.e., ‖ · ‖H10 (Ω) := ‖ · ‖H1(Ω)

and ‖ · ‖L20(Ω) := ‖ · ‖L2(Ω).

The solution of the problem is characterized by the Karush Kuhn Tuckersystem (KKT-system), which reads as follows, cf. [21] and others.

Find (u, p, f, λ, µ) ∈ [H10 (Ω)]d×L2

0(Ω)× [L2(Ω)]d× [H10 (Ω)]d×L2

0(Ω) suchthat

(u, u)L2(Ω) + (∇λ,∇u)L2(Ω) + (µ,∇ · u)L2(Ω) = (uD, u)L2(Ω)

(∇ · λ, p)L2(Ω) = 0

α(f, f)L2(Ω) − (λ, f)L2(Ω) = 0

(∇u,∇λ)L2(Ω) + (p,∇ · λ)L2(Ω) − (f, λ)L2(Ω) = 0

(∇ · u, µ)L2(Ω) = 0

holds for all (u, p, f , λ, µ) ∈ [H10 (Ω)]d×L2

0(Ω)× [L2(Ω)]d× [H10 (Ω)]d×L2

0(Ω).The third line of the KKT-system directly implies f = α−1λ.This allows to eliminate the control f , which leads to the reduced KKT-

system, which reads as follows. Find x := (u, p, λ, µ) ∈ X := [H10 (Ω)]d ×

L20(Ω)× [H1

0 (Ω)]d × L20(Ω) such that

(u, u)L2(Ω) + (∇λ,∇u)L2(Ω) + (µ,∇ · u)L2(Ω) = (uD, u)L2(Ω)

(∇ · λ, p)L2(Ω) = 0

(∇u,∇λ)L2(Ω) + (p,∇ · λ)L2(Ω) − α−1(λ, λ)L2(Ω) = 0

(∇ · u, µ)L2(Ω) = 0

holds for all (u, p, λ, µ) ∈ X.Certainly, the KKT-system can be rewritten as one variational equation as

follows. Find x ∈ X such that

B(x, x) = F(x) for all x ∈ X, (1)

4 Stefan Takacs

where

B((u, p, λ, µ), (u, p, λ, µ)) := (u, u)L2(Ω) + (∇λ,∇u)L2(Ω) + (µ,∇ · u)L2(Ω)

+ (∇ · λ, p)L2(Ω) + (∇u,∇λ)L2(Ω) + (p,∇ · λ)L2(Ω) − α−1(λ, λ)L2(Ω)

+ (∇ · u, µ)L2(Ω) and

F(u, p, λ, µ) := (uD, u)L2(Ω).

We are interested in finding an approximative solution for equation (1).Both, the proposed solution strategy and the convergence analysis, follow theabstract framework introduced in [19]. The conditions, (A1), (A1a), (A3)and (A4), mentioned in the present paper are the same conditions as in [19].

For simplicity, we introduce the following notation.

Notation 1 Throughout this paper, C > 0 is a generic constant, independentof the grid level k and the choice of the parameter α. For any scalars a and b,we write a . b (or b & a) if there is a constant C > 0 such that a < C b. Wewrite a h b if a . b . a.

The following property guarantees existence and uniqueness of the solution.

(A1) The relation

‖x‖X . sup06=x∈X

B(x, x)

‖x‖X. ‖x‖X

holds for all x ∈ X.

In [21] it was shown that condition (A1) is satisfied for X := Y ×Y , whereY := U × P , U := [H1

0 (Ω)]d, P := L20(Ω), equipped with norms

‖x‖2X := ‖(u, p, λ, µ)‖2X := ‖(u, p)‖2Y + α−1‖(λ, µ)‖2Y , (2)

where

‖(u, p)‖2Y := ‖u‖2U + ‖p‖2P , (3)

‖u‖2U := ‖u‖2L2(Ω) + α1/2‖u‖2H1(Ω) and (4)

‖p‖2P := sup06=w∈[H1

0 (Ω)]d

α(p,∇ · w)2L2(Ω)

‖w‖2L2(Ω) + α1/2‖w‖2H1(Ω)

.

Using the following notation, we can express the norms in a nicer way.

Notation 2 For any Hilbert space A, the symbol A∗ denotes its dual spaceequipped with the dual norm

‖u‖A∗ := sup0 6=w∈A

〈u,w〉‖w‖A

,

where 〈u, ·〉 := u(w) denotes the duality pairing.For any Hilbert space A and any scalar a > 0, the symbol aA denotes the

space on the underlying set of the Hilbert space A equipped with the norm

‖u‖2aA := a‖u‖2A.


For any two Hilbert spaces A and B, the symbol A ∩ B denotes the space onthe intersection of the underlying sets, u ∈ A ∩B, equipped with the norm

‖u‖2A∩B := ‖u‖2A + ‖u‖2B ,

and the symbol A+B denotes the space on the algebraic sum of the underlyingsets, u1 + u2 : u1 ∈ A, u2 ∈ B, equipped with the norm

‖u‖2A+B := infu1∈A,u2∈B,u=u1+u2

‖u1‖2A + ‖u2‖2B .

The spaces A∗, aA, A∩B and A+B are Hilbert spaces. The fact that A∗

is a Hilbert space follows directly from the Riesz representation theorem, see,e.g., Theorem 1.2 in [1]. The fact that aA is a Hilbert space is obvious andfor the latter two see, e.g., Lemma 2.3.1 in [4].

We immediately see that the norm on U can be rewritten as follows

‖u‖U = ‖u‖L2(Ω)∩α1/2H1(Ω).

To reformulate the norm ‖ · ‖P , we need a regularity assumption.

(R) Regularity of the generalized Stokes problem. Let f ∈ [L2(Ω)]d and g ∈H1

0 (Ω) ∩ L20(Ω) be arbitrarily but fixed and (u, p) ∈ [H1

0 (Ω)]d × L20(Ω) be

the solution of the Stokes problem, i.e., such that

(∇u,∇u)L2(Ω) + (p,∇ · u)L2(Ω) = (f, u)L2(Ω)

(∇ · u, p)L2(Ω) = (g, p)L2(Ω)

holds for all (u, p) ∈ [H10 (Ω)]d × L2

0(Ω).Then (u, p) ∈ [H2(Ω)]d ×H1(Ω) and

‖u‖2H2(Ω) + ‖p‖2H1(Ω) . ‖f‖2L2(Ω) + ‖g‖2H1(Ω).

This condition is satisfied for convex polygonal domains, see Lemma 2.1in [16] which is a direct consequence of Theorem 2 in [10].

Lemma 1 If (R) is satisfied, then

‖p‖P h ‖p‖αH1(Ω)+α1/2L2(Ω) (5)

holds for all p ∈ L20(Ω).

This lemma was shown in Theorem 3.2 in [13] under a regularity assump-tion, which is weaker than regularity assumption (R).

The discretization of problem (1) is done using standard finite elementtechniques. We assume to have a sequence of girds obtained by uniform re-finement. On each grid level k, we discretize the problem using the Galerkinapproach, i.e., we have finite dimensional spaces Xk ⊆ X and consider thefollowing problem. Find xk ∈ Xk such that

B(xk, xk) = F(xk) for all xk ∈ Xk. (6)

6 Stefan Takacs

Using a nodal basis, we can represent this problem in matrix-vector notationas follows:

Ak xk = fk. (7)

Here and in what follows, any underlined quantity, like xk, is the representationof the corresponding non-underlined quantity, here xk, with respect to a nodalbasis of the corresponding Hilbert space, here Xk.

Existence and uniqueness of the discretized problem is guaranteed by thefollowing condition.

(A1a) The relation

‖xk‖X . sup06=xk∈Xk

B(xk, xk)

‖xk‖X. ‖xk‖X

holds for all xk ∈ Xk.

Due to the fact that the model problem is indefinite, condition (A1) does notimply condition (A1a). For the Stokes problem itself, it is well-known thatsuch a condition (also known as discrete inf-sup condition) can only be guar-anteed if the discretization is chosen appropriately. The same is true for theStokes control problem. Fortunately, we can show the discrete inf-sup condi-tion (A1a) for the Stokes control problem based on pre-existing knowledgeon the discrete inf-sup condition for the Stokes problem. This allows to showthat all discretizations which are suitable for the Stokes flow problem are alsosuitable for the Stokes control problem.

We choose the space Xk as follows:

Xk := Yk × Yk where Yk := Uk × Pkand the choice of Uk ⊆ U = [H1

0 (Ω)]d and Pk ⊆ P = L20(Ω) is discussed below.

Note that Xk has product structure and that the state and the adjoined state(Lagrange multipliers) are discretized the same way. The same has alreadybeen done for optimal control problems with elliptic state equation, cf. [19]and many others.

Due to the fact that the grids are obtained by uniform refinement, thediscrete subsets are nested, i.e., Uk ⊆ Uk+1 and Pk ⊆ Pk+1. Therefore, alsoXk ⊆ Xk+1 holds.

The next step is to show condition (A1a). We have seen that the analysisdone in [21], applied to the infinite dimensional spaces, shows condition (A1).If the analysis done in [21] is applied to the discretized spaces, we obtain that

‖xk‖Xk . sup06=xk∈Xk

B(xk, xk)

‖xk‖Xk. ‖xk‖Xk (8)

is satisfied for all xk ∈ Xk, where

‖xk‖2Xk := ‖(uk, pk, λk, µk)‖2Xk := ‖(uk, pk)‖2Yk + α−1‖(λk, µk)‖2Yk ,‖(uk, pk)‖2Yk := ‖uk‖2U + ‖pk‖2Pk ,

‖pk‖2Pk := sup06=wk∈Uk

α(pk,∇ · wk)2L2(Ω)

‖wk‖2L2(Ω) + α1/2‖wk‖2H1(Ω)

and


‖ · ‖2U is as above.Note that this is not condition (A1a), as the norms ‖ · ‖P and ‖ · ‖Pk are

not equal. For showing condition (A1a), it suffices to show that these twonorms are equivalent which implies also the equivalence of the norms ‖ · ‖Xand ‖ · ‖Xk . This can be shown using the following condition.

(S) The discretization of P is H1-conforming, i.e., Pk ⊆ H1(Ω), and the weakinf-sup condition

sup06=uk∈Uk

(∇ · uk, pk)L2(Ω)

‖uk‖L2(Ω)& ‖∇pk‖L2(Ω)

holds for all pk ∈ Pk.

Lemma 2 Assume that the discretization satisfies condition (S). Then con-dition (A1a) is satisfied for the model problem.

Proof Lemma 2.2 in [12] states (provided that (S) is satisfied) that ‖ · ‖P h‖ · ‖Pk is satisfied. A direct consequence is ‖ · ‖X h ‖ · ‖Xk . Therefore, condi-tion (8) implies condition (A1a). ut

Note that condition (S) is a standard condition which ensures that thechosen discretization is stable for the Stokes problem. In [3,20] it was shownthat condition (S) is satisfied for the Taylor-Hood element (P1−P2-element)for polygonal domains where at least one vertex of each element is located inthe interior of the domain. Here and in what follows we assume that the prob-lem is discretized with the Taylor-Hood element and that the mesh satisfiesthe named condition.

3 An all-at-once multigrid method

The problem shall be solved with an all-at-once multigrid method. The ab-stract algorithm for solving the discretized equation (7) on grid level k reads as

follows. Starting from an initial approximation x(0)k , one iterate of the multigrid

method is given by the following two steps:

– Smoothing procedure: Compute

x(0,m)k := x

(0,m−1)k + A−1k

(fk−Ak x(0,m−1)k

)for m = 1, . . . , ν

with x(0,0)k = x

(0)k . The choice of the matrix A−1k will be discussed below.

– Coarse-grid correction:

– Compute the defect r(1)k := f

k− Ak x(0,ν)k and restrict it to grid level

k − 1 using an restriction matrix Ik−1k :

r(1)k−1 := Ik−1k

(fk−Ak x(0,ν)k

).

8 Stefan Takacs

– Solve the coarse-grid problem

Ak−1 p(1)k−1 = r(1)k−1 (9)

approximatively.– Prolongate p

k−1 to the grid level k using an prolongation matrix Ikk−1and add the result to the previous iterate:

x(1)k := x

(0,ν)k + Ikk−1 p

(1)k−1.

As we have assumed to have nested spaces, the intergrid-transfer matrices Ikk−1and Ik−1k can be chosen in a canonical way: Ikk−1 is the canonical embedding

and the restriction Ik−1k is its transpose.If the problem on the coarser grid is solved exactly (two-grid method), the

coarse-grid correction is given by

x(1)k := x

(0,ν)k + Ikk−1A−1k−1 I

k−1k

(fk−Ak x(0,ν)k

). (10)

In practice the problem (9) is approximatively solved by applying one step(V-cycle) or two steps (W-cycle) of the multigrid method, recursively. On thecoarsest grid level (k = 0) the problem (9) is solved exactly.

To construct a multigrid convergence result based on Hackbusch’s splittingof the analysis into smoothing property and approximation property, we haveto introduce an appropriate framework.

Convergence is shown on the spaces Xk, which are equipped with an L2-likenorms ||| · |||0,k, which are defined a follows:

|||xk|||20,k := ‖xk‖2Lk := (Lkxk, xk)`2 ,

where

Lk :=

Ak

Skα−1Ak

α−1Sk

, (11)

where Ak := diag (MU,k + α1/2KU,k) and Sk := α diag (DkA−1k DT

k ). Here,MU,k and KU,k are mass matrix and stiffness matrix, representing the L2-inner product and the H1-inner product in Uk, respectively. The matrix Dk

represents the bilinear form d(uk, pk) = (∇ · uk, pk)L2(Ω) on Uk × Pk.Note that the matrix Lk is a diagonal matrix whose entries can be com-

puted efficiently.Note that due to standard scaling arguments

Ak is spectrally equivalent to (1 + α1/2h−2k )MU,k and (12)

Sk is spectrally equivalent to α(1 + α1/2h−2k )−1h−2k MP,k (13)

holds. Here, MP,k is the standard mass matrix, representing the L2-inner prod-uct on Pk.


Based on the norm ||| · |||0,k, we can introduce the residual norm ||| · |||2,kusing

|||xk|||2,k := supxk∈Xk

B (xk, xk)

|||xk|||0,k.

Smoothing property and approximation property read as follows.

– Smoothing property:

|||x(0,ν)k − x∗k|||2,k ≤ η(ν)|||x(0)k − x∗k|||0,k (14)

will be shown for some function η(ν) with limν→∞ η(ν) = 0. Here and inwhat follows, x∗k ∈ Xk is the exact solution of the discretized problem (7).

– Approximation property:

|||x(1)k − x∗k|||0,k ≤ CA|||x

(0,ν)k − x∗k|||2,k (15)

will be shown for some constant CA > 0.

It is well known that, if we combine both conditions, we obtain

|||x(1)k − x∗k|||0,k ≤ q(ν)|||x(0)k − x

∗k|||0,k,

where q(ν) = CAη(ν), i.e., that the two-grid method converges for ν largeenough. The convergence of the W-cycle multigrid method can be shown undermild assumptions, see e.g. [9].

The choice of an appropriate smoother is a key issue in constructing sucha multigrid method. Here, we introduce one smoother which is appropriatefor a large class of problems including the model problem: normal equationsmoothers, cf. [6], which read as follows.

x(0,m)k := x

(0,m−1)k + τ L−1k AkL

−1k︸︷︷︸

A−1k :=

(fk−Ak x(0,m−1)k

)for m = 1, . . . , ν.

Here, a fixed τ > 0 has to be chosen such that the spectral radius ρ(τA−1k Ak) isbounded away from 2 on all grid levels k and for all choices of the parameter α.

Using a standard inverse inequality, one can show that

‖xk‖X . |||xk|||0,k (16)

is satisfied for all xk ∈ Xk. Based on this result, we obtain.

Lemma 3 The damping parameter τ > 0 can be chosen independent of gridlevel k and the choice of the parameter α such that

τ ρ(A−1k Ak) ≤ 2− ε < 2. (17)

holds for some constant ε > 0. For this choice of τ , there is a constant CS > 0,independent of the grid level k and the choice of the parameter α, such thatthe smoothing property (14) is satisfied with rate

η(ν) := CSν−1/2. (18)

10 Stefan Takacs

The proof of this lemma is standard, cf. [6]. For completeness, we give a sketchof the proof.

Proof The combination of (A1) and (16) directly implies that ρ(A−1k Ak) =ρ(L−1k AkL

−1k Ak) is uniformly bounded. This implies directly the existence of

a uniform τ satisfying (17).

The smoothing property (14) with η as in (18) is equivalent to

q := ‖L−1/2k Ak(I − τL−1k AkL−1k Ak)νL−1/2k ‖`2 ≤ CSν−1/2. (19)

Defining Q := τ1/2L−1/2k AkL−1/2k , we obtain

q2 = τ−1‖Q(I −QTQ)ν‖2 = τ−1ρ(QTQ(I −QTQ)2ν).

Here, QTQ is a symmetric and positive definite matrix with eigenvalues λ ∈(0, 2 − ε]. We obtain q2 ≤ τ−1 supλ∈(0,2−ε] λ(1 − λ)2ν . This is bounded from

above by max(2ν)2ν(1+2ν)−1−2ν , (2−ε)(1−ε)2ν ≤ max2, (2−ε)/(−2e ln(1−ε))ν−1, where e is Euler’s number. This implies (19) with CS = max2, (2−ε)/(−2e ln(1− ε))1/2τ−1/2 and finishes the proof.

Certainly, the iteration procedure (14) should be efficient-to-apply. Usingthe fact that the mass matrices MU,k and MP,k in (11) and their diagonals arespectrally equivalent under weak assumptions, for the practical realization ofthe smoother these matrices can be replaced by their diagonals.

Remark 1 The convergence result shown in this paper is still valid if the nor-mal equation smoother is replaced by any other smoother that satisfies thesmoothing property (14), like the smoother presented in [17].

4 A convergence proof

The proof of the approximation property is done using the approximation the-orem introduced in [19] which requires besides the conditions (A1) and (A1a)two more conditions (conditions (A3) and (A4)) involving, besides the Hilbertspace X, two more Hilbert spaces X−,k := (X−, ‖ · ‖X−,k) and X+,k :=(X+, ‖ · ‖X+,k

), which are chosen as follows.

As weaker space, we choose X− := Y− × Y−, where Y− := U− × P−,U− := [L2(Ω)]d and P− := [H1

0 (Ω) ∩ L20(Ω)]∗, equipped with norms

‖x‖2X−,k:= ‖(u, p, λ, µ)‖2X−,k

:= ‖(u, p)‖2Y−,k+ α−1‖(λ, µ)‖2Y−,k

,

‖(u, p)‖2Y−,k:= ‖u‖2U−,k

+ ‖p‖2P−,k,

‖u‖2U−,k:= h−2k ‖u‖

2[H1

0 (Ω)+α−1/2L2(Ω)]∗ and

‖p‖2P−,k:= h−2k ‖p‖

2[α−1L2

0(Ω)∩α−1/2H10 (Ω)]∗


Note that dual spaces are (X−)∗ := (Y−)∗ × (Y−)∗, where (Y−)∗ = (U−)∗ ×(P−)∗, (U−)∗ = [L2(Ω)]d and (P−)∗ = H1

0 (Ω) ∩ L20(Ω), equipped with norms

‖F‖2(X−,k)∗:= ‖(f, g, ζ, χ)‖2(X−,k)∗

:= ‖(f, g)‖2(Y−,k)∗+ α‖(ζ, χ)‖2(Y−,k)∗

,

‖(f, g)‖2(Y−,k)∗:= ‖f‖2(U−,k)∗

+ ‖g‖2(P−,k)∗,

‖f‖2(U−,k)∗:= h2k‖f‖2H1

0 (Ω)+α−1/2L2(Ω) and

‖g‖2(P−,k)∗:= h2k‖g‖2α−1L2

0(Ω)∩α−1/2H10 (Ω).

Remark 2 Note that the norm |||·|||0,k was constructed from the norm ‖·‖X (asdefined in (2), (5), (4) and (5)) “just by replacing ‖ · ‖H1(Ω) by h−1k ‖ · ‖L2(Ω)”.This follows the classical approach known from literature. However, the norm‖ · ‖X−,k is obtained by additionally “replacing ‖ · ‖L2(Ω) by h−1k ‖ · ‖[H1

0 (Ω)]∗”.Note that we will show that the approximation property (22), which we willprove in this section, is stronger than the approximation property (15).

As stronger space, we choose X+ := Y+ × Y+, where Y+ := U+ × P+,U+ := [H2(Ω) ∩H1

0 (Ω)]d and P+ := H1(Ω) ∩ L20(Ω), equipped with norms

‖x‖2X+,k:= ‖(u, p, λ, µ)‖2X+,k

:= ‖(u, p)‖2Y+,k+ α−1‖(λ, µ)‖2Y+,k

,

‖(u, p)‖2Y+,k:= ‖u‖2U+,k

+ ‖p‖2P+,k,

‖u‖2U+,k:= h2k‖u‖2H1(Ω)∩α1/2H2(Ω) and

‖p‖2P+,k:= h2k‖p‖2αH2(Ω)+α1/2H1(Ω).

The additional conditions, which we will prove below, read as follows.

(A3) On all grid levels k, the approximation error result

infxk∈Xk

‖x− xk‖X . ‖x‖X+,kfor all x ∈ X+

is satisfied.(A4) For all grid levels k, all F ∈ (X−)∗ the solution xF ∈ X of the problem,

find x ∈ X such that B(x, x) = F(x) for all x ∈ X, (20)

satisfies xF ∈ X+ and the inequality

‖xF‖X+,k. ‖F‖(X−,k)∗ . (21)

Based on these assumptions, the following theorem shows the approxima-tion property.

Theorem 1 Let for k = 0, 1, 2, . . . the symmetric matrices Ak be obtained bydiscretizing problem (6) using a sequence of finite-dimensional nested subspacesXk−1 ⊆ Xk ⊂ X. Let the Hilbert spaces X+ ⊆ X ⊆ X− be as above andassume that the conditions (A1), (A1a), (A3) and (A4) are satisfied. Thenthe coarse-grid correction (10) satisfies the approximation property

‖x(1)k − x∗k‖X−,k ≤ CA sup

xk∈Xk

B(x(0,ν)k − x∗k, xk

)‖xk‖X−,k

, (22)

12 Stefan Takacs

where the constant CA only depends on the constants that appear (implicitly)in the named conditions.

For a proof, see [19], Theorem 4.1.

Theorem 2 Condition (A3) is satisfied.

Proof This proof is analogous to the proof of Theorem 4.2 in [16]. However, tokeep this paper as self-contained as possible, we give a proof of this theorem.

Note that it suffices to show approximation error results for the individualvariables separately. Using a standard interpolation operator Πk : [H2(Ω)]d →Uk, we obtain for the velocity field u

‖u−Πku‖2L2(Ω) . h2k‖u‖2H1(Ω) and ‖u−Πku‖2H1(Ω) . h2k‖u‖2H2(Ω),

for all u ∈ [H2(Ω)]d and therefore

infuk∈Uk

‖u− uk‖2U ≤ ‖u−Πku‖2U = ‖u−Πku‖2L2(Ω) + α1/2‖u−Πku‖2H1(Ω)

. h2k

(‖u‖2H1(Ω) + α1/2‖u‖2H2(Ω)

)= ‖u‖2U+,k

.

The same can be done for the adjoined velocity λ. Also for the pressure dis-tribution p we can do a similar estimate. The estimates

infpk∈Pk

‖p− pk‖2L2(Ω) . h2k‖p‖2H1(Ω) and infpk∈Pk

‖p− pk‖2H1(Ω) . h2k‖p‖2H2(Ω)

are standard approximation error results which imply

infpk∈Pk

‖p− pk‖2P = infpk∈Pk

‖p− pk‖2αH1(Ω)+α1/2L2(Ω)

= infpk∈Pk

q1∈H1(Ω)

q2∈L2(Ω)q1+q2=p−pk

‖q1‖2αH1(Ω) + ‖q2‖2α1/2L2(Ω)

= infp1∈H1(Ω)

p2∈L2(Ω)p1+p2=p

infp1,k∈Pk

‖p1 − p1,k‖2αH1(Ω) + infp2,k∈Pk

‖p2 − p2,k‖2α1/2L2(Ω)

≤ infp1∈H2(Ω)

p2∈H1(Ω)p1+p2=p

infp1,k∈Pk

‖p1 − p1,k‖2αH1(Ω) + infp2,k∈Pk

‖p2 − p2,k‖2α1/2L2(Ω)

. h2k infp1∈H2(Ω)

p2∈H1(Ω)p1+p2=p

‖p1‖2αH2(Ω) + ‖p2‖2α1/2H1(Ω) = h2k‖p‖2αH2(Ω)+α1/2H1(Ω).

The same can be done for the adjoined pressure µ. This finishes the proof. ut

For showing (A4), we recall Theorem 4.6 in [16] on the regularity of thegeneralized Stokes problem. For this purpose, we need a regularity assumptionfor the Poisson problem with homogeneous Neumann boundary conditions.


(R1) Regularity of the Poisson problem. Let g ∈ L2(Ω) and p ∈ H1(Ω) ∩L20(Ω) be such that

(∇p,∇p)H1(Ω) = (g, p)L2(Ω) for all p ∈ H1(Ω) ∩ L20(Ω).

Then p ∈ H2(Ω) and ‖p‖H2(Ω) . ‖g‖L2(Ω).

Such a regularity assumption can be guaranteed for convex polygonal domains(see, e.g., [8]).

Theorem 4.6 in [16] directly implies the following theorem.

Theorem 3 Suppose that the regularity assumptions (R) and (R1) are sat-isfied. Let f ∈ [L2(Ω)]d and g ∈ H1

0 (Ω)∩L20(Ω). The solution of the problem,

find (u, p) ∈ Y such that

α−1/2(u, u)L2(Ω) + (∇u,∇u)L2(Ω) + (p,∇ · u)L2(Ω) = (f, u)L2(Ω)

(∇ · u, p)L2(Ω) = (g, p)L2(Ω)

for all (u, p) ∈ Y , satisfies (u, p) ∈ Y+ and the inequality

‖u‖2α−1/2H1(Ω)∩H2(Ω) + ‖p‖2α1/2H2(Ω)+H1(Ω)

. ‖f‖2α1/2H10 (Ω)+L2(Ω) + ‖g‖2α−1/2L2

0(Ω)∩H10 (Ω)

is satisfied.

Proof We choose the parameter β (which occurs in [16]) to be β := α−1/2. ut

Lemma 4 Suppose that assumptions (R) and (R1) are satisfied. Let F ∈(X−)∗ be arbitrarily but fixed. Then, xF , the solution of (20), satisfies xF ∈X+ and the bound

‖xF‖2X+,k. ‖F‖2(X−,k)∗

+ h2k

(‖uF‖2H1(Ω) + α−1‖λF‖2H1(Ω)

). (23)

Proof Let F(u, p, λ, µ) := (f, u)L2(Ω) + (g, p)L2(Ω) + (ζ, λ)L2(Ω) + (χ, µ)L2(Ω),

where f, ζ ∈ [L2(Ω)]d and g, χ ∈ H10 (Ω) ∩ L2

0(Ω).

Let f := f − uF + α−1/2λF and ζ := ζ + α−1λF + α−1/2uF . Then we canrewrite the KKT-system as follows:

(∇λF ,∇u)L2(Ω) + α−1/2(λF , u)L2(Ω) + (µF ,∇ · u)L2(Ω) = (f , u)L2(Ω)

(∇ · λF , p)L2(Ω) = (g, p)L2(Ω)

and

(∇uF ,∇λ)L2(Ω) + α−1/2(uF , λ)L2(Ω) + (pF ,∇ · λ)L2(Ω) = (ζ, λ)L2(Ω)

(∇ · uF , µ)L2(Ω) = (χ, p)L2(Ω).

14 Stefan Takacs

As f ∈ [L2(Ω)]d, g ∈ H10 (Ω)∩L2

0(Ω), ζ ∈ [L2(Ω)]d and χ ∈ H10 (Ω)∩L2

0(Ω), weobtain using Theorem 3 that xF ∈ X+ and the following bounds are satisfied:

‖λF‖2α−1/2H1(Ω)∩H2(Ω) + ‖µ‖2α1/2H2(Ω)+H1(Ω)

. ‖f − uF + α−1/2λF‖2α1/2H10 (Ω)+L2(Ω) + ‖g‖2α−1/2L2

0(Ω)∩H10 (Ω)

and

‖uF‖2α−1/2H1(Ω)∩H2(Ω) + ‖pF‖2α1/2H2(Ω)+H1(Ω)

. ‖ζ − α−1λF + α−1/2uF‖2α1/2H10 (Ω)+L2(Ω) + ‖χ‖2α−1/2L2

0(Ω)∩H10 (Ω).

We can combine these two estimates and obtain

‖uF‖2H1(Ω)∩α1/2H2(Ω) + ‖pF‖2αH2(Ω)+α1/2H1(Ω)

+ α−1‖λF‖2H1(Ω)∩α1/2H2(Ω) + α−1‖µ‖2αH2(Ω)+α1/2H1(Ω)

. ‖f‖2H10 (Ω)+α−1/2L2(Ω) + ‖g‖2α−1L2

0(Ω)α−1/2∩H10 (Ω)

+ α‖ζ‖2H10 (Ω)+α−1/2L2(Ω) + α‖χ‖2α−1L2

0(Ω)∩α−1/2H10 (Ω)

+ ‖uF‖2H10 (Ω)+α−1/2L2(Ω) + α−1‖λF‖2H1

0 (Ω)+α−1/2L2(Ω).

Note that ‖uF‖H10 (Ω)+α−1/2L2(Ω) ≤ ‖uF‖H1

0 (Ω) = ‖uF‖H1(Ω) holds because of

uF ∈ [H10 (Ω)]d. As the analogous holds also for λF , this finishes the proof. ut

To show condition (A4), we have to bound ‖uF‖2H1(Ω) + α−1‖λF‖2H1(Ω)

from above. For showing such a result, we need some notation.

As H10 (Ω) is dense in L2(Ω), for u ∈ [H2(Ω)]d the function −∆u ∈

[L2(Ω)]d can be approximated by some function wε ∈ [H10 (Ω)]d such that

‖ −∆u− wε‖2L2(Ω) ≤ ε.

So, we can introduce an operator −∆ε : [H2(Ω)]d → [H10 (Ω)]d such that

‖ −∆u− (−∆ε)u‖2L2(Ω) ≤ ε.

Analogously, we introduce the operator ∇ε : H1(Ω)→ [H10 (Ω)]d such that

‖∇p−∇εp‖2L2(Ω) ≤ ε.

Lemma 5 Let F ∈ (X−)∗ and let xF = (uF , pF , λF , µF ) be the solutionof (20). Then xF satisfies the estimate

h2k

(‖uF‖2H1(Ω) + α−1‖λF‖2H1(Ω)

). ‖F‖(X−,k)∗‖xF‖X+,k

. (24)


Proof Let F(u, p, λ, µ) := (f, u)L2(Ω) + (g, p)L2(Ω) + (ζ, λ)L2(Ω) + (χ, µ)L2(Ω),

where f, ζ ∈ [L2(Ω)]d and g, χ ∈ H10 (Ω) ∩ L2

0(Ω).

The idea of this proof is to show that for all ε > 0 there is some xε ∈ Xsuch that

F(xε)− B(xF , xε)

. h−2k ‖F‖(X−,k)∗‖xF‖X+,k− ‖uF‖2H1(Ω) − α

−1‖λF‖2H1(Ω) (25)

+ ε(α1/2 + α−1/2)h−1k (‖xF‖X+,k+ ‖F‖(X−,k)∗) + ε2.

Note that the left-hand-side of the inequality is 0. Therefore, this would be suf-ficient to show the statement of the lemma, as ε > 0 can be chosen arbitrarilysmall.

In the following, we show that (25) is satisfied for the choice xε := (−∆εuF ,−∇·∇εpF , ∆ελF ,∇·∇εµF ). We estimate the individual summands of F(xε)−B(xF , x

ε) separately. For the first one, we obtain

− (uF ,−∆εuF )L2(Ω) ≤ −(uF ,−∆uF )L2(Ω) + ε‖uF‖L2(Ω)

= −(∇uF ,∇uF )L2(Ω) + ε‖uF‖L2(Ω) . −‖uF‖2H1(Ω) + εh−1k ‖xF‖X+,k

due to the fact that uF ∈ [H2(Ω)∩H10 (Ω)]d and due to Friedrichs’ inequality.

The same can be done for α−1(λF , ∆ελF )L2(Ω).

For the next two summands,

− (∇uF ,∇∆ελF )L2(Ω) − (∇λF ,∇(−∆ε)uF )L2(Ω)

= (∆uF , ∆ελF )L2(Ω) − (∆λF , ∆

εuF )L2(Ω)

≤ (∆uF , ∆λF )L2(Ω) − (∆λF , ∆uF )L2(Ω) + ε(‖∆uF‖L2(Ω) + ‖∆λF‖L2(Ω))

≤ ε(‖uF‖H2(Ω) + ‖λF‖H2(Ω)) ≤ εh−1k (α1/4 + α−1/4)‖xF‖X+,k

is satisfied due to the fact that ∆ε maps into [H10 (Ω)]d.

For the next two summands, we obtain

− (∇ · uF ,∇ · ∇εµF )L2(Ω) − (∇ · (−∆ε)uF , µF )L2(Ω)

= (∇∇ · uF ,∇εµF )L2(Ω) − (∆εuF ,∇µF )L2(Ω)

≤ (∇∇ · uF ,∇εµF )L2(Ω) − (∆εuF ,∇εµF )L2(Ω) + ε‖∆εuF‖L2(Ω)

≤ −(∇uF ,∇∇εµF )L2 − (∆uF ,∇εµF )L2 + ε(‖∆uF‖L2 + ‖∇εpF‖L2 + ε)

= −(∇uF ,∇∇εµF )L2 + (∇uF ,∇∇εµF )L2 + ε(‖∆uF‖L2 + ‖∇εpF‖L2 + ε)

≤ ε(‖uF‖H2(Ω) + ‖pF‖H1(Ω) + 2ε) . εh−1k (α−1/4 + α−1/2)‖xF‖X+,k+ ε2.

The same can be done for −(∇ · λF ,−∇ · ∇εpF )L2(Ω) − (∇ ·∆ελF , pF )L2(Ω).

Let f2 ∈ [H10 (Ω)]d and f1 := f − f2. Then

(f1,−∆εuF )L2(Ω) . ‖f1‖α−1/2L2(Ω)‖uF‖α1/2H2(Ω) + εα1/4‖f1‖α−1/2L2(Ω)

16 Stefan Takacs

holds as well as

(f2,−∆εuF )L2(Ω) . (∇f2,∇uF )L2(Ω) + ε‖f2‖L2(Ω)

. ‖f2‖H1(Ω)‖uF‖H1(Ω) + ε‖f2‖H1(Ω).

This implies

(f,−∆εu)L2(Ω)

. ‖f‖H10 (Ω)+α−1/2L2(Ω)‖u‖H1(Ω)∩α1/2H2(Ω)

+ ε(1 + α1/4)‖f‖H10 (Ω)+α−1/2L2(Ω)

. ‖f‖H10 (Ω)+α−1/2L2(Ω)‖u‖H1(Ω)∩α1/2H2(Ω) + εh−1k (1 + α1/4)‖F‖(X−,k)∗ .

Let p2 ∈ H2(Ω) and p1 := pF − p2 ∈ H1(Ω). We have

(g,−∇ · ∇εp1)L2(Ω) = (∇g,∇εp1)L2(Ω)

. ‖g‖α−1/2H1(Ω)‖p1‖α1/2H1(Ω) + ε‖g‖H1(Ω).

Moreover, using g ∈ H10 (Ω), we have also

(g,−∇ · ∇εp2)L2(Ω) = (∇g,∇εp2)L2(Ω) . (∇g,∇p2)L2(Ω) + ε‖g‖H1(Ω)

= −(g,∇ · ∇p2)L2(Ω) + ε‖g‖H1(Ω) ≤ ‖g‖α−1L2(Ω)‖p2‖αH2(Ω) + ε‖g‖H1(Ω)

and therefore

(g,−∇ · ∇εpF )L2(Ω)

. ‖g‖α−1L2(Ω)∩α−1/2H1(Ω)‖pF‖α1/2H1(Ω)+αH2(Ω) + ε‖g‖H1(Ω)

. ‖g‖α−1L2(Ω)∩α−1/2H1(Ω)‖pF‖α1/2H1(Ω)+αH2(Ω) + εh−1k α1/4‖F‖(X−,k)∗

is satisfied. The same can be done for (ζ,∆ελF )L2(Ω) and (χ,∇ ·∇εµF )L2(Ω).Combining these results, we immediately obtain (25), which finishes the

proof. ut

Theorem 4 Condition (A4) is satisfied.

Proof By combining (23) and (24), we obtain

‖xF‖2X+,k≤ C

(‖F‖2(X−,k)∗

+ ‖F‖(X−,k)∗‖xF‖X+,k

)for some constant C > 0 (independent of k and β) which implies

‖xF‖X+,k≤ 1

2

(C +

√4C + C2

)‖F‖(X−,k)∗ ,

i.e. (21), which finishes the proof. ut

So, we have shown condition (A4). So, Theorem 1 implies the approxi-mation property. Note, that we have now shown the approximation propertyin the norm ‖ · ‖X−,k , i.e., (22). The next step is to show the approximationproperty in the norm-pair ||| · |||0,k and ||| · |||2,k, i.e., (15).

To show (15), the following lemma is sufficient.


Lemma 6 The inequality

|||xk|||0,k . ‖xk‖X−,k (26)

is satisfied for all xk ∈ Xk.

Proof The proof of this lemma is based on (12), on (13) and on Lemma 4.7in [16]. Lemma 4.7 states (in the notation of the present paper and for thechoice β := α−1/2) that

α−1/2|||yk|||2Y,0,k . α−1/2‖yk‖2Y−,k

is satisfied for all yk ∈ Yk. As for xk = (yk, ψk) ∈ Xk = Yk × Yk both,

|||(yk, ψk)|||20,k = |||yk|||2Y,0,k + α−1|||ψk|||2Y,0,k

and

‖(yk, ψk)‖2X−,k= ‖yk‖2Y−,k

+ α−1‖ψk‖2Y−,k,

is satisfied by definition, (26) follows immediately. ut

So, we have shown the approximation property (15). So, we obtain thefollowing overall convergence result.

Theorem 5 Assume that

– the regularity assumptions (R) and (R1) are satisfied on the domain Ω,– the problem is discretized using the Taylor-Hood element and– the normal equation smoother introduced above is used as smoother.

Then the two-grid method converges if sufficiently many smoothing stepsare applied, i.e., we have

|||x(1)k − x∗k|||0,k ≤ q(ν)|||x(0)k − x

∗k|||0,k,

with q(ν) := CS CA ν−1/2, where the constants CA and CS are independent of

the grid level k and the choice of the parameter α.

Note that we have shown that the method converges in the norm ||| · |||0,k ifsufficiently many pre-smoothing steps are applied. The application of post-smoothing steps does not derogate the convergence because the proposedsmoother is power-bounded. Moreover, if we assume that only post-smoothingsteps are applied, the combination of smoothing property and approximationproperty (which now have to be combined in the inverse order) leads to con-vergence in the residual norm ||| · |||2,k. Again, due to power-boundedness ofthe smoother, the method stays convergent if, besides sufficiently many post-smoothing steps, also pre-smoothing steps are applied.

For all the mentioned cases, the convergence of the W-cycle multigridmethod follows under weak assumptions, cf. [9].

18 Stefan Takacs

5 Numerical Results

In this section, we illustrate the convergence theory presented within this paperwith numerical experiments.

The domain Ω was chosen to be the unit square Ω := (0, 1)2. As mentionedin Section 2, the weak inf-sup-condition (S) can be shown for the Taylor-Hoodelement only if at least one vertex of each element is located in the interiorof the domain Ω. As this is not satisfied for the standard decomposition ofthe unit square into two triangular elements, we have chosen the coarsest grid(grid level k = 0) to be a decomposition of the domain Ω into 8 triangles, cf.Fig. 1. The grid levels k = 1, 2, . . . were constructed by uniform refinement,i.e., every triangle was decomposed into four subtriangles.

The desired velocity field (desired state) uD was chosen to be

uD(ξ1, ξ2) :=

(ξ2 − 1

212 − ξ1

)for

√(ξ1 − 1

2

)2+(ξ2 − 1

2

)2< 4

5

0 otherwise.

The desired velocity field is visualized in the left-hand-side picture in both,Fig. 2 and 3.

For solving the discretized KKT-system, we have used the proposed W-cycle multigrid method. We have applied ν pre- and ν post-smoothing stepsusing the normal equation smoother. The damping parameter was chosen tobe τ = 0.35 for all grid levels k and all choices of α.

Fig. 1 Discretization on grid levels k = 1 and k = 2, where the squares denote the degreesof freedom of (the components of) u and λ are the the dots denote the degrees of freedomof p and µ

The solution of the optimal control problem can be seen in Fig. 2 and3. Note that the desired velocity field is a general L2-function (due to thejump) and therefore it cannot be reached by the (optimal) velocity field whichis an H1-function and therefore continuous. For the case α = 1, we observethe optimal velocity field and the control to be rather smooth. (The controldoes not take large values). For small values of α, like α = 10−12, the desiredvelocity field is approximated quite well, cf. Fig. 3. This is achieved by ratherlarge values of the control (high forces) which are concentrated on the regionwhere the desired velocity field has its jump. Forces have to be applied with thesame orientation as the desired state, as well as with the opposite orientation.


1.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0001

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.01

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 2 Desired velocity field uD, optimal velocity field u and optimal control f for α = 1on grid level k = 3

1.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

10 000.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 3 Desired velocity field uD, optimal velocity field u and optimal control f for α =10−12 on grid level k = 3

(In the picture mainly the forces with opposite orientation can be seen.) Asmentioned in the introduction, we are interested in a fast linear solver whichalso works well for such small choices of α.

The number of iterations and the convergence rate were measured as fol-

lows: we start with x(0)k = 0 and measure the reduction of the error in each

step using the residual norm ||| · |||2,k. The iteration was stopped when theinitial error was reduced by a factor of ε = 10−6. The convergence rates q isthe mean convergence rate in this iteration, i.e.,

q =

(|||x(n)k − x∗k|||2,k|||x(0)k − x∗k|||2,k

)1/n

,

where n is the number of iterations needed to reach the stopping criterion.

Here, x∗k is the exact solution and x(i)k is the i-th iterate.

In Table 1 we compare for a fixed choice α = 1 the convergence rates forseveral choices of ν, the number of pre- and post-smoothing steps. We see thatthe convergence rates behave approximately like ν−1/2. This is consistent withthe theory which guarantees the convergence rate being bounded by C ν−1/2 asthis only describes the asymptotic behavior. Moreover, we give the CPU-timesof a straight-forward implementation of the proposed algorithm to confirm thatthe overall complexity is almost linear in the number of unknowns.

20 Stefan Takacs

nr. of ν = 1 + 1 ν = 2 + 2 ν = 4 + 4 ν = 8 + 8unknownsn q t n q t n q t n q t

k = 3 4934 63 0.802 0.3 32 0.648 0.3 20 0.488 0.3 14 0.353 0.4

k = 4 19078 61 0.796 1.7 32 0.647 1.6 21 0.507 1.9 15 0.390 2.7

k = 5 75014 60 0.792 8.4 32 0.645 8.1 20 0.498 9.2 15 0.382 13.3

k = 6 297478 58 0.787 37.2 31 0.636 35.7 20 0.490 44.7 14 0.367 57.8

k = 7 1184774 56 0.779 159.7 29 0.620 155.9 18 0.463 186.3 13 0.340 252.6

Table 1 Number of iterations n, convergence rate q and time t (in seconds) to reach theaccuracy goal depending on ν = νpre + νpost, the number of pre- and post-smoothing steps,W-cycle, α = 1

α = 1 α = 10−3 α = 10−6 α = 10−9 α = 10−12

n q n q n q n q n q

k = 3 32 0.648 33 0.651 35 0.673 48 0.749 51 0.760k = 4 32 0.647 32 0.646 33 0.657 46 0.738 73 0.827k = 5 32 0.645 32 0.644 32 0.644 39 0.697 60 0.793k = 6 31 0.636 31 0.636 31 0.635 32 0.647 46 0.739k = 7 29 0.620 29 0.620 29 0.618 29 0.621 42 0.716

Table 2 Number of iterations n and convergence rate q for ν = 2 + 2 pre- and post-smoothing steps, W-cycle

α = 1 α = 10−3 α = 10−6 α = 10−9 α = 10−12

n q n q n q n q n q

k = 3 30 0.631 31 0.637 26 0.587 35 0.671 40 0.705k = 4 32 0.649 33 0.653 29 0.621 31 0.639 49 0.752k = 5 34 0.659 34 0.661 32 0.643 27 0.595 45 0.734k = 6 34 0.665 35 0.668 33 0.653 28 0.603 36 0.678k = 7 35 0.673 36 0.676 34 0.660 30 0.626 32 0.647

Table 3 Number of iterations n and convergence rate q for ν = 3 + 3 pre- and post-smoothing steps, V-cycle

In Table 2 we compare various grid levels k and choices of the parameter αfor the W-cycle. Here, we have used a fixed choice of ν = 2 + 2 pre- and post-smoothing steps. First we observe that the number of iterations seems to bewell-bounded for all grid levels k which yields an optimal convergence behavior.Moreover, we see that the number of iterations is also well-bounded for a widerange of choices of the parameter α, i.e., we observe also robust convergenceas predicted by the convergence theory. In Table 3, we see that the (moreefficient) V-cycle multigrid method converges with rates comparable to theconvergence rates of the W-cycle multigrid method. However, the V-cycle isnot covered by the convergence theory. In Table 4, the relative L2-errors of thevelocity field uk and the pressure distribution pk, both for the final iterates,are given. We observe that for moderate α, the errors in the velocity field are


‖u(n)k−u∗

k‖L2(Ω)

‖u(0)k−u∗

k‖L2(Ω)

‖p(n)k−p∗k‖L2(Ω)

‖p(0)k−p∗

k‖L2(Ω)

α = 1 3.61 10−8 9.68 10−8

α = 10−3 6.36 10−10 1.15 10−9

α = 10−6 1.59 10−8 5.21 10−7

α = 10−9 9.62 10−7 4.04 10−4

α = 10−12 1.28 10−5 4.00 10−2

Table 4 Final relative error of the velocity uk and the pressure pk of the W-cycle multigridmethod with ν = 2 + 2 pre- and post-smoothing steps on grid level k = 5

well bounded, whereas the errors in the pressure distribution p grow mildly asα approaches 0.

α = 1 α = 10−3 α = 10−6 α = 10−9 α = 10−12

n q n q n q n q n q

k = 3 85 0.849 82 0.843 54 0.771 32 0.642 23 0.548k = 4 87 0.853 86 0.851 62 0.797 34 0.664 22 0.529k = 5 88 0.854 90 0.858 66 0.810 40 0.703 22 0.529k = 6 86 0.851 94 0.863 70 0.819 46 0.735 24 0.557k = 7 86 0.849 94 0.863 72 0.823 50 0.758 27 0.599

Table 5 Number of iterations n and convergence rate q for a preconditioned MINRESmethod

In Table 5, the author gives the number of iterations that is required for aMINRES iteration, which was set up as proposed in [21], to reduce the initialerror by a factor of ε = 10−6. Here it has to be mentioned that for each step ofthe MINRES iteration, we have to apply one multigrid cycle for both, the thevelocity u and the adjoined velocity λ, and due to the Cahouet-Chabard-likesplitting (cf. [7]), as proposed in [21], one multigrid cycle and one step of sym-metric Gauss-Seidel for both, the pressure p and the adjoined pressure µ. Themultigrid cycles have been realized as proposed in [21], i.e., as V-cycles withone symmetric Gauss-Seidel iteration for both, pre- and post-smoothing. Al-though the multigrid cycles for these blocks can be computed faster than onemultigrid cycle of the proposed all-at-once multigrid method, due to smalleriteration numbers the proposed method shows comparable overall computa-tional costs. (Only for particularly small values of α the block-preconditionedMINRES approach seems to be faster.) The numerical results presented in thispaper can be significantly improved by a slight modification of the smoother,cf. [17].

6 Conclusions and Further Work

In the present paper we have shown that the construction of an all-at-oncemultigrid method for a Stokes control problem is possible. Here, a precondi-

22 Stefan Takacs

tioned normal equation smoother was chosen. The overall numerical complex-ity of this method seems to be comparable to block-preconditioned MINRESiterations, cf., e.g., Table 4.2 in [21], which shows the number of MINRESiterations needed. (Note that in each MINRES step one multigrid cycle is ap-plied to each component of the overall block-matrix, i.e., to the velocity u, thepressure p, the adjoined velocity λ and the adjoined pressure µ.)

One advantage of the all-at-once multigrid method, introduced in thepresent paper, is the fact that an outer iteration is not necessary, the multigriditeration is an linear iteration scheme which can be directly applied to solve theproblem. As we could show the approximation property for a particular choiceof norms, the construction of other smoothers is of particular interest. Theconvergence rates we have observed in this paper for a multigrid method withnormal equation smoothing are comparable with the convergence rates ob-served in [19] for a multigrid method with normal equation smoothing appliedto an optimal control problem with elliptic state equation. For that problemwe have seen that other smoothers are available which lead to much fasterconvergence rates, cf. [18] and others. Similar improvements were possible forthe generalized Stokes problem, cf. [16]. Therefore, it seems to be reasonableto construct faster smoothers also for the the Stokes control problem.

Acknowledgments. The author thanks Markus Kollmann for providingparts of the code used to compute the numerical results presented in this paper.Moreover, the support of the numerical analysis group of the MathematicalInstitute, University of Oxford, is gratefully acknowledged. Finally, the authorthanks the referees for very valuable remarks that have allowed to improve thepaper significantly.

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This paper has been published inNumerische Mathematik 130 (3), p. 517 - 540, 2015

The final publication is available at Springer viahttp://dx.doi.org/10.1007/s00211-014-0674-5

a robust all-at-once multigrid method for the stokes...

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