Boundary Control Problems in Convective Heat Transfer with ... · Vanka-type smoothers. With thesethe problem is split into small blocks of ﬁnite element subdomains with a small

Commun. Comput. Phys.doi: 10.4208/cicp.130914.230115a

Vol. 18, No. 3, pp. 621-649September 2015

Boundary Control Problems in Convective Heat

Transfer with Lifting Function Approach and

Multigrid Vanka-Type Solvers

Eugenio Aulisa1, Giorgio Bornia1,∗ and Sandro Manservisi2

1 Department of Mathematics and Statistics, Texas Tech University, Lubbock,TX 70409-1042, USA.2 Dipartimento di Ingegneria Industriale, Universita’ di Bologna, Bologna, Italy.

Received 13 September 2014; Accepted (in revised version) 23 January 2015

Abstract. This paper deals with boundary optimal control problems for the heat andNavier-Stokes equations and addresses the issue of defining controls in function spaceswhich are naturally associated to the volume variables by trace restriction. For thisreason we reformulate the boundary optimal control problem into a distributed prob-lem through a lifting function approach. The stronger regularity requirements whichare imposed by standard boundary control approaches can then be avoided. Further-more, we propose a new numerical strategy that allows to solve the coupled optimal-ity system in a robust way for a large number of unknowns. The optimality systemresulting from a finite element discretization is solved by a local multigrid algorithmwith domain decomposition Vanka-type smoothers. The purpose of these smoothersis to solve the optimality system implicitly over subdomains with a small number ofdegrees of freedom, in order to achieve robustness with respect to the regularizationparameters in the cost functional. We present the results of some test cases where tem-perature is the observed quantity and the control quantity corresponds to the bound-ary values of the fluid temperature in a portion of the boundary. The control region forthe observed quantity is a part of the domain where it is interesting to match a desiredtemperature value.

AMS subject classifications: 49J20, 65N55

Key words: Boundary optimal control, multigrid and domain decomposition algorithms, heatand Navier-Stokes equations.

1 Introduction

The optimal control of the heat and Navier-Stokes equations is an interesting subject dueto a wide range of applications. In particular, coupled thermo-fluid mechanics optimal

∗Corresponding author. Email addresses: [email protected] (E. Aulisa), [email protected](G. Bornia), [email protected] (S. Manservisi)

http://www.global-sci.com/ 621 c©2015 Global-Science Press

622 E. Aulisa, G. Bornia and S. Manservisi / Commun. Comput. Phys., 18 (2015), pp. 621-649

control problems are very challenging due to their difficulties both in the theoretical for-mulation and in the construction of the computational algorithm. The implementationof a computationally feasible and robust boundary control algorithm is a nontrivial task.The first nontrivial issue is the definition and numerical implementation of the boundarycontrol in a large functional space which is naturally associated to the volume variables.A control boundary constraint or a cumbersome numerical implementation can seriouslylimit the set of possible solutions. For this reason in many circumstances it is desirableto transform a boundary optimal control problem into a distributed problem throughthe lifting function approach of nonhomogeneous boundary conditions. With the liftingfunction approach, boundary controls are defined in the appropriate function bound-ary spaces restriction of the natural state volume spaces. Another challenging task in thestudy of optimal control problems is the numerical solution of the optimality system. Thecoupled implicit solution of the state, adjoint and control equations may allow a strongerand more robust solution process, as the oscillations that would be induced in the case ofuncoupled solution become negligible. The uncoupled solution of the state and the ad-joint system brings to limits in the choice of the penalty parameter for the tracking termin the objective functional. In fact in the limit of this parameter tending to infinity thesolution tends to lose smoothness and the uncoupled algorithm cannot converge. For astrong and robust optimization the optimality system must be solved in a coupled way.To this end, we propose the use of multigrid algorithms with domain decompositionVanka-type smoothers. With these the problem is split into small blocks of finite elementsubdomains with a small number of degrees of freedom, so that an optimal solution iscomputed by solving the fully coupled state-adjoint system on each subdomain. In thisway large penalty parameters for the tracking term can be used and a better matchingwith the desired state can be attained.

Various works have been presented in literature for this class of problems, see for ex-ample [13–15] and references therein. The analysis of a two-dimensional problem for theBoussinesq equations is treated in [15]. Neumann boundary optimal control problems forthe stationary Boussinesq equations including solid media are considered in [13]. In [14],a linear feedback control problem for the Boussinesq equations is studied. In this paper,we consider a similar optimal control problem for the heat and Navier-Stokes equationsin which velocity influences temperature but the effects of temperature on the velocityand pressure fields, such as buoyancy, are neglected. This type of one-way coupling oc-curs in the case where forced convection is the dominant physical mechanism for heattransfer. The control is performed through the boundary conditions of temperature onall or part of the boundary. These boundary controls are implemented in the form oflifting functions, since this method brings several advantages from theoretical and com-putational points of view [3, 17].

The implementation of boundary controls is a nontrivial task, especially when onewants to search for controls in natural trace spaces. In [18, 19] the trace space H1/2(Γ)is used for the control, but this involves the introduction of a norm on this space that isrealized by means of a stabilized hypersingular boundary integral operator. This con-

E. Aulisa, G. Bornia and S. Manservisi / Commun. Comput. Phys., 18 (2015), pp. 621-649 623

struction is only applicable for elliptic constraint equations and if a fundamental solutionis known. For the discretization, finite element and boundary element approximationsare considered. In [12], wavelet coordinates are used to introduce a representation of anoriginal elliptic abstract control problem. In the wavelet framework, this allows to ob-tain an equivalent cost functional in which the norm of the wavelet representation of thecontrol stems from measuring the original control in the natural space H1/2(Γ). In ourpresent work we penalize the cost functional with lifting functions and we use standardnorms on volume spaces. Our approach is intended to work for general operators andnot to be restricted to elliptic operators.

We briefly introduce some basic notations and definitions for the functional analyticframework. Let Ω be a subset of R

N with Lipschitz continuous boundary Γ. When E⊂Ω,|E| denotes the Lebesgue measure of the set E. When F⊂Γ, σ(F) is the surface measureof F. We denote as Hm(Ω) the usual Sobolev spaces of square integrable functions withrespect to the Lebesgue measure in R

N and with square integrable weak derivatives upto order m. We use the boldface notation Hm(Ω) for vector-valued functions. We denotewith (·,·)m the scalar product in Hm(Ω) and the induced norm is ‖·‖m =

√(·,·)m [1]. Let

Hm0 (Ω) denote the closure of C∞

0 (Ω) with respect to the norm ‖·‖m and H−m(Ω) denotethe dual space of Hm

0 (Ω). Also, we define the spaces

L20(Ω)=

p∈L2(Ω)

∣∣∫

Ωpdx=0

(1.1)

and, given any subset Γs⊂Γ, we denote the space H1Γs(Ω) as

H1Γs(Ω)=u∈H1(Ω) |γΓs u=0, (1.2)

where γΓs is the trace restriction operator on Γs. Let us introduce the bilinear and trilinearforms. We define

aT(T,v)=∫

Ω∇T ·∇vdΩ ∀T,v∈H1(Ω), (1.3)

cT(u,T,v)=∫

Ω(u·∇)T ·vdΩ ∀T,v∈H1(Ω), u∈H1(Ω). (1.4)

For the Navier-Stokes equations, we set

a(u,v)=∫

Ω∇u :∇vdx, d(v,q)=−

∫

Ωq∇·vdx, (1.5)

c(u,v,w)=∫

Ω(u·∇)v ·wdx. (1.6)

Also, let us introduce the nondimensional quantities

Re=ρUL

µ, Pr=

µcp

k, Pe=RePr, (1.7)


where Re is the Reynolds number, Pr is the Prandtl number, Pe is the Peclet number, andρ,U,L,µ,cp,k denote the reference values for density, velocity, length, dynamic viscosity,specific heat at constant pressure and thermal conductivity of the fluid, respectively.

Hence, in the Navier-Stokes equations we pick f ∈H−1(Ω), g∈H1/2(Γ) and we seekfor (u,p)∈H1(Ω)×L2

0(Ω) such that

1

Rea(u,v)+c(u,u,v)+d(v,p)=< f ,v> ∀v∈H1

0(Ω), (1.8)

d(u,q)=0 ∀q∈L20(Ω), (1.9)

u= g on Γ, (1.10)

where g satisfies the compatibility condition∫

Γg·ndΓ=0. Let us consider the temperature

equation. In order to formulate the optimal control problem, for the temperature Dirich-let boundary conditions we subdivide the boundary into two parts, Γd and Γc = Γ\Γd.The Dirichlet datum can be split into two functions hd and hc defined on Γd and Γc re-spectively. With the function hd on Γd we enforce fixed Dirichlet conditions, while thefunction hc is variable in order to perform the control on the system. Therefore, givenfT ∈H−1(Ω) we seek T∈H1(Ω) such that

1

PeaT(T,v)+cT(u,T,v)=< fT,v> ∀v∈H1

0(Ω), (1.11)

T=hd on Γd , T=hc on Γc , (1.12)

where the quantity fT is a heat source that may come from viscous dissipation or otherphysical effects. Let us enforce Dirichlet temperature conditions with a lifting function.So, we simply introduce T0∈H1(Ω) such that

T0=hd on Γd , T0=hc on Γc . (1.13)

Hence, we can express the temperature field as T=T∗+T0 and, for a given T0∈H1(Ω)such that (1.13) is satisfied, we seek T∗∈H1

0(Ω) such that

1

PeaT(T

∗,v)+cT(u,T∗,v)=< fT ,v>−1

PeaT(T0,v)+cT(u,T0,v) ∀v∈H1

0(Ω). (1.14)

The heat equation has the following strong form

−1

Pe∆T∗+(u·∇)T∗= fT+

1

Pe∆T0−(u·∇)T0 , (1.15)

T∗=0 on Γ, (1.16)

T0=hd on Γd , T0=hc on Γc (1.17)

and for the Navier-Stokes equations we can write

(u·∇)u=1

Re∆u−∇p+ f in Ω, (1.18)

∇·u=0 in Ω, (1.19)

u= g on Γ. (1.20)


We remark that in the system (1.15)-(1.20) the coupling between the Navier-Stokes andtemperature equations is a one-way coupling, in the sense that velocity influences tem-perature through convection while we do not consider possible influences of temperaturein the Navier-Stokes equations, such as buoyancy or variable density.

Now, let Uad be the set of admissible targets and T0 be the control. We formulate theoptimal control problem as follows.

Problem 1. Given a desired temperature Td ∈Uad and a pair (u,p) satisfying the system(1.8)-(1.10), find a minimizer of the objective functional

J (T∗,T0)=α

2

∫

Ω|T∗+T0−Td|

2 dΩ+β

2

∫

ΩT2

0 dΩ+γ

2

∫

Ω|∇T0|

2 dΩ, (1.21)

where α, β and γ are positive constants and the variables T∗ and T0 are subject to the stateequation (1.15)-(1.17).

Of course, the target Td need not be a solution of the state equation (1.15). For instance,we may choose Uad = L2(Ω). Clearly, the goal is to match the fluid temperature to thedesired profile Td. The β-terms are used to limit the size of the control T0. The values ofthe constants α,β and γ can be chosen to adjust the relative importance of the terms inthe functional. Large values of α allow for better temperature matching whereas smallvalues result in poorer matching. The presence of the γ-term is motivated by the need tohave T0 in H1 so that the optimal control problem is well-posed.

The lifting function can be considered an extension of the temperature field from theboundary to the interior. The optimal solution is then searched by exploring all possi-ble extended functions. This approach brings several advantages from both a theoreticaland a computational point of view. In fact in this case the boundary control algorithmcan be solved by robust distributed control techniques over the inner part of the domain.Furthermore, the optimal boundary controls can be determined in natural half-integerSobolev spaces as traces of the optimal extended functions. Also, in a direct boundaryapproach one would have to deal with fractional Sobolev norms which are much lessconvenient from a numerical point of view than the H1-norm of the lifting function. Thisapproach has also a disadvantage: the number of controlled points increases since theinterior has much more points than the boundary. This drawback can be limited by re-stricting the controlled area to a small region close to the boundary.

The plan of the paper is as follows. In Section 2 we address the existence results for theoptimal control problem under study, and we illustrate the advantageous features of thelifting function approach for the treatment of boundary controls. Section 3 is devoted tothe description of the local multigrid algorithm for the solution of the optimality system,which is considered in combination with domain decomposition Vanka-type smoothers.Section 4 illustrates the results of several computations aimed at describing the numericalproperties of the proposed algorithm.


2 Existence results

2.1 Existence of a constraint solution

In this paragraph we claim the basic results that are needed for the existence of a mini-mizer and for the subsequent first-order necessary condition. The state system consists inthe temperature equation (1.11), in which the velocity satisfies the Navier-Stokes system(1.8)-(1.10). The existence of a weak solution to (1.8)-(1.10) with nonhomogeneous Dirich-let boundary conditions can be proved by using the Hopf lifting construction [9,20]. Theoperators of the temperature equation (1.11) do not satisfy the coercivity property due tothe presence of the convective terms and therefore the usual Lax-Milgram setting cannotbe applied. This problem falls within the class of non-coercive elliptic partial differentialequations. Coercivity can be obtained only by adding further regularity hypotheses onthe velocity field.

Nevertheless, it is possible to claim the existence of the state solutions for the non-coercive elliptic case if the velocity field u is in L2(Ω) [8]. This existence result is obtainednot in the Lax-Milgram setting but by using a Leray-Schauder Topological Degree argu-ment [7]. We recall the result, which holds for weak solutions of a convection-diffusionequation of the following type

−div(AT∇T)+(u·∇)T+bT= f in Ω, (2.1)

T=T1 on Γ1, (2.2)

AT∇T ·n+λT=T2 on Γ2 . (2.3)

Here, Γ1 and Γ2 are measurable subsets of Γ such that σ(Γ1∩Γ2) = 0, and Γ = Γ1∪Γ2.Condition (2.2) is a Dirichlet boundary condition, while (2.3) is referred to as Fourierboundary condition.

Theorem 2.1. Let N∗=N when N≥3, N∗∈]2,∞[ when N=2. Consider the following hypothe-ses:

1. Γ1 and Γ2 are measurable subsets of Γ such that σ(Γ1∩Γ2)=0 and Γ=Γ1∪Γ2;

2. A :Ω→RN×N is a measurable matrix-valued function which satisfies these two properties:

• ∃αA >0 such that A(x)ξ ·ξ≥αA |ξ|2 for a.e. x∈Ω and for all ξ∈R

N ;

• ∃ΛA >0 such that ‖A(x)‖≤ΛA for a.e. x∈Ω;

3. b∈LN∗/2(Ω),b≥0 a.e. on Ω;

4. λ∈LN∗−1(∂Ω),λ≥0 σ−a.e. on Γ;

5. u∈LN∗(Ω);

6. f ∈ (H1Γ1(Ω))∗ (the superscript ∗ denotes the dual);


7. ∃b0 >0,∃E⊂Ω such that b≥ b0 in E, ∃λ0 >0,∃S⊂Γ2 such that λ≥λ0 in S, and eitherσ(Γ1)>0 or |E|>0 or σ(S)>0.

Then, there exists a unique solution T∈H1(Ω) of (2.1).

Proof. The proof of this result is based on a Leray-Schauder Topological Degree argumentand can be found in [8].

We remark that the hypothesis (7) in the previous theorem is usually referred to as thecoercivity of the non-convective parts in (2.1). Based on the previous result, we can statethe existence of solutions to the state equations (1.14).

Theorem 2.2. The Navier-Stokes system in (1.8) with Dirichlet boundary conditions has at leastone solution (u,p)∈H1(Ω)×L2

0(Ω), which is unique when Re<Rec, for some Rec>0 dependingon the data. Given a velocity field u satisfying (1.8) there exists a unique solution T∗∈H1

0(Ω) to(1.14).

Proof. The existence of a weak solution (u,p) ∈ H1(Ω)×L20(Ω) to (1.8) with nonhomo-

geneous Dirichlet boundary conditions is well-known [9, 20]. Now, it is straightforwardto check that we can fit the temperature state equations (1.14) within the framework of(2.1). First, thanks to the Sobolev compact embeddings H1(Ω) →→ Lq(Ω) which holdfor 1≤q<∞ when N=2 and for 1≤q≤6 when N=3, we have that u∈H1(Ω) verifies thehypothesis (5) both with N=2 and with N=3. Then, for the matrix AT we have AT= 1

Pe I,where I is the identity matrix, so that (2) is trivially verified. Also, we notice that it suf-fices to have a non-zero measure Dirichlet boundary, σ(Γ1)> 0, to avoid the conditions(3) and (4), which would require b> 0 on a set with strictly positive Lebesgue measureand λ>0 on a set with strictly positive surface measure. In our case we have b=0, (1.14)is a particular case of (2.1) and Theorem 2.2 completes the proof.

Remark 2.1. We remark that, if one needs to use the Lax-Milgram setting, existence canbe proven by assuming some condition on the velocity field u. For instance, in the caseof fully Dirichlet boundary conditions (Γ2=∅), one can have coercivity with

−1

2divu+b≥0 in D′(Ω). (2.4)

This condition of additional regularity on the velocity field is not needed in the Leray-Lions setting.

2.2 Existence of an optimal solution

We define the set of admissible controls Cad as

Cad=T0∈H1(Ω) : T0=hd on Γd, (2.5)

where hd ∈H1/2(Γd) is a given function. The set of admissible states Aad is defined as

Aad=T∗∈H10(Ω) : T∗ solves (2.13) with T0∈Cad. (2.6)


Remark 2.2. The functions included in the set Cad can be defined over the subdomainΩc ⊂ Ω arbitrary small that includes the boundary Γd. This can be used to limit thecontrolled area.

Given these definitions, the optimal control problem consists of seeking T∗∈Aad andT0 ∈Cad such that the cost functional (1.21) is minimized for an admissible desired tem-perature profile Td ∈Uad. We now state the existence of a global minimizer in the set ofadmissible solutions Aad×Cad, i.e. the set of admissible states and controls.

Theorem 2.3. Given Td∈Uad, there exists a solution (T∗,T0)∈Aad×Cad of the optimal controlproblem (1).

Proof. The proof follows standard techniques [6, 11]. By Theorem (2.2) the state systemhas a solution, therefore the set of admissible solutions Aad×Cad is nonempty. We define

M := inf(T∗,T0)∈Aad×Cad

J (T∗,T0).

Clearly M exists by the greatest lower bound property of real numbers, since the setAad×Cad is nonempty, and M≥0 since the objective functional J (T∗,T0) is non-negative.Thus let (T∗

n ,T0,n) be a minimizing sequence in Aad×Cad for J (T∗,T0), i.e.,

limn→∞

J (T∗n ,T0,n)=M .

Now we show that the minimizing sequence (T∗n ,T0,n) is uniformly bounded. The

convergence of the sequence J (T∗n ,T0,n) implies that it is bounded in R, hence from the

definition of the objective functional we have that ‖T0,n‖1 is uniformly bounded, i.e., thesequence T0,n is uniformly bounded in Cad. Then by the continuous dependence of thestate solution on the data we also have that ‖T∗

n‖1 is uniformly bounded, i.e. the sequenceT∗

n is uniformly bounded in Aad. Since Aad×Cad is a reflexive Banach space, we can

extract a subsequence (T∗m,T0,m) that converges weakly to some (T∗,T0) in H1(Ω)×

H10(Ω).

Now we pass the weak limit inside the constraint operator in order to show that

(T∗,T0)∈ Aad×Cad, i.e. the weak limit of the subsequences satisfies the constraints. Thisis straightforward due to the linearity of the state equations with respect to the argumentsT∗

m and T0,m. Therefore, the set Aad×Cad is closed in the weak topology. In order to see

that (T∗,T0) is also a solution for the optimal control problem we finally have to show thatit minimizes the functional. In fact, by the weak lower semi-continuity of the functional,we have

M≤J (T∗,T0)≤ liminfm→∞

J (T∗m,T0,m)= lim

m→∞J (T∗

m,T0,m)=M .

Therefore J (T∗,T0) = M, i.e., the infimum M of the functional is attained at the point

(T∗,T0), which is indeed a global minimizer.


Remark 2.3. We remark that, for the previous result to hold, we need to have β and γstrictly positive. This yields the boundedness of the minimizing sequence T0,n in Cad. Theregularization terms can be avoided in the case of inequality constraints on the controlfunction T0.

2.3 Advantages of the lifting function approach

For a better understanding of the proposed formulation for boundary control problems,let us point out the differences occurring between a standard direct boundary controlapproach and the lifting function one. A direct boundary control formulation would beas follows:

Problem 2. Seek for a minimizer of the objective functional

Jb(T,hc)=αb

2

∫

Ω|T−Td|

2 dΩ+βb

2

∫

Γc

h2c dΩ+

γb

2

∫

Γc

|∇shc|2 dΩ, (2.7)

where ∇s denotes the surface Laplacian operator.

This direct implementation presents some disadvantages that can be circumventedwith the lifting function approach. First of all, due to the presence of the βb and γb

penalty boundary integrals in the objective functional (2.7), we see that the solution of theboundary control equation would belong to H1(Γc). Hence, it would be smoother thanthe half-integer space H1/2(Γc), which is a natural trace space for H1(Ω), the space for thetemperature T variable. Therefore, a direct boundary approach would yield a smootherboundary temperature than needed, thus excluding less regular solutions. Instead, withthe lifting function approach one computes an optimal control function T0 in H1(Ω) andthen one can determine the optimal boundary control simply by trace restriction.

Also, the optimality condition for the control variable hc arising from the objectivefunctional (2.7) is an equation defined on the boundary control subregion Γc. When thecontrol region Γc is not single-connected, i.e. consisting of a set of disjoint boundary por-tions, a different control equation should be implemented for each boundary part, lead-ing to a rather cumbersome implementation. On the other hand, with the lifting functionequation one can deal with controls in disjoint parts of the boundary with a single equa-tion defined on the overall fluid domain, thus obtaining an easier representation.

Moreover, the introduction of the surface Laplacian term in (2.7) leads to a differentialrather than algebraic optimality condition, so that boundary conditions on the controlfunction must be enforced. At the conjunction between the Γc and Γd parts, in mostcases the boundary conditions to be imposed on the control function hd are not physicallymeaningful and they must be enforced with some amount of arbitrariness.

Concerning the difference between the functional (2.7) and (1.21), we finally have toremark that the βb and γb terms in (2.7) actually limit the size of the H1(Γ) norm of theboundary control function, while the β and γ terms in (1.21) limit the H1(Ω) norm of thelifting function T0. This means that in the second case not only the degrees of freedom for


the boundary control are involved, but also the degrees of freedom inside the domain,which are used for auxiliary purposes. If the addition of the inner degrees of freedom iscomputationally too expensive, the extension of the boundary conditions can be limitedto any desired subvolume.

Also, we remark that Problem (1) and Problem (2) are not equivalent, in the sense that

a solution (T,hc) of Problem (2) need not be equal to (T∗,γΓc T0). In fact, hc∈H1(Γc) whileγΓc T0 ∈ H1/2(Γc) (the existence of a solution to Problem (2) can be proven much in thesame way as for Problem (1)).

2.4 First order necessary condition and the optimality system

In order to determine an optimality system whose solutions are candidate solutions forthe optimal control problem, we use the well-known technique of Lagrange multipliersfor problems constrained by partial differential equations [10, 21]. To this end, let usconsider the constraint operator M : H1

0(Ω)×H1(Ω)→H−1(Ω) [11]. For given u solvingthe system (1.18), the constraint operator is defined as M(T,T0)= f , where

< f ,v>=1

PeaT(T

∗,v)+cT(u,T∗,v)+1

PeaT(T0,v)+cT(u,T0,v) (2.8)

for all functions v∈H10 (Ω) and boundary conditions on the control T0=hd on Γd. In other

words, the constraints (1.14) may be expressed as M(T,T0)= fT . Since in this work thecoupling is one-way, the function u must be regarded as an external datum, so that (u,p)is a pair of external functions.

The Frechet differential M′ of the operator M is the continuous linear operator M′ :H1

0(Ω)×H1Γd(Ω)→H−1(Ω) such that M′ ·(δT∗,δT0)=q if and only if

<q,v>=1

PeaT(δT∗,v)+cT(u,δT∗,v)+

1

PeaT(δT0,v)+cT(u,δT0,v) (2.9)

for all test functions v∈H10 (Ω) and with a given u solving the system (1.18). Due to the ex-

istence and uniqueness result for the state equations, the operator M′ is an isomorphism,hence surjective, and its range is closed.

After the introduction of the constraint operator, one can deal with the optimal con-trol problem by adopting well-known Lagrange multiplier techniques. To do this, oneintroduces the augmented objective functional

L(T∗,T0,λ)= aJ (T∗,T0)+<λ,M(T∗,T0)>, (2.10)

where λ ∈ H10(Ω) and < ·,·> denotes the duality pairing between H1

0(Ω) and its dualH−1(Ω). Then, one may claim the following first order necessary condition by usingstandard techniques (for instance, see [11]).


Theorem 2.4. Let (T∗,T0) be a local minimizer for the objective functional J (T∗,T0). Thereexists (a,λ)∈R×H1(Ω) such that

L′(T∗,T0,λ)= aJ ′(T∗,T0)·(δT∗,δT0)

+<λ,M′ ·(δT∗,δT0)>=0 ∀(δT∗,δT0)∈ (H10(Ω)×H1

Γd(Ω)), (2.11)

where the superscript ′ denotes Frechet differentiation. Furthermore, since M′(T∗,T0)·(δT,δT0)is surjective, we have a 6=0 so that one may set a=1.

From (2.11) one derives the following adjoint and control equations. First, we noticethat

J ′(T∗,T0)·(δT∗,δT0)=α(T∗+T0−Td,δT∗)+α(T∗+T0−Td,δT0)

+β(T0,δT0)+γ(∇T0,∇δT0). (2.12)

By splitting the independent variations, for the adjoint equation we gather the varia-tions δT∗ and seek λ∈H1

0(Ω) such that

1

PeaT(λ,δT∗)+cT(u,δT∗,λ)+α(T∗+T0−Td,δT∗)=0 ∀δT∗∈H1

0(Ω). (2.13)

The control equation is given by gathering the variations δT0, to obtain

β(T0,δT0)+γaT(T0,δT0)+1

PeaT(λ,δT0)+cT(u,δT0,λ)

+α(T∗+T0−Td,δT0)=0 ∀δT0∈H1Γd(Ω). (2.14)

3 Numerical algorithm

The global system can be split into two parts: the Navier-Stokes equations and the tem-perature optimality system. The Navier-Stokes system is decoupled from the optimal-ity system and can be solved beforehand. Its linearization is performed using the fixedpoint approximation strategy. Once the velocity field is known, it can be explicitly usedinto the optimality system that is therefore linear. For the solution of both the linearizedNavier-Stokes and the temperature optimality systems we intend to use the Local Multi-grid algorithm which is described in Section 3.1. The basic idea is to combine the majorfeatures of geometric multigrid along with a-priori local mesh refinement.

Concerning the optimality system, the Local Multigrid algorithm is an example of col-lective smoothing multigrid (CSMG) approach [4]. The coupling in the optimality systembetween the state, adjoint and control variables is realized at the smoothing step level, inorder to achieve typical multigrid efficiency and robustness with respect to changes inthe regularization constants α,β,γ of the cost functional [4]. For the multigrid smoothing


of both systems, we investigate the use of Vanka-type smoothers, a discussion on whichis addressed in Section 3.2.

The discretization of the optimality system and of the Navier-Stokes equations is per-formed in a finite element framework. We approximate the optimality system variablesT∗, λ and T0 by choosing a finite element space Yh⊂H1(Ω). For the velocity vector u andthe pressure variable p, we choose a pair of finite dimensional spaces (Xh,Mh), such thatXh ⊂H1(Ω) and Mh ⊂ L2(Ω), respectively. The construction of the spaces Yh,Xh,Mh is il-lustrated in the following section within the framework of the Local Multigrid algorithm.We also recall that the pair of spaces Xh and Mh has to be chosen appropriately in sucha way that the discrete inf-sup condition is satisfied. This guarantees the stability of theapproximation of the Navier-Stokes equations [9].

We denote X0,h=Xh∩H10(Ω), M0,h=Mh∩L2

0(Ω), Y0,h=Yh∩H10(Ω); moreover, for any

subset Γs ⊂ Γ, we set YΓs,h =Yh∩H1Γs(Ω). We seek a solution (T∗

h ,λh,T0,h)∈Y0,h×Y0,h×Yh

that satisfies the discrete optimality system

1

PeaT(T

∗h ,wh)+cT(uh,T∗

h ,wh)+1

PeaT(T0,h,wh)+cT(uh,T0,h,wh)=< fT ,wh>, (3.1)

1

PeaT(λh,δT∗

h )+cT(uh,δT∗h ,λh)+α(T∗

h +T0,h−Td,h,δT∗h )=0, (3.2)

β(T0,h,δT0,h)+γaT(T0,h,δT0,h)+1

PeaT(λh,δT0,h)+cT(uh,δT0,h,λh)

+α(T∗h +T0,h−Td,h,δT0,h)=0, (3.3)

T∗h =0 on Γ, (3.4)

λh=0 on Γ, (3.5)

T0,h=hd on Γd , (3.6)

for all test functions (wh,δT∗h ,δT0,h)∈Y0,h×Y0,h×YΓd,h. The pair (uh,ph)∈Xh×M0,h satisfies

the discrete Navier-Stokes equations

a(uh,vh)+c(uh,uh,vh)+d(vh,ph)=< f h,vh>, (3.7)

d(uh,qh)=0, (3.8)

uh= gh on Γ, (3.9)

for all test functions (vh,qh)∈Xh,0×M0,h.We remark that the boundary conditions for T∗

h and λh are Dirichlet homogeneousdue to the temperature decomposition Th = T∗

h +T0,h. The non-homogeneous boundaryconditions, either fixed or variable for the control, are all taken into account with the lift-ing function T0,h. Hence, after solving the optimality system for the variables (T∗

h ,λh,T0,h)one can retrieve the total temperature field Th by the sum Th=T∗

h +T0,h.

3.1 Local Multigrid algorithm

In this section we describe the Local Multigrid algorithm (LMG) used to discretize andsolve the temperature optimality system and the Navier-Stokes system. The domain Ω


Wall

Domain Ω Nonconforming mesh Level l=0

Level l=1 Level l=2 Level l=3

Level l=4 Level l=5 Level n=6Figure 1: Example of a nonconforming mesh built using multilevel triangulations.

is first divided into several non-overlapping subregions, for each of which we introducea finite element discretization. The different subregions may have different mesh sizesand the mesh may not be conforming at the interface between adjacent subregions. Nev-ertheless, continuity for each variable on the common boundaries can be enforced in anatural and straightforward way by an appropriate definition of the spaces in the geo-metric multigrid framework.

In Fig. 1 an example of geometrically nonconforming mesh partitioning is generatedfor a square domain Ω. In the region close to the right and bottom boundary sides a pro-gressive mesh refinement is constructed as follows. By starting at the coarse level l = 0,we discretize the entire domain Ω≡Ω0 into a collection of finite elements (quadrilateralsor triangles in two dimensions; hexahedra, wedges or tetrahedra in three dimensions). Ageometrically conforming coarse triangulation T 0

h is then generated throughout the en-


tire domain Ω0. Based on a simple element midpoint refinement, successive level meshesT l

h are built recursively on the corresponding subregions Ωl up to the top level l=n, untilthe physics of the considered problem is well resolved. Finer subregions are included incoarser ones, namely Ω0 ⊇ Ω1 ⊇ ··· ⊇ Ωn. This choice of local mesh refinement is veryefficient, since only a small number of degrees of freedom is added during the refine-ment process, which drastically reduces the computational cost with respect to the caseof uniform mesh refinement.

Each level mesh T lh is not defined on the whole domain Ω, but only on the subregion

Ωl, where the refinement up to level l is performed. Every triangulation T lh is geomet-

rically conforming within its own domain Ωl. In the example of Fig. 1, the refined sub-regions Ωl are thinner regions closer to the right and bottom sides. If we consider thenonconforming mesh given by the union of all the triangulations T l

h , the nodes of thecoarser mesh are always included in the nodes of the finer mesh on adjacent boundaries.This is the key point for this kind of discretization and it is always true for different levelmeshes generated by using midpoint refinement.

For each level l=0,1,··· ,n, let V lloc⊂H1

0(Ωl) be a continuous finite-element space built

on the triangulation T lh (for instance, Lagrange piecewise-linear or piecewise-quadratic).

Clearly, functions in V lloc have zero trace on ∂Ωl . Let V l be the extension by zero of V l

locover the whole domain Ω. Notice that each element ul ∈V l is continuous due to the zerotrace on the boundary ∂Ωl . Also, for all levels l we have V l ⊂ H1

0(Ω), i.e., these spacesare conforming in H1

0(Ω) by their definition. Moreover, V l−1*V l and V i∩V j 6=0, i, j=0,1,··· ,l, since finer functions can generate coarser ones. We define the space Vl as thealgebraic sum

Vl =l

∑k=0

Vk=

v∣∣v=

l

∑k=0

vk,vi ∈V i, i=0,1,··· ,l

. (3.10)

Clearly, this space is conforming in H10(Ω) as the algebraic sum of conforming subspaces

and it is defined on the geometrically nonconforming triangulation

Th,l =l⋃

k=0

T kh . (3.11)

Therefore we have V0 ⊆V1 ⊆···Vn, Th,0 ⊆Th,1 ···Th,n and for each level l two families ofspaces:

1. V l, built over a geometrically conforming triangulation on Ωl , with zero trace on∂Ωl and extension by zero in Ω\Ωl ;

2. Vl, built over a geometrically nonconforming triangulation up to level l.

In the case of uniform mesh refinement, we have Ω0 ≡Ω1 ≡ ··· ≡Ωn. If we denote theuniformly refined spaces with the superscript ∼, we recover Vl≡V l for all levels, becausethe refinement is performed over the whole domain Ω. Also we have V0⊆V1⊆···Vn [5].


Comparing the local with the uniform case, it is clear that Vl ⊆ Vl for all levels l, whichreflects the reduced number of degrees of freedom obtained with local refinement. Wedefine the index nMG to be the highest level index l for which Vl =V l. Note that nMG ≥0since on the coarsest level (l=0) we built a conforming triangulation all over the domainΩ. Moreover if n= nMG then at each level we have a conforming triangulation, and thestandard geometric multigrid spaces are recovered.

We can now define the inter-space transfer operators as follows. The coarse-to-fineoperator

I ll−1 :Vl−1→Vl (3.12)

is taken to be the natural injection. In other words

I ll−1u=u, ∀ u∈Vl−1. (3.13)

The fine-to-coarse inter-space transfer operator

I l−1l :Vl →Vl−1 (3.14)

is defined to be the transpose of I ll−1, with respect to the inner product (·,·). In other

words

(I l−1l wl,ul−1)=(wl, I

ll−1ul−1) ∀ ul−1∈Vl−1, wl ∈Vl. (3.15)

Note that although the inter-space operators are defined similarly to the intergrid opera-tors used in the standard geometric multigrid method [5], they differ due to the differentspaces involved (Vl instead of Vl).

We also define the prolongation operator

Pl :V l →Vl (3.16)

to be the natural injection, in other words Plv=v,v∈V l , and the restriction operator

Rl :Vl →V l (3.17)

to be the transpose of Pl, with respect to the inner product (·,·) such that

(Rlwl,ul)=(wl,Plu

l) ∀ ul ∈V l , wl ∈Vl. (3.18)

The Local Multigrid algorithm works similarly to a geometric multigrid algorithmwith the exception that the spaces for the unknown variables, the residuals, the errorcorrection and the corresponding test functions are redefined according to the local meshrefinement process [5]. For the same reason the inter-level (fine-to-coarse and coarse-to-fine), the restriction and the prolongation operators are non-standard. Below we presenta schematic description for the solution of the model problem

a(uh,vh)=( f ,vh) ∀vh ∈Vh, (3.19)


where Vh := Vn as defined in (3.10), a(·,·) is a differential operator, f is an externalforce/source term and they both depend on the physics of particular problem at hand.First we describe the lth level iteration, then the full algorithm.

The lth Level Iteration. LMG(l,z0,g) is the approximate solution of

a(z,vl)=(g,vl) ∀vl ∈Vl (3.20)

obtained at the lth level with initial guess z0∈Vl.For l=0, LMG(l,z0,g) is the solution of the problem

a(LMG(l,z0,g),v0)=(g,v0) ∀v0∈V0, (3.21)

obtained with a direct method.For l>0, LMG(l,z0,g) is obtained recursively in 3 steps.

1. Pre-smoothing Step. For 1≤ j≤m1, let

zj = zj−1+Pl

(Sl)−1

Rlrj, (3.22)

where zj ∈Vl and rj ∈Vl satisfies

(rj,vl)=(g,vl)−a(zj−1,vl) ∀vl ∈Vl . (3.23)

Here, Sl : V l →V l is some invertible smoothing operator that will be defined laterand it depends on the specific equation system.

2. Error Correction Step. Let q0=0, g := I l−1l rm1

and q1= LMG(l−1,q0, g). Then

zm1+1= zm1+ I l

l−1q1. (3.24)

3. Post-smoothing Step. For m1+2≤ j≤m1+m2+1, let

zj = zj−1+Pl

(Sl)−1

Rlrj. (3.25)

Then the output of the lth level iteration is

LMG(l,z0,g) := zm1+m2+1 , (3.26)

where m1 and m2 are positive integers.

The Full Local Multigrid algorithm. The solution uh∈Vh of the problem

a(uh,vh)=( f ,vh) ∀vh ∈Vh (3.27)


is obtained with the following nested iterations.

For l=0, u0∈V0 is the solution of

a(u0,v0)=( f ,v0) ∀v0∈V0, (3.28)

obtained with a direct method.

For 1≤ l≤n, the approximate solutions ul ∈Vl are obtained recursively from

ul,0= I ll−1ul−1, (3.29)

ul,j= LMG(l,ul,j−1, f ), 1≤ j≤ r, (3.30)

ul =ul,r. (3.31)

Finally, the solution of the problem (3.19) is uh=un. The spaces Yh for the variables T∗h , λh

and T0,h in the discretized temperature optimality system, as well as the spaces (Xh,Mh)for uh and ph are built up to the finest level n like the space Vn defined by (3.10).

3.2 Domain decomposition Vanka-type smoothing

With the exception of the coarse level where a direct solver is used, the pre-smoothingand the post-smoothing steps in the LMG algorithm are done with Vanka-type itera-tive smoothers, whose purpose is to reduce the residual norm. The Vanka-type class ofsmoothers can be considered as a block Gauss-Seidel method, where each block consistsof a small number of degrees of freedom (for details see [2,16,22,23]). In each step a largenumber of small linear systems is solved. Each system corresponds to all the degrees offreedom associated with a block of elements in a compact subdomain. The blocks areoverlapping and their union covers the whole domain. The Vanka-smoother can be re-garded as an overlapping domain decomposition method, where each subsystem can besolved using the most appropriate direct or iterative solver.

Note that in the pre- and post-smoothing steps we defined the smoothing operator asSl :V l→V l. Since V l is built as a continuous extension by zero of V l

loc to the whole domainΩ, then without loss of generality we can consider the equivalent operator Sl

loc :V lloc→V l

loc.The space V l

loc is associated to the conforming triangulation T lh on Ωl .

For conforming finite elements the Navier-Stokes block may consist of all the ele-ments containing a certain set of pressure vertices. Thus, in this case a smoothing stepwith a Vanka smoother consists of a loop over all the blocks, solving only the equationsinvolving the degrees of freedom (dofs) related to the elements that are around the pres-sure vertices. The velocity and pressure dofs are solved many times in one smoothingstep. Examples of computations with Vanka-smoothers can be found in [2, 22, 23].

Fig. 2 shows an example of a minimal Vanka-block for a 2D structured Quad9/Quad4Taylor-Hood finite element triangulation. We use the same strategy for non-structured2D and 3D triangulations. The solution steps are summarized, with reference to Fig. 2, asfollows:


Pressure dofs not solved for

Velocity dofs solved for

Pressure dofs solved for

Velocity dofs not solved for

Figure 2: Vanka-block for the Navier-Stokes Quad9/Quad4 Taylor-Hood finite element.

1. The internal elements are identified: one darker element located in the middle.

2. A search for all the pressure dofs associated with the internal elements is per-formed: the 4 solid circles.

3. A search for all the velocity dofs associated to the internal elements and the neigh-boring ones (the shaded elements) is done: the 49 solid crosses.

4. The equation system associated with all the found dofs (4+49×2=102) is extractedfrom the global system, and solved using a direct solver.

The equation system associated with the dofs to be solved involves a number of pressureand velocity dofs that are not solved. In this case the 32 dashed circles for the pressure,and the 80 dashed crosses for the velocity are considered known from the previous so-lution. Their contribution to the equation system is taken into account in the right handside of the equation as an explicit term calculated with the last updated values.

The number of internal elements used in each Vanka-block can be fixed depending onthe solution strategy. A larger number of internal elements implies a bigger size of the do-main for each Vanka-block. This in turn yields a smaller number of Vanka-blocks neededto span the whole domain, which improves the convergence properties. However, thenumber of dofs increases linearly with the Vanka-block size, thus the computational timeincreases with a cubic factor, since a direct solver is used. For our two-dimensional simu-lations, we found that a good compromise between these contrasting factors is obtainedwith a number of 256 quadratic internal elements. Also notice that in our algorithm thelimiting case of a unique Vanka block containing the entire level subdomain correspondsto a direct solver smoother for the LMG algorithm.

For the temperature optimality system we solve only for the quadratic dofs associatedto the internal and neighboring elements. We perform the solution of the optimality sys-tem in a fully coupled way block by block. In the case described in Fig. 2, only the 49 solidcrosses must be considered for each unknown, for a total of 49×3= 147 dofs per blockfor the 3 scalar variables (T∗,λ,T0). The coupled implicit solution of the state, costate and


control equations allows a better solution process. In fact, numerical oscillations wouldbe induced in the case of uncoupled solution, even with the use of gradient-type algo-rithms [3]. These oscillations are removed in case of coupled solution for the optimalitysystem.

Remark 3.1. In solving the Vanka-block equation systems we have used direct solvers.Iterative solvers such as GMRES with Schur complement preconditioner (for the Navier-Stokes block), and GMRES with ILU preconditioner (for the temperature optimalityblock) are also possible. The results are somehow comparable in terms of computationaltime, since the dimensions of the systems to be solved are relatively small. Direct solversseem to be more robust with respect to iterative ones but at this moment there is no clearadvantage in using ones rather than the others. In this work we only use direct solversand we plan to investigate the convergence and robustness properties of the iterativesolvers for the Vanka-blocks in future papers.

4 Numerical results

4.1 Multigrid smoother comparison

In this part we compare the performances of two different smoothers in the multigridalgorithm: the Vanka-type smoother previously discussed in Section 3.2 and a GMRESsmoother with ILU preconditioner. Since we focus on the role played by the smoothers,we consider for this series of tests a standard geometric multigrid algorithm, which cor-responds to our Local Multigrid algorithm in the special case of uniform mesh refine-ment [5]. We study the solution of the optimality system for different target regions Ωc

and a given boundary control region Γc. In this way we can also evaluate how the loca-tion of the target region with respect to the boundary control region Γc affects the solutionof the system.

Let us consider a two-dimensional domain Ω=(x,y) : x∈ [0,1],y∈ [0,2] as shown inFig. 3. From now on and unless otherwise stated, we assume unit reference values andrefer to non-dimensional quantities. We perform the calculations for three different cases,denoted as Case A, B and C, corresponding to three target regions shown in Fig. 3 andgiven by Ωc,A=(x,y) :x∈[0.25,0.75],y∈[0.4,0.8], Ωc,B=(x,y) :x∈[0.25,0.75],y∈[0.8,1.2]and Ωc,C = (x,y) : x ∈ [0.25,0.75],y ∈ [1.2,1.6], respectively. Over each of these targetregions we want to steer the temperature profile towards the constant value Td = 0.9.The boundary region where Dirichlet boundary conditions of the temperature field arecontrolled is given by Γc = (x,y) : x∈ [0.25,0.75],y= 0. Concerning the boundary withfixed Dirichlet temperature conditions we set T=T0=0 on y=1 and T=T0=1 elsewhere.For the Navier-Stokes equations, we set an inflow velocity profile g = (0,0.375) on theboundary Γc and a pressure outflow condition on the line y= 1. Hence, Γc acts both asa boundary control region and as a mass injection region. No slip boundary conditionsare enforced on the remaining part of the boundary. In this test we want to study the


Γc

Ωc

x

y

Γc

Ωc

x

y

Γc

Ωc

x

y

Figure 3: Three cases with mass injection from below and different target regions Ωc: y∈ [0.4, 0.8] (Case A),y∈ [0.8, 1.2] (Case B), y∈ [1.2, 1.6] (Case C).

numerical solution of the optimality system for different choices of the position of thecontrol boundary Γc and the target regions Ωc,A, Ωc,B and Ωc,C respectively. The relationbetween the target region and the boundary control region is different for each optimalcontrol problem under study. We set β=γ=1 so that the objective functional becomes

J (T∗,T0)=α

2

∫

ΩχΩc

|T∗+T0−Td|2 dΩ+

1

2

∫

ΩT2

0 dΩ+1

2

∫

Ω|∇T0|

2 dΩ. (4.1)

We remark that, due to the presence of the indicator function χΩcof the domain Ωc, the

integral in the α term is restricted to the target region. On the other hand, the penaltyterms must be extended to the whole domain Ω, in order to regularize the control andlimit its size. This penalty must involve the entire fluid domain, or at least a part of thefluid domain whose boundary contains all the boundary control regions.

The numerical solution of the optimality system is very challenging since many pa-rameters are involved. The computation of a candidate optimal solution for this problemis by no means trivial. A variety of features may be studied of both theoretical and prac-tical nature. For instance, the choice of the values for α, β and γ is very important forvarious reasons. A higher value of α implies a lower error with respect to the targetprofile. The values of these constants lead to different values for the adjoint and controlvariables, so that a different local minimum may be computed. We remark that the γconstant must be set to a non-zero value not only to to have solution in H1, but also inorder to have a sufficient smoothness on the control. Starting at the coarse level l=0, wediscretize the computational domain with a mesh that consists of n0=4×5=20 quadraticelements. The mesh is uniformly refined up to the level l = 5 by using the midpointrefinement strategy. Thus, at each level the number of elements is given by nl =n0×4l .


Table 1: Values of the temperature error ET =∫

Ωc|T−Td|

2dΩ for Cases A, B and C, for various values of α

and for different types of smoothers.

Case A

l=4 l=5

α ET , Vanka ET, GMRES ET, Vanka ET, GMRES

102 9.28649e-04 9.28649e-04 9.26760e-04 9.26760e-04

103 2.86080e-04 2.86080e-04 2.85127e-04 2.85127e-04

104 1.44570e-04 1.44570e-04 1.44511e-04 1.44511e-04

105 1.40525e-04 1.40525e-04 1.40484e-04 1.40484e-04

106 1.33327e-04 1.33327e-04 1.33216e-04 1.33216e-04

107 1.12623e-04 1.12623e-04 none none

108 1.06577e-04 1.06577e-04 none none

Case B

l=4 l=5


104 1.10891e-04 1.10891e-04 1.10418e-04 1.10418e-04

105 2.75524e-05 2.75524e-05 2.75341e-05 2.75341e-05

106 2.54883e-05 2.54883e-05 2.54872e-05 2.54872e-05

107 2.54511e-05 2.54511e-05 2.54500e-05 2.54500e-05

108 2.53119e-05 none 2.53090e-05 none

109 2.40505e-05 none 2.40328e-05 none

1010 none none none none

Case C

l=4 l=5


106 1.49337e-04 1.49337e-04 1.49335e-04 1.49335e-04

107 1.49068e-04 1.49068e-04 1.49068e-04 1.49068e-04

108 1.49065e-04 1.49065e-04 1.49065e-04 1.49065e-04

109 1.49065e-04 none 1.49065e-04 none

1010 1.49063e-04 none 1.49063e-04 none

1011 1.49044e-04 none 1.49043e-04 none

1012 1.48862e-04 none 1.48859e-04 none

In Table 1 we show the values of the temperature error, which constitutes the trackingterm in the cost functional (4.1), namely ET =

∫Ωc|T−Td|

2dΩ, for the three target regionsas a function of the α parameter. We compare the solution errors ET obtained at twodifferent discretization levels, l = 4 and l = 5, and using the two different smoothers,Vanka and GMRES. As a stopping criterion for the solver we have chosen the l∞ normof the difference between two solutions of two consecutive V-cycles to not exceed 10−4.If this target was not achieved in less than 15 V-cycles, we assumed the solver failed to


0 0.5 1 1.5 2

0

2 · 10−2

4 · 10−2

6 · 10−2

8 · 10−2

0.1

0.12

0

A

B

C

y

T ∗

0 0.5 1 1.5 2

0

0.5

1

1.5

0

A

B

C

y

T0

0 0.5 1 1.5 2

0

0.5

1

1.5

0

A

B

C

y

T

0 0.5 1 1.5 2

−10

0

10

20

30

40

0

A B

C

y

λ

Figure 4: Profiles of T∗, T0, T and λ along the longitudinal line x= 0.5 with α= 104 for Cases A, B, C andwithout control (Case 0).

converge. It is clear that the αmax parameter depends on the particular problem underconsideration, being in general dependent on the target function, on the definition ofboth the target region Ωc and the boundary control region Γc, along with all the othergiven data of the problem. A higher value of the α parameter implies a smaller errorET in tracking the desired field. On the other hand a higher value of α brings to a lesssmooth solution, hence to a stiffer problem whose numerical solution eventually fails toconverge. Nevertheless, the maximum value of α cannot be known a priori nor estimated,and must be determined for each case by numerical investigation.

Clearly, each specific type of smoother used in the multigrid algorithm has a differentmaximum value of α before experiencing lack of convergence. We observed that overallthe Vanka smoother performs better than the GMRES one. In particular Table 1 showsthat for the case A the two smoothers perform equally well, while in cases B and C theVanka smoother is more robust, being capable of finding solutions with values of α re-spectively 2 and 4 orders of magnitude larger than the GMRES ones. In order to studythe effects of a change in the position of the target region Ωc with respect to the boundarycontrol region Γc corresponding to Cases A, B and C, in Fig. 4 we plot the profiles of T∗,T0, T and λ along the longitudinal line x=0.5 for Cases A, B and C and without control(Case 0), assuming α=104.


0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

0

A

B

C

x

T0

Figure 5: Boundary controls as restriction of the lifting function T0 on the boundary line y=0 with α=104 forCases A, B, C and without control (Case 0).

0

0.5

1 0

1

2

0

0.5

1

xy

T0

0

0.5

1 0

1

2

0

0.5

1

xy

T0

0

0.5

1 0

1

2

0

0.5

1

xy

T0

0

0.5

1 0

1

2

0

1

xy

T0

Figure 6: Case B. Plots over the fluid domain Ω of the lifting function T0 for α= 103 ,5·103,104,105 (top tobottom, left to right).

In Fig. 5 we show the boundary controls as restriction of the lifting function on theboundary line y=0 for Cases A, B and C with α=104 and temperature boundary valuewithout control. In order to show the dependence on the α parameter, in Fig. 6 we plot


0 0.2 0.4 0.6 0.8 1

0

0.5

1

x

T0

0 0.5 1 1.5 2

−15

−10

−5

0

y

λ

Figure 7: Case B. On the left: profiles of T along the longitudinal line x=0.5 for α=103,5·103,104,105 (circle,triangle, square and pentagon mark, respectively). On the right: a zoom on the target region.

0 0.5 1 1.5 2

0

0.5

1

y

T

0.8 0.9 1 1.1 1.2

0.7

0.8

0.9

1

y

T

Figure 8: Case B. On the left: boundary controls as restriction of the lifting function on the boundary line y=0.On the right: adjoint temperature λ along the longitudinal line x=0.5. In both figures the plots are reportedfor α=103 ,5·103,104,105 (circle, triangle, square and pentagon mark, respectively).

for Case B the lifting function T0 over the domain Ω for α= 103,5·103,104,105. In Fig. 7we show on the left the temperature profiles along the longitudinal line x=0.5 for CaseB and α= 103,5·103,104,105. On the right we plot a zoom of these profiles on the targetregion Ωc. For the same values of α, Fig. 8 also shows the boundary controls on the leftand the corresponding adjoint temperature λ profiles on the right.

4.2 Solution on different meshes

In the second part of this section we compare the solutions obtained using different non-conforming meshes. We consider only Case A as defined in the previous subsection,where the target region Ωc is in between 0.4≤y≤0.8. In Figs. 9 and 10 all the meshes con-sidered in this test are shown. We report the results about the solution of the optimality


l4 l4,3 l4,3,2 l4,3,2,1

Figure 9: Conforming and nonconforming meshes obtained with 4 selective consecutive refinements.

l5 l5,4 l5,4,3 l5,4,3,2

Figure 10: Conforming and nonconforming meshes obtained with 5 consecutive selective refinements.

system in the finite element spaces constructed according to (3.10) on the nonconformingtriangulations considered. We use the notation li for conforming mesh obtained after imidpoint refinements and li,j for 2-level nonconforming mesh obtained using the li con-forming mesh for 0≤ y≤ 0.8 and the lj conforming mesh for 0.8≤ y≤ 2.0. The label li,j,k

is used for 3-level nonconforming mesh obtained using the li,j nonconforming mesh for0≤ y ≤ 1.2 and the lk conforming mesh for 1.2≤ y ≤ 2.0. Finally li,j,k,l denotes a 4-levelnonconforming mesh obtained using the li,j,k nonconforming mesh for 0≤y≤1.6, and thell conforming mesh for 1.6≤y≤2.0.

The solution is computed using the Full Local Multigrid algorithm described in Sec-tion 3.1 together with the Vanka-type smoother illustrated in Section 3.2. In Table 2 wecompare the results obtained with the mesh l4 and with the 3 different nonconformingmeshes l4,3, l4,3,2 and l4,3,2,1. Note that the part of the domain containing the control re-gion Γc and the target region Ωc is discretized for all the meshes at the finest refinement



Ωc|T−Td|

2dΩ and computational times for several values of α,

and for different types of nonconforming meshes.

ET

l4 l4,3 l4,3,2 l4,3,2,1

α nel=5120 nel=2816 nel=2432 nel=2384

102 9.28649e-04 9.28649e-04 9.28649e-04 9.28649e-04

103 2.86080e-04 2.86080e-04 2.86080e-04 2.86080e-04

104 1.44570e-04 1.44570e-04 1.44570e-04 1.44570e-04

105 1.40525e-04 1.40525e-04 1.40525e-04 1.40525e-04

106 1.33327e-04 1.33327e-04 1.33327e-04 1.33327e-04

107 1.12623e-04 1.12623e-04 1.12623e-04 1.12623e-04

108 1.06577e-04 1.06577e-04 1.06577e-04 1.06577e-04

Computational Time

l4 l4,3 l4,3,2 l4,3,2,1

α nel=5120 nel=2816 nel=2432 nel=2384

102 10.46 7.02 5.61 5.27

103 10.46 6.99 5.62 5.22

104 13.9 7.71 5.56 5.23

105 13.91 9.3 5.52 5.25

106 27.63 10.81 6.85 6.53

107 62.28 13.81 10 9.5

108 44.16 16.17 10.25 10

level. Similarly, in Table 3 we compare the results obtained with the mesh l5 and with the3 different nonconforming meshes l5,4, l5,4,3 and l5,4,3,2. Once again the part of the domaincontaining Γc and Ωc is always discretized at the top refinement level.

We compare the error ET and the computational times for several values of the pa-rameter α. As pointed out before, the error ET decreases as the value of the parameterα increases. However, with α increasing the problem becomes numerically stiffer andthe solver eventually fails to converge. In Tables 2 and 3 this behavior is obtained forboth the conforming and the nonconforming cases, but it is more evident for the con-forming ones. In particular as the value of α increases from 102 to 108 the computationaltime increases accordingly, but while for the conforming meshes l4 and l5 (left columnsin the two tables), the computational time quadruples, before eventually breaking, forthe nonconforming meshes l4,3,2,1 and l5,4,3,2 (right columns in the tables) it only doubles.Moreover in none of the considered cases the convergence fails to be achieved. An inter-mediate behavior is observed for the 2 central columns and in both tables.

For given α (left to right in both tables) there is no significant difference in the valuesof the error ET found. However the computational time significantly reduces, since thetotal number of elements, nel, decreases (moving left to right), ranging from 5120 to 2384in Table 2, and from 20480 to 9536 in Table 3. The reduction in the computational time is



Ωc|T−Td|

2dΩ for various values of α and for different types of

smoothers.

ET

l5 l5,4 l5,4,3 l5,4,3,2

α nel=20480 nel=11264 nel=9728 nel=9536

102 9.26760e-04 9.26760e-04 9.26760e-04 9.26760e-04

103 2.85127e-04 2.85127e-04 2.85127e-04 2.85127e-04

104 1.44511e-04 1.44511e-04 1.44511e-04 1.44511e-04

105 1.40484e-04 1.40484e-04 1.40484e-04 1.40484e-04

106 1.33216e-04 1.33216e-04 1.33216e-04 1.33216e-04

107 none none 1.12513e-04 1.12513e-04

108 none none 1.06543e-04 1.06543e-04

Computational time

l5 l5,4 l5,4,3 l5,4,3,2

α nel=20480 nel=11264 nel=9728 nel=9536

102 47.12 30.04 26.22 24.75

103 46.87 30.24 25.9 24.86

104 50.67 33.4 32.72 24.73

105 62.88 40.44 35.05 30.75

106 139.98 87.97 37.26 32.59

107 none none 53.06 49.04

108 none none 124.16 67.77

more evident for increasing values of the parameter α. In particular in Table 3 for the lasttwo values of α (107 and 108), the algorithm does not converge for the meshes l5 and l5,4,while it converges for the two nonconforming meshes l5,4,3 and l5,4,3,2.

5 Conclusions

In this paper we have proposed the lifting function approach for the treatment of bound-ary optimal control problems involving the temperature and Navier-Stokes equations.With the lifting function approach boundary controls are determined in the natural tracefunction spaces of the volume variables, without stronger regularity requirements. Exis-tence results for the state system can be obtained in the framework of non-coercive ellip-tic equations. Also, the numerical implementation is very similar to distributed controlproblems and it permits to avoid some difficulties related to standard direct boundarycontrol approaches. We have proposed the use of a local multigrid algorithm for the nu-merical solution of the optimality system, where the smoothing steps are performed withdomain decomposition Vanka-type smoothers. The numerical results for the optimalitysystem show that candidate optimal solutions can be computed in an effective manner


and with robustness with respect to the regularization parameters of the cost functional.Also, local mesh refinement can be included in order to reduce the computational effort.A convergence analysis of the proposed algorithm will be dealt with in future works.

We are currently working on a cut-off formulation that reduces the size of the domainfor the optimality condition to a subset of Ω containing the boundary control region Γc.This formulation has two advantages. On one hand, it is clear that the degrees of freedomof the lifting function are reduced with a benefit on the computational time. On the otherhand, for fixed values of α, β and γ the penalty integrals are evaluated on a smallerdomain. Then, it is our expectation that a better tracking can be achieved.

Acknowledgments

This work was supported by National Science Foundation grant DMS-1412796.

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