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An Improved FiniteElement Approximationand Superconvergence forTemperature Control ProblemsYuelong Tanga

a Department of Mathematics and ComputationalScience, Hunan University of Science and EngineeringYongzhou 425100, Hunan, China. E-mail:Published online: 24 Dec 2013.

To cite this article: Yuelong Tang (2013) An Improved Finite Element Approximationand Superconvergence for Temperature Control Problems, Mathematical Modelling andAnalysis, 18:5, 631-640

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Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTU

Volume 18 Number 5, November 2013, 631–640 http://www.tandfonline.com/TMMA

http://dx.doi.org/10.3846/13926292.2013.868840 Print ISSN: 1392-6292

c©Vilnius Gediminas Technical University, 2013 Online ISSN: 1648-3510

An Improved Finite Element Approximationand Superconvergence for Temperature ControlProblems∗

Yuelong Tang

Department of Mathematics and Computational Science, Hunan Universityof Science and Engineering

Yongzhou 425100, Hunan, China

E-mail: tangyuelonga@163.com

Received July 22, 2012; revised October 22, 2013; published online December 1, 2013

Abstract. In this paper, we consider an improved finite element approximation fortemperature control problems, where the state and the adjoint state are discretizedby piecewise linear functions while the control is not discretized directly. The nu-merical solution of the control is obtained by a projection of the adjoint state to theset of admissible controls. We derive a priori error estimates and superconvergenceof second-order. Moreover, we present some numerical examples to illustrate ourtheoretical results.

Keywords: optimal control, finite element method, approximation, convergence analysis.

AMS Subject Classification: 49J20; 65M60; 80M10.

1 Introduction

Temperature control problem plays a very important role in our everyday life.For example, a refrigerator works through a cooling system very similar tothe one in an air conditioning unit. It has refrigerant that it cycles througha compressor and evaporator to take heat out of the refrigerator. Normally,the system only cycles on periodically to maintain a temperature range. Thesystem is governed by the following equation:

−div(A∇y) + φ(y) = f, x ∈ Ω,y|∂Ω = 0,

where Ω is a convex domain with a Lipschitz boundary ∂Ω, y is the temper-ature distribution which is maintained equal to zero along the boundary, φ(y)denotes the thermal radiation or positive temperature feedback due to chemicalreactions (see e.g., [16]) and f is a source term.

∗ This work is supported by the Foundation of Hunan Educational Committee (13C338).

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632 Yuelong Tang

Then we may control the temperature distribution y to come close to agiven target yd by acting with an additional distributed source term u. Thecorresponding temperature control problem is formulated as follows:

minu∈K

1

2‖y − yd‖2 +

1

2‖u‖2

,

− div(A∇y) + φ(y) = f +Bu, x ∈ Ω,y|∂Ω = 0,

(1.1)

where K is a closed convex subset in L2(Ω), the coefficient matrix A =(aij)n×n ∈ (W 1,∞(Ω))n×n, such that (Aξ) · ξ ≥ c|ξ|2, ∀ξ ∈ Rn, f ∈ L2(Ω), andB is a linear continuous operator. Moreover, K is defined by

K =v(x) ∈ L2(Ω) : a ≤ v(x) ≤ b, a.e. x ∈ Ω

,

where a and b are constants.The literature on finite element solving optimal control problems is huge.

For a linear elliptic optimal control problem, a priori error estimates wereinvestigated in [10], a posteriori error estimates based on recovery techniqueshave been obtained in [9,17], a posteriori error estimates of residual type havebeen derived in [12], a superconvergence result can be seen in [14], a variationaldiscretization method was introduced in [13], and Chen et al. considered itsLegendre Galerkin spectral approximation in [4].

For semilinear elliptic optimal control problems, a priori error estimateswere investigated in [1], a posteriori error estimates have been obtained in [11],a superconvergence can be seen in [3], Borzı considered a second-order dis-cretization and multigrid solution in [2]. Recently, some adaptive algorithmand superconvergence results can be found in [6,7,15]. Notice that all the con-trol variables are first-order convergence and the superconvergence of the stateand the adjoint state variables are 1.5 order in the above works.

To the best of our knowledge, our work is the first to consider a numericalapproximation for the semilinear elliptic temperature control problem in whichboth the a priori error estimates and the superconvergence are of the second-order.

The paper is organized as follows: In Section 2, we introduce an improvedfinite element approximation for the model problem. In Section 3, we derive apriori error estimates. In Section 4, we obtain the superconvergence property.We present some numerical examples to confirm our theoretical results in thelast section.

2 An Improved Finite Element for the Model Problem

For ease of exposition, we set W = H10 (Ω), U = L2(Ω), ‖ · ‖ = ‖ · ‖L2(Ω),

‖ · ‖m = ‖ · ‖Hm(Ω) and

a(y, w) =

∫Ω

(A∇y) · ∇w, ∀y, w ∈W,

(u,w) =

∫Ω

u · w, ∀u,w ∈ U.

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Superconvergence of Temperature Control Problems 633

It follows from the assumptions on the coefficient matrix A that

a(y, y) ≥ c‖y‖21,∣∣a(y, w)

∣∣ ≤ C‖y‖1‖w‖1, ∀y, w ∈W. (2.1)

Then the standard weak formula for the state equation reads as follows:

a(y, w) + (φ(y), w) = (f +Bu,w), ∀w ∈W, (2.2)

where we assume that the function φ(·) ∈ W 2,∞(−R,R) for any R > 0 andφ′(·) ≥ 0.

Then the model problem (1.1) can be restated as: Find (y, u) ∈ W × K,such that min

u∈K

1

2‖y − yd‖2 +

1

2‖u‖2

,

a(y, w) + (φ(y), w) = (f +Bu,w), ∀w ∈W.(2.3)

It is well known (see e.g., [10]) that the control problem (2.3) has a solution(y, u) ∈ W ×K, and that if the pair (y, u) ∈ W ×K is the solution of (2.3),then there is an adjoint state p ∈W such that the triplet (y, p, u) ∈W×W×Ksatisfies the following optimal conditions:

a(y, w) +(φ(y), w

)= (f +Bu,w), ∀w ∈W, (2.4)

a(q, p) +(φ′(y)p, q

)= (yd − y, q), ∀q ∈W, (2.5)

(u−B∗p, v − u) ≥ 0, ∀v ∈ K, (2.6)

where B∗ is the adjoint operator of B.We introduce the following pointwise projection operator:

Π[a,b](g(x)) = max(a,min

(b, g(x)

)). (2.7)

As in [13,14], it is easy to prove the following lemma:

Lemma 1. Let (y, p, u) be the solution of (2.4)–(2.6). Then we have

u = Π[a,b](B∗p). (2.8)

Let T h be a regular triangulation of Ω, such that Ω = ∪τ∈T h τ . Let h =maxτ∈T hhτ, where hτ denotes the diameter of the element τ . Associated withT h is a finite dimensional subspace Sh of C(Ω), such that χ|τ are polynomials ofm-order(m ≤ 1) for all χ ∈ Sh and τ ∈ T h. Let Wh =

vh ∈ Sh : vh|∂Ω = 0

.

It is easy to see that Wh ⊂W .Then an improved finite element approximation scheme of (2.3) is as follows:

Find (yh, uh) ∈Wh ×K, such that minuh∈K

1

2‖yh − yd‖2 +

1

2‖uh‖2

,

a(yh, wh) +(φ(yh), wh

)= (f +Buh, wh), ∀wh ∈Wh.

(2.9)

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634 Yuelong Tang

It follows (see e.g., [11]) that the control problem (2.9) has a solution (yh, uh) ∈Wh × K, and that if the pair (yh, uh) ∈ Wh × K is the solution of (2.9),then there is an adjoint state ph ∈ Wh such that the triplet (yh, ph, uh) ∈Wh ×Wh ×K satisfies the following optimal conditions:

a(yh, wh) +(φ(yh), wh

)= (f +Buh, wh), ∀wh ∈Wh, (2.10)

a(qh, ph) +(φ′(yh)ph, qh

)= (yd − yh, qh), ∀qh ∈Wh, (2.11)

(uh −B∗ph, v − uh) ≥ 0, ∀v ∈ K. (2.12)

Similar to Lemma 1, as in [13,14], it is easy to show the following lemma:

Lemma 2. Let (yh, ph, uh) be the solution of (2.10)–(2.12). Then we have

uh = Π[a,b](B∗ph). (2.13)

3 A Priori Error Estimates

We introduce some intermediate variables. Let (y(uh), p(uh)) satisfy the fol-lowing system:

a(y(uh), w

)+(φ(y(uh)

), w)

= (f +Buh, w), ∀w ∈W, (3.1)

a(q, p(uh)

)+(φ′(y(uh)

)p(uh), q

)=(yd − y(uh), q

), ∀q ∈W. (3.2)

Just for ease of exposition, let

J(u) =1

2‖y − yd‖2 +

1

2‖u‖2,

and J ′(u) is the Frechet derivative of J(u) at u. It is easy to prove that(J ′(u), v

)= (u−B∗p, v), ∀v ∈ K,(

J ′(uh), v)

=(uh −B∗p(uh), v

), ∀v ∈ K.

Lemma 3. [5] Let πh be the standard Lagrange interpolation operator. Form = 0 or 1, q > n

2 and ∀v ∈W 2,q(Ω), we have

|v − πhv|Wm,q(Ω) ≤ Ch2−m|v|W 2,q(Ω).

Lemma 4. Let (yh, ph, uh) and (y(uh), p(uh)) be the solutions of (2.10)–(2.12)and (3.1)–(3.2), respectively. Assume that p(uh), y(uh) ∈ H2(Ω). Then thereexists a constant C independent of h such that∥∥y(uh)− yh

∥∥1

+∥∥p(uh)− ph

∥∥1≤ Ch. (3.3)

Proof. According to (2.1), (2.10), (3.1), φ′(·) ≥ 0 and Cauchy’s inequalitywith ε, we have

c∥∥y(uh)− yh

∥∥21

≤ a(y(uh)− yh, y(uh)− yh

)+(φ(y(uh)

)− φ(yh), y(uh)− yh

)= a

(y(uh)− πhy(uh), y(uh)− yh

)+(φ(y(uh)

)− φ(yh), y(uh)− πhy(uh)

)≤ C

∥∥y(uh)− yh∥∥1

∥∥y(uh)− πhy(uh)∥∥1

+ C∥∥y(uh)− yh

∥∥∥∥πhy(uh)− yh∥∥

≤ 2ε∥∥y(uh)− yh

∥∥21

+C

2ε

∥∥y(uh)− πhy(uh)∥∥21. (3.4)

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Superconvergence of Temperature Control Problems 635

Note that y(uh) ∈ H2(Ω), by using Lemma 3, we obtain∥∥y(uh)− πhy(uh)∥∥1≤ Ch

∥∥y(uh)∥∥2. (3.5)

Let 2ε < c, it follows from (3.4)–(3.5) that∥∥y(uh)− yh∥∥1≤ Ch. (3.6)

Similarly, we can prove that∥∥p(uh)− ph∥∥1≤ Ch. (3.7)

Then (3.3) follows from (3.6)–(3.7). ut

We introduce the following auxiliary problems:

−div(A∗∇ξ) + Φξ = F1, in Ω, ξ|∂Ω = 0, (3.8)

−div(A∇ζ) + φ′(y(uh))ζ = F2, in Ω, ζ|∂Ω = 0, (3.9)

where

Φ =

φ(y(uh))− φ(yh)

y(uh)− yh, y(uh) 6= yh,

φ′(yh), y(uh) = yh.

From the regularity estimates (see e.g., [16]), we obtain

‖ξ‖2 ≤ C‖F1‖, ‖ζ‖2 ≤ C‖F2‖.

By using the Aubin–Nitsche technique, it is easy to prove the following lemma:

Lemma 5. Let (yh, ph, uh) and (y(uh), p(uh)) be the solutions of (2.10)–(2.12)and (3.1)–(3.2), respectively. Assume that p(uh), y(uh) ∈ H2(Ω). Then thereexists a constant C independent of h such that∥∥y(uh)− yh

∥∥+∥∥p(uh)− ph

∥∥ ≤ Ch2. (3.10)

Lemma 6. Let (y, p, u) and (yh, ph, uh) be the solutions of (2.4)–(2.6) and(2.10)–(2.12), respectively. Assume that all the conditions in Lemma 5 arevalid. Then there exists a constant C independent of h such that

‖u− uh‖ ≤ Ch2. (3.11)

Proof. It is clear that(J ′(v)− J ′(u), v − u

)≥ c‖v − u‖2, ∀v, u ∈ K. (3.12)

By using (2.6) and (2.12), we have

c‖u− uh‖2 ≤(J ′(u)− J ′(uh), u− uh

)= (u−B∗p, u− uh)−

(uh −B∗p(uh), u− uh

)≤(B∗p(uh)−B∗ph, u− uh

)≤ C

∥∥p(uh)− ph∥∥‖u− uh‖. (3.13)

From (3.10) and (3.13), we derive (3.11). ut

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636 Yuelong Tang

By combining Lemmas 3–6, we can prove the following main result.

Theorem 1. Let (y, p, u) and (yh, ph, uh) be the solutions of (2.4)–(2.6) and(2.10)–(2.12), respectively. Assume that all the conditions in Lemmas 4–6 arevalid. Then, we have

‖u− uh‖+ ‖y − yh‖+ ‖p− ph‖ ≤ Ch2. (3.14)

4 Superconvergence Property

We introduce an elliptic projection operator Ph : W → Wh, which satisfies:for any φ ∈W

a(φ− Phφ,wh) = 0, ∀wh ∈Wh. (4.1)

It has the following approximation property (see e.g., [18]):

‖φ− Phφ‖ ≤ Ch2‖φ‖2, ∀φ ∈ H2(Ω). (4.2)

Then we consider the superconvergence property between elliptic projec-tions and numerical solutions of the state and the adjoint state.

Theorem 2. Let (y, p, u) and (yh, ph, uh) be the solutions of (2.4)–(2.6) and(2.10)–(2.12), respectively. Then there exists a constant C independent of hsuch that

‖Phy − yh‖1 + ‖Php− ph‖1 ≤ Ch2. (4.3)

Proof. From (2.4) and (2.10), we get

a(y − yh, wh) +(φ(y)− φ(yh), wh

)=(B(u− uh), wh

), ∀wh ∈Wh.

According to the definition of Ph, we have

a(Phy − yh, wh) +(φ(y)− φ(yh), wh

)=(B(u− uh), wh

), ∀wh ∈Wh.

By selecting wh = Phy − yh and using (2.1) and the assumed conditions onφ(·), we obtain

c‖Phy − yh‖21 ≤ a(Phy − yh, Phy − yh) +(φ(Phy)− φ(yh), Phy − yh

)=(φ(Phy)− φ(y), Phy − yh

)+(B(u− uh), Phy − yh

)≤ C

(‖Phy − y‖+ ‖u− uh‖

)‖Phy − yh‖

≤ Ch2‖Phy − yh‖.

Hence,

‖Phy − yh‖1 ≤ Ch2. (4.4)

Similarly, we can prove that

‖Php− ph‖1 ≤ Ch2. (4.5)

Then (4.3) follows from (4.4)–(4.5). ut

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Superconvergence of Temperature Control Problems 637

Table 1. Numerical results, Example 1.

Mesh ‖u− uh‖ ‖y − yh‖ ‖p− ph‖ ‖Phy − yh‖1 ‖Php− ph‖1

10× 10 3.10200e−02 5.27023e−02 5.03458e−02 5.43093e−02 6.16469e−0220× 20 7.78194e−03 1.38042e−02 1.31083e−02 1.41742e−02 1.62494e−0240× 40 2.00103e−03 3.49306e−03 3.31053e−03 3.58245e−03 4.12064e−0380× 80 5.01440e−04 8.75888e−04 8.29739e−04 8.98110e−04 1.03390e−03160× 160 1.25846e−04 2.19143e−04 2.07566e−04 2.24679e−04 2.58711e−04

1 1.5 2 2.5−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

log10

(sqrt(dofs))

log

10(e

rro

r)

||u−uh||

||y−yh||

||p−ph||

||y−Phy||

1

||p−Php||

1

=−0.5

Figure 1. The order of convergence, Example 1.

5 Numerical Experiment

In this section, optimal control problems were solved numerically with codesdeveloped by using AFEPack. This package is freely available and the detailscan be found in [8]. The discretization has been described in Section 2.

Let Ω = [0, 1]× [0, 1] and B = I, we solve the following semilinear optimalcontrol problem:

minu∈K

1

2‖y − yd‖2 +

1

2‖u‖2

,

−div(A∇y) + φ(y) = f +Bu, x ∈ Ω,y|∂Ω = 0.

Example 1. The data are as follows:

A = I, φ(y) = y3, a = −0.5, b = 0.5,

p(x) = sin(2πx1)sin(2πx2), y(x) = p(x),

u(x) = min(0.5,max

(−0.5, p(x)

)),

f(x) = −div(A∇y(x)

)+ φ

(y(x)

)− u(x),

yd(x) = y(x)− div(A∗∇p(x)

)+ φ′

(y(x)

)p(x).

Some numerical results based on a sequence of uniformly refined meshes areshown in Table 1.

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638 Yuelong Tang

0

0.5

1

0

0.5

1

−0.5

0

0.5

Figure 2. The exact solution u (left) and the error u− uh (right), Example 1.

Table 2. Numerical results, Example 2.

Mesh ‖u− uh‖ ‖y − yh‖ ‖p− ph‖ ‖Phy − yh‖1 ‖Php− ph‖1

10× 10 2.65120e−02 5.30353e−02 5.38607e−02 5.59887e−02 5.59896e−0220× 20 6.23877e−03 1.39308e−02 1.41608e−02 1.40492e−02 1.41264e−0240× 40 1.56441e−03 3.52693e−03 3.52612e−03 3.56435e−03 3.58241e−0380× 80 3.92635e−04 8.64530e−04 8.69441e−04 8.76090e−04 8.80613e−04160× 160 9.82587e−05 2.11316e−04 2.15043e−04 2.19025e−04 2.20152e−04

We show the relationship between log10(error) and log10(sqrt(dofs)) in Fig-ure 1. It is clear that ‖u− uh‖ = O(h2), ‖y − yh‖ = O(h2), ‖p− ph‖ = O(h2),‖y − Phy‖1 = O(h2), ‖p − Php‖1 = O(h2). In Figure 2, we plot the exactsolution u and the error u− uh.

Example 2. The data are as follows:

A = I, φ(y) = ey, a = −1, b = 0.5,

p(x) = (1− 4x1x2)sin(2πx1)sin(2πx2), y(x) = p(x),

u(x) = min(0.5,max

(−1, p(x)

)),

f(x) = −div(A∇y(x)

)+ φ

(y(x)

)− u(x),

yd(x) = y(x)− div(A∗∇p(x)

)+ φ′

(y(x)

)p(x).

In Table 2, we have shown some numerical results on a sequential meshes.The second - order convergence of errors ‖u−uh‖, ‖y−yh‖, ‖p−ph‖, ‖y−Phy‖1,and ‖p − Php‖1 can be seen in Figure 3. The exact solution u and the erroru− uh are shown in Figure 4.

The above numerical examples confirm ‖u − uh‖ = O(h2), ‖y − Phy‖1 =O(h2), ‖p − Php‖1 = O(h2). But in [3], the authors have just derived the

estimates ‖uh − Ghuh‖ = O(h32 ), ‖y − Phy‖1 = O(h

32 ), ‖p − Php‖1 = O(h

32 ).

Hence, our method is much more efficient.

6 Conclusions

We have investigated an improved finite element approximation for semilineartemperature control problem and have obtained a priori error estimates and

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Superconvergence of Temperature Control Problems 639

1 1.5 2 2.5−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

log10

(sqrt(dofs))

log

10(e

rro

r)

||u−uh||

||y−yh||

||p−ph||

||y−Phy||

1

||p−Php||

1

=−0.5

Figure 3. The order of convergence, Example 2.

0

0.5

1

0

0.5

1

−1

−0.5

0

0.5

Figure 4. The exact solution u (left) and the error u− uh (right), Example 2.

superconvergence of the second-order. Our analysis for the semilinear temper-ature control problem seems to be new, and these results can be extended togeneral convex problems.

References

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[2] A. Borzı. High-order discretization and multigrid solution of elliptic nonlin-ear constrained control problems. J. Comput. Appl. Math., 200:67–85, 2005.http://dx.doi.org/10.1016/j.cam.2005.12.023.

[3] Y. Chen and Y. Dai. Superconvergence for optimal control problems gonvernedby semi-linear elliptic equations. J. Sci. Comput., 39:206–221, 2009.http://dx.doi.org/10.1007/s10915-008-9258-9.

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