Error analysis of variational discretization solving temperature control problems

Tang Journal of Inequalities and Applications 2013, 2013:450http://www.journalofinequalitiesandapplications.com/content/2013/1/450

RESEARCH Open Access

Error analysis of variational discretizationsolving temperature control problemsYuelong Tang*

*Correspondence:[email protected] of Mathematics andComputational Science, HunanUniversity of Science andEngineering, Yongzhou, Hunan425100, China

AbstractIn this paper, we consider variational discretization solving temperature controlproblems with pointwise control constraints, where the state and the adjoint state areapproximated by piecewise linear finite element functions, while the control is notdirectly discretized. We derive a priori error estimates of second-order for the control,the state and the adjoint state. Moreover, we obtain a posteriori error estimates. Finally,we present some numerical algorithms for the control problem and do somenumerical experiments to illustrate our theoretical results.

Keywords: variational discretization; finite element; optimal control problems; apriori error estimates; a posteriori error estimates

1 IntroductionWe are interested in a material plate defined in a two-dimensional convex domain � witha Lipschitz boundary ∂�. For the state y of thematerial, we choose the temperature distri-butionwhich ismaintained equal to zero along the boundary.Wedenote thermal radiationor positive temperature feedback due to chemical reactions by the term φ(y) (see, e.g., [])and assume that there exists a source f ∈ L(�). This system is governed by the followingequation:{

–div(A(x)∇y(x)) + φ(y(x)) = f (x), x ∈ �,y(x) = , x ∈ ∂�.

The setting above suggests that we may control the temperature distribution y to comeclose to a given target by acting with an additional distributed source term u, namely thecontrol function. The corresponding optimal control problem is formulated as follows:⎧⎪⎨

⎪⎩minu∈K {∫

�(g(y) + h(u))dx},

–div(A(x)∇y(x)) + φ(y(x)) = f (x) + Bu(x), x ∈ �,y(x) = , x ∈ ∂�,

(.)

where g(·) and h(·) are strictly convex continuous differentiable functions, h(u) → +∞ as‖u‖L(�) → ∞, A(x) = (aij(x))× ∈ (W ,∞(�̄))× such that (A(x)ξ ) · ξ ≥ c | ξ |, ∀ξ ∈ R

,f (x) ∈ L(�), B is a linear continuous operator, and K is defined by

K ={v(x) ∈ L(�) : a ≤ v(x)≤ b, a.e. x ∈ �

},

where a and b are two constants.

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Tang Journal of Inequalities and Applications 2013, 2013:450 Page 2 of 15http://www.journalofinequalitiesandapplications.com/content/2013/1/450

Optimal control problems have been extensively used in many aspects of the modernlife such as social, economic, scientific and engineering numerical simulation []. Finiteelement approximation seems to be the most widely used method in computing optimalcontrol problems. A systematic introduction of finite element method for PDEs and opti-mal control problems can be found in [–]. Concerning elliptic optimal control problems,a priori error estimates were investigated in [, ], a posteriori error estimates based onrecovery techniques have been obtained in [–], a posteriori error estimates of resid-ual type have been derived in [–], some error estimates and superconvergence resultshave been established in [–], and some adaptive finite element methods can be foundin [–]. For parabolic optimal control problems, a priori error estimates are estab-lished in [–], a posteriori error estimates of residual type are investigated in [, ].Recently, error estimates of spectral method for optimal control problems have been de-rived in [, ], and numerical methods for constrained elliptic control problems withrapidly oscillating coefficients are studied in [].For a constrained optimal control problem, the control has lower regularity than the

state and the adjoint state. Somost researchers considered using piecewise linear finite ele-ment functions to approximate the state and the adjoint state and using piecewise constantfunctions to approximate the control. They constructed a projection gradient algorithmwhere the a priori error estimates of the control is first-order in [, ]. Recently, Borzìconsidered a second-order discretization and multigrid solution of elliptic nonlinear con-strained control problems in [], Hinze introduced a variational discretization conceptfor optimal control problems and derived a priori error estimates for the control which issecond-order in [, ]. The purpose of this paper is to consider variational discretiza-tion for convex temperature control problems governed by nonlinear elliptic equationswith pointwise control constraints.In this paper, we adopt the standard notation Wm,q(�) for Sobolev spaces on � with

the norm ‖ · ‖Wm,q(�) and seminorm | · |Wm,q(�). We set H(�) ≡ {v ∈H(�) : v|∂� = } and

denote Wm,(�) by Hm(�). In addition, c or C denotes a generic positive constant.The paper is organized as follows: In Section , we introduce a variational discretiza-

tion approximation scheme for the model problem. In Section , we derive a priori errorestimates. In Section , we derive sharp a posteriori error estimates of residual type. Wepresent some numerical algorithms and do some numerical experiments to verify our the-oretical results in the last section.

2 Variational discretization approximation for themodel problemWe now consider a variational discretization approximation for the model problem (.).For ease of exposition, we set V = H

(�), U = L(�), ‖ · ‖ = ‖ · ‖L(�), ‖ · ‖,� = ‖ · ‖H(�),‖ · ‖,� = ‖ · ‖H(�) and

a(y,w) =∫

�

(A(x)∇y

) · ∇w, ∀y,w ∈ V ,

(u,w) =∫

�

u ·w, ∀u,w ∈U .

It follows from the assumptions on A(x) that

a(y, y) ≥ c‖y‖,�,∣∣a(y,w)∣∣ ≤ C‖y‖,�‖w‖,�, ∀y,w ∈ V . (.)



Then the standard weak formula for the state equation is

a(y,w) +(φ(y),w

)= (f + Bu,w), ∀w ∈ V , (.)

where we assume that the function φ(·) ∈W ,∞(–R,R) for any R > , φ′(·) ≥ and φ′(y) ∈L(�) for any y ∈H(�). Thus, the equation above has a unique solution.Throughout the paper, we impose the following assumptions:(A) g ′(·) and h′(·) are Lipschitz continuous, namely,

∣∣g ′(y) – g ′(y)∣∣ ≤ C|y – y|, ∀y, y ∈ L(�),∣∣h′(u(x)) – h′(u(x))∣∣ ≤ C|x – x|, ∀u ∈ K ,x,x ∈ �̄.

(A) There exists a positive constantm such that

h′′(u) ≥m, ∀u ∈ K .

Then the model problem (.) can be restated as

{minu∈K {∫

�(g(y) + h(u))dx},

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

It is well known (see, e.g., []) that the control problem (.) has a solution (y,u) ∈ V ×K ,and that if the pair (y,u) ∈ V ×K is the solution of (.), then there is an adjoint state p ∈ Vsuch that the triplet (y,p,u) ∈ V ×V ×K satisfies the following optimality conditions:

a(y,w) +(φ(y),w

)= (f + Bu,w), ∀w ∈ V , (.)

a(q,p) +(φ′(y)p,q

)=

(g ′(y),q

), ∀q ∈ V , (.)(

h′(u) + B∗p, v – u) ≥ , ∀v ∈ K , (.)

where B∗ is the adjoint operator of B.

Lemma . Suppose that assumptions (A)-(A) are satisfied. Let p ∈ V be the solution of(.)-(.). Then the following equation:

h′(s(x)) + B∗p(x) = , (.)

admits a unique solution s(x) and s(x) ∈ C,(�̄).

Proof It follows from h′′(u) ≥ m > that (.) has a unique solution. Note that g ′(y) ∈L(�). From the regularity theory of patrial differential equations (see, e.g., []), we have

p(x) ∈H(�)∩W ,(�).

Because � is a two-dimension convex domain, according to embedding theorem, we get

p(x) ∈ C,(�̄).



From (A) and (.), we get

m∣∣s(x) – s(x)

∣∣≤

∣∣∣∣∫

h′′(θs(x) + ( – θ )s(x)

)dθ · (s(x) – s(x)

)∣∣∣∣=

∣∣h′(s(x)) – h′(s(x))∣∣=

∣∣–B∗p(x) + B∗p(x)∣∣

≤ C|x – x|.

Consequently, we complete the proof of equation (.). �

We introduce the following pointwise projection operator:

�[a,b](g(x)

)= max

(a,min

(b, g(x)

)). (.)

It is clear that �[a,b](·) is Lipschitz continuous with constant . As in [], it is easy to provethe following lemma.

Lemma . Let (y,p,u) and s(x) be the solutions of (.)-(.) and (.), respectively. As-sume that assumptions (A)-(A) are satisfied. Then

u(x) = �[a,b](s(x)

). (.)

Remark . We should point out that (.) and (.) are equivalent. This theory can beused tomore complex situation, for example,K is characterized by a bound on the integralon u over �, namely,

∫�u(x)dx≥ , we have similar results.

Let T h be a regular triangulation of �, such that �̄ =⋃

τ∈T h τ̄ . Let h = maxτ∈T h{hτ },where hτ denotes the diameter of the element τ . Associatedwith T h is a finite dimensionalsubspace Sh of C(�̄), such that χ |τ are polynomials of m-order (m ≥ ) for all χ ∈ Sh andτ ∈ T h. Let Vh = {vh ∈ Sh : vh|∂� = }. It is easy to see that Vh ⊂ V .Then a possible variational discretization approximation scheme of (.) is as follows:

{minuh∈K {∫

�(g(yh) + h(uh))dx},

a(yh,wh) + (φ(yh),wh) = (f + Buh,wh), ∀wh ∈ Vh.(.)

It is well known (see, e.g., []) that control problem (.) has a solution (yh,uh) ∈ Vh ×K ,and that if the pair (yh,uh) ∈ Vh ×K is the solution of (.), then there is an adjoint stateph ∈ Vh such that the triplet (yh,ph,uh) ∈ Vh × Vh × K satisfies the following optimalityconditions:

a(yh,wh) +(φ(yh),wh

)= (f + Buh,wh), ∀wh ∈ Vh, (.)

a(qh,ph) +(φ′(yh)ph,qh

)=

(g ′(yh),qh

), ∀qh ∈ Vh, (.)(

h′(uh) + B∗ph, v – uh) ≥ , ∀v ∈ K . (.)

Similar to Lemma ., it is easy to show the following lemma.



Lemma . Suppose that assumptions (A)-(A) are satisfied. Let (yh,ph,uh) be the solu-tion of (.)-(.), and sh(x) is the solution of the following equation:

h′(sh(x)) + B∗ph(x) = . (.)

Then we have

uh(x) = �[a,b](sh(x)

). (.)

Remark . Inmany applications, the objective functional is uniform convex near the so-lution u, which is assumed in many studies on numerical methods of the problem, see, forexample, [, ]. In this paper, we assumed that g(·) and h(·) are strictly convex continu-ous differentiable functions, for instance, h(u) = α

∫�u, which is frequently met, then the

exact solution of the variational inequality (.) is uh(x) = max(a,min(b, – αB∗ph(x))), and

for numerically solving the problem, we can replace uh(x) by max(a,min(b, – αB∗ph(x))) in

our program.

3 A priori error estimatesWe now derive a priori error estimates of the variational discretization approximationscheme. Just for ease of exposition, let

J(u) =∫

�

(g(y) + h(u)

)dx,

and J ′(u) is the Fréchet derivative of J(u) at u. Similarly to (.)-(.), we can prove that

(J ′(u), v

)=

(h′(u) + B∗p, v

), ∀v ∈ K ,(

J ′(uh), v)=

(h′(uh) + B∗p(uh), v

), ∀v ∈ K ,

where p(uh) satisfies the following system:

a(y(uh),w

)+

(φ(y(uh)

),w

)= (f + Buh,w), ∀w ∈ V , (.)

a(q,p(uh)

)+

(φ′(y(uh))p(uh),q) = (

g ′(y(uh)),q), ∀q ∈ V . (.)

Let πh : C(�̄) → Vh be the standard Lagrange interpolation operator such that for anyv ∈ C(�̄), πhv(Pi) = v(Pi) for all Pi ∈ P, where P is the vertex set associated with the tri-angulation T h, and n is the dimension of the domain �, we have the following result:

Lemma . [] Let πh be the standard Lagrange interpolation operator. For m = or ,q > n

and ∀v ∈W ,q(�), we have

|v – πhv|Wm,q(�) ≤ Ch–m|v|W,q(�).

Lemma. Let (yh,ph,uh) and (y(uh),p(uh)) be the solutions of (.)-(.) and (.)-(.),respectively. Assume that p(uh), y(uh) ∈ H(�) and φ′(·) is locally Lipschitz continuous.Then there exists a constant C independent of h such that

∥∥y(uh) – yh∥∥,� +

∥∥p(uh) – ph∥∥,� ≤ Ch. (.)



Proof From φ′(·)≥ , (.), (.) and embedding theorem ‖v‖L(�) ≤ C‖v‖H(�), we have

c∥∥p(uh) – ph

∥∥,�

≤ a(p(uh) – ph,p(uh) – ph

)+

(φ′(y(uh))(p(uh) – ph

),p(uh) – ph

)= a

(p(uh) – πhp(uh),p(uh) – ph

)+

(φ′(y(uh))(p(uh) – ph

),p(uh) – πhp(uh)

)+

(g ′(y(uh)) – g ′(yh),πhp(uh) – ph

)+

((φ′(yh) – φ′(y(uh)))ph,πhp(uh) – ph

)≤ C

∥∥p(uh) – ph∥∥,�

∥∥p(uh) – πhp(uh)∥∥,� +C

∥∥y(uh) – yh∥∥∥∥πhp(uh) – ph

∥∥+C

∥∥φ′(y(uh))∥∥∥∥p(uh) – ph∥∥L(�)

∥∥p(uh) – πhp(uh)∥∥L(�)

+C∥∥y(uh) – yh

∥∥‖ph‖L(�)∥∥πhp(uh) – ph

∥∥L(�)

≤ C(δ)∥∥p(uh) – πhp(uh)

∥∥,� +C(δ)

∥∥y(uh) – yh∥∥

+Cδ(∥∥p(uh) – ph

∥∥,� +

∥∥πhp(uh) – ph∥∥,�

)≤ C(δ)

∥∥p(uh) – πhp(uh)∥∥,� +C(δ)

∥∥y(uh) – yh∥∥ +Cδ

∥∥p(uh) – ph∥∥,�. (.)

Note that p(uh) ∈H(�), by using Lemma ., we obtain

∥∥p(uh) – ph∥∥,� ≤ C

∥∥p(uh) – πh(p(uh)

)∥∥,� +C

∥∥y(uh) – yh∥∥

≤ Ch∥∥p(uh)∥∥,� +C

∥∥y(uh) – yh∥∥

≤ Ch +C∥∥y(uh) – yh

∥∥. (.)

Similarly, we can prove that

∥∥y(uh) – yh∥∥,� ≤ Ch

∥∥y(uh)∥∥,� ≤ Ch. (.)

Then (.) follows from (.)-(.). �

In order to derive sharp a priori estimates, we introduce the following auxiliary prob-lems:

–div(A∗∇ξ

)+ ξ = F, in �, ξ |∂� = , (.)

–div(A∇ζ ) + φ′(y(uh))ζ = F, in �, ζ |∂� = , (.)

where

={

φ(y(uh))–φ(yh)y(uh)–yh

, y(uh) �= yh,φ′(yh), y(uh) = yh.

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

‖ξ‖,� ≤ C‖F‖, ‖ζ‖,� ≤ C‖F‖.



Lemma . Let (yh,ph,uh) be the solution of (.)-(.). Suppose that y(uh),p(uh) ∈H(�) and φ′(·) is locally Lipschitz continuous. Then there exists a constant C indepen-dent of h such that

∥∥y(uh) – yh∥∥ +

∥∥p(uh) – ph∥∥ ≤ Ch. (.)

Proof Let F = y(uh) – yh and ξh = πhξ . We have

∥∥y(uh) – yh∥∥ = a

(y(uh) – yh, ξ

)+

( ξ , y(uh) – yh

)= a

(y(uh) – yh, ξ – ξh

)+

( ξ , y(uh) – yh

)–

(φ(y(uh)

)– φ(yh), ξh

)= a

(y(uh) – yh, ξ – ξh

)+

(φ(y(uh)

)– φ(yh), ξ – ξh

)≤ C

∥∥y(uh) – yh∥∥,�‖ξ – ξh‖,� +C

∥∥y(uh) – yh∥∥‖ξ – ξh‖

≤ C∥∥y(uh) – yh

∥∥,�‖ξ – ξh‖,�. (.)

Note that

‖ξ – ξh‖,� ≤ Ch‖ξ‖,� ≤ Ch∥∥y(uh) – yh

∥∥. (.)

Thus,

∥∥y(uh) – yh∥∥ ≤ Ch

∥∥y(uh) – yh∥∥,� ≤ Ch. (.)

Similarly, let F = p(uh) – ph and ζh = πhζ , we obtain

∥∥p(uh) – ph∥∥ ≤ Ch. (.)

From (.) and (.), we get (.). �

Lemma . Let (y,p,u) and (yh,ph,uh) be the solutions of (.)-(.) and (.)-(.), re-spectively. Assume that all the conditions in Lemma . are valid. Then there exists a con-stant C independent of h such that

‖u – uh‖ ≤ Ch. (.)

Proof It is clear that

(J ′(v) – J ′(u), v – u

) ≥ c‖v – u‖, ∀v,u ∈ K . (.)

By using (.) and (.), we have

c‖u – uh‖ ≤ (J ′(u) – J ′(uh),u – uh

)=

(h′(u) + B∗p,u – uh

)–

(h′(uh) + B∗p(uh),u – uh

)≤ –

(h′(uh) + B∗ph,u – uh

)+

(B∗ph – B∗p(uh),u – uh

)



≤ (B∗ph – B∗p(uh),u – uh

)≤ C

∥∥ph – p(uh)∥∥‖u – uh‖. (.)

From (.) and (.), we derive (.). �

Now we combine Lemmas .-. to come up with the following main result.

Theorem . Let (y,p,u) and (yh,ph,uh) be the solutions of (.)-(.) and (.)-(.),respectively. Assume that all the conditions in Lemmas .-. are valid. Then we have

‖u – uh‖ + ‖y – yh‖ + ‖p – ph‖ ≤ Ch. (.)

Proof Note that

‖p – ph‖ ≤ ∥∥p – p(uh)∥∥ +

∥∥p(uh) – ph∥∥, (.)

‖y – yh‖ ≤ ∥∥y – y(uh)∥∥ +

∥∥y(uh) – yh∥∥. (.)

From (.)-(.), (.)-(.) and the regularity estimates, we have

∥∥p – p(uh)∥∥ ≤ ∥∥p – p(uh)

∥∥,� ≤ C

∥∥y – y(uh)∥∥, (.)∥∥y – y(uh)

∥∥ ≤ ∥∥y – y(uh)∥∥,� ≤ C‖u – uh‖. (.)

Then, (.) follows from (.), (.) and (.)-(.). �

4 A posteriori error estimatesWenowderiveaposteriori error estimates for the variational discretization approximationscheme. The following lemmas are very important in deriving a posteriori error estimatesof residual type.

Lemma . [] ∀v ∈W ,q(�), ≤ q < ∞,

‖v‖W,q(∂τ ) ≤ C(h– q

τ ‖v‖W,q(τ ) + h–

qτ |v|W ,q(τ )

). (.)

Lemma . Let (y,p,u) and (yh,ph,uh) be the solutions of (.)-(.) and (.)-(.), re-spectively. Then we have

‖u – uh‖ ≤ C∥∥ph – p(uh)

∥∥, (.)

where p(uh) is defined in (.).



Proof It follows from (.) and (.) that

c‖u – uh‖ ≤ (J ′(u),u – uh

)–

(J ′(uh),u – uh

)≤ –

(J ′(uh),u – uh

)= –

(h′(uh) + B∗ph,u – uh

)+

(B∗ph – B∗p(uh),u – uh

)≤ C(δ)

∥∥ph – p(uh)∥∥ + δ‖u – uh‖. (.)

Let δ be small enough, then (.) follows from (.). �

Lemma. Let (yh,ph,uh) and (y(uh),p(uh)) be the solutions of (.)-(.) and (.)-(.),respectively. Assume that φ′(·) is locally Lipschitz continuous. Then there exists a positiveconstant C independent of h such that

∥∥yh – y(uh)∥∥,� +

∥∥ph – p(uh)∥∥,� ≤ C

(η + η

), (.)

where

η =

∑τ∈T h

hτ∫

τ

(g ′(yh) + div

(A∗∇ph

)– φ′(yh)ph

) dx + ∑l∩∂��=∅

hl∫l

[A∗∇ph · n] ds,

η =

∑τ∈T h

hτ∫

τ

(div(A∇yh) – φ(yh) + f + Buh

) dx + ∑l∩∂��=∅

hl∫l[A∇yh · n] ds,

where hl is the size of the face l = τ̄ l ∩ τ̄

l , and τ l , τ

l are two neighboring elements in T h,[A∇yh · n]l , and [A∗∇ph · n]l are the A-normal and A∗-normal derivative jumps over theinterior face l, respectively, defined by

[A∇yh · n]l = (A∇yh|τ l –A∇yh|τl ) · n,[A∗∇ph · n]

l =(A∗∇ph|τ l –A∗∇ph|τl

) · n,

where n is the normal vector on l = τ l ∩ τ

l outwards τ l . For later convenience, we defined

[A∇yh · n]l = and [A∗∇ph · n]l = when l ⊂ ∂�.

Proof Let ep = ph – p(uh) and epI = πhep, it follows from the Green formula, embeddingtheorem ‖v‖L(�) ≤ C‖v‖H(�), Lemma ., (.) and (.) that

c∥∥ph – p(uh)

∥∥,�

≤ a(ep,ph – p(uh)

)+

(φ′(y(uh))(ph – p(uh)

), ep

)= a

(ep – epI ,ph – p(uh)

)+

(φ′(yh)ph – φ′(y(uh))p(uh), ep – epI

)+ a

(epI ,ph – p(uh)

)+

(φ′(yh)ph – φ′(y(uh))p(uh), epI ) + (

φ′(y(uh))ph – φ′(yh)ph, ep)

=∑τ∈T h

∫τ

(g ′(yh) + div

(A∗∇ph

)– φ′(yh)ph

)(epI – ep

)dx +

(g ′(yh) – g ′(y(uh)), ep)



+∑τ∈T h

∫∂τ

(A∗∇ph · n)(

ep – epI)ds +

(φ′(y(uh))ph – φ′(yh)ph, ep

)

≤ Cη +C(δ)

∥∥yh – y(uh)∥∥,� + δ

∥∥ph – p(uh)∥∥,�. (.)

Similarly, we obtain

c∥∥yh – y(uh)

∥∥,�

≤ a(yh – y(uh), ey

)+

(φ(yh) – φ

(y(uh)

), ey

)=

(A∇(

yh – y(uh)),∇(

ey – eyI))

+(φ(yh) – φ

(y(uh)

), ey – eyI

)=

∑τ∈T h

∫τ

(div(A∇yh) – φ(yh) + f + Buh

)(eyI – ey

)dx +

∑τ∈T h

∫∂τ

(A∇yh · n)(ey – eyI)ds

≤ C(δ)η + δ

∥∥yh – y(uh)∥∥,�. (.)

From (.) and (.), we derive (.). �

Theorem . Let (y,p,u) and (yh,ph,uh) be the solutions of (.)-(.) and (.)-(.),respectively. Assume that all the conditions in Lemmas .-. are valid. Then there existsa constant C independent of h such that

‖u – uh‖ + ‖y – yh‖,� + ‖p – ph‖,� ≤ C(η + η

), (.)

where η and η are defined in Lemma ..

Proof Note that

‖p – ph‖,� ≤ ∥∥p – p(uh)∥∥,� +

∥∥ph – p(uh)∥∥,�, (.)

‖y – yh‖,� ≤ ∥∥y – y(uh)∥∥,� +

∥∥yh – y(uh)∥∥,�, (.)

and

∥∥p – p(uh)∥∥,� ≤ ∥∥p – p(uh)

∥∥,� ≤ C

∥∥y – y(uh)∥∥, (.)∥∥y – y(uh)

∥∥,� ≤ ∥∥y – y(uh)

∥∥,� ≤ C‖u – uh‖. (.)

Then (.) follows from (.), (.) and (.)-(.). �

5 Numerical experimentsFor a constrained optimization problem

minu∈K⊂U

J(u),

where J(u) is a convex functional onU and K is a convex subset ofU , the iterative schemereads (n = , , , . . .):

{b(un+

, v) = b(un, v) – ρn(J ′(un), v), ∀v ∈U ,

un+ = PbK (un+

),

(.)



where b(·, ·) is a symmetric and positive definite bilinear form, and similarly to [], theprojection operator Pb

K :U → K is defined: For given w ∈ U find PbKw ∈ K such that

b(PbKw –w,Pb

Kw –w)= min

u∈K b(u –w,u –w).

The bilinear form b(·, ·) provides a suitable precondition for the projection gradient al-gorithm. Let Uh = {vh ∈ L(�),a ≤ vh ≤ b : vh|τ = constant,∀τ ∈ T h}. For an acceptableerror tol and a fixed step size ρn, by applying (.) to the discretized nonlinear elliptic op-timal control problem, we introduce the following projection gradient algorithm (see, e.g.,[, ]), for ease of exposition, we have omitted the subscript h.

Algorithm . (Projection gradient algorithm)Step . Initialize u;Step . Solve the following equations:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

b(un+ , v) = b(un, v) – ρn(h′(un) + B∗pn, v), un+

,un ∈Uh,∀v ∈ Uh,

a(yn,w) + (φ(yn),w) = (f + Bun,w), yn ∈ Vh,∀w ∈ Vh,a(q,pn) + (φ′(yn)pn,q) = (yn – yd,q), pn ∈ Vh,∀q ∈ Vh,un+ = Pb

K (un+ );

(.)

Step . Calculate the iterative error: en+ = ‖un+ – un‖;Step . If en+ ≤ tol, stop; else, go to Step .

According to the preceding analysis, we construct the following variational discretiza-tion algorithm.

Algorithm . (Variational discretization algorithm)Step . Initialize u;Step . Solve the following equations:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

b(un+ , v) = b(un, v) – ρn(h′(un) + B∗pn, v), un+

,un ∈U ,∀v ∈ U ,

a(yn,w) + (φ(yn),w) = (f + Bun,w), yn ∈ Vh,∀w ∈ Vh,a(q,pn) + (φ′(yn)pn,q) = (yn – yd,q), pn ∈ Vh,∀q ∈ Vh,un+ =�[a,b](un+

).

(.)

Step . Calculate the iterative error: en+ = ‖un+ – un‖;Step . If en+ ≤ tol, stop; else, go to Step .

It is well known that there are four major types of adaptive finite element methods,namely, the h-methods (mesh refinement), the p-methods (order enrichment), the r-methods (mesh redistribution) and the hp-methods (the combination of h-method andp-method). For an acceptable error Tol, by using a posteriori error estimator η

and η as

the mesh refinement indicator and the Algorithm ., we present the following adaptivevariational discretization algorithm.



Algorithm . (Adaptive variational discretization algorithm)Step . Solve the discretized optimization problem with Algorithm . on the current

mesh obtain the numerical solution u′n and calculate the error estimators η and η;

Step . Adjust the mesh by using estimators η and η, then update the numerical solu-tion u′

n and obtain the new numerical solution u′n+ on new mesh;

Step . If ‖u′n+ – u′

n‖ ≤ Tol, stop; else, go to Step .

All of the following numerical examples were solved numerically with codes developedbased on AFEPack which provided a general tool of finite element approximation forPDEs. The package is freely available and the details can be found in [].We consider the following optimal control problems:

⎧⎪⎨⎪⎩

minu∈K {∫�(g(y(x)) + h(u(x)))dx},

–div(A(x)∇y(x)) + φ(y(x)) = f (x) + Bu(x), x ∈ �,y(x) = , x ∈ ∂�,

where

g(y(x)

)=

[y(x) – y(x)

],h(u(x)

)=

[u(x) – u(x)

],and

K ={v(x) ∈ L(�) : a≤ v(x)≤ b,x ∈ �

},

the domain � is the unit square [, ]× [, ] and B = I .

Example In the first example, we compare the convergence order of ‖u – uh‖ in Algo-rithm . with that in Algorithm .. The data are as follows:

A(x) = E, φ(y) = y, a = , b = .,

p(x) = xx sin(πx) sin(πx),

y(x) = p(x),

u(x) = + sin(πx) sin(πx),

u(x) = min(.,max

(,u(x) – p(x)

)),

f (x) = –div(A(x)∇y(x)

)+ φ

(y(x)

)– u(x),

y(x) = y(x) + div(A∗(x)∇p(x)

)– φ′(y(x))p(x).

The numerical results are listed in Table and Table .In Figure , we see clearly that in the projection gradient algorithm, ‖u – uh‖ = O(h),

while in the variational discretization algorithm, ‖u – uh‖ = O(h). In Figure , we showthe profiles of the exact solution u alongside the solution error.



Table 1 Algorithm 5.1, Example 1

Mesh ‖u – uh‖ ‖y – yh‖ ‖p – ph‖16× 16 5.98051E-02 4.92490E-02 4.76241E-0232× 32 2.84008E-02 1.25725E-02 1.21182E-0264× 64 1.39765E-02 3.15952E-03 3.04294E-03128× 128 6.96692E-03 7.90942E-04 7.61570E-04256× 256 3.48077E-03 1.97800E-04 1.90445E-04

Table 2 Algorithm 5.2, Example 1

Mesh ‖u – uh‖ ‖y – yh‖ ‖p – ph‖16× 16 4.31006E-02 4.92551E-02 4.76397E-0232× 32 1.09812E-02 1.25730E-02 1.21193E-0264× 64 2.77425E-03 3.15960E-03 3.04301E-03128× 128 6.92439E-04 7.90991E-04 7.61575E-04256× 256 1.73211E-04 1.97847E-04 1.90445E-04

Figure 1 The convergence order of ‖u – uh‖.

Figure 2 The exact solution u (left) and the error uh – u (right).

Example In order to illustrate the reliability and efficiency of the a posteriori error esti-mates in Theorem ., we use Algorithm . to solve this example. The data are as follows:

φ(y) = y, a = –., b = ,

A(x) ={E, x + x ≥ , · E, x + x < ,



Table 3 Numerical results, Example 2.

Mesh Nodes Sides Elements Dofs ‖u – uh‖Uniform mesh 513 1456 944 513 6.19419E-02Adaptive mesh 145 392 248 145 6.04841E-02

Figure 3 The exact solution u (left) and the error uh – u (right).

p(x) ={ sin(πx) sin(πx), x + x ≥ ,sin(πx) sin(πx), x + x < ,

y(x) = p(x),

u(x) = ,

u(x) = min(,max

(–.,u(x) – p(x)

)),

f (x) = –div(A(x)∇y(x)

)+ φ

(y(x)

)– u(x),

y(x) = y(x) + div(A∗(x)∇p(x)

)– φ′(y(x))p(x).

The numerical results based on adaptive mesh and uniform mesh are presented in Ta-ble . In Figure , we show the profiles of the exact solution u alongside the solution error.From Table , it is clear that the adaptive mesh generated via the error indicators in Theo-rem . are able to save substantial computational work, in comparison with the uniformmesh. Our numerical results confirm our theoretical results.

Competing interestsThe author declares that he has no competing interests.

AcknowledgementsThis work is supported by the Scientific Research Project of Department of Education of Hunan Province (13C338).

Received: 23 March 2013 Accepted: 10 September 2013 Published: 07 Nov 2013

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10.1186/1029-242X-2013-450Cite this article as: Tang: Error analysis of variational discretization solving temperature control problems. Journal ofInequalities and Applications 2013, 2013:450


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Error analysis of variational discretization solving temperature control problems