Superconvergence for Optimal Control Problems

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J Sci Comput (2009) 39: 206–221 ... y − yd. 2 +. 1. 2 u 2. } ,. (1.1). −div(A∇y) + φ(y) = f + Bu, x ∈ ,. (1.2) y|∂ = 0,. (1.3). This work is supported by National ...

J Sci Comput (2009) 39: 206–221 DOI 10.1007/s10915-008-9258-9

Superconvergence for Optimal Control Problems Governed by Semi-linear Elliptic Equations Yanping Chen · Yongquan Dai

Received: 17 March 2008 / Revised: 15 August 2008 / Accepted: 28 November 2008 / Published online: 10 December 2008 © Springer Science+Business Media, LLC 2008

Abstract In this paper, we will investigate the superconvergence of the finite element approximation for quadratic optimal control problem governed by semi-linear elliptic equations. The state and co-state variables are approximated by the piecewise linear functions and the control variable is approximated by the piecewise constant functions. We derive the superconvergence properties for both the control variable and the state variables. Finally, some numerical examples are given to demonstrate the theoretical results. Keywords Finite element approximation · Superconvergence · Semi-linear elliptic equation · Optimal control problem · Interpolate operator 1 Introduction In this paper, we shall study the superconvergence of the finite element approximation for the following optimal control problem   1 1 2 2 (1.1) min y − yd  + u , u∈K 2 2 −div(A∇y) + φ(y) = f + Bu,

x ∈ ,

y|∂ = 0,

(1.2) (1.3)

This work is supported by National Science Foundation of China, the National Basic Research Program under the Grant 2005CB321703, Scientific Research Fund of Hunan Provincial Education Department, and Hunan Provincial Innovation Foundation for Postgraduate (No. S2008yjscx04). Y. Dai Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, People’s Republic of China e-mail: [email protected] Y. Chen () School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, People’s Republic of China e-mail: [email protected]

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where  is a convex bounded domain in R2 with smooth boundary. The details will be specified later. Problem (1.1)–(1.3) appears, for example, in temperate control problems, see [13]. Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. There have been extensively studies on this aspect, see, for example, [1, 2, 6–11, 14, 16, 18, 23, 24, 29, 30]. A systematic introduction of finite element method for PDEs and optimal control problem can be found in, for example, [12, 15, 27], and [28]. For optimal control problem governed by semi-linear elliptic state equations, a priori error estimates of finite element approximation were studied in, for example, [3] and [23]. A posteriori error estimates for this problem were discussed by Huang et al. [17]. In [20, 26], superconvergence properties of the control variable for the linear elliptic control problem are presented. Yang and Chang in [31] showed the superconvergence properties for optimal control problem of bilinear type. The purpose of this paper is to extend the superconvergence property of [20] to the above semi-linear control problem and also to get the superconvergence properties for the state variables. The paper is organized as follows: In Sect. 2, we shall give a brief review on the finite element method and then construct the approximation schemes for the model optimal control problem. In Sect. 3, we shall give some intermediate error estimates which is useful to derive the superconvergence. In Sect. 4, superconvergence results for both control and state variables were derived. In Sect. 5, we will give some applications of the results derived in Sect. 4. In Sect. 6, some numerical examples are given to demonstrate our theoretical results. In the last section we briefly discuss some possible future work. In this paper we adopt the standard notation W m,p () for Sobolev spaces on  with a norm  · m,p given by  p D α φLp () , φpm,p = |α|≤m

a semi-norm | · |m,p given by |φ|pm,p =



p

D α φLp () .

|α|=m

We set m,p

W0

() = {φ ∈ W m,p () : φ|∂ = 0}.

For simplicity, we denote H m () = W m,2 () and  · m =  · m,2 when p = 2. In addition, c and C denotes a general positive constant independent of h.

2 Preliminaries In this section we will study the finite element approximation of (1.1)–(1.3). In the rest of the paper, we shall take the state space V = H01 (), the control space U = L2 (). Let B be a linear continuous operator from U to H , Let K denotes the admissible set of the control variable defined by K = {u ∈ U : u ≥ 0}. Let  a(y, w) = (A∇y) · ∇w, ∀y, w ∈ V , 

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 (u, v) =

uv,

∀(u, v) ∈ U × U,



A(·) = (ai,j (·))n×n ∈ (W 1,∞ ())n×n , satisfying that there is a constant c > 0 such that for any vector X ∈ Rn , Xt AX ≥ cX2Rn . It follows from the assumptions on A that there are constants c, C > 0 such that ∀y, w ∈ V a(y, y) ≥ cy2V ,

|a(y, w)| ≤ CyV wV .

Then the standard weak formula for the state equation reads: find y(u) ∈ V such that a(y(u), w) + (φ(y(u)), w) = (f + Bu, w),

∀w ∈ V ,

where we assume that the function φ(·) ∈ W 2,∞ (−R, R) for any R > 0, φ (y) ∈ L2 () for any y ∈ H 1 (), and φ ≥ 0. Thus the above equation has a unique solution. We recast (1.1)–(1.3) in the following weak form: find (y, u) ∈ V × U such that   1 1 2 2 y − yd  + u , min (2.1) u∈K⊂U 2 2 a(y, w) + (φ(y), w) = (f + Bu, w),

∀w ∈ V = H01 ().

(2.2)

It is well known (see, e.g., [22]) that the control problem (2.1)–(2.2) has a unique solution (y, u) and that if a pair (y, u) is the solution of (2.1)–(2.2), then there is a co-state p ∈ V such that the triplet (y, p, u) satisfies the following optimality conditions: a(y, w) + (φ(y), w) = (f + Bu, w),

a(q, p) + (φ (y)p, q) = (y − yd , q), (u + B ∗ p, v − u) ≥ 0,

∀w ∈ V = H01 (),

(2.3)

∀q ∈ V =

(2.4)

H01 (),

∀v ∈ K ⊂ U = L2 (),

(2.5)

where B ∗ is the adjoint operator of B. In the following we construct the finite element approximation for the optimal control problem (2.1)–(2.2). For ease of exposition we will assume that  is a polygon. Let Th be a quasi-uniform partition of  into triangles. And let h be the maximum diameter of T in Th . Moreover, we set ˜ T is constant on T ∈ Th }, Uh = {u˜ ∈ U : u| Vh = {yh ∈ C() : yh |T ∈ P1 , ∀T ∈ Th }, where P1 is the polynomials of degree less than or equal to 1 for triangulation. Let Kh ⊂ K ∩ Uh be a closed convex set. Now, the finite element approximation of the problem (2.1)– (2.2) is as follows:   1 1 yh − yd 2 + uh 2 , min (2.6) uh ∈Kh 2 2 a(yh , wh ) + (φ(yh ), wh ) = (f + Buh , wh ),

∀wh ∈ Vh ⊂ H01 ().

(2.7)

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The control problem (2.6)–(2.7) has a unique solution (yh , uh ), and a pair (yh , uh ) is the solution of (2.6)–(2.7) if and only if their is a co-state ph such that the triplet (yh , ph , uh ) satisfying the following optimal conditions: a(yh , wh ) + (φ(yh ), wh ) = (f + Buh , wh ),

∀wh ∈ Vh ⊂ H01 (),

(2.8)

∀qh ∈ Vh ⊂

(2.9)



a(qh , ph ) + (φ (yh )ph , qh ) = (yh − yd , qh ), ∗

(uh + B ph , vh − uh ) ≥ 0,

H01 (),

∀vh ∈ Kh ⊂ L (). 2

(2.10)

3 Intermediate Error Estimates First, we shall use some intermediate variables. For any u˜ ∈ K, let (y(u), ˜ p(u)) ˜ be the solution of the following equations: a(y(u), ˜ w) + (φ(y(u)), ˜ w) = (f + B u, ˜ w),

∀w ∈ V ,

˜ u), ˜ q) = (y(u) ˜ − yd , q), a(q, p(u)) ˜ + (φ (y(u))p(

∀q ∈ V .

(3.1) (3.2)

Then, for any u˜ ∈ K, let (yh (u), ˜ ph (u)) ˜ be the solution of the following equations: a(yh (u), ˜ wh ) + (φ(yh (u)), ˜ wh ) = (f + B u, ˜ wh ),

∀wh ∈ Vh ,

˜ + (φ (yh (u))p ˜ h (u), ˜ qh ) = (yh (u) ˜ − yd , qh ), a(qh , ph (u))

∀qh ∈ Vh .

(3.3) (3.4)

Thus, we have (y, p) = (y(u), p(u)), (yh , ph ) = (yh (uh ), ph (uh )). We introduce the standard L2 ()-orthogonal projection Qh : U → Uh , which satisfies: for any ψ ∈ U (ψ − Qh ψ, μh ) = 0,

∀μh ∈ Uh ,

(3.5)

and the elliptic projection Rh : V → Vh , which satisfies: for all w ∈ V a(w − Rh w, wh ) = 0,

wh ∈ Vh .

(3.6)

We have the following approximation properties: ψ − Qh ψ−s ≤ Ch1+s |ψ|1 , w − Rh w ≤ Ch w2 , 2

s = 0, 1,

(3.7)

for w ∈ H ().

(3.8)

2

Lemma 3.1 Let u be the solution of (2.3)–(2.4), for h sufficiently small, there exists a positive constant C such that y(Qh u) − y(u)1 ≤ Ch2 ,

(3.9)

p(Qh u) − p(u)1 ≤ Ch .

(3.10)

2

Proof Choose u˜ = Qh u and u˜ = u in (3.1)–(3.2), respectively, then we have the following error equations a(y(Qh u) − y(u), w) + (φ(y(Qh u)) − φ(y(u)), w) = (B(Qh u − u), w), (3.11) a(q, p(Qh u) − p(u)) + (φ (y(Qh u))p(Qh u) − φ (y(u))p(u), q) = (y(Qh u) − y(u), q),

(3.12)

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for any w ∈ V and q ∈ V . First, choose w = y(Qh u) − y(u) in (3.11), we have a(y(Qh u) − y(u), y(Qh u) − y(u)) + (φ(y(Qh u)) − φ(y(u)), y(Qh u) − y(u)) = (B(Qh u − u), y(Qh u) − y(u)).

(3.13)

Now, we estimate the right hand side of (3.13). Using the continuity of B and (3.7), we have (B(Qh u − u), y(Qh u) − y(u)) = (Qh u − u, B ∗ (y(Qh u) − y(u))) ≤ CB ∗ (y(Qh (u)) − y(u))1 · Qh u − u−1 ≤ Ch2 u1 · y(Qh u) − y(u)1 .

(3.14)

From (3.13), (3.14) and the assumption of A and φ(·), we have cy(Qh u) − y(u)21 ≤ a(y(Qh u) − y(u), y(Qh u) − y(u)) + (φ(y(Qh u)) − φ(y(u)), y(Qh u) − y(u)) = (B(Qh u − u), y(Qh u) − y(u)) ≤ Ch2 y(Qh u) − y(u)1 ,

(3.15)

then, (3.9) can be obtained from (3.15). Choose q = p(Qh u) − p(u) in (3.12), we have a(p(Qh u) − p(u), p(Qh u) − p(u)) + (φ (y(Qh u))p(Qh u) − φ y(u)p(u), p(Qh u) − p(u)) = (y(Qh u) − y(u), p(Qh u) − p(u)),

(3.16)

namely, a(p(Qh u) − p(u), p(Qh u) − p(u)) + (φ (y(Qh u))(p(Qh u) − p(u)), p(Qh u) − p(u)) = (y(Qh u) − y(u), p(Qh u) − p(u)) + (p(u)(φ (y(u)) − φ (y(Qh u))), p(Qh u) − p(u)).

(3.17)

Notice that (y(Qh u) − y(u), p(Qh u) − p(u)) ≤ Cy(Qh u) − y(u) · p(Qh u) − p(u) ≤ Ch2 p(Qh u) − p(u)1 .

(3.18)

Using the assumption of φ(·) and (3.9), we have (p(u)(φ (y(u)) − φ (y(Qh u))), p(Qh u) − p(u)) ≤ Cp(u)0,4 φ (y(u)) − φ (y(Qh u)) · p(Qh u) − p(u)0,4 ≤ Cp(u)1 φW 2,∞ y(u) − y(Qh u) · p(Qh u) − p(u)1 ≤ Ch2 p(Qh u) − p(u)1 ,

(3.19)

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where we used the embedding v0,4 ≤ Cv1 . Then, using (3.17), (3.18), (3.19) and the assumption of φ(·), we have cp(Qh u) − p(u)21 ≤ a(p(Qh u) − ph (u), p(Qh u) − p(u)) + (φ (y(Qh u)(p(Qh u) − p(u)), p(Qh u) − p(u)) = (y(Qh u) − y(u), p(Qh u) − p(u)) + (p(u)(φ (y(u)) − φ (y(Qh u))), p(Qh u) − p(u)) ≤ Ch2 p(Qh u) − p(u)1 ,

(3.20) 

which implies (3.12). Lemma 3.2 For any u˜ ∈ K, if the intermediate solution satisfies y(u), ˜ p(u) ˜ ∈ H 2 (), and  is convex, then, we have ˜ − Rh y(u) ˜ 1 ≤ Ch2 , yh (u)

(3.21)

˜ − Rh p(u) ˜ 1 ≤ Ch2 . ph (u)

(3.22)

Proof According to Theorem 8.2.9 in [25], the following estimate hold ˜ + p(u) ˜ − ph (u) ˜ ≤ Ch2 . y(u) ˜ − yh (u)

(3.23)

From (3.1)–(3.2) and (3.3)–(3.4), we have the following error equations: ˜ − y(u), ˜ wh ) + (φ(yh (u)) ˜ − φ(y(u)), ˜ wh ) = 0, a(yh (u)

(3.24)

˜ − p(u)) ˜ + (φ (yh (u))p ˜ h (u) ˜ − φ (y(u))p( ˜ u), ˜ qh ) a(qh , ph (u) ˜ − y(u), ˜ qh ), = (yh (u)

(3.25)

for any wh ∈ Vh and qh ∈ Vh . Using the definition of Rh , the above equation can be restated as ˜ − Rh y(u), ˜ wh ) = (φ(y(u)) ˜ − φ(yh (u)), ˜ wh ), a(yh (u)

(3.26)



˜ − Rh p(u)) ˜ + (φ (yh (u))(p ˜ ˜ − Rh p(u)), ˜ qh ) a(qh , ph (u) h (u) ˜ − y(u), ˜ qh ) = (yh (u) ˜ − φ (yh (u))), ˜ qh ) + (φ (yh (u))(p( ˜ u) ˜ − Rh p(u)), ˜ qh ). + (p(u)(φ ˜ (y(u)) ˜ − Rh y(u) ˜ in (3.26), we have First, let wh = yh (u) ˜ − Rh y(u) ˜ 21 cyh (u) ≤ a(yh (u) ˜ − Rh y(u), ˜ yh (u) ˜ − Rh y(u)) ˜ ˜ yh (u) ˜ − Rh y(u)) ˜ = (φ(y(u)) ˜ − φ(yh (u)),

(3.27)

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≤ CφW 1,∞ y(u) ˜ − yh (u) ˜ · yh (u) ˜ − Rh y(u) ˜ ˜ 2 · yh (u) ˜ − Rh y(u) ˜ ≤ Ch2 φW 1,∞ y(u) ˜ − Rh y(u) ˜ 1, ≤ Ch2 yh (u)

(3.28)

which implies (3.21). ˜ − Rh p(u) ˜ in (3.27). Notice that Then, let qh = ph (u) (yh (u) ˜ − y(u), ˜ ph (u) ˜ − Rh p(u)) ˜ ≤ yh (u) ˜ − y(u) ˜ · ph (u) ˜ − Rh p(u) ˜ ˜ 2 · ph (u) ˜ − Rh p(u) ˜ ≤ Ch2 y(u) ˜ − Rh p(u) ˜ 1. ≤ Ch2 ph (u)

(3.29)

Using the assumption of φ(·), we have (p(u)(φ ˜ (y(u)) ˜ − φ (yh (u))), ˜ ph (u) ˜ − Rh p(u)) ˜ ˜ − φ (yh (u)) ˜ · ph (u) ˜ − Rh p(u) ˜ 0,4 ≤ Cp(u) ˜ 0,4 φ (y(u)) ≤ Ch2 p(u) ˜ 1 · φW 2,∞ y(u) ˜ 2 · ph (u) ˜ − Rh p(u) ˜ 1 ≤ Ch2 ph (u) ˜ − Rh p(u) ˜ 1,

(3.30)

where we used the embedding v0,4 ≤ Cv1 . Then, using the definition of Rh and the assumption of φ(·), we have ˜ u) ˜ − Rh p(u)), ˜ ph (u) ˜ − Rh p(u)) ˜ (φ (yh (u))(p( ˜ − Rh p(u) ˜ · ph (u) ˜ − Rh p(u) ˜ ≤ CφW 1,∞ p(u) ˜ 2 · ph (u) ˜ − Rh p(u) ˜ ≤ Ch2 φW 1,∞ p(u) ˜ − Rh p(u) ˜ 1. ≤ Ch2 ph (u)

(3.31)

From (3.27) and (3.29)–(3.31), we have cph (u) ˜ − Rh p(u) ˜ 21 ≤ a(ph (u) ˜ − Rh p(u), ˜ ph (u) ˜ − Rh p(u)) ˜ ˜ ˜ − Rh p(u)), ˜ ph (u) ˜ − Rh p(u)) ˜ + (φ (yh (u))(p h (u) ˜ − y(u), ˜ ph (u) ˜ − Rh p(u)) ˜ + (p(u)(φ ˜ (y(u)) ˜ − φ (yh (u))), ˜ ph (u) ˜ − Rh p(u)) ˜ = (yh (u) ˜ u) ˜ − Rh p(u)), ˜ ph (u) ˜ − Rh p(u)) ˜ + (φ (yh (u))(p( ˜ − Rh p(u) ˜ 1, ≤ Ch2 ph (u) which implies (3.22). Let y(u) and yh (uh ) be the solution of (2.2) and (2.7), respectively. Set   1 1 2 2 y − yd  + u , J (u) = 2 2   1 1 yh (uh ) − yd 2 + uh 2 . Jh (uh ) = 2 2

(3.32) 

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Then, the reduced problems of (2.1) and (2.6) read as min{J (u)},

(3.33)

min {Jh (uh )},

(3.34)

u∈K

and uh ∈Kh

respectively. It can be shown that (J (u), v) = (u + B ∗ p, v), (J (uh ), v) = (uh + B ∗ p(uh ), v), (J (Qh u), v) = (Qh u + B ∗ p(Qh u), v), (Jh (uh ), v) = (uh + B ∗ ph , v), where p(uh ) and p(Qh u) are solutions of (3.1)–(3.2) with u˜ = uh and u˜ = Qh u, respectively. In many application, J (·) is uniform convex near the solution u. The convexity of J (·) is closely related to the second order sufficient conditions of the control problem, which are assumed in many studies on numerical methods of the problem. Then, there is a constant c > 0, independent of h, such that (J (Qh u) − J (uh ), Qh u − uh ) ≥ cQh u − uh 2 ,

(3.35)

where u and uh are solutions of (3.33) and (3.34) respectively, Qh u is the orthogonal projection of u which is defined in (3.5). We shall assume the above inequality throughout this paper. More discussion of this can be found in, for example, [4] and [5].

4 Superconvergence Properties In this section, we will discuss the superconvergence for both the control variable and the state variables. Let π c be the average operator defined in [20]. Let + = {∪τ : τ ⊂ , u|τ > 0}, 0 = {∪τ : τ ⊂ , u|τ = 0}, b = \(+ ∪ 0 ). In this paper, we assume that u and Th are regular such that meas(b ) ≤ Ch. Theorem 4.1 Let u be the solution of (2.3)–(2.5) and uh be the solution of (2.8)–(2.10). We assume that the exact control and state solution satisfy u, u + B ∗ p ∈ W 1,∞ (), and y(u), p(u) ∈ H 2 ().

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Then, we have 3

Qh u − uh  ≤ Ch 2 .

(4.1)

Proof Let v = uh in (2.5) and vh = Qh u in (2.10), then add the two inequalities we have (uh + B ∗ ph − u − B ∗ p, Qh u − uh ) + (u + B ∗ p, Qh u − u) ≥ 0.

(4.2)

Hence (Qh u − uh , Qh u − uh ) = (u − uh , Qh u − uh ) ≤ (B ∗ ph − B ∗ p, Qh u − uh ) + (u + B ∗ p, Qh u − u).

(4.3)

For the first term of (4.3), we divide it into three parts, (B ∗ ph − B ∗ p, Qh u − uh ) = (B ∗ ph − B ∗ p(uh ), Qh u − uh ) + (B ∗ p(uh ) − B ∗ p(Qh u), Qh u − uh ) + (B ∗ p(Qh u) − B ∗ p(u), Qh u − uh ),

(4.4)

then, from (4.3)–(4.4), we have (Qh u − uh , Qh u − uh ) − (B ∗ p(uh ) − B ∗ p(Qh u), Qh u − uh ) ≤ (B ∗ ph − B ∗ p(uh ), Qh u − uh ) + (B ∗ p(Qh u) − B ∗ p(u), Qh u − uh ) + (u + B ∗ p, Qh u − u).

(4.5)

According to (3.23), (B ∗ p(uh ) − B ∗ ph , Qh u − uh ) ≤ CB ∗ (p(uh ) − ph ) · Qh u − uh  ≤ Ch2 Qh u − uh .

(4.6)

For the second term of (4.3) (u + B ∗ p, Qh u − u) =

 +

 +

 0

+

b

(u + B ∗ p, Qh u − u)dx.

Obviously, (Qh u − u)|0 = 0. From (2.5), we have pointwise a.e. (u + B ∗ p) ≥ 0, we choose ˜ \+ = u, so that (u + B ∗ p, u)|+ ≤ 0. Hence, (u + B ∗ p)|+ = 0. Then u| ˜ + = 0 and u| (u + B ∗ p, Qh u − u) = (u + B ∗ p, Qh u − u)b ≤ (u + B ∗ p − π c (u + B ∗ p), Qh u − u)b ≤ Ch2 u + B ∗ p1,b u1,b ≤ Ch2 u + B ∗ p1,∞ u1,∞ · meas(b ) ≤ Ch3 .

(4.7)

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According to (3.35), the left hand of (4.5) can be restated as: (Qh u − uh , Qh u − uh ) − (B ∗ p(uh ) − B ∗ p(Qh u), Qh u − uh ) = (Qh u + B ∗ p(Qh u), Qh u − uh ) − (uh + B ∗ p(uh ), Qh u − uh ) = (J (Qh u) − J (uh ), Qh u − uh ) ≥ cQh u − uh 2 .

(4.8)

Then, combining Lemma 3.1 and (4.5)–(4.8), we have 3

Qh u − uh  ≤ Ch 2 , 

which complete the proof of Theorem 4.1.

Theorem 4.2 Let u be the solution of (2.3)–(2.5), uh be the solution of (2.8)–(2.10) and  is convex. We assume that the exact control and state solution satisfy u, u + B ∗ p ∈ W 1,∞ (), and y(u), p(u) ∈ H 2 (). Then, we have 3

yh − Rh y1 ≤ Ch 2 ,

(4.9)

3 2

ph − Rh p1 ≤ Ch .

(4.10)

Proof From (2.3)–(2.4) and (2.8)–(2.9). We have the following error equations a(yh − y, wh ) + (φ(yh ) − φ(y), wh ) = (B(uh − u), wh ),



a(qh , ph − p) + (φ (yh )ph − φ (y)p, qh ) = (yh − y, qh ),

∀wh ∈ Vh ,

(4.11)

∀qh ∈ Vh .

(4.12)

Using the definition of Rh , we have a(yh − Rh y, wh ) + (φ(yh ) − φ(Rh y), wh ) = (B(uh − u), wh ) + (φ(y) − φ(Rh y), wh ),

(4.13)



a(qh , ph − Rh p) + (φ (yh )(ph − Rh p), qh ) = (yh − y, qh ) + (φ (yh )(p − Rh p), qh ) + (p(φ (y) − φ (yh )), qh ),

(4.14)

for any wh ∈ Vh and qh ∈ Vh . First, take wh = yh − Rh y in (4.13) and using the assumption of φ(·), we have cyh − Rh y21 ≤ a(yh − Rh y, yh − Rh y) + (φ(yh ) − φ(Rh y), yh − Rh y) = (B(uh − Qh u), yh − Rh y) + (B(Qh u − u), yh − Rh y) + (φ(y) − φ(Rh y), yh − Rh y) ≤ C(uh − Qh u · yh − Rh y + Qh u − u−1 B ∗ (yh − Rh y)1

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+ φ1,∞ y − Rh y · yh − Rh y) 3

≤ C(h 2 yh − Rh y1 + h2 u1 yh − Rh y1 + h2 φ1,∞ y2 yh − Rh y1 ) 3

≤ Ch 2 yh − Rh y1 ,

(4.15)

which implies (4.9). Then, we take qh = ph − Rh p in (4.14). Notice that (yh − y, ph − Rh p) = (yh − Rh y, ph − Rh p) + (Rh y − y, ph − Rh p) 3

≤ Ch 2 ph − Rh p.

(4.16)

Using the definition of Rh and the assumption of φ(·), we have (φ (yh )(p − Rh p), ph − Rh p) ≤ Ch2 φ1,∞ p2 ph − Rh p ≤ Ch2 ph − Rh p,

(4.17)

and (p(φ (y) − φ (yh )), ph − Rh p) ≤ Cφ2,∞ (p(y − yh ), ph − Rh p) ≤ Cφ2,∞ y − yh  · p0,4 ph − Rh p0,4 ≤ Cφ2,∞ (y − Rh y + Rh y − yh )p1 ph − Rh p1 3

≤ Ch 2 ph − Rh p1 .

(4.18)

From (4.14) and (4.16)–(4.18), we have cph − Rh p21 ≤ a(ph − Rh p, ph − Rh p) + (φ (yh )(ph − Rh p), ph − Rh p) = (yh − y, ph − Rh p) + (φ (yh )(p − Rh p), ph − Rh p) + (p(φ (y) − φ (yh )), ph − Rh p) 3

≤ Ch 2 ph − Rh p1 ,

(4.19) 

which implies (4.10). 5 Application

In this section, some applications of the results derived in Sect. 4 will be presented. 2 which is presented by First, we will introduce a higher order interpolation operator I2h Lin and Zhu [21] and Ih be the Lagrangian interpolate operator. They satisfy the following properties: 2 v1 ≤ Ch2 v3 , v − I2h

∀v ∈ H 3 (),

2 2 Ih = I2h , I2h 2 v1 I2h

Using these properties we have

≤ Cv1 ,

(5.1) (5.2)

∀v ∈ Vh .

(5.3)

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Theorem 5.1 Suppose that all the conditions of Theorem 4.2 are valid. Moreover, we will assume that y, p ∈ H 3 . Then, we have 3

2 y − I2h yh 1 ≤ Ch 2 , 3 2

2 p − I2h ph 1 ≤ Ch .

(5.4) (5.5)

Proof Obviously, by (5.2) and (5.3), we have 2 2 2 2 y − I2h yh = y − I2h y + I2h (Ih y − Rh y) + I2h (Rh y − yh ),

(5.6)

2 2 y − I2h yh 1 ≤ y − I2h y1 + CIh y − Rh y1 + CRh y − yh 1 .

(5.7)

According to Theorem 2.1.1 in [21], we have Ih y − Rh y1 ≤ Ch2 y3 .

(5.8)

Combining with (4.9) and (5.1), we complete the proof of (5.4). Similarly, we have (5.5).  Then, let us construct the recovery operator Gh . Let Gh v be a continuous piecewise linear function (without zero boundary constraint). The value of Gh v on the nodes are defined by lease-squares argument on an element patches surrounding the nodes, the details can be refer to the definition of Rh in [20]. Theorem 5.2 Let u and uh be the solutions of (2.3)–(2.5) and (2.8)–(2.10), respectively. Assume that u ∈ W 1,∞ () and  is convex. Then, 3

u − Gh uh  ≤ Ch 2 .

(5.9)

Proof Let Qh u be defined in (3.5). Then, u − Gh uh  ≤ u − Gh u + Gh u − Gh Qh u + Gh Qh u − Gh uh .

(5.10)

According to Lemma 4.2 in [20], we have 3

u − Gh u ≤ Ch 2 .

(5.11)

Using the definition of Gh , we have Gh u = Gh Qh u,

(5.12)

Gh Qh u − Gh uh  ≤ CQh u − uh .

(5.13)

and

Combining (4.1) and (5.10)–(5.13), we complete the proof of Theorem 5.2.



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Table 1 The error of Example 1 on a sequential uniform refined meshes Resolution

u − uh 

Qh u − uh 

u − Gh uh 

16 × 16

2.551E-2

3.598E-3

2.362E-2

32 × 32

1.295E-2

1.340E-3

9.459E-3

64 × 64

6.516E-3

4.677E-4

3.525E-3

128 × 128

3.267E-3

1.699E-4

1.268E-3

6 Numerical Examples In this section, we carry out some numerical examples to demonstrate our theoretical results. The optimal problem was solved numerically by a precondition projection algorithm, see for instance [17], with codes developed based on AFEPack [19]. The discretization was already described in Sect. 2: the state and co-state equations were approximated by piecewise linear functions, whereas the control function u is discretized by piecewise constant functions. In our examples, we choose the domain  = [0, 1] × [0, 1] and B = I . Example 1 The example is to solve the following two dimension semi-linear elliptic control problem   1 1 min y − yd 2 + u − u0 2 , u∈K 2 2 (6.1) s.t.

− y + y 3 = f + u,

u ≥ 0,

where y = sin(πx1 ) sin(πx2 ), p = sin(πx1 ) sin(πx2 ), πx2 πx1 u0 = 1.0 − sin − sin , 2 2 u = max(u0 − p, 0),

(6.2)

f = 2π y + y − u, 2

3

yd = y − 2π 2 p − 3y 2 p. The dual equation of the state equation is − p + 3y 2 p = y − yd .

(6.3)

In Table 1, the error u − uh , Qh u − uh  and u − Gh uh  obtained on a sequence of uniformly refined meshes are shown. In Fig. 1, the profile of the numerical solution of u on the 64 × 64 mesh grid is plotted. Example 2 We consider the following semi-linear elliptic optimal control problem   1 1 y − yd 2 + u − u0 2 , min u∈K 2 2 s.t. − y + y 3 = f + u, u ≥ 0,

(6.4)

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219

Fig. 1 The profile of the numerical solution of Example 1 on 64 × 64 triangle mesh

where yd = sin(2πx1 ) + sin(2πx2 ), y = yd , u0 = max(4π 2 yd , 0), u = u0 ,

(6.5)

f = 4π y + y − u, 2

3

p = 0. The dual equation of the state equation is − p + 3y 2 p = y − yd .

(6.6)

Table 2 shows the numerical results of both u − uh , Qh u − uh  and u − Gh uh  on a sequential uniformly refined meshes. Figure 2 shows the numerical solution on the 64 × 64 mesh grid.

7 Conclusion In this paper, we present the superconvergence analysis of the finite element approximation for optimal control problems governed by semi-linear elliptic equations. Our superconvergence analysis for the semi-linear elliptic equations by standard finite element method seems

220

J Sci Comput (2009) 39: 206–221

Fig. 2 The profile of the numerical solution of Example 2 on 64 × 64 triangle mesh

Table 2 The error of Example 2 on a sequential uniform refined meshes Resolution

u − uh 

Qh u − uh 

u − Gh uh 

16 × 16

6.390E-1

4.559E-2

4.701E-1

32 × 32

3.214E-1

1.608E-2

1.689E-1

64 × 64

1.611E-1

5.681E-3

5.973E-2

128 × 128

8.065E-2

2.008E-3

2.106E-2

to be new, and these results can be extended to general convex problems. We shall study the superconvergence for optimal control problems governed by semi-linear parabolic equations. Furthermore we shall study the superconvergence of these problems by mixed finite element method.

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