A PRIORI ERROR ESTIMATE AND SUPERCONVERGENCE ANALYSIS FOR AN OPTIMAL CONTROL PROBLEM OF BILINEAR TYPE *

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1 Journal of Computational Mathematics, Vol.??, No.??, 2008, A PRIORI ERROR ESTIMATE AND SUPERCONVERGENCE ANALYSIS FOR AN OPTIMAL CONTROL PROBLEM OF BILINEAR TYPE * Danping Yang Department of Mathematics, East China Normal University, Shanghai , China dpyang@math.ecnu.edu.cn Yanzhen Chang School of Mathematics and System Science, Shandong University, Jinan , China changyanzhen@mail.sdu.edu.cn Wenin Liu KBS, University of Kent, Canterury, CT2 7PE, England W.B.Liu@kent.ac.uk Astract In this paper, we investigate a priori error estimates and superconvergence properties for a model optimal control prolem of ilinear type, which includes some parameter estimation application. The state and co-state are discretized y piecewise linear functions and control is approximated y piecewise constant functions. We derive a priori error estimates and superconvergence analysis for oth the control and the state approximations. We also give the optimal L 2 -norm error estimates and the almost optimal L -norm estimates aout the state and co-state. The results can e readily used for constructing a posteriori error estimators in adaptive finite element approximation of such optimal control prolems. Mathematics suject classification: 49J20, 65N30. Key words: Bilinear control prolem, Finite element approximation, Superconvergence, A priori error estimate, A posteriori error estimator 1. Introduction The finite element approximation of optimal control prolems has een extensively studied in the literature. There have een extensive studies in convergence of the standard finite element approximation of optimal control prolems, see, some examples in [2, 3, 9, 10, 18, 20, 23], although it is impossile to give even a very rief review here. For optimal control prolems governed y linear state equations, a priori error estimates of the finite element approximation were estalished long ago; see, e.g., [9, 10]. But it is more difficult to otain such error estimates for nonlinear control prolems. For some classes of nonlinear optimal control prolems, a priori error estimates were estalished in [4, 11, 17]. The optimal control prolem of ilinear type considered in this paper includes a useful model prolem of parameter estimation, and there does not seem to exist systematical studies in the literature on its finite element approximation and error analysis, except [14] where a posteriori error estimates were presented. Furthermore superconvergence analysis is an important topic for finite element approximation of PDEs. Due to the lower regularity of the constrained optimal control (normally only H 1 () W 1, ()), only a half order of convergence rate can e expected to gain y using * Received Octoer 30, 2006 / Revised version received Octoer 8, 2007 / Accepted Novemer 14, 2007 /

2 2 D.P. YANG, Y.Z. CHANG AND W.B. LIU the standard recovery techniques, see [26]. Very recently Meyer and Rösch in [19] showed that in fact one order can e gained via using a special projection, which was unique to the linear optimal control prolem they studied. This is a quite interesting result considering the low regularity of the optimal control. It is useful to estalish such a superconvergence property for our model control prolem, which is normally difficult to compute with higher accuracy. In this paper we firstly study a priori error estimates, superconvergence analysis of the control prolem with a projection and interpolator different from [19]. The plan of the paper is as follows. In Sections 2-3, we descrie the control prolem and give the finite element approximation. In Section 4, we derive a priori error estimates. In Section 5, superconvergence analysis is carried out. Then some applications and super-convergent result are discussed in Section Optimal Control Prolem In this section, we formulate the ilinear optimal control prolem. Let e a ounded convex polygon in R 2 with oundary. We adopt the standard notation W m,q () for Soolev spaces on with the norm m,q, and the seminorm m,q,. We set W m,q 0 () {w W m,q () : w = 0} and denote W m,2 () (W m,2 0 ()) y H m () (H0 m ()) with the norm m, and the seminorm m,. We shall take the state space V = H0(), 1 the control space U = L 2 (), and the oservation space Y = L 2 (). Define the control constraint set K U: K = {v U : v 0}. We are interested in the following optimal control prolem of ilinear type : (a) min v K {1 2 y y d 2 0, + α 2 v 2 0,}, () div(a y) + vy = f in, y = 0, (2.1) where α is a positive constant. f L 2 () and A( ) = (a ij ( )) 2 2 [W 1, ()] 2 2 is a symmetric positive definite matrix. This prolem can e interpreted as a model of estimating the true parameter u via the measured data y d using the least square formulation. To consider the finite element approximation of the aove optimal control prolem, we have to give a weak formula for the state equation. Let a(y, w) = (A y) w y, w V, (v, w) = vw v, w U. We assume that there are constants a 0 > 0 and C 0 > 0 such that a 0 y 2 V a(y, y), a(y, w) C 0 y V w V, y, w V. (2.2) Then the standard weak formula for the state equation reads as follows: find y(v) V such that a(y(v), w) + (vy(v), w) = (f, w) w V. (2.3) Introduce a cost function J(v) = 1 2 y(v) y d 2 0, + α 2 v 2 0,. (2.4)

3 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem 3 Then the aove control prolem can e restated as follows, which we shall lael OCP: find u K such that J(u) = min J(v), (2.5) v K associated with a(y(v), w) + (vy(v), w) = (f, w), w V. (2.6) It is well known (see, for example [16]) that the control prolem OCP has at least one solution (y, u), and that if the pair (y, u) is a solution of OCP then there is a co-state p V such that the triplet (y, p, u) satisfies the following optimality conditions, which we shall lael OCP-OPT: (a) a(y, w) + (uy, w) = (f, w), w V, () a(q, p) + (up, q) = (y y d, q), q V, (c) (αu yp, v u) 0, v K. (2.7) The inequality (2.7)(c) is equivalent to u = max{0, 1 yp}. (2.8) α 3. Finite Element Approximation In this section, we study the finite element approximation of the ilinear optimal control prolem OCP. Here we consider only n-simplex elements, as they are among the most widely used ones. Also we consider only conforming finite elements. Let h e a polygonal approximation to. Then we know that h = in this paper. Let T h e a partitioning of h into disjoint regular n-simplices τ, such that h = τ T h τ. Each element has at most one face on h, and τ and τ have either only one common vertex or a whole edge or face if τ and τ T h. Associated with T h is a finite-dimensional suspace S h of C( h ), such that χ τ are polynomials of m-order (m 1) for each χ S h and τ T h. Let V h = S h H0(). 1 It is easy to see that V h V. Let TU h e another partitioning of h into disjoint regular n-simplices τ U, such that h = τ U T τ U h U. Assume that τ U and τ U have either only one common vertex or a whole face or are disjoint if τ U and τ U T U h. Associated with T U h is another finite-dimensional suspace W U h of L 2 (), such that χ τu are polynomials of order m (m 0) for each χ W h and τ U TU h. In this paper, we will only consider the simplest finite element spaces, i.e., m = 1 for V h and m = 0 for U h. Let h τ (h τu ) denote the maximum diameter of the element τ (τ U ) in T h (TU h), let h = max τ T h{h τ}, h U = max τu TU h{h τ U }. Then a possile finite element approximation of OCP, which we shall lael OCP h, is: find u h K h such that J(u h ) = min J(v h ), (3.1) v h K h associated with a(y h, w h ) + (v h y h, w h ) = (f, w h ), w h V h, (3.2) where K h is a closed convex set in U h. This is a finite-dimensional optimization prolem and may e solved y existing mathematical programming methods such as the steepest descent method, conjugate gradient method, trust domain method, and SQP.

4 4 D.P. YANG, Y.Z. CHANG AND W.B. LIU It follows that the control prolem OCP h has at least one solution (y h, u h ) and that if the pair (y h, u h ) V h K h is a solution of OCP h then there is a co-state p h V h such that the triplet (y h, p h, u h ) V h V h K h satisfies the following optimality conditions, which we shall lael OCP OPT h : (a) a(y h, w h ) + (u h y h, w h ) = (f, w h ), w h V h, () a(q h, p h ) + (u h p h, q h ) = (y h y d, q h ), q h V h, (c) (αu h y h p h, v h u h ) 0, v h K h. (3.3) Introduce an averaging operator πh c from U onto Uh such that (πh c v) τ = 1 v, τ TU h τ, (3.4) where τ is the measure of τ. The inequality (3.3)(c) is equivalent to τ u h = max{0, 1 α πc h(y h p h )}. (3.5) In the next section, we will analyze convergence and convergent rate of the finite element approximation. 4. Convergence and a Priori Error Estimates In this section, we are interested in deriving some error estimates for the finite element approximation of the control prolem. To this end, we firstly need to show the strong convergence of the approximation. We will proceed in two steps: first to show the weak convergence and then the strong convergence Weak convergence Theorem 4.1. Let (y h, p h, u h ) e the solutions of (3.3). Then there exists a susequence (y hk, p hk, u hk ) such that (y hk, p hk, u hk ) weakly converge to a solution (y, p, u) of the system (2.7) in H 1 () H 1 () L 2 () as k tends to infinity such that h k tends to zero. Proof. The proof is divided into two steps. In the first step, we prove there exists one weakly convergent susequence. In the second step, we prove the limit of the susequence is a solution of the prolem (2.7). First step of proof. Taking w h = y h and q h = p h in (3.3), we have y h L2 () C f L2 (), p h L2 () C y h y d L2 (). (4.1) These imply that y h and p h are ounded in H0 1 (). Furthermore, we see that u h 2 L 2 () = u h 2 = u h 2 τ T h U τ C h d ( y h p h ) 2 C ( y h 4 ) 1/2 ( p h 4 ) 1/2 τ TU h τ τ TU h τ τ C[ y h 4 ] 1/2 [ p h 4 ] 1/2 C y h 2 H 1 () p h 2 H 1 (), (4.2)

5 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem 5 where we have used the fact that w L4 () C w H1 () for each w H 1 (). This means that u h is uniformly ounded in L 2 (). It follows from the emedding theorems that there exists a susequence (y hk, p hk, u hk ) such that (y hk, p hk ) is weakly convergent in H0 1() H1 0 () and strongly convergent in H s () for 0 s < 1, and u hk is weakly convergent in L 2 (). Let the limit of (y hk, p hk, u hk ) e (y, p, u). Second step of proof. We prove (y, p, u) is a solution of (2.7). Let w Ih V h and q Ih V h e the interpolations of w V and q V. Note that and a(y, w) + (uy, w) =(f, w) + a(y y hk, w) + (u(y y hk ), w) + (u u hk, (y hk y)w) + (u u hk, yw) + a(y hk, w w Ihk ) + (u hk y hk, w w Ihk ) + (f, w Ihk w) (4.3) a(q, p) + (up, q) =(y y d, q) + a(q, p p hk ) + (u(p p hk ), q) + (u u hk, (p hk p)q) + (u u hk, pq) + a(q q Ihk, p hk ) + (u hk p hk, q q Ihk ) + (y y d, q Ihk q). (4.4) By use of weak convergence of (y hk, p hk ) in H 1 () H 1 () and weak convergence of u hk in L 2 (), we have lim [ a(y y h k, w) + a(q, p p hk ) + ((u u hk ), yw) + ((u u hk ), pq) ] = 0. h k 0 By use of strong convergence of (y hk, p hk ) in H s () for 0 s < 1, we otain [ lim (u(y yhk ), w) + (u u hk, (y y hk )w) h k 0 Since we have and + (u(p p hk ), q) + (u u hk, (p p hk )q) ] = 0. lim [ w w I hk H h k 0 1 () + q q Ihk H 1 ()] = 0, a(y hk, w w Ihk ) + (f, w Ihk w) y hk L 2 () (w w Ihk ) L 2 () + f L 2 () w Ihk w L 2 () 0, as h k 0 a(q q Ihk, p hk ) + (y y d, q Ihk q) (q q Ihk ) L 2 () p hk L 2 () + y y d L 2 () q Ihk q L 2 () 0, as h k 0. Noting that (u hk y hk, w w Ihk ) Ch u hk y hk L 2 () w H 1 () and u hk y hk 2 L 2 () = u hk 2 y hk 2 [ y hk 4 ] 1/2 [ u hk 4 ] 1/2 [ y hk 4 ] 1/2 [h d (h d y hk p hk ) 4 ] 1/2 τ TU h τ [ h d/2 [ y hk 4 ] 1/2 [h d (h d/2 ( τ T h U y hk 4 ] 1/2 [ τ T h U τ y hk 4 ) 1/4 ( p hk 4 ) 1/4 ) 4 ] 1/2 τ [ y hk 4 p hk 4 ] 1/2 τ τ

6 6 D.P. YANG, Y.Z. CHANG AND W.B. LIU and using v L4 () C v H1 () for each v H 1 (), we infer Similarly, we have (u hk y hk, w w Ihk ) Ch 1 d/4 0. (u hk p hk, q q Ihk ) Ch 1 d/4 0. Sustituting aove estimates into (4.3) and (4.4) leads to Finally, we see that for each v K, (a) a(y, w) + (uy, w) = (f, w), w V, () a(q, p) + (up, q) = (y y d, q), q V. (αu yp, v u hk ) = (α(u u hk ) (yp y hk p hk ), v u hk ) + (αu hk y hk p hk, (v π c h k v)) + (αu hk y hk p hk, π c h k v u hk ) (α(u u hk ), v u hk ) (yp y hk p hk, v u hk ) + (αu hk y hk p hk, (v π c h k v)). By using the weak convergence of u hk to u in L 2 (), the strong convergence of y hk p hk to yp and the strong convergence of πh c v to v, we infer (αu yp, v u) 0, v K. We thus have proved that (y, p, u) is a solution of (2.7). The proof of Theorem 4.1 is then completed Strong convergence Next we prove a strong convergence result. To this end, introduce two auxiliary operators: for each v U, (y(v), p(v)) V V is the solution of the equations: (a) a(y(v), w) + (vy(v), w) = (f, w), w V, () a(q, p(v)) + (vp(v), q) = (y(v) y d, q), q V, (4.5) and (y h (v), p h (v)) V h V h is the solution of the equations: (a) a(y h (v), w h ) + (vy h (v), w h ) = (f, w h ), w h V h, () a(q h, p h (v)) + (vp h (v), q h ) = (y h (v) y d, q h ), q h V h. (4.6) Lemma 4.1. For each v U, if there exists a constant α 0 > 0 such that α 0 w 2 H 1 () a(w, w) + (vw, w), w V, (4.7) then (4.5) and (4.6) have unique solutions respectively. Moreover, y h (v) and p h (v) strongly converge to y(v) and p(v) respectively. If the domain is convex, then there hold a priori error estimates (a) y(v) y h (v) L 2 () + h (y(v) y h (v)) L 2 () Ch 2 y(v) H 2 (), () p(v) p h (v) L 2 () + h (p(v) p h (v)) L 2 () Ch 2 p(v) H 2 (). (4.8)

7 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem 7 Proof. It is clear that (a) a(y(v) y h (v), w h ) + (u(y(v) y h (v)), w h ) = 0, w h V h, () a(q h, p(v) p h (v)) + (u(p(v) p h (v)), q h ) = (y(v) y h (v), q h ), q h V h, (4.9) such that (a) (y(v) y h (v)) L2 () C y(v) y Ih (v) H1 (), () (p(v) p h (v)) L 2 () C{ p(v) p Ih (v) H 1 () + y(v) y Ih (v) L 2 ()}. (4.10) These mean that (y h (v), p h (v)) strongly converges to (y(v), p(v)). If is convex, then y(v) and p(v) are in H 2 (). Using the standard analysis of finite element methods, we can otain the error estimate (4.8). Lemma 4.2. For v, w U, we have (y(v) y(w)) L2 () + (p(v) p(w)) L2 () C v w H 1 (), (4.11) (y h (v) y h (w)) L 2 () + (p h (v) p h (w)) L 2 () C v w H 1 (). (4.12) Furthermore, for the domain is convex, then y(v) y(w) L () + p(v) p(w) L () C v w L2 (). (4.13) Proof. It is clear that for any w, v V, (a) a(y(v) y(w), w) + (v(y(v) y(w)), w) = (w v, wy(w)), () a(q, p(v) p(w)) + (v(p(v) p(w)), q) = (y(v) y(w), q) + (w v, qp(w)). (4.14) As a result of (4.14), we otain (4.11) and (4.12). On the other hand, we see that (a) div(a (y(v) y(w))) + v(y(v) y(u)) = (w v)y(w), () div(a (p(v) p(w))) + v(p(v) p(w)) = (w v)p(w) + y(v) y(w). (4.15) For a convex domain, we have that y(v) y(w) H 2 () and p(v) p(w) H 2 () such that (a) y(v) y(w) H 2 () v w L 2 (), () p(v) p(w) H2 () v w L2 () + y(v) y(w) L2 (). (4.16) By virtue of the emedding theory, we know that v L () C v H 2 (). So the estimate (4.13) is derived. Theorem 4.2. Let (y h, p h, u h ) e the sequence of solutions of (3.3) weakly converging to a solution (y, p, u) of the system (2.7) in H 1 () H 1 () L 2 (). Then (y h, p h, u h ) strongly converges to (y, p, u) in H 1 () H 1 () L 2 (). Proof. Using the emedding theory, we know that (y h, p h ) is strongly convergent in H s () H s () for 0 s < 1. Furthermore, since hence (y h, p h ) is strongly convergent in L 4 () L 4 (). v L 4 () C v H 1 2 () v H 1 2 (), (4.17)

8 8 D.P. YANG, Y.Z. CHANG AND W.B. LIU Firstly, we prove that u h is strongly convergent in L 2 (). It is clear that Hence we have Similarly to (4.2), we have Hence, u u h = max{0, 1 α yp} max{0, 1 α πc h(y h p h )} max{0, 1 α yp} max{0, 1 α πc h(yp)} + 1 α πc h(yp y h p h ) 1 α yp πc h (yp) + 1 α πc h (yp y hp h ). u u h L 2 () C{ yp π c h(yp) L 2 () + π c h(yp y h p h ) L 2 ()}. π c h (yp y hp h ) 2 L 2 () C[ y h y 4 ] 1/2 [ p h p 4 ] 1/2 0, as h 0. (4.18) u u h L2 () 0, as h 0. Then we prove that (y h, p h ) is strongly convergent in H 1 () H 1 (). It follows from the Lemma 4.1 that (y h (u), p h (u)) strongly converges to (y, p) = (y(u), p(u)) in H 1 () H 1 (). On the other hand, from the Lemma 4.2, we know that (y h y h (u), p h p h (u)) strongly converges to (0, 0) in H 1 () H 1 (), since u u h L2 () tends to zero. We thus have proved that (y h, p h ) strongly converges to (y, p) in H 1 () H 1 () A priori error estimates Now we are in the position of giving a priori error estimates. To this end, we need some facts as follows. Firstly, it is a matter of calculation to show that Similarly, for each v h U h, there holds Furthermore we have (J (v), w) = (αv y(v)p(v), w), w U. (4.19) (J h(v h ), w h ) = (αv h y h (v h )p h (v h ), w h ), w h U h. (4.20) Lemma 4.3. Let u e a solution of (2.7). If there exist constants ǫ > 0 and c 0 > 0 such that for all v L 2 () and w L 2 () satisfying v u L2 () ǫ and w u L2 () ǫ, then the following inequality holds: c 0 v w 2 L 2 () (J (v) J (w), v w). (4.21) The proof of Lemma 4.3 is referred to [5], [6] and [14]. Now we can derive a priori error estimates. Theorem 4.3. (Error Estimate) Let (y h, p h, u h ) e the sequence of solutions of (3.3) strongly converging to a solution (y, p, u) of the system (2.7) in H 1 () H 1 () L 2 (). Then, there holds the following a priori error estimate: u u h L2 () + y y h L2 () + p p h L2 () C(h 2 + h U ). (4.22)

9 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem 9 Proof. Since the domain is convex, hence y H 2 (), p H 2 () and u H 1 () L (). Note that u h strongly converges to u in L 2 (). It follows from the definition of J(u), Lemma 4.3 that we have where c 0 u u h 2 L 2 () (J (u) J (u h ), u u h ) = (αu, u u h ) (αu h, u u h ) (yp y(u h )p(u h ), u u h ) (yp, u u h ) + (y h p h, u h u) + (y h p h, u π c hu) + (αu h, π c h u u) (yp y(u h)p(u h ), u u h ) = (αu h y h p h, π c hu u) + (y h p h y(u h )p(u h ), u h u) = I 1 + I 2 + I 3 + I 4, I 1 = (yp y(u h )p(u h ), π c hu u), I 2 = (y(u h )p(u h ) y h p h, π c h u u), I 3 = (y h p h y(u h )p(u h ), u h u), I 4 = (yp, π c h u u) = (yp πc h (yp), πc h u u). Now let us to estimate term y term. It is easy to show that (4.23) (4.24) I 1 C u πhu c 2 L 2 () + c 0 4 u u h 2 L 2 (), (4.25) I 2 C[ u πhu c 2 L 2 () + p h p(u h ) 2 L 2 () + y h y(u h ) 2 L 2 () ], (4.26) I 3 c 0 4 u u h 2 L 2 () + C[ p h p(u h ) 2 L 2 () + y h y(u h ) 2 L 2 ()], (4.27) I 4 u π c h u 2 L 2 () + c 0 4 yp πc h (yp) 2 L 2 (). (4.28) Thus, y using the estimates aove and Lemma 4.1, we have u u h L 2 () C[ p h p(u h ) L 2 () + y h y(u h ) L 2 () Then, it follows from the Lemma 4.2 that + u π c h u L 2 () + yp π c h (yp) L 2 () ] C(h 2 + h U ). (4.29) y y h L 2 () + p p h L 2 () y(u) y h (u) L 2 () + p(u) p h (u) L 2 () The proof of Theorem 4.3 is completed. + y h (u) y h (u h ) L 2 () + p h (u) p h (u h ) L 2 () Ch 2 + u u h H 1 () C(h 2 + h U ). 5. Superconvergence Analysis In this section, we will provide some super-close results. Let us start from the superconvergence analysis for the control u. Let We decompose h into three parts: + = {x : u(x) > 0}, 0 = {x : u(x) = 0}. h + = {τ U : τ U +, }, h 0 = {τ U : τ U 0 }, h = h \ G h, G h = h + h 0.

10 10 D.P. YANG, Y.Z. CHANG AND W.B. LIU 5.1. Super-close result for u In this section, we assume that + is piecewise smooth and has a finite length such that meas( h ) Ch U. (5.1) We also assume that p, y H 2 () W 1, (), and then u W 1, (). Introduce a function: π h cu, in τ U G h ; ũ = 0, in τ U h and u h = 0; (5.2) α 1 πh c(yp), in τ U h and u h > 0; and we have the following result. Theorem 5.1. Assume the condition (5.1) holds. Let u e the solution of (2.7) and u h e the solution of (3.3) strongly converging to u. Then, To prove Theorem 5.1, we need to prove two lemmas. u h ũ L 2 () Ch 2 U. (5.3) Lemma 5.1. Let u and u h e the solutions of (2.7) and (3.3), respectively. Then, (αũ yp, ũ u h ) 0 (5.4) and (αu h y h p h, u h ũ) 0. (5.5) Proof. It is clear that { α 1 yp yp 0, u = 0 yp < 0; so that πh c(u) = α 1 πh c(yp) in h + and πh c(u) = 0 in h 0. We have πh c(yp)(ũ u h) in h + ; 0 πh c αũ(ũ u h ) = (yp)(ũ u h) in h 0; (yp 0, ũ = 0) πh c (yp)(ũ u h) in h and u h > 0; 0 = πh c(yp)(ũ u h) in h and u h = 0; or equivalently, (αũ, ũ u h ) (πh(yp), c ũ u h ). Noting (αũ yp, ũ u h ) = (αũ πh c(yp), ũ u h), we thus have derived (5.4). Similarly, noting that { α 1 πh c u h = (y hp h ) πh c(y hp h ) 0; 0 πh c(y hp h ) < 0; and ũ 0 if u h = 0, we have { π c h (y h p h )(u h ũ) πh c αu h (u h ũ) = (y hp h ) 0; 0 πh c(yp)(u h ũ) πh c(y hp h ) < 0; (u h = 0)

11 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem 11 or equivalently, (αu h, u h ũ) (π c h (y hp h ), u h ũ). This leads to (5.5). Lemma 5.2. Assume the condition (5.1) holds. Then, y h (ũ) y h (u) H1 () + p h (ũ) p h (u) H1 () Ch 2 U. (5.6) Proof. It follows from the definition of y h (u) and y h (ũ) that for each w h V h, a(y h (u), w h ) + (uy h (u), w h ) = (f, w h ) and a(y h (ũ), w h ) + (ũy h (ũ), w h ) = (f, w h ). Thus a(y h (ũ) y h (u), w h ) + (u(y h (ũ) y h (u)), w h ) =(u ũ, (y h (ũ) y(ũ))w h ) + (u ũ, y(ũ)w h ). (5.7) Taking w h = y h (ũ) y h (u) in (5.7), we have y h (ũ) y h (u) 2 H 1 () C (u ũ, y(ũ)(y h(ũ) y h (u))) C[ (u ũ, y(ũ)(y h (ũ) y h (u)) πh c (y(ũ)(y h(ũ) y h (u)))) G h + (u ũ, y(ũ)(y h (ũ) y h (u))) h ] C[h 2 U u H 1 () y h (ũ) y h (u) H 1 () + (u ũ, y(ũ)(y h (ũ) y h (u))) h ]. (5.8) By using the trace theory of Soolev space v L 1 ( +) C v H 1 (), and noting that (u ũ, y(ũ)(y h (ũ) y h (u))) h (u ũ, y(ũ)(y h (ũ) y h (u)) Q(y(ũ)(y h (ũ) y h (u)))) h + (u ũ, Q(y(ũ)(y h (ũ) y h (u)))) h Ch 2 U u W 1, ()[ y(ũ)(y h (ũ) y h (u)) H1 ( h ) + Q(y(ũ)(y h (ũ) y h (u))) L 1 ( h )] + Ch 2 U u W 1, () y h (ũ) y h (u) H1 (), (5.9) where Q is an orthogonal projection from H 1 ( h ) onto L1 ( + ), we have y h (ũ) y h (u) H 1 () Ch 2 U. (5.10) From the definition of p h (u) and p h (ũ), we have that for each w h V h, a(q h, p h (u)) + (up h (u), q h ) = (y h (u) y d, q h ) and a(q h, p h (ũ)) + (ũp h (ũ), q h ) = (y h (ũ) y d, q h ).

12 12 D.P. YANG, Y.Z. CHANG AND W.B. LIU So that for each q h V h, a(q h, p h (ũ) p h (u)) + (u(p h (ũ) p h (u)), q h ) =(u ũ, (p h (ũ) p(ũ))q h ) + (u ũ, p(ũ)q h ) + (y h (ũ) y h (u), q h ). (5.11) Taking q h = p h (ũ) p h (u) in (5.11), we have p h (ũ) p h (u) 2 H 1 () C[ (u ũ, p(ũ)(p h (ũ) p h (u))) + (y h (ũ) y h (u), p h (ũ) p h (u)) ] C[ (u ũ, p(ũ)(p h (ũ) p h (u)) π c h(p(ũ)(p h (ũ) p h (u)))) G h + (u ũ, p(ũ)(p h (ũ) p h (u))) h + y h (ũ) y h (u) 2 L 2 () ] C[h 2 U u H 1 () p h (ũ) p h (u) H 1 () + y h (ũ) y h (u) 2 L 2 () + (u ũ, p(ũ)(p h (ũ) p h (u))) h ]. (5.12) By using (5.10) and ounding the last term on the right-hand side of (5.12), similarly to (5.9), we also can show that p h (ũ) p h (u) H 1 () Ch 2 U. (5.13) Thus (5.6) is derived. This ends the proof of Lemma 5.2. Now we are in the position of proving Theorem 5.1. Proof of Theorem 5.1. Note that u h and ũ strongly converge to u in L 2 (). It follows from Lemma 4.3 and Lemma 5.1 that where Then it gives c 0 u h ũ 2 L 2 () (J h (u h) J h (ũ), u h ũ) =α(u h ũ, u h ũ) (y h p h y h (ũ)p h (ũ), u h ũ) (y h p h, u h ũ) (yp, u h ũ) (y h p h y h (ũ)p h (ũ), u h ũ) =(y h (ũ)p h (ũ) yp, u h ũ) = R 1 + R 2, (5.14) R 1 = (y h (u)p h (u) yp, u h ũ), R 2 = (y h (ũ)p h (ũ) y h (u)p h (u), u h ũ). and R 1 ((y h (u) y)p, u h ũ) U + (y h (u)(p h (u) p), u h ũ) c 0 4 u h ũ 2 L 2 () + C( y h(u) y 2 L 2 () + p h(u) p 2 L 2 () ) c 0 4 u h ũ 2 L 2 () + Ch4 ( y 2 H 2 () + p 2 H 2 () ) (5.15) Thus, we have R 2 (y h (ũ)(p h (ũ) p h (u)), u h ũ) + ((y h (ũ) y h (u))p h (u), u h ũ) c 0 4 u h ũ 2 L 2 () + C( y h(ũ) y h (u) 2 H 1 () + p h(ũ) p h (u) 2 H 1 ()). (5.16) u h ũ L2 () C( y h (ũ) y h (u) H1 () + p h (ũ) p h (u) H1 ()). (5.17) Applying Lemma 5.2 to (5.17) leads to (5.3). This ends the proof of Theorem 5.1.

13 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem Super-close results for y and p Theorem 5.2. Assume that all the conditions in Theorem 5.1 are valid. Let y h, p h e the solutions of (3.3). Then, Proof. Noting that y h y h (u) H 1 () + p h p h (u) H 1 () C(h 2 + h 2 U). (5.18) where a 0 y h y h (u) 2 H 1 () a(y h y h (u), y h y h (u)) + (u(y h y h (u)), y h y h (u)) = I 1 + I 2 + I 3, (5.19) Now we have I 1 = ((u h ũ)y h, y h y h (u)), I 2 = (ũ u, y h (y h y h (u))) h, I 3 = (ũ u, y h (y h y h (u)) πh(y c h (y h y h (u)))) G h. I 2 (u ũ, y h (y h y h (u)) Q(y h (y h y h (u)))) h + (u ũ, Q(y h (y h y h (u)))) h =: I 21 + I 22, (5.20) (5.21) where I 21 and I 22 satisfy I 21 Ch 2 U u W 1, ( U ) y h y h (u) H 1 ( h ), I 22 Ch 2 U u W 1, () Q(y h (y h y h (u))) L 1 ( +). (5.22) Consequently, I 2 Ch 2 U u W 1, () y h y h (u) H1 (). (5.23) Using standard techniques, it is easy to estimate I 1 + I 3 where we omit the details. It gives Noting that y h y h (u) H1 () C(h 2 U + h2 ). a 0 p h p h (u) 2 H 1 () a(p h p h (u), p h p h (u)) + (u(p h p h (u)), p h p h (u)) = (y h y h (u), p h p h (u)) ((u h u)p h, p h p h (u)), (5.24) and similarly to estimate y h y h (u) H 1 (), we also have The proof of Theorem 5.2 is completed. p h p h (u) H1 () C(h 2 U + h2 ). In the next section we will discuss a superconvergence result for u and some applications. 6. Superconvergence Result for u and Some Applications As a consequence of super-close results in the section 5, we can derive the following optimal a priori error estimates in L 2 -norm.

14 14 D.P. YANG, Y.Z. CHANG AND W.B. LIU 6.1. L 2 -norm and L -norm error estimates Theorem 6.1. Assume that all the conditions in Theorem 5.1 and 5.2 are valid. Then there holds the following a priori error estimate: y y h L2 () + p p h L2 () C(h 2 U + h2 ). (6.1) Proof. It follows from results in Theorem 5.2 that y y h L 2 () + p p h L 2 () y y h (u) L2 () + p p h (u) L2 () + y h (u) y h L2 () + p h (u) p h L2 () C(h 2 U + h 2 ). This completes the proof of Theorem 6.1. Furthermore, we can have the following almost optimal a priori error estimates in L -norm. Theorem 6.2. Assume that all the conditions in Theorems 5.1 and 5.2 are valid. If y, p W 2, (), then there hold the following a priori error estimates: y y h L () + p p h L () C(h 2 U + h2 ) lnh 1 2 (6.2) and u u h L () C(h + h U ). (6.3) Proof. In the 2-d case, it follows from the known result w h L () C lnh 1 2 wh L 2 () w h V h, and results in Theorem 5.2 that we have y y h L () + p p h L () y y h (u) L () + p p h (u) L () + C lnh 1 2 [ (yh (u) y h ) L2 () + (p h (u) p h ) L2 () ] C lnh 1 2 (h 2 + h 2 U). This is (6.2). On the other hand, y using the inverse property of finite element spaces, v h L () Ch 1 U v h L 2 () v h U h and the results in Theorem 5.1, we have u u h L () u ũ L () + ũ u h L () u ũ L () + Ch 1 U ũ u h L 2 () C(h + h U ). This is (6.3). The proof of Theorem 6.2 is then completed.

15 A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Prolem Superconvergence result for u and a posteriori error estimators As a consequence of the optimal L 2 -norm estimation for y h, p h, we have the following superconvergence result for u. Theorem 6.3. Assume that all the conditions in Theorems 5.1 and 5.2 hold. Let There holds a superconvergence error estimate: û h = max(0, 1 α y hp h ). Proof. Since max is Lipschitz continuous, hence By using Theorem 6.1, we derive (6.4). u û h L 2 () C(h 2 U + h 2 ). (6.4) u û h = max{0, 1 α yp} max{0, 1 α y hp h } 1 α yp y hp h C[ p p h + y y h ]. As another application of the results in Theorems 5.1 and 5.2, we derive super-convergence for the states y, p and the optimal control u. To this end, let us adopt the same recovery operator G h v = (R h v x, R h v y ) as in [26], where R h is the recovery operator defined for the recovery of u in [26], v x = v/ x and v y = v/ y. It should e noted that G h is same as the Z-Z gradient recovery (see, e.g., [28, 29]) in our piecewise linear case. Then we have: Theorem 6.4. Suppose that all the conditions in Theorems 5.1 and 5.2 are valid. Moreover, we assume that y, p H 3 (). Then, Proof. First note that G h y h y L2 () + G h p h p L2 () C(h 2 + h 2 U ). (6.5) G h y h y L 2 () G h y h G h y h (u) L 2 () + G h y h (u) y L 2 (). (6.6) It follows from Theorem 5.2 that G h y h G h y h (u) L 2 () C (y h y h (u)) L 2 () C(h 2 + h 2 U). (6.7) It has een proved in [27] (Remark 3.2 and Theorem 3.2) that G h v I = v on τ if v is a quadratic function on the neighorhood of τ U ( τ τ {τ }). Then, it follows from the standard interpolation error estimate technique (see, e.g., [7]) that G h y h (u) y L2 () G h y h (u) G h y I L2 () + G h y I y L2 () Ch 2 ( y 3, + y 2, ). (6.8) Here y I is the linear interpolation of y. Therefore, it follows from (6.6)-(6.8) that Similarly, it can e proved that G h y h y L2 () C(h 2 + h 2 U ). (6.9) G h p h p L 2 () C(h 2 + h 2 U). (6.10)

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