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A characteristic-mixed finite element method for time-dependent convection–diffusion optimal control problem
Applied Mathematics and Computation 218 (2011) 3430–3440
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A characteristic-mixed finite element method for time-dependent convection–diffusion optimal control problem Hongfei Fu a,⇑, Hongxing Rui b a b
Department of Computational and Applied Mathematics, China University of Petroleum, Qingdao 266555, China School of Mathematics, Shandong University, Jinan 250100, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection–diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart–Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Optimal control Characteristic-mixed finite element Convection–diffusion equations A priori error estimates
1. Introduction Optimal control problems governed by convection–diffusion equations arise in many scientific and engineering applications, such as atmospheric pollution control problems [1,2]. Efficient numerical methods are essential to successful applications of such optimal control problems. To the best of our knowledge, there are only a few published results on optimal control problems governed by steady-state convection–diffusion equations; see [3] of standard finite element discretizations with stabilization based on local projection method [4] of symmetric stabilization method [5] of SUPG method [6] of edge-stabilization method, and [7] of the application of RT mixed DG scheme. For the approximation of constrained optimal control problems governed by time-dependent convection–diffusion equations, it is much more complicated, see [8,9]. Systematic introductions of the finite element method for PDEs and optimal control problems can be found in, for example [10–14]. In many optimal control problems, the objective functional contains not only the state variable, but also its gradient. For example, in the flow control problem, the gradient stands for Darcy velocity and it is an important physics variable, or, in the temperature control problem, large temperature gradients during cooling or heating may lead to its destruction. Therefore, in these cases people pay their special attention on the gradient of the primal state variable, see Refs. [15–17]. In this paper, we are interested in the following convex optimal control problem with pointwise control constraints:
min
u2U ad
1 2
Z 0
T
Z
ðy yd Þ2 þ
X
Z X
ðp pd Þ2 þ
Z
u2 dt
XU
⇑ Corresponding author. E-mail addresses: [email protected] (H. Fu), [email protected] (H. Rui). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.087
ð1:1Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3431
subject to
8 yt þ divðby þ pÞ ¼ f þ Bu; > > > < p ¼ Ary; > y ¼ 0; > > : yð0Þ ¼ y0 ;
in X ð0; T; in X ð0; T; on @ X ð0; T;
ð1:2Þ
in X;
and
n1 6 uðx; tÞ 6 n2 ;
a:e: in XU ð0; T;
ð1:3Þ
@y ; @t
where yt ¼ b ¼ bðx; tÞ denotes a velocity field in the flow control, A = A(x) is a diffusion coefficient, f = f(x, t) accounts for external sources and sinks, and y0 = y0(x) is a prescribed initial data. The details will be specified in the next section. In the optimal control problem (1.1)–(1.3), the state equation is usually convection-dominated. It is well known that the standard finite element discretization applied to the convection–diffusion equation (1.2) leads to strong oscillation when the diffusion is small. The methods of characteristic finite element [18–20] are proved effectively to reduce the oscillatory behavior. These methods combine the time derivative term and the convection term of divergence-free form in the governing equations to carry out the temporal discretization in a Lagrange coordinate. They symmetrize the governing equation and stabilize the numerical approximations. Besides, they generate accurate numerical solutions and significantly reduce the numerical diffusion and grid-orientation effect present in upwind methods, even if large time steps and coarse spatial meshes are used. Mixed finite element method [21–23] has been proved to be an effective numerical method for solving fluid problems and optimal control problems. It has an advantage of approximating the unknown scaler variable and its diffusive flux simultaneously. Besides, the method can approximate the unknown variable and its flux to a same order of accuracy. Recently, there are some research articles on this method for optimal control problems, see [15,16], for example. In this work, we combine the characteristic finite element method with the mixed finite element method, and present a characteristicmixed finite element scheme to the quadratic optimal control problems governed by convection–diffusion equations. The characteristic approximation is applied to handle the convection term, and the lowest-order Raviart–Thomas mixed finite element spatial approximation is adopted to deal with the diffusion term. We obtain a priori error estimates for both the state and control approximations. The rest of the paper is organized as follows: in Section 2, we first give a precise description of the optimal control problem and then derive the continuous optimality conditions. In Section 3, we construct a characteristic-mixed finite element approximation scheme for the optimal control problem. In Section 4, main error estimates are derived for the control problem with obstacle constraints. Section 5 contains concluding remarks. The approach developed in this work is applicable to more general convex control problems. Let X and XU be bounded open sets in R2 , with Lipschitz boundaries oX and oXU. Just for simplicity of presentation, we assume that X and XU are convex polygon. In this paper, we employ the usual notion for Lebesgue and Sobolev spaces, see [10] for details. In addition C and d denote a generic constant and small positive number which are independent of the discrete parameters and may have different values in different circumstances, respectively. 2. Optimal control problem and optimality conditions In this section we briefly discuss the precise formulation of the optimization problem under consideration. Furthermore, we derive the optimality conditions accordingly. Let
V ¼ Hðdiv; XÞ ¼ fv 2 L2 ðXÞ2 : divv 2 L2 ðXÞg endowed with the norm given by
kv kV ¼ kv kHðdiv;XÞ ¼ ðkv k2 þ kdivv k2 Þ1=2 and
W ¼ L2 ðXÞ;
U ¼ L2 ðXU Þ:
Partition (0, T] into 0 ¼ t 0 < t1 < t2 < < t NT ¼ T, with time steps ki ¼ ti t i1 ; i ¼ 1; 2; . . . ; N T ; k ¼ max16i6NT ki . Let fi = f(x, ti). We define, for 1 6 p < 1, the discrete time-dependent norms
kf klp ð0;T;XÞ ¼
NT X
!1p ki kf i kpX
i¼1
and the standard modification for p = 1. Let
n o p l ð0; T; XÞ :¼ f : kf klp ð0;T;XÞÞ < 1 ;
1 6 p 6 1:
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To set up a weak formulation of the state equation, we shall take the state space V ¼ L2 ð0; T; VÞ; W ¼ L2 ð0; T; WÞ, the control space U ¼ L2 ð0; T; UÞ, and the observation space H ¼ L2 ð0; T; HÞ with H = L2(X) to fix the idea. Let B be a linear continuous operator from U to H. To formulate the optimal control problem, we introduce the admissible set Uad collecting the inequality constraints (1.3) as
~ 2 U : n1 6 u ~ ðx; tÞ 6 n2 ; a:e: in XU ð0; Tg; U ad ¼ fu
ð2:1Þ
where the bounds n1 ; n2 2 R fulfill n1 < n2. In the optimal control problem (1.1)-(1.2), yd 2 H1(0, T; L2(X)), pd 2 H1(0, T; L2(X)2), f 2 L2(0, T; L2(X)), y0 2 L2(X), and ÞÞ22 such that there are positive constants c⁄ and c⁄ satisfying AðxÞ ¼ ðai;j ðxÞÞ22 2 ðC 1 ðX
c jXj2 6 X T AX 6 c jXj2 ;
8X 2 R2 :
Þ2 Þ and is divergence-free, i.e., The velocity field vector b = (b1(x, t), b2(x, t))T lies in the function space L1 ð0; T; W 1;1 ðX
divb ¼ 0;
8x 2 X;
t 2 ð0; T:
ð2:2Þ
To avoid technical boundary difficulties associate with the method of characteristics, we assume that X is a rectangle and the state equation is X-periodic, i.e., we assume all functions in Eq. (1.2) are spatially X-periodic; see [18,24] for example. Other assumption can be found in [19] or [20]. Let
/ðx; tÞ :¼ ðjbj2 þ 1Þ1=2 ¼ ðb1 ðx; tÞ2 þ b2 ðx; tÞ2 þ 1Þ1=2 and let the characteristic direction associated with the operator yt + b ry be denoted by s = s(x, t), where
/ys :¼ /
@y ¼ yt þ b ry: @s
ð2:3Þ
Then the first equation in (1.2) can be written in the following form:
/ys þ divp ¼ f þ Bu:
ð2:4Þ
For the given control set Uad, the above-mentioned convex optimal control problem (1.1)–(1.3) can be restated as follows: (CCP)
min
u2U ad
1 2
Z 0
T
Z
ðy yd Þ2 þ
Z
X
ðp pd Þ2 þ
Z
X
u2 dt;
ð2:5Þ
XU
where ðp; y; uÞ 2 V W U such that
8 > < ð/ys ; wÞ þ ðdivp; wÞ ¼ ðf þ Bu; wÞ; 8w 2 W; ðap; v Þ ðy; divv Þ ¼ 0; 8v 2 V; t 2 ð0; T; > : yð0Þ ¼ y0 ;
t 2 ð0; T; ð2:6Þ
where a = A1(x) and the inner product in L2(X) or L2(X)2 is indicated by (, ). It is well known (see, e.g. [11]) that the control problem (CCP) has a unique solution (p, y, u), and that a triplet (p, y, u) is the solution of (CCP) if and only if there is a co-state ðq; zÞ 2 V W such that (p, y, q, z, u) satisfies the following optimality conditions: (CCP-OPT)
8 ð/ys ; wÞ þ ðdivp; wÞ ¼ ðf þ Bu; wÞ; 8w 2 W; > > > < ðap; v Þ ðy; divv Þ ¼ 0; 8v 2 V; t 2 ð0; T; > > > : yð0Þ ¼ y0 ;
t 2 ð0; T;
8 ð/zs ; wÞ þ ðdivq; wÞ ¼ ðy yd ; wÞ; 8w 2 W; t 2 ½0; TÞ; > > < ðaq; v Þ ðz; divv Þ ¼ ðp pd ; v Þ; 8v 2 V; t 2 ½0; TÞ; > > : zðTÞ ¼ 0; Z 0
T
~ uÞU dt P 0; ðu þ B z; u
8u~ 2 U ad U ¼ L2 ð0; T; UÞ;
where B⁄ is the adjoint operator of B, and (, )U is the inner product of U. Remark 2.1. In fact, the co-state solution z of Eq. (2.8) satisfies the following equation
zt divðbz þ Aðrz þ p pd ÞÞ ¼ y yd :
ð2:7Þ
ð2:8Þ
ð2:9Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
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Then it follows from Propositions 2.1–2.3 of paper [25] and Lemmas 5.7–5.9 of paper [19] that
y; z 2 L2 ð0; T; H2 ðXÞÞ \ H1 ð0; T; L2 ðXÞÞ,!Cð½0; T; H10 ðXÞÞ and
u 2 L2 ð0; T; W 1;p ðXÞÞ \ H1 ð0; T; L2 ðXÞÞ \ L1 ðð0; TÞ XÞ for any p < 1. Remark 2.2 [11]. Inequality (2.9) is equivalent to the following:
u þ B z > 0; u ¼ n1 ;
u þ B z < 0; u ¼ n2 ;
u þ B z ¼ 0; n1 < u < n2 :
ð2:10Þ
Using a pointwise projection on the admissible set Uad,
PUad : U ! U ad ;
P Uad ðf Þðx; tÞ ¼ maxðn1 ; minðn2 ; f ðx; tÞÞÞ;
the optimality condition (2.9) can also be expressed as
u ¼ PUad ðB zÞ:
ð2:11Þ
3. Characteristic-mixed finite element approximation In this section, we consider the method of characteristic-mixed finite element for the approximation of convex control problem (CCP). The approximation scheme is also applicable to the control problem with more general convex objective functionals. Here we only consider the n-simplex Lagrange elements which are most widely used in practical computations. Let G(x⁄, t⁄; t) be an approximate characteristic curve passing through point x⁄ at time t⁄, which is defined by
Gðx ; t ; tÞ :¼ x bðx ; t Þðt tÞ:
ð3:1Þ
We denote by x ¼ Gðx; t i ; t i1 Þ be the foot at time ti1 of the characteristic curve with head x at time ti, and x represents the head of the characteristic curve with foot x at time ti1, namely,
x ¼ Gðx; ti ; t i1 Þ: Let f ðxÞ ¼ f ð xÞ. We approximate
/i
ð3:2Þ @yi @s
ðxÞ ¼ @y ðx; ti Þ by a backward difference quotient in the s-direction, i.e., @s
i1 @yi yi y : ’ @s ki
ð3:3Þ
We remark that, since the problem is X-periodic, G is a differential homeomorphism of X onto itself for sufficiently small ki i1 is always defined; the tangent to the characteristics (i.e., the s-segment) cannot cross a boundary to an (see [24]). Thus, y undefined location. ¼ [ hs Let T h and T hU be quasi-uniform regular triangulations of X and XU, respectively, such that X s2T ; XU ¼ [sU 2T hU sU . Let h ¼ maxs2T h hs ; hU ¼ maxsU 2T h hsU , where hs and hsU denote the diameters of the element s and sU, respectively. U Let Vh Wh V W denote the lowest order Raviart–Thomas space [23] associated with the triangulations T h . Let h 2 U U = L (XU) consist of piecewise constant functions on T hU of XU due to the limited regularity of the optimal control. It is clear that Vh V, Wh W and Uh U. Let U had be a closed convex set in Uh, that is,
~ h 2 U h : n1 6 u ~ h 6 n2 ; a:e: in XU g: U had ¼ fu
ð3:4Þ
Then the corresponding fully discrete characteristic-mixed finite element approximation of (CCP), which will be labeled as (CCP)hk, is defined as follows:
min
uih 2U had
Z Z Z NT i 2 i 2 1X ; ki yh yd ðx; t i Þ þ ðpih pd ðx; t i ÞÞ2 þ uh 2 i¼1 X X XU
ð3:5Þ
where pih ; yih ; uih 2 V h W h U h ; i ¼ 1; 2; . . . ; N T satisfies
8 i i1 yh y i > i h > ; w ; w Þ þ Bu ; w þ divp ; ¼ f ðx; t i h h h > h h k < i i i h aph ; v h yh ; divv h ¼ 0; 8v h 2 V ; > > > : 0 yh ¼ yh0
8wh 2 W h ; ð3:6Þ
and yh0 2 W h is an approximation of y0 which will be specified later on. hk again a unique solution pih ; yih ; uih , and that a triplet i It is well known (see, e.g.hk[11]) that the control problem (CCP) i1 has h h i i i1 i i1 i ph ; yh ; uh is the solution of (CCP) if and only if there is a co-state qh ; zh 2 V W such that ph ; yih ; qi1 h ; zh ; uh satisfies the following optimality conditions: (CCP-OPT)hk
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8 i i1 yh y i > i h > ; w ; w Þ ¼ f ðx; t Þ þ Bu ; w þ ð divp ; i h h h > h h k < i i i h aph ; v h yh ; divv h ¼ 0; 8v h 2 V ; > > > : 0 yh ¼ yh0 ;
8wh 2 W h ; ð3:7Þ
8 zi1 zih J i > > h > ; w þ divqhi1 ; wh ¼ yih yid ; wh ; 8wh 2 W h ; h > ki < i1 aqh ; v h zhi1 ; divv h ¼ pih pid ; v h ; 8v h 2 V h ; > > > > : NT zh ¼ 0;
ð3:8Þ
i i ~ uh þ B zi1 h ; uh uh U P 0;
ð3:9Þ
8u~ h 2 U had ;
1 denotes the determinant of the Jacobian transformation from G to x. Similar to where zih ¼ zih ð xÞ; and J i ¼ det @Gðx;t@xi ;ti1 Þ 2 the discussion in [8], we know that for sufficiently small ki, the Jacobian J i ¼ 1 þ O ki . 4. A priori error estimates In this section, we are able to derive some a priori error estimates for the optimal control problem (CCP-OPT) and its characteristic-mixed finite element approximation (CCP-OPT)hk with the control approximated by piecewise constant elements. To obtain a priori error estimates for the proposed scheme (3.7)–(3.9), we introduce intermediate variables i h h 2 i1 ph ðuÞ; yih ðuÞ; qi1 h ðuÞ; zh ðuÞ 2 ðV W Þ ; i ¼ 1; 2; . . . ; N T ; associate with the control u as follows:
8 i i1 ðuÞ y ðuÞy > > ; wh þ ðdivpih ðuÞ; wh Þ ¼ ðf i þ Bui ; wh Þ; > h ki h < i aph ðuÞ; v h yih ðuÞ; divv h ¼ 0; 8v h 2 V h ; > > > : 0 yh ðuÞ ¼ yh0 ;
8wh 2 W h ;
8 i zi1 ðuÞzi ðuÞJ i > i > > h k h ; wh þ divqi1 8wh 2 W h ; > h ðuÞ; wh ¼ yh ðuÞ yd ; wh ; i < i1 i i aqh ðuÞ; v h zi1 8v h 2 V h ; > h ðuÞ; divv h ¼ ph ðuÞ pd ; v h ; > > > : NT zh ðuÞ ¼ 0:
ð4:1Þ
ð4:2Þ
For simplicity of illustration, set
hj ¼ jh jh ðuÞ; fk ¼ kh kh ðuÞ; h0y
gj ¼ j jh ðuÞ; for j ¼ p; y; nk ¼ k kh ðuÞ; nNz T
for k ¼ q; z:
fNz T
It is clear that ¼ 0 and ¼ ¼ 0. Before deriving the main error estimates for the characteristic-mixed finite element approximation of optimal control problem governed by convection–diffusion equation, some lemmas are prepared. Þ2 Þ. Then for i = 0, . . . , NT and f 2 L2(X) it holds that Lemma 4.1 [20]. Suppose that b 2 L1 ð0; T; W 1;1 ðX
kf i k2 6 ð1 þ Cki Þkf i k2 : Lemma 4.2. Let (ph, yh, qh, zh) and (ph(u), yh(u), qh(u), zh(u)) be the solutions of (3.7), (3.8) and (4.1), (4.2), respectively. Then the following estimates hold
kph ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kyh yh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cku uh kl2 ð0;T;L2 ðXU ÞÞ ;
ð4:3Þ
kqh qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kzh zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cku uh kl2 ð0;T;L2 ðXU ÞÞ :
ð4:4Þ
Proof. We first prove Eq. (4.3). It follows from Eqs. (3.7) and (4.1) that
8 i i1 > < hy khy ; wh þ divhip ; wh ¼ B uih ui ; wh ; i > : ahip ; v h hiy ; divv h ¼ 0; 8v h 2 V h :
8wh 2 W h ;
ð4:5Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3435
By selecting wh ¼ hiy ; v h ¼ hip as test functions and adding the two equations in (4.5), we obtain
hiy hyi1 i ; hy ki
!
þ ahip ; hip ¼ B uih ui ; hiy : 2
It follows from the inequality aða bÞ P 12 ða2 b Þ and Lemma 4.1 that
2 2 1 i 2 i1 2 2 2 hy hy þ a hip 6 C ui uih þ C khiy k þ khyi1 k : 2ki
where a⁄ = 1/c⁄. Multiplying both sides of the above inequality by 2ki and summing over i from 1 to N(6NT). Noting h0y ¼ 0, we obtain that
2 N 2 N 2 N X X X 2 N 2 ki hip 6 C ki hiy þ hyi1 þ C ki ui uih : hy þ 2a i¼1
i¼1
i¼1
The application of the discrete Gronwall’s lemma leads to the first assertion of Lemma 4.2. Similarly, by subtracting Eq. (4.2) from Eq. (3.7) we have
8 i1 i i i > < fz kfz J ; wh þ divfi1 8wh 2 W h ; q ; wh ¼ hy ; wh ; i > : afqi1 ; v h fzi1 ; divv h ¼ hip ; v h ; 8v h 2 V h : Choosing wh ¼ fi1 z ;
ð4:6Þ
v h ¼ fqi1 in Eq. (4.6) and adding the two equations we also obtain
fzi1 fiz i1 ; fz ki
!
þ a
fqi1 ; fi1 q
! fi fi J i i1 i i1 i i1 z z : ; fz ¼ hy ; f z hp ; fq ki
2 Recalling J i ¼ 1 þ O ki , so we infer according to Lemma 4.1 that
! 2 fi fi J i 1 i1 2 i 2 i1 2 i i1 i i1 i z i1 z ; fz fz fz þ a fq 6 hy ; fz þ hp ; fq þ C fz þ 2ki ki 2 2 2 2 2 6 C fiz þ fzi1 þ C hiy þ C hip þ d fqi1 ;
where d is a small positive number. Similarly, multiplying both sides of the above inequality by 2ki and summing over i from NT to M + 1(P1), we obtain by taking d = a⁄/2 that
NT NT NT X X X M 2 i1 2 i 2 i1 2 i 2 i 2 f þ a þ C k f 6 C k f þ f k h þ hp ; i i i z z z y q i¼Mþ1
since
fNz T
i¼Mþ1
i¼Mþ1
¼ 0. Thus it follows from the discrete Gronwall’s lemma that
kfz kl1 ð0;T;L2 ðXÞÞ þ kfq kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðkhy kl2 ð0;T;L2 ðXÞÞ þ khp kl2 ð0;T;L2 ðXÞ2 Þ Þ:
ð4:7Þ
Therefore Eq. (4.4) is proved from (4.7) and (4.3). h Lemma 4.3. Let (p, y, q, z, u) and (ph, yh, qh, zh, uh) be the solutions of (CCP-OPT) and (CCP-OPT)hk, respectively. Let Ph be the standard L2(XU)-orthogonal projection from U to Uh such that
~ Ph u ~; u ~ h ÞU ¼ 0; ðu
8u~ h 2 U h :
ð4:8Þ
Assume that u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU)), z 2 L2(0, T; H1(X)) \ H1(0, T; L2(X)). Then
ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Cðk þ hU þ kz zh ðuÞkl2 ð0;T;L2 ðXÞÞ Þ;
ð4:9Þ
where zh(u) is defined in Eq. (4.2). Proof. It follows from the definition (4.8) that Ph ui 2 U had , and we can prove (see, e.g.[10]) that for u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU))
ku Ph ukl2 ð0;T;L2 ðXU ÞÞ 6 ChU kukl2 ð0;T;H1 ðXU ÞÞ :
ð4:10Þ
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H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
Moreover, if u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU)) and z 2 L2(0, T; H1(X)), we have NT X
ki ðui þ B zi ; Ph ui ui ÞU ¼
i¼1
NT X
ki ððui þ B zi Þ Ph ðui þ B zi Þ; Ph ui ui Þ
i¼1
6 ku þ B z Ph ðu þ B zÞkl2 ð0;T;L2 ðXU ÞÞ kPh u ukl2 ð0;T;L2 ðXU ÞÞ 2 6 ChU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ :
ð4:11Þ
Recalling the inequalities (2.9) and (3.9) that
ku uh k2l2 ð0;T;L2 ðXU ÞÞ ¼
NT NT X X ki ui ; ui uih U ki uih ; ui uih U i¼1
¼
NT X
i¼1 NT NT X X i i i i ki ui þ B zi ; ui uih U þ ki uih þ B zi1 ki uih þ B zi1 h ; uh Ph u U þ h ; Ph u u U
i¼1
i¼1
i¼1
NT X i i i þ ki B zi1 h z ; u uh U i¼1
60þ0þ
NT NT NT X X X ki uih ui ; Ph ui ui U þ ki ðui þ B zi ; Ph ui ui ÞU þ ki ðB ðzi1 zi Þ; Ph ui ui ÞU i¼1
þ
NT X
i¼1
i¼1
NT X i i i i1 ki ðB ðzi1 zi1 ki B zi1 ; u Ph ui U h ðuÞÞ; u Ph u ÞU þ h ðuÞ zh
i¼1
i¼1
NT NT NT X X X i i i1 i i1 þ ki B zi1 ki B zi1 ki B ðzi1 zi Þ; ui uih U : ; u uih U þ h zh ðuÞ ; u uh U þ h ðuÞ z i¼1
i¼1
i¼1
ð4:12Þ First, we bound the sixth non-zero term on the right-hand side of Eq. (4.12). By making use of Eq. (4.5) with i1 wh ¼ fi1 and Eq. (4.6) with wh ¼ hiy ; v h ¼ hip we have z ; v h ¼ fq NT NT X X i i i i ki B zhi1 zi1 ki zhi1 zi1 h ðuÞ ; u uh U ¼ h ðuÞ; Bðu uh Þ i¼1
i¼1
¼
¼
i¼1
hiy hi1 y ; fi1 z ki
NT X
fi1 z
NT X
i¼1
ki ki
!
fiz J i i ; hy ki
!
NT X ki divhip ; fzi1 i¼1
NT NT X X ki divfqi1 ; hiy ki hip ; hip i¼1
i¼1
NT NT X X ki hiy ; hiy ki hip ; hip ¼ khy k2l2 ð0;T;L2 ðXÞÞ khp k2l2 ð0;T;L2 ðXÞ2 Þ 6 0: ¼ i¼1
ð4:13Þ
i¼1
Thus we obtain from Eqs. (4.10)–(4.13), Lemma 4.2 and Cauchy–Schwarz inequality that
2 2 ku uh k2l2 ð0;T;L2 ðXU ÞÞ 6 CðdÞhU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ þ CðdÞk kzt k2L2 ð0;T;L2 ðXÞÞ þ CðdÞkz zh ðuÞk2l2 ð0;T;L2 ðXÞÞ 2
þ Cdkzh ðuÞ zh k2l2 ð0;T;L2 ðXÞÞ þ Cdku uh k2l2 ð0;T;L2 ðXU ÞÞ 6 Ck kzt k2L2 ð0;T;L2 ðXÞÞ 2 þ ChU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ þ Ckz zh ðuÞk2l2 ð0;T;L2 ðXÞÞ þ Cdku uh k2l2 ð0;T;L2 ðXU ÞÞ :
ð4:14Þ
We finish the proof by choosing Cd = 1/2 in Eq. (4.14). h Lemma 4.4. Let (p, y, q, z) and (ph(u), yh(u), qh(u), zh(u)) be the solutions of (2.7), (2.8) and (4.1), (4.2), respectively. Assume that y; z 2 L1 ð0; T; H10 ðXÞ \ H2 ðXÞÞ \ H1 ð0; T; H1 ðXÞÞ \ H2 ð0; T; L2 ðXÞÞ and p, q 2 L2(0, T; H1(X)2) \ H1(0, T; L2(X)2). Then the following estimates hold
kp ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ ky yh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hÞ;
ð4:15Þ
kq qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kz zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hÞ:
ð4:16Þ
The proof of this lemma needs the following intermediate results which are summarized in Lemma 4.5.
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3437
Lemma 4.5. For each t 2 (0, T], we introduce the mixed finite element projection ðRh y; Rh pÞ of (y, p) as follows:
(
8wh 2 W h ;
ðdivðp Rh pÞ; wh Þ ¼ 0;
ðaðp Rh pÞ; v h Þ ðy Rh y; divv h Þ ¼ 0;
ð4:17Þ
8v h 2 V h :
Then there exists a unique pair ðRh y; Rh pÞ such that
kp Rh pkl2 ð0;T;L2 ðXÞ2 Þ 6 Chkpkl2 ð0;T;H1 ðXÞ2 Þ ; 2
ky Rh ykl1 ð0;T;L2 ðXÞÞ þ hky Rh ykl1 ð0;T;H1 ðXÞÞ 6 Ch kykl1 ð0;T;H2 ðXÞÞ ; @ ðy Rh yÞ 6 ChkykH1 ð0;T;H1 ðXÞÞ : 2 @t L ð0;T;L2 ðXÞÞ
ð4:18Þ
Proof. For the proof of the unique solution of Eq. (4.17) and the approximate results (4.18), we refer the reader to [10,26], for example. h The proof of Lemma 4.4 We first proof Eq. (4.15). Decomposing the error gj = j jh(u) as gj ¼ ðj Rh jÞþ ðRh j jh ðuÞÞ ¼ lj þ mj , with j = y, p. Since the estimate for lj is known, we need only to derive an estimate for mj. It follows from Eqs. (4.1), (4.17) and (2.6), we obtain an error equation on mj:
8 i i1 i i1 < my my ; wh þ divm i ; wh ¼ ðri ; wh Þ ly l y ; wh ; p
ki
:
ki
ðam ip ; v h Þ ðmiy ; divv h Þ ¼ 0; i
i
ki
ð4:19Þ
8v h 2 V h ;
i1
where ri ¼ /i @y y kyi . @s For an L2-norm error estimate, we choose wh ¼ miy ; together to obtain
miy myi1
8wh 2 W h ;
! i y
;m
i p;
þ am m
i p
¼
i
i y
r ;m
v h ¼ mip as test functions in system (4.19) and add the two equations
liy lyi1 ki
! i y
;m
yi1 li1 l y ki
! i y
;m
:
ð4:20Þ
Multiplying both sides of (4.20) by ki and summing on 1 6 i 6 N. Similar to the estimates in Lemma 4.2, we derive
N 2 N 2 N N 2 N 2 X X X X X 1 N 2 2 1 1 yi1 þC ki m ip 6 C ki miy þ mi1 ki kri k2 þ C ki liy lyi1 þ C ki lyi1 l my þ a ; y 2 i¼1 i¼1 i¼1 i¼1 i¼1 ð4:21Þ here we have omitted the term Noting that
m0y by choosing the initial approximate value yh0 ¼ Rh y0 .
kri k2 6 Cki kyss k2L2 ðti1 ;ti ;L2 ðXÞÞ ; 2 @ ly 2 i 2 6 Cki h kyk2H1 ðti1 ;ti ;H1 ðXÞÞ ; ly lyi1 6 ki @t 2 L ðt i1 ;t i ;L2 ðXÞÞ i1 i1 2 2 2 2 2 2 ly ly 6 Cki krli1 y k 6 Cki h kykl1 ð0;T;H2 ðXÞÞ :
ð4:22Þ
Then we have
2 N N 2 X X N 2 2 2 2 þ Ck kyss k2L2 ð0;T;L2 ðXÞÞ þ Ch kyk2H1 ð0;T;H1 ðXÞÞ þ Ch kyk2l1 ð0;T;H2 ðXÞÞ : ki km ip k2 6 C ki miy þ mi1 my þ 2a y i¼1
i¼1
ð4:23Þ Applying the discrete Gronwall’s lemma to Eq. (4.23) yields that
kmy kl1 ð0;T;L2 ðXÞÞ þ km p kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðk þ hÞ:
ð4:24Þ
Combining (4.24) with the well-known estimates (4.18) for ly and lp, we finish the proof of (4.15). For the proof of Eq. (4.16), we write nk ¼ k kh ðuÞ ¼ ðk Rh kÞ þ ðRh k kh ðuÞÞ ¼ qk þ pk for k = z and q. Here ðRh z; Rh qÞ 2 W h V h are defined the same as ðRh y; Rh pÞ in Eq. (4.17), and of course qk (k = q, z) is bounded in the desired way. In order to estimate pk, we write
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H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
8 i1 i pz pz > ; wh þ divpqi1 ; wh > ki > > > i i1 i i i i i i > > q q q q J p p J > > ¼ ðvi1 ; wh Þ þ z ki z ; wh z ki z ; wh z ki z ; wh > > > i i i > < þ z kzi J ; wh þ yi yih ðuÞ; wh ðyi yi1 ; wh Þ þ yid yi1 d ; wh ; > > > 8wh 2 W h ; > > > > > i1 i1 > > > apq ; v h pz ; divv h > > : ¼ ðpi pih ðuÞ; v h Þ þ ðpi pi1 ; v h Þ pid pdi1 ; v h ; 8v h 2 V h ; i1
i1
ð4:25Þ
i
where vi1 ¼ /i1 @z@s z kiz . i1 We choose wh ¼ pi1 and add the two equations in (4.25) to obtain z ; v h ¼ pq
i pi1 p z z ki
;p
i1 z
i1 q ;
þ ap
i1 q
p
¼
i1
v ;p
i1 z
þ
p iz p iz Ji
qiz qzi1
i1 z
;p
ki !
qiz q iz ki !
i1 z
;p
q iz q iz Ji ki
! ;p
i1 z
zi zi J i ; pzi1 þ yi yih ðuÞ; pi1 z ki ki i yi yi1 ; pi1 þ p pi1 ; pi1 pi pih ðuÞ; pi1 q z q i1 i i1 i1 þ ðyid yi1 : d ; pz Þ pd pd ; pq
; pi1 z
þ
ð4:26Þ
Multiplying both sides of Eq. (4.26) by ki and summing on i from NT to M + 1. Similar to the estimates in Lemma 4.2, we also obtain NT NT NT NT 2 X X X X 2 2 1 1 2 pM 2 þ a þC ki kpi1 ki piz þ pi1 ki kvi1 k2 þ C ki qiz qzi1 z q k 6 C z 2 i¼Mþ1 i¼Mþ1 i¼Mþ1 i¼Mþ1
þC
NT X i¼Mþ1
þC
NT X
NT NT 2 2 X X 1 1 i 1 i i i i 2 þ C i Ji ki qiz q k q q þ C ki p i z z z z pz J i¼Mþ1
i¼Mþ1
1
ki kzi zi J i k2 þ Cky yh ðuÞk2l1 ð0;T;L2 ðXÞÞ þ Ckp ph ðuÞk2l2 ð0;T;L2 ðXÞ2 Þ
i¼Mþ1
þC
X
NT X
v ¼y;yd
i¼Mþ1
ki kv i v i1 k2 þ C
X
NT X
v ¼p;pd
i¼Mþ1
ki kv i v i1 k2 þ d
NT X i¼Mþ1
2 ki pqi1 :
ð4:27Þ
Following the estimates in Eqs. (4.22), we have at once
kvi1 k2 6 Cki kzss k2L2 ðti1 ;ti ;L2 ðXÞÞ ; @ qz 2 2 i i1 2 kqz qz k 6 ki 6 Cki h kzk2H1 ðti1 ;ti ;H1 ðXÞÞ ; @t L2 ðti1 ;ti ;L2 ðXÞÞ i q q i 2 6 Ck2 rqi 2 6 Ck2 h2 kzk21 i i z z z l ð0;T;H2 ðXÞÞ ; kv i v i1 k2 6 ki kv t k2L2 ðti1 ;ti ;L2 ðXÞÞ ; kv v i
i1 2
k 6 ki kv
2 t kL2 ðt ;t ;L2 ðXÞ2 Þ ; i1 i
ð4:28Þ
v ¼ y; yd ; v ¼ p; pd :
2 Besides, since J ¼ 1 þ O ki , then we have i
i q i J i k2 6 Ck2 kqi k2 6 Ck2 h2 kzk21 kq z z z i i l ð0;T;H1 ðXÞÞ ; i J i k2 6 Ck2 kpi k2 ; i p kp i z z z 4 kzi zi J i k2 6 Cki kzi k2 :
ð4:29Þ
Inserting these estimates (4.15), (4.28)-(4.29) into Eq. (4.27), it thus follows by taking d = a⁄/2 that NT NT 2 X X X M 2 2 2 2 p þ a ki kpi1 ki piz þ pzi1 þ Ck kv ss k2L2 ð0;T;L2 ðXÞÞ þ kzk2l2 ð0;T;L2 ðXÞÞ z q k 6 C i¼Mþ1
v ¼y;z
i¼Mþ1
þ Ck
2
X v ¼y;yd
þ Ch
2
kv
2 t kL2 ð0;T;L2 ðXÞÞ
þ
X v ¼p;pd
kv
!
2 t kL2 ð0;T;L2 ðXÞ2 Þ
X 2 kv k2H1 ð0;T;H1 ðXÞÞ þ kv k2l1 ð0;T;H2 ðXÞÞ þ Ch kpk2l1 ð0;T;H1 ðXÞ2 Þ :
v ¼y;z
!
ð4:30Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3439
From which we infer
kpz kl1 ð0;T;L2 ðXÞÞ þ kpq kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðk þ hÞ:
ð4:31Þ
Incorporating (4.31) with the well-known estimates for qk (k = q, z), we finish the proof of (4.16). Thus Lemma 4.4 is proved. h Collecting the bounds given by Lemmas 4.2, 4.3, 4.4, we have the following main result. Theorem 4.6. Suppose that (p, y, q, z) and (ph, yh, qh, zh) are the solutions of (CCP-OPT) and (CCP-OPT)hk, respectively. Assume that all conditions of Lemmas 4.2, 4.3, 4.4 are valid. Then
kp ph kl2 ð0;T;L2 ðXÞ2 Þ þ ky yh kl1 ð0;T;L2 ðXÞÞ þ kq qh kl2 ð0;T;L2 ðXÞ2 Þ þ kz zh kl1 ð0;T;L2 ðXÞÞ þ ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Cðk þ hU þ hÞ:
ð4:32Þ
Proof. It follows from Eqs. (4.9) and (4.16) that
ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Ckkzt kL2 ð0;T;L2 ðXÞÞ þ ChU kzkl2 ð0;T;H1 ðXÞÞ þ kukl2 ð0;T;H1 ðXU ÞÞ þ Ckz zh ðuÞkl2 ð0;T;L2 ðXÞÞ 6 Cðk þ hU þ hÞ:
ð4:33Þ
Moreover, it follows from Lemmas 4.2, 4.4 and (4.33) that
kp ph kl2 ð0;T;L2 ðXÞ2 Þ þ ky yh kl1 ð0;T;L2 ðXÞÞ þ kq qh kl2 ð0;T;L2 ðXÞ2 Þ þ kz zh kl1 ð0;T;L2 ðXÞÞ 6 kph ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kyh yh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kqh qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kzh zh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kp ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ ky yh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kq qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kz zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hU þ hÞ: ð4:34Þ Thus, Theorem 4.6 follows immediately from (4.33) and (4.34). h
5. Concluding remarks In this paper, we have derived a priori error estimates for the characteristic-mixed finite element discretization of optimal control problem governed by time-dependent convection–diffusion equations, where bilateral pointwise inequality constraints on the control variable are considered. The a posteriori error estimates and numerical experiments will be addressed in the future work. In this area there are still many important issues to be addressed, such as final state constrained optimal control problems, and more complicated practical problems. Acknowledgements The authors thank the editor and the anonymous referee for their valuable comments and suggestions on an earlier version of this paper. This research was supported by the National Basic Research Program of China (No. 2007CB814906), the National Natural Science Foundation of China (Nos. 11171190, 11101431) and the Natural Science Foundation of Shandong Province (Nos. ZR2010AL020, ZR2011AQ003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
D. Parra-Guevara, Y.N. Skiba, Elements of the mathematical modelling in the control of pollutants emissions, Ecol. Model. 167 (2003) 263–275. J. Zhu, Q. Zeng, A mathematical theoretical frame for control of air pollution, Sci. China Ser. D 32 (2002) 864–870. R. Becker, B. Vexler, Optimal control of the convection–diffusion equation using stabilized finite element methods, Numer. Math. 106 (2007) 349–367. M. Braack, Optimal control in fluid mechanics by finite elements with symmetric stabilization, SIAM J. Control Optim. 48 (2009) 672–687. S. Collis, M. Heinkenschloss, Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems, CAAM TR02-01, 2002. N.N. Yan, Z.J. Zhou, A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation, J. Comput. Appl. Math. 223 (2009) 198–217. N.N. Yan, Z.J. Zhou, A RT mixed FEM/DG scheme for optimal control governed by convection diffusion equations, J. Sci. Comput. 41 (2009) 273–299. H.F. Fu, H.X. Rui, A priori error estimates for optimal control problems governed by transient advection-diffusion equations, J. Sci. Comput. 38 (2009) 290–315. H.F. Fu, A characteristic finite element method for optimal control problems governed by convection–diffusion equations, J. Comput. Appl. Math. 235 (2010) 825–836. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. W.B. Liu, N.N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Series in Information and Computational Science, vol. 41, Science Press, Beijing, 2008. O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984.
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[14] D. Tiba, Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland, 1995. [15] Y.P. Chen, W.B. Liu, Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Int. J. Numer. Anal. Model. 3 (2006) 311–321. [16] Y.P. Chen, Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Eng. 199 (2010) 1415–1423. [17] J. Zhou, Y.P. Chen, Y. Dai, Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint, Appl. Math. Comput. 217 (2010) 2057–2066. [18] J. Douglas, T.F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982) 871–885. [19] P. Houston, E. Süli, Adaptive Lagrange–Galerkin methods for unsteady convection–diffusion problems, Math. Comput. 70 (2000) 77–106. [20] H.X. Rui, M. Tabata, A second order characteristic finite element scheme for convection–diffusion problems, Numer. Math. 92 (2002) 161–177. [21] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991. [22] J. Douglas, J.E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comput. 44 (1985) 39–52. [23] P.A. Raviart, J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Mathematics, vol. 606, Springer, Berlin, 1977. pp. 292–315. [24] R.E. Ewing, T.F. Russell, M.F. Wheeler, Convergence analysis of anapproximation of miscible displacement in displacement inporous media by mixed finite elements and a modified method of characteristics, Comput. Methods Appl. Mech. Eng. 47 (1984) 73–92. [25] D. Meidner, B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints, SIAM J. Control Optim. 47 (2008) 1301–1319. [26] M.F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973) 723– 759.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A characteristic-mixed finite element method for time-dependent convection–diffusion optimal control problem Hongfei Fu a,⇑, Hongxing Rui b a b
Department of Computational and Applied Mathematics, China University of Petroleum, Qingdao 266555, China School of Mathematics, Shandong University, Jinan 250100, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection–diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart–Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Optimal control Characteristic-mixed finite element Convection–diffusion equations A priori error estimates
1. Introduction Optimal control problems governed by convection–diffusion equations arise in many scientific and engineering applications, such as atmospheric pollution control problems [1,2]. Efficient numerical methods are essential to successful applications of such optimal control problems. To the best of our knowledge, there are only a few published results on optimal control problems governed by steady-state convection–diffusion equations; see [3] of standard finite element discretizations with stabilization based on local projection method [4] of symmetric stabilization method [5] of SUPG method [6] of edge-stabilization method, and [7] of the application of RT mixed DG scheme. For the approximation of constrained optimal control problems governed by time-dependent convection–diffusion equations, it is much more complicated, see [8,9]. Systematic introductions of the finite element method for PDEs and optimal control problems can be found in, for example [10–14]. In many optimal control problems, the objective functional contains not only the state variable, but also its gradient. For example, in the flow control problem, the gradient stands for Darcy velocity and it is an important physics variable, or, in the temperature control problem, large temperature gradients during cooling or heating may lead to its destruction. Therefore, in these cases people pay their special attention on the gradient of the primal state variable, see Refs. [15–17]. In this paper, we are interested in the following convex optimal control problem with pointwise control constraints:
min
u2U ad
1 2
Z 0
T
Z
ðy yd Þ2 þ
X
Z X
ðp pd Þ2 þ
Z
u2 dt
XU
⇑ Corresponding author. E-mail addresses: [email protected] (H. Fu), [email protected] (H. Rui). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.087
ð1:1Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3431
subject to
8 yt þ divðby þ pÞ ¼ f þ Bu; > > > < p ¼ Ary; > y ¼ 0; > > : yð0Þ ¼ y0 ;
in X ð0; T; in X ð0; T; on @ X ð0; T;
ð1:2Þ
in X;
and
n1 6 uðx; tÞ 6 n2 ;
a:e: in XU ð0; T;
ð1:3Þ
@y ; @t
where yt ¼ b ¼ bðx; tÞ denotes a velocity field in the flow control, A = A(x) is a diffusion coefficient, f = f(x, t) accounts for external sources and sinks, and y0 = y0(x) is a prescribed initial data. The details will be specified in the next section. In the optimal control problem (1.1)–(1.3), the state equation is usually convection-dominated. It is well known that the standard finite element discretization applied to the convection–diffusion equation (1.2) leads to strong oscillation when the diffusion is small. The methods of characteristic finite element [18–20] are proved effectively to reduce the oscillatory behavior. These methods combine the time derivative term and the convection term of divergence-free form in the governing equations to carry out the temporal discretization in a Lagrange coordinate. They symmetrize the governing equation and stabilize the numerical approximations. Besides, they generate accurate numerical solutions and significantly reduce the numerical diffusion and grid-orientation effect present in upwind methods, even if large time steps and coarse spatial meshes are used. Mixed finite element method [21–23] has been proved to be an effective numerical method for solving fluid problems and optimal control problems. It has an advantage of approximating the unknown scaler variable and its diffusive flux simultaneously. Besides, the method can approximate the unknown variable and its flux to a same order of accuracy. Recently, there are some research articles on this method for optimal control problems, see [15,16], for example. In this work, we combine the characteristic finite element method with the mixed finite element method, and present a characteristicmixed finite element scheme to the quadratic optimal control problems governed by convection–diffusion equations. The characteristic approximation is applied to handle the convection term, and the lowest-order Raviart–Thomas mixed finite element spatial approximation is adopted to deal with the diffusion term. We obtain a priori error estimates for both the state and control approximations. The rest of the paper is organized as follows: in Section 2, we first give a precise description of the optimal control problem and then derive the continuous optimality conditions. In Section 3, we construct a characteristic-mixed finite element approximation scheme for the optimal control problem. In Section 4, main error estimates are derived for the control problem with obstacle constraints. Section 5 contains concluding remarks. The approach developed in this work is applicable to more general convex control problems. Let X and XU be bounded open sets in R2 , with Lipschitz boundaries oX and oXU. Just for simplicity of presentation, we assume that X and XU are convex polygon. In this paper, we employ the usual notion for Lebesgue and Sobolev spaces, see [10] for details. In addition C and d denote a generic constant and small positive number which are independent of the discrete parameters and may have different values in different circumstances, respectively. 2. Optimal control problem and optimality conditions In this section we briefly discuss the precise formulation of the optimization problem under consideration. Furthermore, we derive the optimality conditions accordingly. Let
V ¼ Hðdiv; XÞ ¼ fv 2 L2 ðXÞ2 : divv 2 L2 ðXÞg endowed with the norm given by
kv kV ¼ kv kHðdiv;XÞ ¼ ðkv k2 þ kdivv k2 Þ1=2 and
W ¼ L2 ðXÞ;
U ¼ L2 ðXU Þ:
Partition (0, T] into 0 ¼ t 0 < t1 < t2 < < t NT ¼ T, with time steps ki ¼ ti t i1 ; i ¼ 1; 2; . . . ; N T ; k ¼ max16i6NT ki . Let fi = f(x, ti). We define, for 1 6 p < 1, the discrete time-dependent norms
kf klp ð0;T;XÞ ¼
NT X
!1p ki kf i kpX
i¼1
and the standard modification for p = 1. Let
n o p l ð0; T; XÞ :¼ f : kf klp ð0;T;XÞÞ < 1 ;
1 6 p 6 1:
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H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
To set up a weak formulation of the state equation, we shall take the state space V ¼ L2 ð0; T; VÞ; W ¼ L2 ð0; T; WÞ, the control space U ¼ L2 ð0; T; UÞ, and the observation space H ¼ L2 ð0; T; HÞ with H = L2(X) to fix the idea. Let B be a linear continuous operator from U to H. To formulate the optimal control problem, we introduce the admissible set Uad collecting the inequality constraints (1.3) as
~ 2 U : n1 6 u ~ ðx; tÞ 6 n2 ; a:e: in XU ð0; Tg; U ad ¼ fu
ð2:1Þ
where the bounds n1 ; n2 2 R fulfill n1 < n2. In the optimal control problem (1.1)-(1.2), yd 2 H1(0, T; L2(X)), pd 2 H1(0, T; L2(X)2), f 2 L2(0, T; L2(X)), y0 2 L2(X), and ÞÞ22 such that there are positive constants c⁄ and c⁄ satisfying AðxÞ ¼ ðai;j ðxÞÞ22 2 ðC 1 ðX
c jXj2 6 X T AX 6 c jXj2 ;
8X 2 R2 :
Þ2 Þ and is divergence-free, i.e., The velocity field vector b = (b1(x, t), b2(x, t))T lies in the function space L1 ð0; T; W 1;1 ðX
divb ¼ 0;
8x 2 X;
t 2 ð0; T:
ð2:2Þ
To avoid technical boundary difficulties associate with the method of characteristics, we assume that X is a rectangle and the state equation is X-periodic, i.e., we assume all functions in Eq. (1.2) are spatially X-periodic; see [18,24] for example. Other assumption can be found in [19] or [20]. Let
/ðx; tÞ :¼ ðjbj2 þ 1Þ1=2 ¼ ðb1 ðx; tÞ2 þ b2 ðx; tÞ2 þ 1Þ1=2 and let the characteristic direction associated with the operator yt + b ry be denoted by s = s(x, t), where
/ys :¼ /
@y ¼ yt þ b ry: @s
ð2:3Þ
Then the first equation in (1.2) can be written in the following form:
/ys þ divp ¼ f þ Bu:
ð2:4Þ
For the given control set Uad, the above-mentioned convex optimal control problem (1.1)–(1.3) can be restated as follows: (CCP)
min
u2U ad
1 2
Z 0
T
Z
ðy yd Þ2 þ
Z
X
ðp pd Þ2 þ
Z
X
u2 dt;
ð2:5Þ
XU
where ðp; y; uÞ 2 V W U such that
8 > < ð/ys ; wÞ þ ðdivp; wÞ ¼ ðf þ Bu; wÞ; 8w 2 W; ðap; v Þ ðy; divv Þ ¼ 0; 8v 2 V; t 2 ð0; T; > : yð0Þ ¼ y0 ;
t 2 ð0; T; ð2:6Þ
where a = A1(x) and the inner product in L2(X) or L2(X)2 is indicated by (, ). It is well known (see, e.g. [11]) that the control problem (CCP) has a unique solution (p, y, u), and that a triplet (p, y, u) is the solution of (CCP) if and only if there is a co-state ðq; zÞ 2 V W such that (p, y, q, z, u) satisfies the following optimality conditions: (CCP-OPT)
8 ð/ys ; wÞ þ ðdivp; wÞ ¼ ðf þ Bu; wÞ; 8w 2 W; > > > < ðap; v Þ ðy; divv Þ ¼ 0; 8v 2 V; t 2 ð0; T; > > > : yð0Þ ¼ y0 ;
t 2 ð0; T;
8 ð/zs ; wÞ þ ðdivq; wÞ ¼ ðy yd ; wÞ; 8w 2 W; t 2 ½0; TÞ; > > < ðaq; v Þ ðz; divv Þ ¼ ðp pd ; v Þ; 8v 2 V; t 2 ½0; TÞ; > > : zðTÞ ¼ 0; Z 0
T
~ uÞU dt P 0; ðu þ B z; u
8u~ 2 U ad U ¼ L2 ð0; T; UÞ;
where B⁄ is the adjoint operator of B, and (, )U is the inner product of U. Remark 2.1. In fact, the co-state solution z of Eq. (2.8) satisfies the following equation
zt divðbz þ Aðrz þ p pd ÞÞ ¼ y yd :
ð2:7Þ
ð2:8Þ
ð2:9Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
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Then it follows from Propositions 2.1–2.3 of paper [25] and Lemmas 5.7–5.9 of paper [19] that
y; z 2 L2 ð0; T; H2 ðXÞÞ \ H1 ð0; T; L2 ðXÞÞ,!Cð½0; T; H10 ðXÞÞ and
u 2 L2 ð0; T; W 1;p ðXÞÞ \ H1 ð0; T; L2 ðXÞÞ \ L1 ðð0; TÞ XÞ for any p < 1. Remark 2.2 [11]. Inequality (2.9) is equivalent to the following:
u þ B z > 0; u ¼ n1 ;
u þ B z < 0; u ¼ n2 ;
u þ B z ¼ 0; n1 < u < n2 :
ð2:10Þ
Using a pointwise projection on the admissible set Uad,
PUad : U ! U ad ;
P Uad ðf Þðx; tÞ ¼ maxðn1 ; minðn2 ; f ðx; tÞÞÞ;
the optimality condition (2.9) can also be expressed as
u ¼ PUad ðB zÞ:
ð2:11Þ
3. Characteristic-mixed finite element approximation In this section, we consider the method of characteristic-mixed finite element for the approximation of convex control problem (CCP). The approximation scheme is also applicable to the control problem with more general convex objective functionals. Here we only consider the n-simplex Lagrange elements which are most widely used in practical computations. Let G(x⁄, t⁄; t) be an approximate characteristic curve passing through point x⁄ at time t⁄, which is defined by
Gðx ; t ; tÞ :¼ x bðx ; t Þðt tÞ:
ð3:1Þ
We denote by x ¼ Gðx; t i ; t i1 Þ be the foot at time ti1 of the characteristic curve with head x at time ti, and x represents the head of the characteristic curve with foot x at time ti1, namely,
x ¼ Gðx; ti ; t i1 Þ: Let f ðxÞ ¼ f ð xÞ. We approximate
/i
ð3:2Þ @yi @s
ðxÞ ¼ @y ðx; ti Þ by a backward difference quotient in the s-direction, i.e., @s
i1 @yi yi y : ’ @s ki
ð3:3Þ
We remark that, since the problem is X-periodic, G is a differential homeomorphism of X onto itself for sufficiently small ki i1 is always defined; the tangent to the characteristics (i.e., the s-segment) cannot cross a boundary to an (see [24]). Thus, y undefined location. ¼ [ hs Let T h and T hU be quasi-uniform regular triangulations of X and XU, respectively, such that X s2T ; XU ¼ [sU 2T hU sU . Let h ¼ maxs2T h hs ; hU ¼ maxsU 2T h hsU , where hs and hsU denote the diameters of the element s and sU, respectively. U Let Vh Wh V W denote the lowest order Raviart–Thomas space [23] associated with the triangulations T h . Let h 2 U U = L (XU) consist of piecewise constant functions on T hU of XU due to the limited regularity of the optimal control. It is clear that Vh V, Wh W and Uh U. Let U had be a closed convex set in Uh, that is,
~ h 2 U h : n1 6 u ~ h 6 n2 ; a:e: in XU g: U had ¼ fu
ð3:4Þ
Then the corresponding fully discrete characteristic-mixed finite element approximation of (CCP), which will be labeled as (CCP)hk, is defined as follows:
min
uih 2U had
Z Z Z NT i 2 i 2 1X ; ki yh yd ðx; t i Þ þ ðpih pd ðx; t i ÞÞ2 þ uh 2 i¼1 X X XU
ð3:5Þ
where pih ; yih ; uih 2 V h W h U h ; i ¼ 1; 2; . . . ; N T satisfies
8 i i1 yh y i > i h > ; w ; w Þ þ Bu ; w þ divp ; ¼ f ðx; t i h h h > h h k < i i i h aph ; v h yh ; divv h ¼ 0; 8v h 2 V ; > > > : 0 yh ¼ yh0
8wh 2 W h ; ð3:6Þ
and yh0 2 W h is an approximation of y0 which will be specified later on. hk again a unique solution pih ; yih ; uih , and that a triplet i It is well known (see, e.g.hk[11]) that the control problem (CCP) i1 has h h i i i1 i i1 i ph ; yh ; uh is the solution of (CCP) if and only if there is a co-state qh ; zh 2 V W such that ph ; yih ; qi1 h ; zh ; uh satisfies the following optimality conditions: (CCP-OPT)hk
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8 i i1 yh y i > i h > ; w ; w Þ ¼ f ðx; t Þ þ Bu ; w þ ð divp ; i h h h > h h k < i i i h aph ; v h yh ; divv h ¼ 0; 8v h 2 V ; > > > : 0 yh ¼ yh0 ;
8wh 2 W h ; ð3:7Þ
8 zi1 zih J i > > h > ; w þ divqhi1 ; wh ¼ yih yid ; wh ; 8wh 2 W h ; h > ki < i1 aqh ; v h zhi1 ; divv h ¼ pih pid ; v h ; 8v h 2 V h ; > > > > : NT zh ¼ 0;
ð3:8Þ
i i ~ uh þ B zi1 h ; uh uh U P 0;
ð3:9Þ
8u~ h 2 U had ;
1 denotes the determinant of the Jacobian transformation from G to x. Similar to where zih ¼ zih ð xÞ; and J i ¼ det @Gðx;t@xi ;ti1 Þ 2 the discussion in [8], we know that for sufficiently small ki, the Jacobian J i ¼ 1 þ O ki . 4. A priori error estimates In this section, we are able to derive some a priori error estimates for the optimal control problem (CCP-OPT) and its characteristic-mixed finite element approximation (CCP-OPT)hk with the control approximated by piecewise constant elements. To obtain a priori error estimates for the proposed scheme (3.7)–(3.9), we introduce intermediate variables i h h 2 i1 ph ðuÞ; yih ðuÞ; qi1 h ðuÞ; zh ðuÞ 2 ðV W Þ ; i ¼ 1; 2; . . . ; N T ; associate with the control u as follows:
8 i i1 ðuÞ y ðuÞy > > ; wh þ ðdivpih ðuÞ; wh Þ ¼ ðf i þ Bui ; wh Þ; > h ki h < i aph ðuÞ; v h yih ðuÞ; divv h ¼ 0; 8v h 2 V h ; > > > : 0 yh ðuÞ ¼ yh0 ;
8wh 2 W h ;
8 i zi1 ðuÞzi ðuÞJ i > i > > h k h ; wh þ divqi1 8wh 2 W h ; > h ðuÞ; wh ¼ yh ðuÞ yd ; wh ; i < i1 i i aqh ðuÞ; v h zi1 8v h 2 V h ; > h ðuÞ; divv h ¼ ph ðuÞ pd ; v h ; > > > : NT zh ðuÞ ¼ 0:
ð4:1Þ
ð4:2Þ
For simplicity of illustration, set
hj ¼ jh jh ðuÞ; fk ¼ kh kh ðuÞ; h0y
gj ¼ j jh ðuÞ; for j ¼ p; y; nk ¼ k kh ðuÞ; nNz T
for k ¼ q; z:
fNz T
It is clear that ¼ 0 and ¼ ¼ 0. Before deriving the main error estimates for the characteristic-mixed finite element approximation of optimal control problem governed by convection–diffusion equation, some lemmas are prepared. Þ2 Þ. Then for i = 0, . . . , NT and f 2 L2(X) it holds that Lemma 4.1 [20]. Suppose that b 2 L1 ð0; T; W 1;1 ðX
kf i k2 6 ð1 þ Cki Þkf i k2 : Lemma 4.2. Let (ph, yh, qh, zh) and (ph(u), yh(u), qh(u), zh(u)) be the solutions of (3.7), (3.8) and (4.1), (4.2), respectively. Then the following estimates hold
kph ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kyh yh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cku uh kl2 ð0;T;L2 ðXU ÞÞ ;
ð4:3Þ
kqh qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kzh zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cku uh kl2 ð0;T;L2 ðXU ÞÞ :
ð4:4Þ
Proof. We first prove Eq. (4.3). It follows from Eqs. (3.7) and (4.1) that
8 i i1 > < hy khy ; wh þ divhip ; wh ¼ B uih ui ; wh ; i > : ahip ; v h hiy ; divv h ¼ 0; 8v h 2 V h :
8wh 2 W h ;
ð4:5Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
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By selecting wh ¼ hiy ; v h ¼ hip as test functions and adding the two equations in (4.5), we obtain
hiy hyi1 i ; hy ki
!
þ ahip ; hip ¼ B uih ui ; hiy : 2
It follows from the inequality aða bÞ P 12 ða2 b Þ and Lemma 4.1 that
2 2 1 i 2 i1 2 2 2 hy hy þ a hip 6 C ui uih þ C khiy k þ khyi1 k : 2ki
where a⁄ = 1/c⁄. Multiplying both sides of the above inequality by 2ki and summing over i from 1 to N(6NT). Noting h0y ¼ 0, we obtain that
2 N 2 N 2 N X X X 2 N 2 ki hip 6 C ki hiy þ hyi1 þ C ki ui uih : hy þ 2a i¼1
i¼1
i¼1
The application of the discrete Gronwall’s lemma leads to the first assertion of Lemma 4.2. Similarly, by subtracting Eq. (4.2) from Eq. (3.7) we have
8 i1 i i i > < fz kfz J ; wh þ divfi1 8wh 2 W h ; q ; wh ¼ hy ; wh ; i > : afqi1 ; v h fzi1 ; divv h ¼ hip ; v h ; 8v h 2 V h : Choosing wh ¼ fi1 z ;
ð4:6Þ
v h ¼ fqi1 in Eq. (4.6) and adding the two equations we also obtain
fzi1 fiz i1 ; fz ki
!
þ a
fqi1 ; fi1 q
! fi fi J i i1 i i1 i i1 z z : ; fz ¼ hy ; f z hp ; fq ki
2 Recalling J i ¼ 1 þ O ki , so we infer according to Lemma 4.1 that
! 2 fi fi J i 1 i1 2 i 2 i1 2 i i1 i i1 i z i1 z ; fz fz fz þ a fq 6 hy ; fz þ hp ; fq þ C fz þ 2ki ki 2 2 2 2 2 6 C fiz þ fzi1 þ C hiy þ C hip þ d fqi1 ;
where d is a small positive number. Similarly, multiplying both sides of the above inequality by 2ki and summing over i from NT to M + 1(P1), we obtain by taking d = a⁄/2 that
NT NT NT X X X M 2 i1 2 i 2 i1 2 i 2 i 2 f þ a þ C k f 6 C k f þ f k h þ hp ; i i i z z z y q i¼Mþ1
since
fNz T
i¼Mþ1
i¼Mþ1
¼ 0. Thus it follows from the discrete Gronwall’s lemma that
kfz kl1 ð0;T;L2 ðXÞÞ þ kfq kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðkhy kl2 ð0;T;L2 ðXÞÞ þ khp kl2 ð0;T;L2 ðXÞ2 Þ Þ:
ð4:7Þ
Therefore Eq. (4.4) is proved from (4.7) and (4.3). h Lemma 4.3. Let (p, y, q, z, u) and (ph, yh, qh, zh, uh) be the solutions of (CCP-OPT) and (CCP-OPT)hk, respectively. Let Ph be the standard L2(XU)-orthogonal projection from U to Uh such that
~ Ph u ~; u ~ h ÞU ¼ 0; ðu
8u~ h 2 U h :
ð4:8Þ
Assume that u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU)), z 2 L2(0, T; H1(X)) \ H1(0, T; L2(X)). Then
ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Cðk þ hU þ kz zh ðuÞkl2 ð0;T;L2 ðXÞÞ Þ;
ð4:9Þ
where zh(u) is defined in Eq. (4.2). Proof. It follows from the definition (4.8) that Ph ui 2 U had , and we can prove (see, e.g.[10]) that for u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU))
ku Ph ukl2 ð0;T;L2 ðXU ÞÞ 6 ChU kukl2 ð0;T;H1 ðXU ÞÞ :
ð4:10Þ
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Moreover, if u 2 L2(0, T; H1(XU)) \ C([0, T]; L2(XU)) and z 2 L2(0, T; H1(X)), we have NT X
ki ðui þ B zi ; Ph ui ui ÞU ¼
i¼1
NT X
ki ððui þ B zi Þ Ph ðui þ B zi Þ; Ph ui ui Þ
i¼1
6 ku þ B z Ph ðu þ B zÞkl2 ð0;T;L2 ðXU ÞÞ kPh u ukl2 ð0;T;L2 ðXU ÞÞ 2 6 ChU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ :
ð4:11Þ
Recalling the inequalities (2.9) and (3.9) that
ku uh k2l2 ð0;T;L2 ðXU ÞÞ ¼
NT NT X X ki ui ; ui uih U ki uih ; ui uih U i¼1
¼
NT X
i¼1 NT NT X X i i i i ki ui þ B zi ; ui uih U þ ki uih þ B zi1 ki uih þ B zi1 h ; uh Ph u U þ h ; Ph u u U
i¼1
i¼1
i¼1
NT X i i i þ ki B zi1 h z ; u uh U i¼1
60þ0þ
NT NT NT X X X ki uih ui ; Ph ui ui U þ ki ðui þ B zi ; Ph ui ui ÞU þ ki ðB ðzi1 zi Þ; Ph ui ui ÞU i¼1
þ
NT X
i¼1
i¼1
NT X i i i i1 ki ðB ðzi1 zi1 ki B zi1 ; u Ph ui U h ðuÞÞ; u Ph u ÞU þ h ðuÞ zh
i¼1
i¼1
NT NT NT X X X i i i1 i i1 þ ki B zi1 ki B zi1 ki B ðzi1 zi Þ; ui uih U : ; u uih U þ h zh ðuÞ ; u uh U þ h ðuÞ z i¼1
i¼1
i¼1
ð4:12Þ First, we bound the sixth non-zero term on the right-hand side of Eq. (4.12). By making use of Eq. (4.5) with i1 wh ¼ fi1 and Eq. (4.6) with wh ¼ hiy ; v h ¼ hip we have z ; v h ¼ fq NT NT X X i i i i ki B zhi1 zi1 ki zhi1 zi1 h ðuÞ ; u uh U ¼ h ðuÞ; Bðu uh Þ i¼1
i¼1
¼
¼
i¼1
hiy hi1 y ; fi1 z ki
NT X
fi1 z
NT X
i¼1
ki ki
!
fiz J i i ; hy ki
!
NT X ki divhip ; fzi1 i¼1
NT NT X X ki divfqi1 ; hiy ki hip ; hip i¼1
i¼1
NT NT X X ki hiy ; hiy ki hip ; hip ¼ khy k2l2 ð0;T;L2 ðXÞÞ khp k2l2 ð0;T;L2 ðXÞ2 Þ 6 0: ¼ i¼1
ð4:13Þ
i¼1
Thus we obtain from Eqs. (4.10)–(4.13), Lemma 4.2 and Cauchy–Schwarz inequality that
2 2 ku uh k2l2 ð0;T;L2 ðXU ÞÞ 6 CðdÞhU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ þ CðdÞk kzt k2L2 ð0;T;L2 ðXÞÞ þ CðdÞkz zh ðuÞk2l2 ð0;T;L2 ðXÞÞ 2
þ Cdkzh ðuÞ zh k2l2 ð0;T;L2 ðXÞÞ þ Cdku uh k2l2 ð0;T;L2 ðXU ÞÞ 6 Ck kzt k2L2 ð0;T;L2 ðXÞÞ 2 þ ChU kzk2l2 ð0;T;H1 ðXÞÞ þ kuk2l2 ð0;T;H1 ðXU ÞÞ þ Ckz zh ðuÞk2l2 ð0;T;L2 ðXÞÞ þ Cdku uh k2l2 ð0;T;L2 ðXU ÞÞ :
ð4:14Þ
We finish the proof by choosing Cd = 1/2 in Eq. (4.14). h Lemma 4.4. Let (p, y, q, z) and (ph(u), yh(u), qh(u), zh(u)) be the solutions of (2.7), (2.8) and (4.1), (4.2), respectively. Assume that y; z 2 L1 ð0; T; H10 ðXÞ \ H2 ðXÞÞ \ H1 ð0; T; H1 ðXÞÞ \ H2 ð0; T; L2 ðXÞÞ and p, q 2 L2(0, T; H1(X)2) \ H1(0, T; L2(X)2). Then the following estimates hold
kp ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ ky yh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hÞ;
ð4:15Þ
kq qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kz zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hÞ:
ð4:16Þ
The proof of this lemma needs the following intermediate results which are summarized in Lemma 4.5.
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
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Lemma 4.5. For each t 2 (0, T], we introduce the mixed finite element projection ðRh y; Rh pÞ of (y, p) as follows:
(
8wh 2 W h ;
ðdivðp Rh pÞ; wh Þ ¼ 0;
ðaðp Rh pÞ; v h Þ ðy Rh y; divv h Þ ¼ 0;
ð4:17Þ
8v h 2 V h :
Then there exists a unique pair ðRh y; Rh pÞ such that
kp Rh pkl2 ð0;T;L2 ðXÞ2 Þ 6 Chkpkl2 ð0;T;H1 ðXÞ2 Þ ; 2
ky Rh ykl1 ð0;T;L2 ðXÞÞ þ hky Rh ykl1 ð0;T;H1 ðXÞÞ 6 Ch kykl1 ð0;T;H2 ðXÞÞ ; @ ðy Rh yÞ 6 ChkykH1 ð0;T;H1 ðXÞÞ : 2 @t L ð0;T;L2 ðXÞÞ
ð4:18Þ
Proof. For the proof of the unique solution of Eq. (4.17) and the approximate results (4.18), we refer the reader to [10,26], for example. h The proof of Lemma 4.4 We first proof Eq. (4.15). Decomposing the error gj = j jh(u) as gj ¼ ðj Rh jÞþ ðRh j jh ðuÞÞ ¼ lj þ mj , with j = y, p. Since the estimate for lj is known, we need only to derive an estimate for mj. It follows from Eqs. (4.1), (4.17) and (2.6), we obtain an error equation on mj:
8 i i1 i i1 < my my ; wh þ divm i ; wh ¼ ðri ; wh Þ ly l y ; wh ; p
ki
:
ki
ðam ip ; v h Þ ðmiy ; divv h Þ ¼ 0; i
i
ki
ð4:19Þ
8v h 2 V h ;
i1
where ri ¼ /i @y y kyi . @s For an L2-norm error estimate, we choose wh ¼ miy ; together to obtain
miy myi1
8wh 2 W h ;
! i y
;m
i p;
þ am m
i p
¼
i
i y
r ;m
v h ¼ mip as test functions in system (4.19) and add the two equations
liy lyi1 ki
! i y
;m
yi1 li1 l y ki
! i y
;m
:
ð4:20Þ
Multiplying both sides of (4.20) by ki and summing on 1 6 i 6 N. Similar to the estimates in Lemma 4.2, we derive
N 2 N 2 N N 2 N 2 X X X X X 1 N 2 2 1 1 yi1 þC ki m ip 6 C ki miy þ mi1 ki kri k2 þ C ki liy lyi1 þ C ki lyi1 l my þ a ; y 2 i¼1 i¼1 i¼1 i¼1 i¼1 ð4:21Þ here we have omitted the term Noting that
m0y by choosing the initial approximate value yh0 ¼ Rh y0 .
kri k2 6 Cki kyss k2L2 ðti1 ;ti ;L2 ðXÞÞ ; 2 @ ly 2 i 2 6 Cki h kyk2H1 ðti1 ;ti ;H1 ðXÞÞ ; ly lyi1 6 ki @t 2 L ðt i1 ;t i ;L2 ðXÞÞ i1 i1 2 2 2 2 2 2 ly ly 6 Cki krli1 y k 6 Cki h kykl1 ð0;T;H2 ðXÞÞ :
ð4:22Þ
Then we have
2 N N 2 X X N 2 2 2 2 þ Ck kyss k2L2 ð0;T;L2 ðXÞÞ þ Ch kyk2H1 ð0;T;H1 ðXÞÞ þ Ch kyk2l1 ð0;T;H2 ðXÞÞ : ki km ip k2 6 C ki miy þ mi1 my þ 2a y i¼1
i¼1
ð4:23Þ Applying the discrete Gronwall’s lemma to Eq. (4.23) yields that
kmy kl1 ð0;T;L2 ðXÞÞ þ km p kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðk þ hÞ:
ð4:24Þ
Combining (4.24) with the well-known estimates (4.18) for ly and lp, we finish the proof of (4.15). For the proof of Eq. (4.16), we write nk ¼ k kh ðuÞ ¼ ðk Rh kÞ þ ðRh k kh ðuÞÞ ¼ qk þ pk for k = z and q. Here ðRh z; Rh qÞ 2 W h V h are defined the same as ðRh y; Rh pÞ in Eq. (4.17), and of course qk (k = q, z) is bounded in the desired way. In order to estimate pk, we write
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8 i1 i pz pz > ; wh þ divpqi1 ; wh > ki > > > i i1 i i i i i i > > q q q q J p p J > > ¼ ðvi1 ; wh Þ þ z ki z ; wh z ki z ; wh z ki z ; wh > > > i i i > < þ z kzi J ; wh þ yi yih ðuÞ; wh ðyi yi1 ; wh Þ þ yid yi1 d ; wh ; > > > 8wh 2 W h ; > > > > > i1 i1 > > > apq ; v h pz ; divv h > > : ¼ ðpi pih ðuÞ; v h Þ þ ðpi pi1 ; v h Þ pid pdi1 ; v h ; 8v h 2 V h ; i1
i1
ð4:25Þ
i
where vi1 ¼ /i1 @z@s z kiz . i1 We choose wh ¼ pi1 and add the two equations in (4.25) to obtain z ; v h ¼ pq
i pi1 p z z ki
;p
i1 z
i1 q ;
þ ap
i1 q
p
¼
i1
v ;p
i1 z
þ
p iz p iz Ji
qiz qzi1
i1 z
;p
ki !
qiz q iz ki !
i1 z
;p
q iz q iz Ji ki
! ;p
i1 z
zi zi J i ; pzi1 þ yi yih ðuÞ; pi1 z ki ki i yi yi1 ; pi1 þ p pi1 ; pi1 pi pih ðuÞ; pi1 q z q i1 i i1 i1 þ ðyid yi1 : d ; pz Þ pd pd ; pq
; pi1 z
þ
ð4:26Þ
Multiplying both sides of Eq. (4.26) by ki and summing on i from NT to M + 1. Similar to the estimates in Lemma 4.2, we also obtain NT NT NT NT 2 X X X X 2 2 1 1 2 pM 2 þ a þC ki kpi1 ki piz þ pi1 ki kvi1 k2 þ C ki qiz qzi1 z q k 6 C z 2 i¼Mþ1 i¼Mþ1 i¼Mþ1 i¼Mþ1
þC
NT X i¼Mþ1
þC
NT X
NT NT 2 2 X X 1 1 i 1 i i i i 2 þ C i Ji ki qiz q k q q þ C ki p i z z z z pz J i¼Mþ1
i¼Mþ1
1
ki kzi zi J i k2 þ Cky yh ðuÞk2l1 ð0;T;L2 ðXÞÞ þ Ckp ph ðuÞk2l2 ð0;T;L2 ðXÞ2 Þ
i¼Mþ1
þC
X
NT X
v ¼y;yd
i¼Mþ1
ki kv i v i1 k2 þ C
X
NT X
v ¼p;pd
i¼Mþ1
ki kv i v i1 k2 þ d
NT X i¼Mþ1
2 ki pqi1 :
ð4:27Þ
Following the estimates in Eqs. (4.22), we have at once
kvi1 k2 6 Cki kzss k2L2 ðti1 ;ti ;L2 ðXÞÞ ; @ qz 2 2 i i1 2 kqz qz k 6 ki 6 Cki h kzk2H1 ðti1 ;ti ;H1 ðXÞÞ ; @t L2 ðti1 ;ti ;L2 ðXÞÞ i q q i 2 6 Ck2 rqi 2 6 Ck2 h2 kzk21 i i z z z l ð0;T;H2 ðXÞÞ ; kv i v i1 k2 6 ki kv t k2L2 ðti1 ;ti ;L2 ðXÞÞ ; kv v i
i1 2
k 6 ki kv
2 t kL2 ðt ;t ;L2 ðXÞ2 Þ ; i1 i
ð4:28Þ
v ¼ y; yd ; v ¼ p; pd :
2 Besides, since J ¼ 1 þ O ki , then we have i
i q i J i k2 6 Ck2 kqi k2 6 Ck2 h2 kzk21 kq z z z i i l ð0;T;H1 ðXÞÞ ; i J i k2 6 Ck2 kpi k2 ; i p kp i z z z 4 kzi zi J i k2 6 Cki kzi k2 :
ð4:29Þ
Inserting these estimates (4.15), (4.28)-(4.29) into Eq. (4.27), it thus follows by taking d = a⁄/2 that NT NT 2 X X X M 2 2 2 2 p þ a ki kpi1 ki piz þ pzi1 þ Ck kv ss k2L2 ð0;T;L2 ðXÞÞ þ kzk2l2 ð0;T;L2 ðXÞÞ z q k 6 C i¼Mþ1
v ¼y;z
i¼Mþ1
þ Ck
2
X v ¼y;yd
þ Ch
2
kv
2 t kL2 ð0;T;L2 ðXÞÞ
þ
X v ¼p;pd
kv
!
2 t kL2 ð0;T;L2 ðXÞ2 Þ
X 2 kv k2H1 ð0;T;H1 ðXÞÞ þ kv k2l1 ð0;T;H2 ðXÞÞ þ Ch kpk2l1 ð0;T;H1 ðXÞ2 Þ :
v ¼y;z
!
ð4:30Þ
H. Fu, H. Rui / Applied Mathematics and Computation 218 (2011) 3430–3440
3439
From which we infer
kpz kl1 ð0;T;L2 ðXÞÞ þ kpq kl2 ð0;T;L2 ðXÞ2 Þ 6 Cðk þ hÞ:
ð4:31Þ
Incorporating (4.31) with the well-known estimates for qk (k = q, z), we finish the proof of (4.16). Thus Lemma 4.4 is proved. h Collecting the bounds given by Lemmas 4.2, 4.3, 4.4, we have the following main result. Theorem 4.6. Suppose that (p, y, q, z) and (ph, yh, qh, zh) are the solutions of (CCP-OPT) and (CCP-OPT)hk, respectively. Assume that all conditions of Lemmas 4.2, 4.3, 4.4 are valid. Then
kp ph kl2 ð0;T;L2 ðXÞ2 Þ þ ky yh kl1 ð0;T;L2 ðXÞÞ þ kq qh kl2 ð0;T;L2 ðXÞ2 Þ þ kz zh kl1 ð0;T;L2 ðXÞÞ þ ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Cðk þ hU þ hÞ:
ð4:32Þ
Proof. It follows from Eqs. (4.9) and (4.16) that
ku uh kl2 ð0;T;L2 ðXU ÞÞ 6 Ckkzt kL2 ð0;T;L2 ðXÞÞ þ ChU kzkl2 ð0;T;H1 ðXÞÞ þ kukl2 ð0;T;H1 ðXU ÞÞ þ Ckz zh ðuÞkl2 ð0;T;L2 ðXÞÞ 6 Cðk þ hU þ hÞ:
ð4:33Þ
Moreover, it follows from Lemmas 4.2, 4.4 and (4.33) that
kp ph kl2 ð0;T;L2 ðXÞ2 Þ þ ky yh kl1 ð0;T;L2 ðXÞÞ þ kq qh kl2 ð0;T;L2 ðXÞ2 Þ þ kz zh kl1 ð0;T;L2 ðXÞÞ 6 kph ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kyh yh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kqh qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kzh zh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kp ph ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ ky yh ðuÞkl1 ð0;T;L2 ðXÞÞ þ kq qh ðuÞkl2 ð0;T;L2 ðXÞ2 Þ þ kz zh ðuÞkl1 ð0;T;L2 ðXÞÞ 6 Cðk þ hU þ hÞ: ð4:34Þ Thus, Theorem 4.6 follows immediately from (4.33) and (4.34). h
5. Concluding remarks In this paper, we have derived a priori error estimates for the characteristic-mixed finite element discretization of optimal control problem governed by time-dependent convection–diffusion equations, where bilateral pointwise inequality constraints on the control variable are considered. The a posteriori error estimates and numerical experiments will be addressed in the future work. In this area there are still many important issues to be addressed, such as final state constrained optimal control problems, and more complicated practical problems. Acknowledgements The authors thank the editor and the anonymous referee for their valuable comments and suggestions on an earlier version of this paper. This research was supported by the National Basic Research Program of China (No. 2007CB814906), the National Natural Science Foundation of China (Nos. 11171190, 11101431) and the Natural Science Foundation of Shandong Province (Nos. ZR2010AL020, ZR2011AQ003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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