A stabilized finite element method for the convection dominated diffusion optimal control problem

Transcription

1 Applicable Analysis An International Journal ISSN: (Print) X (Online) Journal homepage: A stabilized finite element method for the convection dominated diffusion optimal control problem Zhifeng Weng, Jerry Zhijian Yang & Xiliang Lu To cite this article: Zhifeng Weng, Jerry Zhijian Yang & Xiliang Lu (2015): A stabilized finite element method for the convection dominated diffusion optimal control problem, Applicable Analysis, DOI: / To link to this article: Published online: 26 Nov Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [University of Macau Library], [Xiliang Lu] Date: 27 November 2015, At: 04:19

2 APPLICABLE ANALYSIS, A stabilized finite element method for the convection dominated diffusion optimal control problem Zhifeng Weng a,b, Jerry Zhijian Yang a,c and Xiliang Lu a,c a School of Mathematics and Statistics, Wuhan University, Wuhan, P.R. China; b School of Mathematics Science, Huaqiao University, Quanzhou, P.R. China; c Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, P.R. China ABSTRACT In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results. 1. Introduction ARTICLE HISTORY Received 4 May 2015 Accepted 26 October 2015 COMMUNICATED BY C. Bacuta KEYWORDS Convection dominated diffusion equation; stabilized finite element method; optimal control problem; variational discretization AMS SUBJECT CLASSIFICATIONS 65M60; 76D07; 65M12 The convection dominated diffusion equations arise in many engineering applications. When the diffusion coefficient is very small, the convection dominated diffusion equations have a multiscale behavior between the diffusion and the advection, which brings enormous challenges in the endeavor of numerical approximation. The standard Galerkin finite element methods for the convection dominated diffusion problems may produce approximate solution with large nonphysical oscillations unless the mseh size is very small which depends on the diffusion coefficient. A lot of methods have been developed to overcome this problem, for example, the Galerkin least square method,[1,2] streamline upwind Petrov Galerkin (SUPG) method,[3,4] the residual-free bubbles method,[5,6] the local projection stabilization.[7 9]Burmanetal.[10] proposed an edge-stabilization Galerkin method to approximate the convection dominated diffusion equations. Knobloch [11] introduces a generalization of the local projection stabilization for the convection diffusion reaction equations which allows us to use local projection spaces defined on overlapping sets. Wu et al. [12] used the streamline upwind Petrov Galerkin to solve the convection dominated problems in order to eliminate overshoots and undershoots produced by the convection term in the meshless local Petrov Galerkin method. Song et. al. [13] presented variational multiscale method based on bubble functions for convection dominated diffusion equation. However, for the optimal control problem governed by convection dominated diffusion equations, some stabilization techniques are not straightforward to apply. One reason is that the two common approaches for PDE-constraint optimization problem, i.e. first-optimize-then-discretize and CONTACT Xiliang Lu 2015 Taylor & Francis xllv.math@whu.edu.cn

3 2 Z. WENG ET AL. first-discretize-then-optimize, are not equivalent if the stabilized term is not properly chosen. For example, the streamline upwind Galerkin method (SUPG) was not well suited for the duality techniques applied in optimal control when one considers the method for discretizing the state and the adjoint equation in the optimality system. Becker et al. [14] considered the stabilization finite element discretization based on local projections for the convection diffusion equation, where the symmetrical penalty terms were applied. Then finding the optimality system to the optimal control problem on the continuous level and then discretizing the optimality system appropriately is equivalent to considering the optimal control problem on the discrete level with stabilization term. Recently, Yan et al. [15] studied a priori and a posteriori error estimates of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. Zhou et al. [16] presented the local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations. Yücel et al. [17] investigated distributed optimal control of convection diffusion reaction equations using discontinuous Galerkin methods. Akman et al. [18] developed a priori error analysis of the upwind symmetric interior penalty Galerkin method for the optimal control problems governed by unsteady convection diffusion equations. Influenced by the work mentioned above, we will apply the bubble stabilization Galerkin method of Song et al. [13] to the discretization of optimal control problems governed by convection dominated diffusion equations. Firstly, we obtain the continuous optimality system, which contains the state equation, the adjoint equation, and the optimality condition, which is given in terms of a variational inequality. Then we give the discrete optimal control problem by using the bubble stabilization Galerkin method to approximate the state equation, whose optimality system then coincides with that obtained by discretizing the state and adjoint state in the continuous optimality system by finite elements with bubble stabilization, and the control is approximated by piecewise linear functions. However, the above approach cannot recover the full accuracy for the control variable. To obtain a better approximation to the control variable, Hinze introduced a variational discretization concept for optimal control problems and derived an optimal a priori error estimate for the control in [19,20]. The variational discretization concept for optimal control problems with control constraints utilizes the first-order optimality conditions and the discretization of the state and adjoint equations, then the control variable is obtained by the projection of the adjoint state. In this paper, we will combine variational discretization and the stabilized finite element method for the designed control problems. The half-order higher approximation for the control variable can be obtained. The remainders of this paper is organized as follows. In Section 2, we present the model problem and derive the optimality system. The bubble stabilization Galerkin method based on two local Gauss integrations for the optimal control problem is given in Section 3. In Section 4, a priori error estimate for the discrete control variables is derived and In Section 5, we prove the a priori error estimate using variational discretization. Then in Section 6, numerical experiments are shown to verify the theoretical results. Finally, in Section 7, we conclude with a summary and possible extensions. The standard Sobolev space W m,p ( ) is equipped with a norm m,p. For p = 2, let H m ( ) = W m,2 ( ) and write m = m,2 for m 0. (, ) is the inner product in L 2 ( ) or [L 2 ( )] 2.We use c or C to denote a generic positive constant whose value may change from place to place, but remains independent of the mesh size h and the diffusion coefficient ε. 2. Model problem Consider the following optimal control problem governed by convection dominated diffusion equations: min u K J(y, u) = 1 2 y y d α 2 u 2 0. (1)

4 APPLICABLE ANALYSIS 3 subject to: β y ε y + sy = f + Bu, in, (2) y = 0, on, (3) where R 2 is a bounded domain with Lipschitz boundary, α > 0 is a positive constant, β is either a constant vector or divergence free velocity field, ε is small diffusion coefficient, f L 2 ( ) and s s 0 > 0 is the reaction coefficient. The target state y d L 2 ( ), and the control variable u is supported in a sub-domain ω. The operator B : L 2 (ω) L 2 ( ) is the extension-by-zero operator, hence its adjoint operator B : L 2 ( ) L 2 (ω) is the restriction operator and B B is identity operator in L 2 (ω). The admissible set K L 2 (ω) is given by K ={u L 2 (ω); c u c, a.e.inω}, where c < c are two constants for the box constraints. Define a continuous bilinear forms a(, ) on H0 1( ) H1 0 ( ) by a(y, v) = ε( y, v) + (β y, v) + (sy, v), then the weak formulation of the state Equations (2) (3) is: to find y H0 1 ( ) such that a(y, v) = (f + Bu, v), v H 1 0 ( ). To find the optimality system, one needs the adjoint equation β p ε p + sp = y y d,in, p = 0, on, and its weak formulation reads: to find p H0 1 ( ) such that where b(p, w) = (y y d, w), w H 1 0 ( ), b(p, w) = ε( p, w) (β p, w) + (sp, w). Denote by A : K H0 1 ( ) the solution operator of the state Equations (2)and(3) and introduce the reduced cost functional j : K R by : j(u) = J(Au, u), then we can eliminate the state equation and to reformulate the optimization problem as: Minimize j(u), u K. The reduced cost functionalj is quadratic, its first- and second-order derivatives satisfies: (see [14,21]): j (u)(δu) = (B p + αu, δu), (4) j (u)(δu, δu) α δu 2 0, δu K.

5 4 Z. WENG ET AL. Since K is closed and convex, there exists a unique solution to optimal control problem (1) (3), and there exists an adjoint function (i.e. Lagrange multiplier) p H0 1 ( ),suchthat(y, p, u) satisfies the following optimality system (see e.g. [14,22]: a(y, v) = (f + Bu, v), v H0 1, (5) b(p, w) = (y y d, w), w H0 1, (6) (αu + B p, ū u) 0, ū K. (7) For the boxed constraints, the variational inequality (7) can be represented by the pointwise projection: ( B ) p u = Proj [c,c], α where the projection operator is defined by Proj [c,c] = max (c,min(c, u)). When diffusion coefficient ε is very small, the P 1 nodal element discretization for the state Equation (5) is unstable. In order to improve the computational stability and accuracy, one should adopt the stabilized methods. In this work, we will use the bubble stabilization Galerkin scheme based on two local Gauss integrations (see e.g. [23])to deal with the state equation and the adjoint equation in the optimality system (5) (7). 3. Bubble stabilized finite element methods based on two local Gauss integrations Let T h be the quasi-uniform triangulations of the domain with the mesh size h = max{diam(t) : T T h } as in [24]. The finite element spaces are defined by M h ={y h C 0 ( ) y h T P 1 (T), T T h }, Mh b ={v h C 0 ( ) v h T P 1 (T) B(T), T T h }, where P 1 (T) is the space of first-order polynomials on T and B(T) denotes the space of bubble functions. The bubble function defined as follows: B(T) ={v h C(T) v h Span{λ 0 λ 1 λ 2 }}, T T h, where λ i are area coordinates on T, i = 0, 1, 2. The area coordinate is also known as a triangle barycentre coordinate, where the three components (λ 0, λ 1, λ 2 ) are of the ratio between the area of the three triangles and the area of the mother triangle. We will also need the piecewise constant vector space R 0 ={V h (L 2 ( )) 2 V h T (P 0 (T)) 2, T T h }, where P 0 (T) is piecewise constant on T. We introduce a stabilized term G(, ) that defines the inner product in Mh b as: G(y h, v h ) = ν(( y h, v h ) ( y h, v h )), (8) where ν is a nonnegative function depending on the mesh size h,and : (L 2 ( )) 2 R 0 is the L 2 orthogonal projection. This stabilization formulation can be casted in the framework of variational multiscale method (see [13]). From the definition of, one can obtain ( y h y h, v h ) = 0. (9)

6 APPLICABLE ANALYSIS 5 An attractive feature of this stabilization approach is the flexibility in that the stabilized operator can be represented by the difference of local Gauss integration (see e.g. [13]): G(y h, v h ) = ν T T h T,k y h v h d T,1 y h v h d y h, v h Mh b, where T,i g(x) indicates an approximation Gauss integral over T which is exact for polynomials of degree i, i = 1, k, k 2. To prove convergence of stabilized solutions, we first notice that [24] v C v, v (L 2 ( )) 2, (10) v v Ch v 1, v (H 1 ( )) 2. (11) Using the stabilization finite element method as above, a discretization of the optimal control problem (1) (3) can be defined as follows: subject to: min u h K h J(y h, u h ). (12) a(y h, v h ) + G(y h, v h ) = (f + Bu h, v h ), v h M b h, (13) where K h is the discrete control space. For the standard method, K h = M h K, and for the variational discretization method K h = K. Next we give a discrete solution operator A h : K Mh b defined by: and the discrete reduced cost functional a(a h u, v h ) + G(A h u, v h ) = (f + Bu, v h ), v h M b h. (14) j h (u) = J(A h u, u). Similar as the continuous case, by introducing the discrete adjoint variable p h (which satisfies Equation (18)), we can find the first- and second-order derivative of j h : j h (u)(δu) = (B p h + αu, δu), (15) j h (u)(δu, δu) α δu 2 0. δu K. (16) Moreover, since K h is also convex and closed, there exists a unique solution (y h, u h ) to (12) (13)and a corresponding adjoint variable (Lagrange multiplier) p h M b h,suchthat(y h, p h, u h ) satisfies the optimality system: a(y h, v h ) + G(y h, v h ) = (f + Bu h, v h ), v h Mh b, (17) b(p h, w h ) + G(p h, w h ) = (y h y d, w h ), w h Mh b, (18) (αu h + B p h, ū h u h ) 0, ū h K h, (19) Remark 3.1: In our setting the approaches discretize-then-optimize and optimize-thendiscretize coincide due to the fact that G(y h, v h ) is a symmetric bilinear form.

7 6 Z. WENG ET AL. Recall that by standard approximation theory, for a regular family of elements, there exists an interpolation P h : H0 1( ) H2 ( ) Mh b with the following properties in [24] P h v 0 c v 0, (20) v P h v 0 + h (v P h v) 1 ch 2 v 2. (21) Next, we will give the estimate of the term G(P h v, P h v). Lemma 3.1: For any v H0 1( ) H2 ( ) and the following estimate holds Proof: G(P h v, P h v) Cνh 2 v 2 2. (22) From the definition of the term G(P h v, P h v), We can obtain G(P h v, P h v) = G(v + P h v v, v + P h v v) 2(G(v, v) + G(v P h v, v P h v)). For the first term, we can get from the interpolation property of (11) G(v, v) = ν v v 2 0 Cνh2 v 2 2. For the second term, we can obtain by the stability of (10) The proof has been completed. G(v P h v, v P h v) Cν v P h v 2 0 Cνh2 v A priori error analysis for the standard method In this section, we consider a priori error estimates for the optimal control problem (5) (7) and its bubble stabilization Galerkin approximation (17) (19) for the standard method. For the sake of simplicity, we choose ω = and hence B = I is an identity operator. To prove the a priori error estimate, we first introduce a norm: v 2 = s 1/2 0 v ε1/2 v G(v, v). In the following lemma, we give an error estimate for the discretization of the state equation with an additional perturbation in the right hand side. Lemma 4.1: Let for u K, y = Au H0 1( ) H2 ( ) be the associated solution of the state Equations (5)and(7), and for z K, y h = A h z Mh b be the associated discrete solution, i.e: a(y h, v h ) + G(y h, v h ) = (f + z, v h ), v h M b h. (23) Then the following estimate holds: y y h u z 0 + cτh y 2. (24) where Proof: From (5)and(23), we can have τ = ε 1/2 + h + ν 1/2 + ε 1/2 h. (25) a(y y h, v h ) = G(y h, v h ) + (u z, v h ), v h M b h. (26)

8 APPLICABLE ANALYSIS 7 Let y y h = δ + η with δ = y P h y, η = P h y y h. By (21), we get: δ 0 ch 2 y 2. (27) and δ ch 2 y 2 + cε 1/2 h y 2 + cν 1/2 h y 2. (28) For η Mh b and from (26), we can obtain η 2 = a(η, η) + G(η, η) = (u z, η) + G(P h y, η) a(δ, η). For the first term, applying Cauchy Schwarz inequality, we get: (u z, η) C u z 0 η 0 u z 0 η. For the second term, by Cauchy Schwarz inequality and (22), we can have For the third term we derive: where Therefore we can obtain G(P h y, η) (G(P h y, P h y)) 1/2 (G(η, η)) 1/2 cν 1/2 h y 2 η. a(δ, η) δ η + (β δ, η), (β δ, η) = (δ, β η) cε 1/2 δ ε 1/2 η cε 1/2 η δ 0. η u z 0 + cε 1/2 δ 0 + cν 1/2 h y 2 + δ. Applyingthe triangleinequalityand using (27) and(28), we can obtain (24). Remark 4.1: The above estimation involves the diffusion coefficient ε. It is not easy to obtain the ε-free estimation in the convection dominated case, see the discussion in [10,13] and the references cited there. In our numerical tests, the mesh size h is independent of the diffusion coefficient ε.the stabilization parameter ν is chosen as the scale of O(h) in order to stabilize the convective term appropriately. By a similar argument as in Lemma 4.1, we derive the error estimate for the adjoint equation. Lemma 4.2: Let for u K, p H0 1( ) H2 ( ) be the associated solution of the state Equation (5) and (7), and for z K, p h (z) Mh b denote the associated adjoint discrete solution, i.e: b(p h, w h ) + G(p h, w h ) = (y h y d, w h ), w h Mh b, then the following estimate holds: p p h u z 0 + cτh( y 2 + p 2 ). (29) Remark 4.2: when u = z, The error estimates of bubble-stabilization Galerkin method for convection-dominated diffusion equation are derived from (24)and(29) y y h cτh y 2. (30)

9 8 Z. WENG ET AL. and p p h cτh p 2. (31) In addition, we introduce the inactive set in optimum: I = x : c <u(x) < c. Note, that u = p/α on I and therefore u I H 2 ( I ). Using this subdomain I, we define a norm that will be needed later in the paper u 2,ad = ( u 2 W 1, ( ) + u 2 L 2 ( I )) 1/2. Then from [14], it has constructed a special interpolation u I K h of the solution u W 1, ( ), which fulfills the following conditions: j (u)(r u I ) 0, r K, (32) u u I 0 c α h3/2 u 2,ad. (33) Theorem 4.1: Let (y, p, u) and (y h, p h, u h ) denote the solutions to (5) (7)and(17) (19), respectively. Assume that y, p H 2 ( ). Then we have: and where τ is defined as in (25). Proof: From (16), we can get u u h 0 ch 3/2 u 2,ad + cτh( y 2 + p 2 ). (34) y y h + p p h ch 3/2 u 2,ad + cτh( y 2 + p 2 ). (35) By (19)and(32) with r = u h.weobtain: Hence, α u I u h 2 j h (u h)(u I u h, u I u h ) = j h (u I)(u I u h ) j h (u h)(u I u h ). j h (u h)(u I u h ) 0 j (u)(u I u h ). α u I u h 2 j h (u I)(u I u h ) j (u)(u I u h ). Using (4)and(15) and Cauchy Schwarz inequality, we can obtain: α u I u h 2 (p h p, u I u h ) + α(u I u, u I u h ) p h p u I u h +α u I u u I u h. (36) where p is the associated adjoint state to u and p h is the associated discrete adjoint state to u I.From (29), we have: α u I u h p h p +α u I u (1 + α) u u I 0 + cτh( y 2 + p 2 ). (37)

10 APPLICABLE ANALYSIS 9 Then, using (37)and(33) and applying the triangle inequality, we have u u h 1 + α α 2 ch 3/2 u 2,ad + c α hτ( y 2 + p 2 ). we have completed the proof of (34). Then let A h u and p h (u) be the bubble stabilization Galerkin solution of the following equation a(a h u, v h ) + G(A h u, v h ) = (f + u, v h ), v h Mh b, (38) b(p h (u), w h ) + G(p h (u), w h ) = (A h u y d, w h ), w h Mh b. (39) From (18) and(39), we can deduce that b(p h p h (u), w h ) + G(p h p h (u), w h ) = (y h A h u, w h ), w h M b h. Let w h = p h p h (u), we have that p h p h (u) C y h A h u. (40) Similarly, using (17)and(38) and setting v h = y h A h u, it can be proved that y h A h u C u u h 0. (41) Using (30)and(41) and applying the triangle inequality, we can obtain Similarly, combining (31), (40), and (41), we derive y y h y A h u + y h A h u Cτh y 2 + C u u h. (42) p p h p p h (u) + p h p h (u) Cτh p 2 + C u u h. (43) Summing up, (34), (42)and(43) prove the theoretical result (35). Remark 4.3: For the piecewise linear case, the optimal order of control variable is O(h 2 ). However, its the convergence order is only O(h 3/2 ), which is caused by the fact that u may not be smooth near the free boundary even if y and p are smooth there. 5. A priori error estimates for the variational discretization method Next, we consider a priori error estimates for the optimal control problem (5) (7) and its bubble function stabilization Galerkin approximation (17) (19) for the variational discretization method, i.e. K h = K. Theorem 5.1: Let (u, p, y) and (u h, p h, y h ) denote the solutions to (5) (7)and(17) (19)withK h = K, respectively. Assume that y, p H 2 ( ), then we have: u u h 0 + y y h + p p h cτh( y 2 + p 2 ), (44) where τ = ε 1/2 + h + ν 1/2 + ε 1/2 h.

11 10 Z. WENG ET AL. Proof: Let ū = u in (19)andū = u h in (7), then add the resulting inequalities. We can obtain α u u h 2 0 (p p h, u h u) = (p p h (u), u h u) + (p h (u) p h, u h u) = I + II where I = (p p h (u), u h u), II = (p h (u) p h, u h u) and p h (u) is the solution of the following equation: b(p h (u), q h ) + G(p h (u), q h ) = (y(u) y d, q h ), q h M b h. (45) By Cauchy Schwarz inequality and Young s inequality, we can obtain that I = (p p h (u), u h u) (46) p p h (u) 0 u h u 0 1 2α p p h(u) α 2 u h u 2 0. Choosing v h = p h p h (u) in (17)and(14) respectively, and subtracting (17) from (14) implies that II = a(a h u y h, p h p h (u)) + G(A h u y h, p h p h (u)). By definition and using (18)and(45) we have Combining (46)and(47)itgives II = a(a h u y h, p h p h (u)) + G(A h u y h, p h p h (u)) = b(p h p h (u), A h u y h ) + G(p h p h (u), A h u y h ) = (y y h, y h A h u) = (y y h, y h y) + (y y h, y A h u) 1 2 y y h y A hu 2 0. (47) α u u h y y h α p p h(u) y A hu 2 0. From above inequality and the a priori estimation for PDE, e.g. (30)and(31), we can get u u h 0 c( p p h (u) 0 + y A h u 0 ) cτh( y 2 + p 2 ). Finally we need to estimate y y h and p p h. From the definition of A h u in (18)andp h (u) in (45) and the stability of the Galerkin method we deduce that y h A h u u u h 0 (48) and p h p h (u) C y y h 0 C y y h. (49) Therefore the triangle inequality combined with (30), (31), (48)and(49)gives(44). Remark 5.1: IncomparisonwithTheorems4.1, 5.1 givestheconvergenceorderofcontrolvariable is O(h 3/2 ) in theory. But for variational discretization, the estimate of control variable is O(h 2 ) from

12 APPLICABLE ANALYSIS 11 numerical experiments because the method is not to discretize the space of admissible controls but to implicitly utilize the first-order optimality conditions and the relation between co-state and control for the discretization of the control. For more information on the features of u, please refer to [25]. 6. Numerical experiments In this section, we present two numerical examples to demonstrate our theoretical results. The governing equation in the first and the second examples are the linear and the nonlinear state equations, respectively. The state and the adjoint variable are approximated by Mh b and the control variable is approximated by K h. The stabilization parameter is given by ν = Ch and the choice of C accords with the discussion in [26]. The discrete optimality system (17) (19) is solved by the semismooth Newton method, see e.g. [25,27,28]. Example 6.1: 1 min u K (y y d ) 2 dx + α u 2 dx, (50) 2 2 subject to: and the adjoint equation as: β y ε y + y = f + u, in, (51) y = 0, on. (52) β p ε p + p = y y d,in, p = 0, on. Case a. Consider problem (50) (52) with β = (1, 2), α = 1andε = Computational domain =[0, 1] [0, 1] and the admissible set is K ={u L 2 ( ), 3 u 6}. To validate the theoretical results we consider the following given solution: y = sin (2πx 1 ) sin (2πx 2 ), p = π 2 sin (2πx 1 ) sin (2πx 2 ), u = min{ 3, max{6, p}}, and the corresponding f and y d are obtained by inserting y, p, u into the optimality system. In this case, the numerical solutions are computed on a series of triangular meshes, which are created from consecutive global refinement of an initial coarse mesh. At each refinement, every triangle is divided into four congruent triangles. Tables 1 and 2 shows the error of the given schemes. Table 1. Error estimates with ν = 0.1 h for standard method. 1/h y y h Rate p p h Rate u u h 0 Rate e e e e e e e e e e e e e e e e e e

13 Z. WENG ET AL. Table 2. Error estimates with ν = 0.1 h for variational discretization. 1/h y y h Rate p p h Rate u u h 0 Rate e e e e e e e e e e e e e e e e e e (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 1. Plots of exact solution y, p, u (top) and numerical solution y h, p h, u h (middle) for the discrete control variables and numerical solution y h, p h, u h for variational discretization (bottom) with linear case. One can find that two schemes obtain the similar order for the primal and the adjoint states, but the variational discretization get half order higher than the standard method, see the discussion in [20].

14 APPLICABLE ANALYSIS y y h p p h u u h y y h p p h u u h log2(error) log2(error) (a) log2(h) (b) log2(h) Figure 2. (a) L-shaped domain, (b) convergence orders of y, p, u in energy norm, and (c) convergence orders of y, p, u in energy norm for variational discretization. (a) (b) (c) (d) (e) (f) Figure 3. Plots of numerical solution y h, p h, u h for for the discrete control variables (top) and plots of numerical solution y h, p h, u h for variational discretization (bottom) withν = 0.01 h. (c) Moreover, we give the plots of exact and numerical solutions of two methods at the mesh h = 1/80 in Figure 1 for the detail. From these figures, we can see that numerical solutions approximate the exact ones well. Case b. We consider the L-shaped domain in [0, 1] [0, 1/2] [0, 1/2] [1/2, 1]. The domain is divided by the triangulations of mesh size h = 1/16 in Figure 2(a). The coefficients β = (1, 2), α = 1 and the admissible set is K ={u L 2 ( ), 4 u 4}. The true solution is choosing the same one

15 Z. WENG ET AL. (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 4. Plots of numerical solution y h, p h, u h for the discrete control variables (top) and plots of numerical solution y h, p h, u h for variational discretization (bottom) with nonlinear case. as in the Case a. The convergence orders are shown by slopes in Figure 2(b) and (c), which validate the theoretical results. Case c. Furthermore, we consider a problem without exact solutions. The parameters are β = (1, 3), α = 0.1, and ε = The admissible set is K ={u L 2 ( ),0.5 u 8}. The functions are f = 1 and y d = 1. To examine the stable properties of the discrete scheme, Figure 3 shows the numerical solutions by two methods on the mesh T h with h = 1/80. Example 6.2: In this example, we consider the following nonlinear optimal control problem: 1 min u K 2 (y y d ) 2 dx + α 2 u 2 dx, (53)

16 APPLICABLE ANALYSIS 15 Table 3. Error estimates ν = 0.1 h for standard method. 1/h y y h Rate p p h Rate u u h 0 Rate e e e e e e e e e e e e e e e e e e e e e Table 4. Error estimates ν = 0.1 h for standard method. 1/h y y h Rate p p h Rate u u h 0 Rate e e e e e e e e e e e e e e e e e e e e e subject to: and the co-state elliptic equation as: β y ε y + y + y 3 = f + u, in, (54) y = 0, on. (55) β p ε p + 3y 2 p = y y d,in, p = 0, on. Consider problem (53) (55) with β = (2, 3), α = 0.1 andε = The computational domain is [0, 1] [0, 1] and the admissible set is K ={u L 2 ( ), 10 u 5}. The exact solutions are taken as: y = 100(1 x 1 )x 1 (1 x 2 )x2 2, p = 50(1 x 1 )x 1 (1 x 2 )x2 2, u = min{ 10, max{5, p/α}}, and f and y d are obtained by inserting y, p, u into the associated optimality system. In this example, the numerical solutions are computed on a series of triangular meshes, which are created on uniform mesh. The errors are presented in Tables 3 and 4 for two methods, and two plots of exact and numerical solutions of two methods are given at the mesh h = 1/80 in Figure Conclusions In this paper, we discussed the bubble-stabilization Galerkin method for the constrained optimal control problem governed by convection dominated diffusion equations. The main feature of our stabilization method is using two local Gauss integrations to replace the projection operator, which introduces no additional variables. The discussion shows that our stabilization only depends on

17 16 Z. WENG ET AL. the bubble functions. Moreover, we obtain a priori error estimates of the standard method and the variational discretization method. The numerical examples are presented to demonstrate our theoretical results. There are several possible extensions for this research. This method can be extended to the optimal control problem governed by nonlinear convection dominated diffusion equations or the Navier Stokes equation. And the diffusion parameter independent analysis for some special case deserves further investigations. Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions which helped us to improve the manuscript. Disclosure statement No potential conflict of interest was reported by the authors. Funding The work of Z F Weng is partially supported by the Scientific Research Foundation of Huaqiao University [grant number 15BS307]. The work of J Z Yang is partially supported by National Natural Science Foundation of China [grant number ], [grant number ], and the research of X. Lu is partially supported by National Natural Science Foundation of China [grant number ], [grant number ]. References [1] Hughes TJR, Francea LP, Hulbert GM. A new finite element formulation for computational fluid dynamics: Viii. The Galerkin-least-square method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 1989;73: [2] Baiocchi C, Brezzi F, Francea LP. Virtual bubbles and Galerkin-least-square type methods. Comput. Methods Appl. Mech. Eng. 1993;105: [3] Brooks AN, Hughes TJR. Streamline upwind Petrov Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier Stokes equations. Comput. Methods Appl. Mech. Eng. 1982;32: [4] Russo A. Stream-upwind Petrov/Galerkin method (SUPG) vs residual-free bubbles (RFB). Comput. Methods Appl. Mech. Eng. 2006;195: [5] Brezzi F, Russo A. Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 1994;4: [6] Franca LP, Russo A. Deriving upwinding mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett. 1996;9: [7] Braack M, Burman E. Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 2006;43: [8] Matthies G, Skrzypacz P, Tobiska L. A unified convergence analysis for local projection stabilisations applied to the Oseen problem. M2AN Math. Model. Numer. Anal. 2007;41: [9] Zheng HB, Hou YR, Shi F. Adaptive variational multiscale methods for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 2010;229: [10] Burman E, Hansbo P. Edge stabilization for Galerkin approximations of convection-diffusion reaction problems. Comput. Method. Appl. Mech. Eng. 2004;193: [11] Knobloch P. A generalization of the local projection stabilization for convection diffusion reaction equations. SIAM J. Numer. Anal. 2010;48: [12] Wu X, Dai Y, Tao W. MLPG/SUPG method for convection-dominated problems. Numer. Heat Transfer B. 2012;61: [13] Song LN, Hou YR, Zheng HB. A variational multiscale method based on bubble functions for convectiondominated convection-diffusion equation. Appl. Math. Comput. 2010;217: [14] Becker R, Vexler B. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 2007;106: [15] Yan N, Zhou Z. 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. 2009;223:

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