A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems

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1 Zhou Fu Boundary Value Problems 4, 4:7 R S A R C H Open Access A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems Zhaojie Zhou * Hongfei Fu * Correspondence: zhouzhaojie@hotmail.com School of Mathematical Sciences, Shong Normal University, Jinan, 54, China Full list of author information is available at the end of the article Abstract In this paper a posteriori error estimate for continuous interior penalty Galerkin approximation of transient convection dominated diffusion optimal control problems with control constraints is presented. The state equation is discretized by the continuous interior penalty Galerkin method with continuous piecewise linear polynomial space the control variable is approximated by implicit discretization concept. By use of the elliptic reconstruction technique proposed for parabolic equations, a posteriori error estimates for state variable, adjoint state variable control variable are proved, which can be used to guide the mesh refinement in the adaptive algorithm. Keywords: transient convection diffusion optimal control problem; continuous interior penalty Galerkin method; elliptic reconstruction; a posteriori error estimate Introduction Transient convection diffusion optimal control problems are widely used to model some engineering problems, for example, air pollution problem [, ] waste water treatment [3]. In recent years the numerical approximations of this kind of problems form a hot topic, many works are contributed to developing effective numerical methods algorithms. For stabilization methods, we refer to [4 7] for discontinuous Galerkin methods, we refer to [8, 9]. For more literature, one can refer to the references cited therein. It is well known that the solutions to convection diffusion problems may have boundary layers with small widths where their gradients change rapidly. Therefore, only using the stable methods to solve convection diffusion optimal control problems is generally not enough. One approach to improve the quality of a numerical solution is to exploit special mesh which is locally refined near the boundary layers, for example, Shishkin-type mesh or adaptive mesh. Note that a priori knowledge of the locations of the boundary layers is necessary to construct Shishkin-type mesh. Using adaptive mesh to resolve the boundary layers seems to be more natural. As we know the key problem of the adaptive finite element method is the a posteriori error estimate. Compared with a posteriori error estimates for stationary convection diffusion optimal control problems see, [7, ]), the 4 Zhou Fu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly credited.

2 Zhou Fu Boundary Value Problems 4, 4:7 Page of 9 works devoted to a posteriori error estimates for transient convection diffusion optimal control problems are much fewer. In [3]the authors discuss adaptive characteristic finite element approximation of transient convection diffusion optimal control problems with a general diffusion coefficient, where a posteriori error estimates in L, T; L )) norm are derived by dual argument skill for the state adjoint state variables. The primary interest of this paper is to derive a posteriori error estimates for the following transient convection diffusion optimal control problem with dominance convection: min Jy, u)= yx, t) yd x, t) ) γ dxdt u x, t) dxdt.) u U ad T T subject to y t β y αy ε y = f u, yx, t)=, yx,)=y x), x, t) T =, T), x, t) Ɣ T =, T), x, t)..) The details will be specified in the next section. In order to improve the quality of the numerical solutions, the continuous interior penalty Galerkin method CIP Galerkin method) is used to solve the state equation.). This method was firstly proposed in [4]. In [7, 5]theCIPGalerkinmethodwasusedto approximate stationary convection diffusion optimal control problems, where a posteriori error estimates in L ) energy norm were derived. In [6]theCIPGalerkinmethod combined with Crank-Nicolson scheme was used to solve transient convection diffusion optimal control problems without constraints a priori error estimates were deduced. In the present paper, we apply the CIP Galerkin method combined with the backward uler method to solve control constrained transient convection diffusion optimal control problems.)-.), where the control is discretized by the implicit discretization method developed in [7], the state is approximated by piecewise linear finite element space. Due to the existence of boundary layer or interior layer for the state adjoint state as well as limited regularity of control variable, we derive a posteriori error estimates for the state adjoint state, which can be utilized to guide the mesh refinements in the adaptive algorithm. In contrast to [3], here we use the elliptic reconstruction technique developed in [8] for parabolic problems instead of dual argument skill to deduce the a posterior error estimates for the state adjoint state. By use of this technique we can take full advantage of the well-established a posteriori error estimates for stationary convection diffusion optimal control problems in [7, 5] toderivetheaposteriorerrorestimatefor transient convection diffusion optimal control problems. The paper is organized as follows. In Section we describe the continuous interior penalty Galerkin scheme for the constrained optimal control problem. In Section 3 aposteriori error estimates are derived. Finally, we briefly summarize the method used, results obtained possible future extensions challenges. Throughout this paper C > denotes a generic constant independent of mesh parameters may be different at different occurrence. We use the expression a b to st for a Cb.

3 Zhou Fu Boundary Value Problems 4, 4:7 Page 3 of 9 The CIP Galerkin approximation scheme. Problems formulation Consider the following transient convection diffusion optimal control problems: min Jy, u).) u U ad subject to y β y αy ε y = f u, x, t) T, yx, t)=, x, t) Ɣ T, yx,)=y x), x..) Here is a bounded domain in R with boundary. f L T )y x) H ) is the initial value. U ad = {u L T ):a ux, t) b a.e. in T } is a bounded convex set with two constants satisfying a < b. α > is the reaction coefficient, < ε isasmall diffusion coefficient, β W, )) is a velocity field. We assume that the following coercivity condition holds: α β α >. To consider the CIP Galerkin approximation of the above optimal control problem, we first derive a weak formulation for the state equation. Let A, )bethebilinearformgiven by Ay, w)=ε y, w)β y, w)αy, w), y, w H )..3) It is easy to check Ay, y) y,.4) where y = ε y, α y, ) /. Then the weak formulation of state equation.)readsas ) y, w Ay, w)=f u, w), w H ). The variational formulation of optimal control problem.)-.)thencanbewrittenas min Jy, u).5) u U ad subject to ) y, w Ay, w)=f u, w), w H )..6)

4 Zhou Fu Boundary Value Problems 4, 4:7 Page 4 of 9 The existence uniqueness of solutions to.5)-.6) can be guaranteed by the theory in [9]. Moreover, by using the Lagrange functional, the first-order necessary also sufficient here) optimality condition of.5)-.6) can be characterized by y, w)ay, w)=f u, w), w H ), z, w)aw, z)=y y d, w), w H ), γ u z, v u), v U ad..7) From the second equation in.7), we have that the adjoint state z satisfies transient convection diffusion equations with the strong form z ε z βz)αz = y y d, x, t) T, zx, t)=, x, t) Ɣ T, zx, T)=, x..8) In contrast to the state equation, the velocity field of the adjoint equation is β. By the pointwise projection on U ad, P Uad : L T ) U ad, P Uad v)=max a, min vx, t), b )),.9) the optimal condition in.7) simplifies to u = P Uad γ v )..). Semi-discrete discretization Let T h be a regular triangulation of,sothat = K.Leth K T h K denote the diameter of the element K.AssociatedwithT h is a finite dimensional subspace W h of C ) H ), consisting of piecewise linear polynomials. To control the convective derivative of the discrete solution sufficiently, a symmetric stabilization form S see, e.g.,[4]) on W h W h was introduced as follows: Sv h, w h )=σ h [ v h n][ w h n] ds, where σ > is the stabilization parameter. h denotes the collection of interior edges of the elements in T h. h is the size of the edge. [q] denotesthejumpofq across for h defined by [ qx) ] = lim s qx sn) qx sn) ), with n being the outward unit normal. Using the above stabilization form, a semi-discrete CIP Galerkin approximation of optimal control problem.)-.)isdefinedby min Jy h, u h ).) u h U ad

5 Zhou Fu Boundary Value Problems 4, 4:7 Page 5 of 9 subject to { y h, w h)ay h, w h )Sy h, w h )=f u h, w h ), w h W h, y h ) = y h W h..) Here the control variable was approximated by variational discrete concept see [7]). u h in general is not a finite element function associated with the space mesh T h. By stard argument it can be shown that the following first-order optimality condition holds: y h, w h)ay h, w h )Sy h, w h )=f u h, w h ), w h W h, z h, q h)aq h, z h )Sq h, z h )=y h y d, q h ), q h W h,.3) γ u h z h, v h u h ), v h U ad, y h ) = y h, z ht)=. By the pointwise projection operator P Uad,wehave u h = P Uad ) γ z h..4).3 Fully discrete scheme To define a fully discrete scheme, we introduce a time partition. Let = t < t < < t N < t N = T be a time grid with = t n t n, n =,,...,N.Set =t n, t n ]. Using variational discretization concept, the fully discrete CIP Galerkin scheme for.)-.) reads as follows: min J h y n u n h, u n ) h U h = ad y n h y n d, γ u n ) h,.5) subject to { yn h yn h, w h )Ay n h, w h)sy n h, w h)=f n u n h, w h), w h W h, y h = yh W h..6) Similar to a semi-discrete scheme, we can derive the discrete first-order optimality condition: yn h yn h, w h )Ay n h, w h)sy n h, w h)=f n u n h, w h), w h W h, zn h z h n, q h )Aq h, zh n )Sq h, zh n )=y n h yn d, q h), q h W h,.7) γ u n h zn h, v h u n h ), v h U ad, y h = yh, zn h =, n =,,...,N. Again by the pointwise projection operator P Uad,weobtain u n h = P U ad ) γ zn h. We can see that u n h is a piecewise constant function in time.

6 Zhou Fu Boundary Value Problems 4, 4:7 Page 6 of 9 For n =,,...,N,let Y h tn,t n ] = l n t)y n h l n t)y n h, Z h tn,t n ] = l n t)zh n l n t)zh n, U h tn,t n ] = u n h, where l n t)= t t n, l n t)= t n t. For ψ C, T; L )), t t n, t n ], we set ψ = ψx, t n ), ψ = ψx, t n ). Note that Y h = yn h yn h, Z h = zn h zn h for t t n, t n ]. Then the above optimality conditions can be rewritten as Y h, w h)aŷ h, w h )SŶ h, w h )=f n U h, w h ), w h W h, Z h, q h)aq h, Z h )Sq h, Z h )=Ŷ h y n d, q h), q h W h, γ U h Z h, v h U h ), v h U ad..8) 3 A posteriori error estimates The objective of this section is to derive a posteriori error estimates for the state, adjoint state control. 3. The estimate for control To obtain the estimate for control, we introduce an auxiliary problem. For given U h,let yu h ), zu h )) be a solution of the following system: yu h), w)ayu h ), w)=f U h, w), w H ), yu h )x,)=y x), x, zu h), q)aq, zu h )) = yu h ) y d, q), q H ), zu h )x, T)=, x. 3.) Lemma 3. Let y, z, u) Y h, Z h, U h ) be the solutions of.7).8), respectively. Then the following estimate u U h L,T;L )) C zuh ) Z L h,t;l )) Z ) h Z h L,T;L )) holds. Proof It follows from.7).8)that γ u U h L,T;L )) T = γ uu U h ) T γ U h u U h )

7 Zhou Fu Boundary Value Problems 4, 4:7 Page 7 of 9 = = T T T T T zu h u) T z Z h )U h u) γ U h u U h ) T γ U h Z h )u U h ) z zuh ) ) T ) U h u) zuh ) Z h Uh u) γ U h Z h )u U h ) z zuh ) ) T ) U h u) zuh ) Z h Uh u). Here the last inequality was fulfilled due to the implicit discretization of the control variable. By.7)3.)wehave T z zuh ) ) u U h ) = T = = T T y yu h )) z zu h )) y yuh ) ), z zuh ) ) T A y yu h ), z zu h ) ) y yuh ) ) T A y yu h ), z zu h ) ) where yu h )) = y x)zu h )T) = was used. Thus we arrive at γ u U h L,T;L )) T ) zuh ) Z h Uh u) Cδ) zuh ) Z h L,T;L )) Cδ) Z h Z h L,T;L )) Cδ u U h L,T;L )). Choosing δ = γ yields the theorem result. C 3. The estimate for the state adjoint state In this section we shall adopt the elliptic reconstruction technique proposed in [8, ]to derive a posteriori error estimates for the state adjoint state. To this end we first introduce the following elliptic reconstruction definitions for state adjoint state. Definition 3. For n =,,...,N, we define the elliptic reconstruction ν n H ) ω n H ) satisfying the following elliptic problems: A ν n, w ) = f n u nh yn h ) yn h, w, w H ) 3.)

8 Zhou Fu Boundary Value Problems 4, 4:7 Page 8 of 9 A q, ω n ) = y nh ynd zn h ) zn h, q, q H ). 3.3) Noticing that the CIP Galerkin approximation of ν n can be defined as A ν n h, w h Then we have ) ) S ν n h, w h = f n u n h yn h ) yn h, w h, w h Wh n τ. n A y n h νn h, w h) S y n h ν n h, w h) =, which implies y n h = νn h. We can observe a similar property for the CIP Galerkin approximation of ω n. Using the above convention, we define νt) ωt) as νt)=l n t)ν n l n t)ν n ωt)=l n t)ω n l n t)ω n for t n [, N]. We decompose the error as follows: yu h ) Y h t)=yu h ) νt) Y h t) νt) ) := ρ y ξ y zu h ) Z h t)=zu h ) ωt) Z h t) ωt) ) := ρ z ξ z. Nextly we shall derive the estimates of ρ ξ. For simplicity, we introduce the following notations: R n K,y = f n u n h Y h R n,y = [ y n h n], R n K,z = yn h yn d Z h R n,z = [ zh n n ], t ϕ n = ϕn ϕ n β y n h αyn h, βz h n ) αz n h,, t ϕn = t ϕ n t ϕ n. Let A s h Aa h be the discrete operators associated with the state adjoint state, which are defined by the following for v W h : A s h v, w h = Av, wh )Sv, w h ), w h W h, A a h v, w h = Awh, v)sw h, v), w h W h.

9 Zhou Fu Boundary Value Problems 4, 4:7 Page 9 of 9 Thetimeerrorestimatorsarecharacterizedby θ y,n = { t f n t u n h t yn h ), n N, f u h t y h As h y h, n = θ z,n = { t y n h t y n d t zn h ), n N, y N h yn d t zh N Aa h zn h, n = N. Moreover, let α K = min{α, ε h K }, α = min{α, ε h }. By the stard techniques used in a posteriori error estimate for stationary convection diffusion optimal control problems [7, 5], we obtain the following results. Lemma 3.3 Let ν n y n h be the solutions to 3.).6). Then the following a posteriori error estimate holds: ξ n y η y,n, where ηy,n = αk K T h R n K,y,K ε α εr n,y, α h R n,y,. Lemma 3.4 Let ω n zh n posteriori error estimate holds: ξ n z η z,n, be the solutions to 3.3).7). Then the following a where ηz,n = αk R n K,z,K ε α εr n,z, α h R n,z K T h,. Inthefollowingweshalldeducetheestimatesofρ y ρ z.by3.) Definition 3. we can derive the following error equations for ρ y ρ z. Lemma 3.5 Given t, we deduce ) ) ρy, ψ ξy Aρ y, ψ)=, ψ A ν n νt), ψ ) f f n, ψ ), ψ H ) 3.4) ) ) ρz, ψ ξz Aρ z, ψ)=, ψ A ψ, ω n ωt) ) y d y n d, ψ) yu h ) y n h, ψ), ψ H ). 3.5)

10 Zhou Fu Boundary Value Problems 4, 4:7 Page of 9 Proof Note that yuh ) ), ψ A yu h ), ψ ) =f U h, ψ), ψ H ) A ν n, ψ ) = f n U h Y ) h, ψ, ψ H ). Then we have ) ρy, ψ Aρ y, ψ) yuh ) = =f U h, ψ) ), ψ A yu h ), ψ ) νt) ) ξy, ψ Yh, ψ ), ψ A νt), ψ ) ) A νt), ψ ) = f n U h Y ) h, ψ ξy, ψ ) ξy =, ψ f f n, ψ ) A ν n νt), ψ ). ) f f n, ψ ) A νt), ψ ) Similarly we can deduce the error equation for ρ z. Before deriving the estimates for ρ y ρ z, we first introduce the following lemma with respect to a Clément-type interpolation operator. The proof can be found in [, ]. Lemma 3.6 Let I h be a quasi-interpolation operator of Clément type. The following estimates hold for all elements K, all faces all functions v H ): v I h v,k α K v,nk), v I h v, ε 4 α v,n), I h v,k v,nk), where NK) N) denote the union of all elements that share at least one point with K. Then we arrive at the following. Lemma 3.7 The following estimate holds: max t [,T] ρ y T ρy ) ρ y N θ y,n ) f t) f n )

11 Zhou Fu Boundary Value Problems 4, 4:7 Page of 9 [ N K T h α K ) α h ] t R n,y,. t R n K,y,K α ε 4 ε t R n,y, Proof Setting ψ = ρ y in 3.4)leadsto d ρy t) Aρ y, ρ y ) dt ) ξy =, ρ y A ν n ) νt), ρ y f f n ), ρ y. Integrating in time from to T gives ρy T) ρy ) ) T Aρ y, ρ y ) T ) T ξy =, ρ y A ν n ) T νt), ρ y f f n ), ρ y T ) ξy T, ρ y A ν n ) T νt), ρ y ) f f n, ρ y. Assume that ρ y t m ) = max t [,T] ρ y. Again integrating in time from to tm results in ) ρy t m ρy ) ) t m Aρ y, ρ y ) t ) m t ξy m =, ρ y A ν n ) t m νt), ρ y f f n ), ρ y T ) T A ν n νt), ρ y ) T ) f f n, ρ y ξy, ρ y. Combining the above two inequalities yields ) ρy t m ρy ) ) T Aρ y, ρ y ) T ) T A ν n νt), ρ y ) T f f n, ρ y ξy y), ρ := 3 T i. 3.6) i=

12 Zhou Fu Boundary Value Problems 4, 4:7 Page of 9 In the following we shall derive the estimates of T i. By the definition of the elliptic reconstruction, we can bound T as follows: T = which implies l n t) θ y,n, ρ y ) l n t) θ y,n ρ y θ y,n ) ρy t, m ) T Cδ) θ y,n δ ) ρ y t m with an arbitrarily positive constant δ. For the second term T, we can bound it as follows: T f t) f n ρy ) N ρy t m f t) f n N Cδ) ) f t) f n δ ρy t m). Now it remains to estimate T 3.Notethat T 3 = ν n ν n y n h ) yn h, ρ y. This term can be estimated by the techniques used in a posterior error estimates for the stationary problem. To this end we introduce an auxiliary problem { ε φ βφ)αφ = ρ y, φ =, in, on. For the above auxiliary problem, the following stability estimates see, e.g.,[3]) hold: 3.7) ε 3 φ ε φ φ C ρ y. 3.8) Using the above auxiliary problem, we have ν n ν n y n h ) yn h, ρ y = A ν n ν n y n h yn h, φ ).

13 Zhou Fu Boundary Value Problems 4, 4:7 Page 3 of 9 By the definitions of ν n y n n, we can deduce ν n ν n y n h ) yn h, ρ y = f n u n h t y n h f n u n A y n h yn h h t y n h, φ ) A y n h yn h, φ I h φ ), I h φ ) S y n h yn h, I h φ ) S y n h yn h, I h φ ) = f n u n h t y n h f n u n h t y n h, φ I h φ ) A y n h yn h, φ I h φ ) S y n h yn h, I h φ ), where I h φ denotes the Clément interpolation of φ.further,wehave ν n ν n y n h ) yn h, ρ y = t f n t u n h t yn h t β y n h αy n h), φ Ih φ ) K T h K [ε t y h n]i h φ φ) ds S t y h, I h φ) t R n K,y,K φ I h φ,k ε t R n,y, I h φ φ, K T h σ h t R n [,y, n Ih φ) ],. Note that [ I h φ) ], Ch I h φ),n), [ I h φ) ], Ch 3 I hφ,n). Then we derive or h t R n [,y, n Ih φ) ], C 3 h h t R n [,y, n Ih φ ) ], C h t R n Ih,y, φ),n) t R n,y, I h φ,n). This implies h t R n [,y, n Ih φ ) ], C α h t R n,y, I h φ,n).

14 Zhou Fu Boundary Value Problems 4, 4:7 Page 4 of 9 It follows from Lemma )that w n w n y n h ) yn h, ρ y C α K t R n K,y,K α ε 4 ε t R n,y K T h α h t R n,y, ) ρ y,. 3.9), Thus we arrive at T 3 C K T h α K ) α h t R n ),y, ρ y t m [ N Cδ) t R n K,y,K α ε 4 ε t R n,y K T h α K, t R n K,y,K α ε 4 ε t R n,y ) α h ] t R n,y, Cδ ) ρ y t m., Inserting the estimates of T, T T 3 into 3.6)settingδ small enough leads to the theorem results. Theorem 3.8 Let yu h ) Y h be the solutions of.7).3), respectively. Then the following estimate holds: T yuh ) Y h ρy ) Proof Note that T ) θ y,n f t) f n K T h α K ) α h t R n,y, 3 t R n K,y,K α ε 4 ε t R n,y ξ n y ξ n y, T T T T yu h ) Y h ρ y ξ y ρ y ξ y T ρ y 3 ξy n ξ y n ). ) ). 3.) Then by Lemmas we can deduce the estimates of T yu h) Y h.

15 Zhou Fu Boundary Value Problems 4, 4:7 Page 5 of 9 Remark 3.9 Note that yuh ) Y h ρy ξ y. 3.) The second term is the elliptic reconstruction error, which can bounded as follows for t : ξ y = ln ξy n l n ξy n max ξ n, ξ n ) y y. 3.) Then yuh ) Y L h,t;l )) max ρy t) max ξ y max ρy t) max ξ n. t t t n y Combining Lemmas , we can deduce yuh ) Y h L,T;L )) ρy ) θ y,n f t) f n K T h α K t R n K,y,K α ε 4 ε t R n,y ) α h t R n,y, max η y,n. 3.3) n [,N], Now we turn our attention to estimate zu h ) Z h. The argument skills are similar to those used in the estimate of yu h ) Y h. Therefore we just sketch the proof. Setting ψ = ρ z in 3.5)leadsto Let d dt ρz t) Aρ z, ρ z )= ξz, ρ z yu h ) y n h, ρ z). ) A ρ z, ω n ωt) ) y d y n d, ρ ) z max ρ z = ) ρ z t n. t [,T] Then integrating the above equation from tn to T to T,respectively,leadsto ) ρz t n ρz T) ) T Aρ z, ρ z ) T ) ξz T, ρ z A ρ z, ω n ω ) T yu h ) y n h, ρ z). T y d y n d, ρ z) In an analogous way to Lemma 3.7,wecerivetheestimateforρ z.

16 Zhou Fu Boundary Value Problems 4, 4:7 Page 6 of 9 Lemma 3. We have max ρ z t [,T] T ρz T) ρ z [ N N θ z,n ) yd y n d K T h α K ) t R n K,z,K α ε 4 ε t R n,z, ) α h ] N t R n,z, Yh y ) n h yuh ) Y h L,T;L )). Collecting Lemmas using similar arguments to Theorem 3.8 yields the following. Theorem 3. Let zu h ) Z h be the solutions of.7).3), respectively. Then the following estimates hold: T zu h ) Z h ) ρz T) N N τ θz,n n yd y n d Yh y n h [ N t R n K,z,K α ε 4 ε t R n,z, ] α h t R n,z, 3 α K K T h ξz n ξ z n ) ) yu h ) Y L h,t;l )). Remark 3. Similar to Remark 3.9, we can also derive the posteriori error estimates of zu h ) Z L h,t;l )) ρ z T) θ z,n y d y n d Y h y n h [ N t R n K,z,K α ε 4 ε t R n,z, ] α h t R n,z, α K K T h max n [,N] η z,n yuh ) Y h L,T;L )). 3.3 The main results By.7)3.) we can also derive T y yu h ) u U h L,T;L 3.4) ))

17 Zhou Fu Boundary Value Problems 4, 4:7 Page 7 of 9 T z zuh ) u U h L,T;L )). 3.5) Therefore, combining Lemma 3., Theorems3.8, 3., 3.4) 3.5), we can deduce the following estimates. Theorem 3.3 Let y, p, u) Y h, Z h, U h ) be the solutions of.7).3), respectively. Then the following estimate T ) T u U h L,T;L )) y Y h z Z h ) ηy η z holds, where η y = ρ y ) θ y,n f t) f n K T h α K t R n K,y,K α ε 4 ε t R n,y ) α h t R n,y, Y h y n h 3 η z = ρ z T) [ N, ξy n ξ y n ) ) θ z,n y d y n d Z h Z h L,T;L )) α K K T h ] α h t R n,z, 3 t R n K,z,K α ε 4 ε t R n,z, ξ n z ξ n z Remark 3.4 It follows from.7)3.)that y yu h ) L,T;L )) u U h L,T;L )) ) ). z zu h ) L,T;L )) u U h L,T;L )) Using the above estimate Lemma 3., we can derive the posteriori error estimates of u U h L,T;L )) y Y h L,T;L )) z Z h L,T;L )).

18 Zhou Fu Boundary Value Problems 4, 4:7 Page 8 of Conclusion In this paper a posteriori error estimates were established for time-dependent convection diffusion optimal control problems by the elliptic reconstruction technique. By introducing the elliptic reconstruction, we can take full advantage of the well-established a posteriori error estimates for stationary convection diffusion optimal control problems. There are still many issues needed to be addressed, such as optimal control problems with state constraints pointwisely imposed control problems. The applications of our approach to these settings will be postponed to our future work. Competing interests The authors declare that they have no competing interests. Authors contributions All authors read approved the final manuscript. Author details School of Mathematical Sciences, Shong Normal University, Jinan, 54, China. Department of Computational Applied Mathematics, China University of Petroleum, Qingdao, 6658, China. Acknowledgements The authors would like to thank the referees for their valuable comments suggestions. This work is supported by the National Natural Science Foundation of China Grant: 33, 485), the Science Technology Development Planning Project of Shong Province No. GGB98). Received: 5 March 4 Accepted: August 4 References. Parra-Guevara, D, Skiba, Y: lements of the mathematical modelling in the control of pollutants emissions. col. Model. 67, ). Zhu, J, Zeng, QC: A mathematical formulation for optimal control of air pollution. Sci. China, Ser. D 46, 994-3) 3. Martínez, A, Rodríguez, C, Vázquez-Méndez, M: Theoretical numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim. 38, ) 4. Becker, R, Vexler, B: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 6, ) 5. Hinze, M, Yan, NN, Zhou, ZJ: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 7, ) 6. Scott Collis, S, Heinkenschloss, M: Analysis of the Streamline Upwind/Petrov Galerkin Method applied to the solution of optimal control problems. CAAM TR-, Rice University, Houston ) 7. Yan, NN, Zhou, ZJ: A priori a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. J. Comput. Appl. Math. 3, ) 8. Leykekhman, D, Heinkenschloss, M: Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 54), -38 ) 9. Yan, NN, Zhou, ZJ: The local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations. Int. J. Numer. Anal. Model. 7, ). Dede, L, Quarteroni, A: Optimal control numerical adaptivity for advection diffusion-equations. Math. Model. Numer. Anal. 39,9-4 5). Micheletti, S, Perotto, S: The effect of anisotropic mesh adaptation on PD-constrained optimal control problems. SIAM J. Control Optim. 49, ). Yan, NN, Zhou, ZJ: A priori a posteriori error estimates of streamline diffusion finite element method for optimal control problem governed by convection dominated diffusion equation. Numer. Math., Theory Methods Appl., ) 3. Fu, HF, Rui, HX: Adaptive characteristic finite element approximation of convection-diffusion optimal control problem. Numer. Methods Partial Differ. qu. 9, ) 4. Burman,, Hansbo, P: dge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. ng. 93, ) 5. Yan, NN, Zhou, ZJ: A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations. Front. Math. China 3, ) 6. Burman, : Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations. Commun. Math. Sci. 9, ) 7. Hinze, M: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 3, ) 8. Makridakis, C, Nochetto, RH: lliptic reconstruction a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 4, ) 9. Lions, JL: Optimal Control of Systems Governed by Partial Differential quations. Springer, Berlin 97). Lakkis, O, Makridakis, C: lliptic reconstruction a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75, )

19 Zhou Fu Boundary Value Problems 4, 4:7 Page 9 of 9 Scott, LR, Zhang, S: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, ). Verfürth, R: A posteriori error estimators for convection-diffusion equations. Numer. Math. 8, ) 3. Navert, U: A finite element method for convection diffusion problems. PhD thesis, Chalmers Inst. of Tech. 98) doi:.86/s Cite this article as: Zhou Fu: A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems. Boundary Value Problems 4 4:7.