A priori error analysis of stabilized mixed finite element method for reaction-diffusion optimal control problems

Transcription

1 F et al. Bondary Vale Problems :23 DOI /s R E S E A R C H Open Access A priori error analysis of stabilized mixed finite element metod for reaction-diffsion optimal control problems Hongfei F 1*,HiGo 1,JianHo 2 and Jnlong Zao 1 * Correspondence: ongfeif@pc.ed.cn 1 College of Science, Cina University of Petrolem, Qingdao, , Cina Fll list of ator information is available at te end of te article Abstract In tis paper, we propose a novel stabilized mixed finite element metod for te approximation of optimal control problems governed by reaction-diffsion eqations. Compared wit te classical mixed finite element metods, te main contribtions of tis paper are as follows. First, te novel metod ses only one stabilization parameter wic is not mes-dependent, and, te new mixed bilinear formlation is coercive and continos. Second, te novel metod is easy to be implemented on a compter sing te standard Lagrange finite element. Tird, te soltions of te novel metod to te optimal control problems reqire low reglarities. Fort, te Ladyzenkaya-Babska-Brezzi LBB or te inf-sp condition for te mixed element spaces is nnecessary. Based on te novel metod, we derive bot continos and discrete optimality systems for te corresponding constrained optimal control problems, and ten apriorierror analysis in a weigted norm is discssed. Finally, nmerical experiments are given to confirm te efficiency and reliability of te novel stabilized metod. Keywords: optimal control; reaction-diffsion eqation; stabilized mixed finite element; apriorierror analysis; nmerical experiments 1 Introdction Optimal control problems and teir finite element soltions are attracting increasingly attentions of scientists and engineers. For systematic introdctions of finite element metods and teir applications in solving optimal control problems, we refer te reader to [1 6]. In tis paper, we are interested in te following distribted type optimal control problem: 1 min y d U ad 2 y σ d 2 σ 2 + γ 2, sbject to a first-order mixed type reaction-diffsion eqation { div σ + cy = f + B, σ + A y =0, in, in, F et al. Tis article is distribted nder te terms of te Creative Commons Attribtion 4.0 International License ttp://creativecommons.org/licenses/by/4.0/, wic permits nrestricted se, distribtion, and reprodction in any medim, provided yo give appropriate credit to te original ators and te sorce, provide a link to te Creative Commons license, and indicate if canges were made.

2 F et al. Bondary Vale Problems :23 Page 2 of 20 and combined wit omogeneos Diriclet bondary condition y =0, on. 1.3 Here γ > 0 is a penalty parameter, it is sed to measre te relative importance of te terms appearing in te definition of te cost fnctional; B is a linear operator; U ad is an admissible convex control set. Detailed information as regards tis problem in a fnctional analysis setting will be discssed later. Tis model problem plays an important role in many scientific and engineering applications. For example, it can represent an optimal control of Darcy flows, were te primal state variable y means te pressre, te flx state variable σ stands for te velocity field, and te control variable is an external force. Apriorierror estimates of finite element approximations for optimal control problems governed by linear elliptic eqations were stdied extensively; see, for example, Refs. [5, 7 10]. Recently, mixed finite element metod as also been fond very sefl in solving sc optimal control problems wic contain te flx state variable, see Refs. [11 16]. Bt alltese worksneed a matcedmixed finiteelementspacesfortestatevariableanditsflx,i.e., LBB stability condition mst be strictly satisfied. One of teir best coices is te Raviart-Tomas RT element, see [17], and terefore, te widely sed Lagrange elements are exclded. Inspired by te idea of so-called Galerkin least sqares metod, proposed by Hges et al. [18] and teoretically analyzed by Franca and Stenberg [19], andalsoteideaof so-called nsal stabilized finite element metod [20, 21] for advection-diffsion eqations, we propose ere a first-order mixed type stabilized finite element metod for te elliptic optimal control problems. Te novel metod is proved to be efficient and reliable bot in teoretical error analysis and nmerical tests. It also embodies some advantages compared wit te former work [22]: It ses only one stabilization parameter wic is not mes-dependent, owever, te corresponding mixed bilinear formlation is still coercive and continos; it is easier to be implemented sing te standard Lagrange finite elements; it as relatively lower reglarity reqirements of te soltions to optimal control problems; and it can be extended to solve optimal control problems governed by oter type partial differential eqations. Frtermore, compared wit te classical mixed finite element metod, it also as a free coice of mixed element spaces witot te reqirement of LBB stability condition, and less degree of freedoms DoFs are adopted. Ts, it is more competitive in te practical comptation and especially in te ig-dimensional case. Terestoftepaperisorganizedasfollows:InSection2, wefirstproposetenovel stabilized metod wic ses only one stabilization parameter. Ten some preliminary reslts are presented and an optimality system at te continos level is dedced for te optimal control problem. In te end of tis section, explicit soltions of different cases of te admissible control set are discssed. In Section 3, discretization sing standard Lagrange element is discssed, and discrete optimality system is also derived. In Section 4, an optimal-order weigted norm error analysis wit a low reglarity reqirement for te optimal control problem is discssed. In Section 5, we condct some nmerical experiments to verify te teoretical analysis. In te last section, some conclding remarks are given.

3 F et al. Bondary Vale Problems :23 Page 3 of 20 2 Optimal control problems and optimality system Let be a bonded domain in R d d 3 wit Lipscitz bondary.lett be a family of reglar trianglation of, sctat = T T T. Denote T te diameter of te element T in T,andset = max T T T.Intispaper,wesallemploytesalnotion for Lebesge and Sobolev spaces; see Ref. [1] for details. Trogot, let C denote a strictly positive generic constant, not necessarily te same at eac occrrence, bt always independent of te mes size. To give a detailed description of te model optimal control problem, let s define te state spaces Y = H 1 0, = L2 d. 2.1 and te control space U = L Frtermore, let U ad beaclosedconvexsbsetoftecontrolspaceu. In te following context, we will discss some different cases on te coice of U ad. In te following, we are ready to propose a simple bt stable mixed nmerical metod. To tis aim, we assme A =a i,j x d d is symmetric and positive definite, and tere exist positive constants α and β sc tat α X 2 X T AX β X 2, X R d. Besides, sppose c = cx is bonded below and p by two constants c and c sc tat 0 c c c. We now first revisit te classical mixed formlation of problem : Find y, σ L 2 Hdiv; sc tat for any v, τ L 2 Hdiv; A 1 σ, τ y, div τ+div σ, v+cy, v=f + B, v. 2.3 Ten followed by integration by parts, and inspired by te stabilized finite element metod [18 20], we propose a novel stabilized mixed weak formlation for : find y, σ Y sc tat A y, σ ; v, τ=f + B, v, v, τ Y, 2.4 were te mixed bilinear formlation A y, σ ; v, τ= A 1 σ, τ + y, τ σ, v+cy, v A 1 σ + y, τ A v T. 2.5 T T T Here T is an elementwise mes-free constant parameter, and, T denotes te inner prodct in L 2 T d.

4 F et al. Bondary Vale Problems :23 Page 4 of 20 Define te corresponding stabilization norm {y, σ } 2 = A 1 σ, σ + T T T A y, y T +cy, y. 2.6 Ten, we can easily derive te following coercive and bonded reslts. Proposition 2.1 Coercivity and bondedness For 0< 0 T 1 <1,we ave A y, σ ; y, σ c {y, σ } 2, y, σ Y. 2.7 Moreover, for all y, σ ; v, τ Y 2 A y, σ ; v, τ C {y, σ } {v, τ}. 2.8 Here i i =0,1are two positive constants, c =1 1 and C = max{4, 2 + 1/ 0 }. Proof On te one and, it follows from te definition tat A y, σ ; y, σ = A 1 σ, σ +cy, y T T T A 1 σ, σ T + T T T A y, y T. 2.9 Hence, for T 1 <1,tebilinearforma, ;, iscoerciveony, and it satisfies 2.7. On te oter and, we conclde from 2.5tat A y, σ ; v, τ= A 1 σ, τ + y, τ σ, v+cy, v A 1 σ, τ T + Note tat i a i b i T T σ, v T T T T T T y, τ T + T A y, v T T T T T i a 2 i b 2 i. Ten for 0 < 0 T 1 <1,weave i A y, σ ; v, τ [ 4 A 1 σ, σ +A y, y+2 ] 1 2 T A y, y T +cy, y T T [ 4 A 1 τ, τ +A v, v+2 T A v, v T +cv, v T T ] 1 2

5 F et al. Bondary Vale Problems :23 Page 5 of 20 [ 4 A 1 σ, σ ] 1 2 T A y, y T +cy, y 0 T T [ 4 A 1 τ, τ T A v, v T +cv, v 0 T T C {y, σ } {v, τ}, 2.11 ] 1 2 wic proves te bondedness reslt 2.8. Proposition 2.2 Existence and niqeness Assme te condition in Proposition 2.1 is valid. Frtermore, let f L 2. Ten for given control L 2, problem 2.4 admits a niqe soltion y, σ Y. Proof Proposition 2.1 implies tat te mixed bilinear form A, ;, iscoerciveand bonded in a weigted norm 2.6. Ten te Lax-Milgram lemma implies te existence and niqeness of te soltion pair y, σ Y. Frtermore, sppose =0inproblem2.4, we ten ave te following stability reslt wit respect to te rigt-and term f. Proposition 2.3 Stability Let f L 2. If te condition in Proposition 2.1 is valid, ten we ave {y, σ } C f L 2, 2.12 were te constant C depends on te Poincaré constant C, and te reciprocal of c and 0. Proof Let v, τ=y, σ inproblem2.4. Ten it follows from Proposition 2.1,tePoincaré ineqality in Lemma 4.3, and te definition of te stabilization norm tat c {y, σ } 2 f L 2 y L 2 C f L 2 y L 2 C 1/2 f L 2 T y, y T, 0 T T wic implies te conclsion. Denote J y, σ, = 1 2 y y d σ σ d 2 + γ 2 were y = yandσ = σ are-dependent. Ten for te given control set U ad, we reformlate te optimal control problem as follows: OCP 2, J y, σ, = min U ad J y, σ,

6 F et al. Bondary Vale Problems :23 Page 6 of 20 sc tat y, σ, Y U and A y, σ ; v, τ=f + B, v, v, τ Y. It ten follows from Ref. [3] tat te optimal control problem OCP as a niqe soltion y, σ, Y U ad,andy, σ, is te soltion of OCP if and only if tere is a pair of adjoint state z, ω Y, sctaty, σ, z, ω, Y 2 U ad satisfies te following optimality system: OS State eqation: A y, σ ; v, τ = f + B, v, v, τ Y Adjoint state eqation: A v, τ; z, ω = y y d, v σ σ d, τ, v, τ Y Optimality condition: γ B z, 0, U ad Here B is te adjoint operator of B,wicsatisfies Bv, w= v, B w, v, w L 2 H 1 0. Remark 2.4 For te stabilization parameter T being cosen as a constant in te wole domain,teadjointstatesz and ω in 2.14 satisfy te following strong forms: { z = y y d divσ σ d, in, z =0, on, 2.16 and ω = z σ σ d, in In te following, we introdce z and ω be te soltions of te dal problem sc tat A v, τ; z, ω= g, v q, τ, v, τ Y Ten, similar to te proof in Proposition 2.3, we ave te following stability reslt. Proposition 2.5 Stability Let g L 2 and q L 2 d in problem Assme te condition in Proposition 2.1 is valid. Ten we ave {z, ω} C g L 2 + q L 2 d, 2.19 were te constant C depends on te Poincaré constant C, and te reciprocal of c and 0.

7 F et al. Bondary Vale Problems :23 Page 7 of 20 Remark 2.6 Let f, L 2. If te condition in Proposition 2.1 is valid, ten from Proposition 2.3 we ave { y, σ } C f L 2 + L Frtermore, let y d L 2 andσ d L 2 d. Ten from Proposition 2.5 and te above conclsion we ave { z, ω } C y y d L 2 + σ σ d L 2 d C f L 2 + L 2 + y d L 2 + σ d L 2 d In te end of tis section, we pay special attention on te soltion of te variational ineqality It depends eavily on te strctre of te convex set U ad. For some cases see, e.g. [3, 4], we ave te following explicit reslts. Case I. U ad = U Ten te soltion is x= 1 γ B z x Case II. U ad = { U : 0, a.e. in } Ten te soltion is { x=max 0, 1 } γ B z x Case III. U ad = { U : a b,a.e.in } were te bonds a, b R flfill a < b. Ten te soltion is { { x=max a, min b, 1 }} γ B z x Case IV. U ad = { U : 0} Ten te soltion is { x=max 0, 1 } γ B z x + 1 γ B z x, 2.25 were B z x= B z x 1. 3 Stabilized mixed finite element approximation In tis section, we sall consider te approximation of problem OCP based on te novel stabilized mixed finite element metod. As te bilinear form A, ;, is coercive, tere is no need to coose te classical mixed element spaces, for example, RT elements [17, 23], as te flx fnction space. Instead, te widely sed Lagrange finite element spaces, for example, te case of piecewise linear elements, can be adopted.

8 F et al. Bondary Vale Problems :23 Page 8 of 20 In tis work, we consider te approximations of te state and flx state variables in te following finite element spaces: Y = { v C 0 :v T P 1 T, T T, v =0on } Y, = { τ L 2 :τ i T P 1 T, i =1,2,...,d, T T }. 3.1 Here P k denotes polynomials of total degree at most k. Frtermore, we consider piecewise constant elements for te approximation of te control variable, tat is, U = { U : T P 0 T, T T }. 3.2 Let U ad = U U ad be te discrete admissible control set. It is apparently so tat U ad U ad. For te finite element spaces defined above, te stabilized mixed finite element approximation of OCP, wic will be labeled as OCP, can be described as follows: J y, σ, = min J y, σ,, 3.3 U ad were y, σ, Y U satisfies A y, σ ; v, τ =f + B, v, v, τ Y. 3.4 It is again well known see Ref. [3] tat te optimal control problem OCP as a niqe soltion y, σ, Y U ad,andtatatriplety, σ, is te soltion of OCP if and only if tere is a pair of adjoint states z, ω Y,sctaty, σ, z, ω, Y 2 U ad satisfies te following discrete optimality system: OS State eqation: A y, σ ; v, τ = f + B, v, v, τ Y. 3.5 Adjoint state eqation: A v, τ ; z, ω = y y d, v σ σ d, τ, v, τ Y. 3.6 Optimality condition: γ B z, 0, Uad. 3.7 Remark 3.1 Te coercivity of te mixed bilinear formlation A, ;, leadstoapositive definite linear algebraic eqation in te discrete level, and terefore, te discrete state and adjoint state eqations can be solved qickly sing te poplar solvers, sc as te conjgate gradient CG solver and algebraic mlti-grid AMG solver. Let P be an L 2 -projection from U = L 2 tou sc tat for any U P, φ=0, φ U. 3.8

9 F et al. Bondary Vale Problems :23 Page 9 of 20 It is a matter of calclation tat for te optimal control U ad, te projection P belongs to Uad. For example, we can give a proof for Case IV. In fact, let φ 1 U in eqation 3.8, ten we ave P = 0, 3.9 wic proves te conclsion. Te proofs of te oter cases are similar. Besides, it is easy to ceck tat P T = 1, T T, 3.10 T T were T is te area of element T. Finally, similar to te explicit soltions to te variational ineqality 2.15 for different cases. Te soltion of 3.7 can also be described explicitly for te corresponding cases; see [3, 4]. We smmarize tem as below. Case I. Uad = U Ten te soltion is = 1 γ P B z Case II. Uad = { U : T 0, T T } Ten te soltion is = max {0, 1 γ P B z } Case III. U ad = { U : a b,a.e.in } Ten te soltion is { {a, = max min b, 1 γ P B z }} Case IV. U ad = { U : 0} Ten te soltion is = max {0, 1 γ B z } + 1 γ P B z Apriorierror estimates In tis section, we sall give apriorierror estimates for te proposed novel stabilized mixed finite element metod of optimal control problem. Before tat let s first recall te following interpolation and projection reslts. Lemma 4.1 [1] Let P be te L 2 -projection defined in 3.8. Ten for H 1, we ave P L 2 C H

10 F et al. Bondary Vale Problems :23 Page 10 of 20 Lemma 4.2 Let I be te standard Lagrange interpolation operator defined in Ref. [1]. Ten tere is a constant C >0sc tat v I v L 2 T + T v I v H 1 T C 2 T v H 2 T, 4.2 for v H 2 T, T T. Lemma 4.3 Poincaré ineqality Tere is a positive constant C wic depends only on te domain sc tat v L 2 C v L 2 d, v H In tis paper, we aim to demonstrate te following main conclsion between te optimal soltions of OS and te stabilized mixed finite element soltions of OS. Teorem 4.4 Sppose tat y, σ, z, ω, and y, σ, z, ω, are te soltions of OS and OS, respectively. Assme tat te soltions {y, z } H0 1 H2, {σ, ω } H 1 d, and H 1. Ten tere is a positive constant C sc tat L 2 + { y y, σ } σ + { z z, } ω ω [ C + H1 v=y,z;τ=σ,ω v H 2 + τ H 1 d ]. 4.4 Remark 4.5 Compared wit Ref. [22], a same optimal-order convergence between te exact soltions and nmerical soltions is obtained. However, te reqirement of reglarities for te flx state σ and adjoint flx state ω are bot redced from H 2 d to H 1 d. Tis appears to be a more realistic assmption if te original state eqation is only H 2 -reglar, and if te given data f, y d, σ d and te optimal control all belong to L 2. In particlar, for Cases I and IV, we can predict tat te optimal control C asif te given data are sfficiently smoot. Indeed, from 2.22 and2.25we can observe tat te reglarity of te optimal control agrees wit tat of te adjoint state z. To derive te above main reslt, we introdce y, σ, z, ω Y 2 astediscreteintermediatevariables.teyareassociatedwitteoptimalcontrol soltion U ad and satisfy A y, σ ; v, τ = f + B, v, 4.5 A v, τ ; z, ω = y y d, v σ σ d, τ, 4.6 for any v, τ Y. For simplicity of presentation, below let s denote tose soltions of corresponding to te optimal control ỹ, σ, z, ω y, σ, z, ω. Now we are ready to prove Teorem 4.4 in tree steps. First, we prove a direct reslt between te intermediate variables and te nmerical soltions.

11 F et al. Bondary Vale Problems :23 Page 11 of 20 Lemma 4.6 Let y, σ, ω, z and ỹ, σ, z, ω be te soltions of and , respectively. Ten te following estimates old: { y ỹ, σ σ } + { z z, ω ω } C L Proof On te one and, we conclde from 3.5and4.5tat A y ỹ, σ σ ; v, τ = B, v, v, τ Y. 4.8 Selecting v = y ỹ and τ = σ σ in 4.8. It ten follows from te stability reslt in Proposition 2.3 tat { y ỹ, σ σ } C L On te oter and, we dedce from 3.6and4.6tat A v, τ ; z z, ω ω = y ỹ, v σ σ, τ, 4.10 for any v, τ Y. Let v = z z and τ = ω ω. Following te stability reslt in Proposition 2.5 and te conclsion 4.9, we derive { z z, ω ω } C { y ỹ, σ σ } C L Terefore, te proof of Lemma 4.6 is ended. Ten we trn to te validation of an optimal-order convergence between te intermediate variables and te exact soltions. Lemma 4.7 Let y, σ, z, ω and ỹ, σ, z, ω be te soltions of and , respectively. Frtermore, assme te soltions {y, z } H0 1 H2 and {σ, ω } H 1 d. Ten te following estimates old: { } y ỹ, σ σ C y H 2 + σ H 1, 4.12 d { } z z, ω ω C v H 2 + τ H d v=y,z;τ=σ,ω Proof Firstofall,itiscleartat4.5 is te mixed finite element approximation of Terefore, te reslt 4.12 can easily be proved by te interpolation estimation teory in Lemma 4.2. Second, we obtain by sbtracting 4.6 from2.14 tat A v, τ ; z z, ω ω = y ỹ, v σ σ, τ, 4.14 for any v, τ Y.

12 F et al. Bondary Vale Problems :23 Page 12 of 20 Eqivalently, 4.14canbeexpressedas A v, τ ; I z z, I ω ω = A v, τ ; I z z, I ω ω y ỹ, v σ σ, τ, 4.15 were I z Y and I ω are te standard Lagrange piecewise linear interpolation of z and ω,respectively. Let v = I z z and τ = I ω ω in eqation It ten follows from Proposition 2.1, Lemma 4.3, and te Cacy-Scwarz ineqality tat c { I z z, I ω ω } 2 CC, 0, C { I z z, I ω ω } + { y ỹ, σ σ } { I z z, I ω ω }, wic implies { I z z, I ω ω } C { I z z, I ω ω } + { y ỹ, σ σ } Ten te reslt 4.13 can be derived by te triangle ineqality, Lemma 4.3, 4.12 and 4.16; we ave { z z, ω ω } { I z z, I ω ω } + { I z z, I ω ω } C { I z z, I ω ω } + { } y ỹ, σ σ C v H 2 + τ H d v=y,z;τ=σ,ω Ts, Lemma 4.7 is proved. Smmarizing Lemmas wit a simple application of te triangle ineqality, it is easy to obtain { y y, σ } σ + { z z, } ω ω { y ỹ, σ σ } + { y ỹ, σ σ } + { } z z, ω ω + { z z, ω ω } [ C L 2 + v H 2 + τ ] H 1 d v=y,z;τ=σ,ω Terefore, te last step is to concentrate on estimating te L 2 -norm errors between te continos and discrete optimal control.

13 F et al. Bondary Vale Problems :23 Page 13 of 20 Proof of Teorem 4.4 Note tat as proved in 3.8 tatp Uad for all different cases of U ad. It ten follows from te optimality conditions 2.15and3.7tat γ B z, 0, γ B z, P Ts, we ave γ 2 L 2 = γ, γ, = γ B z, + γ B z, P + γ B z, P + B z z, γ B z, P + z z, B In te following, we try to estimate te two terms on te rigt-and of First, it follows from te definition of P in 3.8tat γ B z, P = B z z, P B z P B z, P = z z, B P + z z, B P B z P B z, P Ts, we conclde from te Cacy-Scwarz ineqality, te bondedness of B, andlem- mas 4.1, 4.3 tat z z, B P C z z 2 L 2 + C P 2 L 2 CC, 0 { } z z, ω ω 2 + C P 2 L 2 [ C 2 2 H 1 + v 2 H 2 + τ 2 ] H 1 d, v=y,z;τ=σ,ω z z, B P Cε { z z, ω ω } 2 + C P 2 L 2 Cε 2 L 2 + C2 2 H 1, B z P B z, P C z P z 2 L 2 + C P 2 L 2 C 2[ 2 H 1 + z 2 H 1 ]. Here and ereafter ε is a small positive constant. Similarly, for te second term on te rigt-and side of 4.20weave z z, B = z z, B + z z, B, 4.22

14 F et al. Bondary Vale Problems :23 Page 14 of 20 were z z, B C { } z z, ω ω 2 + Cε Cε [ 2 L 2 + C2 2 H 1 + and following 4.8and4.10weave 2 L 2 v=y,z;τ=σ,ω z z, B = A y ỹ, σ σ ; z z, ω ω = y ỹ, y ỹ σ σ, σ σ 0. v 2 H 2 + τ 2 H 1 d ], Finally, collecting all tese bonds togeter wit ε being small enog, we ave [ L 2 C H 1 + v=y,z;τ=σ,ω v H 2 + τ H 1 d ] Frtermore, inserting te above estimate into 4.18 we directly ave { y y, σ } σ + { z z, } ω ω [ C + v H1 H 2 + ] τ H 1 d v=y,z;τ=σ,ω Ts, Teorem 4.4followsimmediately from 4.23and Nmerical experiments In tis section, we present some nmerical reslts of te novel stabilized mixed finite element metod defined in Section 3 wic confirm te teoretical analysis of te previos section. As for te constrained optimal control problem, te states and control are te main concern of te practical problem. Terefore, in te following tests, we mostly focs on te reslts of te state y,teflxstateσ, and te control. Let =[0,1] [0, 1]. We are ready to test te following type optimal control problem: 1 min y d U ad 2 y σ d 2 σ , sbject to { y = f +, σ = y, in, in. 5.2 For simplicity, we coose = 0.8 as te stabilized parameter. Frtermore, to solve te constrained optimal control problem nmerically, we adopt te following iterations sing amatlabenvironment.

15 F et al. Bondary Vale Problems :23 Page 15 of 20 Algoritm Step 1. Give an initial control soltion 0 U ad and tolerance TOL =1.0E-06. For n =1,2,...,Nmax =100 Step 2. Solve y n, σ n Y as follows: Step 2.1. Given initial σ n,0 and tolerance TOL =1.0E-03. For k =1,2,...,Nmax =20 Step 2.2. Solve y n,k Y sc tat y n,k, v = f + n 1 Step 2.3. Solve σ n,k σ n,k, τ = y n,k, τ. Step 2.4. Stop ntil σ n,k y n, v +1 σ n,k 1, v. sc tat = yn,k, σ n = σ n,k. σ n,k 1 L 2 2 < TOL is satisfied, and let Step 3. Solve z n, ωn Y as follows: Step 3.1. Given initial ω n,0 and tolerance TOL. For k =1,2,...,Nmax Step 3.2. Solve z n,k z n,k, v = y n Step 3.3. Solve ω n,k Y sc tat y d, v 1 ω n,k 1, v. sc tat ω n,k, τ = z n,k 1, τ σ n σ d, τ. 1 Step 3.4. Stop ntil ω n,k z n = zn,k, ω n = ωn,k. ω n,k 1 L 2 2 < TOL is satisfied, and let Step 4. Solve n n zn Step 5. Stop if n U ad, v n 0. sc tat n 1 L 2 < TOL is satisfied, and let = n, y = y n, σ = σ n, z = z n, ω = ω n. Example 1 For te first example, we coose σ d = σ and consider Case III wit a =0,b = 0.5. Te corresponding analytical soltions of te optimal control problem are as follows: yx=sinπx 1 sinπx 2,

16 F et al. Bondary Vale Problems :23 Page 16 of 20 Table 1 Errors, convergence orders and CPU time for Example 1 Mes L2 Order {y y, σ σ } Order CPU time 8*8 16*16 32*32 64* E E E E E E E E s 26s 127s 761s Figre 1 Te nmerical control and corresponding error for Example 1. Figre 2 Te nmerical state y and corresponding error for Example 1. σ x = y, zx = sinπx sinπx, ωx = z, x = max, min{., z}, were te sorce fnction f can be compted sing., wile te desired state yd is determined by..

17 F et al. Bondary Vale Problems :23 Page 17 of 20 Figre 3 Te first component of nmerical flx state σ and corresponding error for Example 1. Figre 4 Te second component of nmerical flx state σ and corresponding error for Example 1. We can see tat te soltions of tis example are -independent. Table displays te errors and convergence orders wit respect to te decreasing niform mes size for te control in L -norm, te states y and σ in weigted norm. Te main CPU time for te comptation exclding te mes generation part is also listed. It ses no more tan five cycles for te iteration algoritm. Figre sows te nmerical soltion and nmerical error of te control wen mes size = /, wile te nmerical soltions and errors of te state and flx state are presented in Figres -, respectively. It can be observed tat te nmerical reslts are in agreement wit te analytical soltions very well. Example For te second example, we consider a nonomogeneos Diriclet bondary problem. Let s take σd = and consider Case IV wit z =. Te corresponding soltions of te optimal control problem are as follows: yx = x x + x x, σ x = y, zx = x x x x,

18 F et al. Bondary Vale Problems :23 Page 18 of 20 Table 2 Errors, convergence orders and CPU time for Example 2 Mes L2 Order {y y, σ σ } Order CPU time 8*8 16*16 32*32 64* E E E E E E E E s 37s 172s 973s Figre 5 Te nmerical control and corresponding error for Example 2. Figre 6 Te nmerical state y and corresponding error for Example 2. ωx = z σ σd, x = z, were te sorce fnction f and te desired states yd can also be determined by. and., respectively. Tis is a -dependent example, i.e., te adjoint flx state ω depends on te parameter. In Table, we also list te corresponding errors, convergence orders and CPU time wit respect to different meses. Figres - sow te plots of te nmerical control, state, and flx state, respectively for = /. We can observe tat te convergence orders are

19 F et al. Bondary Vale Problems :23 Page 19 of 20 Figre 7 Te first component of nmerical flx state σ and corresponding error for Example 2. Figre 8 Te second component of nmerical flx state σ and corresponding error for Example 2. consistent wit Teorem.. Besides, te nmerical reslts are also well matced wit te analytical soltions. 6 Conclding remarks In tis paper, we discss a novel stabilized mixed finite element metod for te approximation of reaction-diffsion optimal control problem. Compared wit or previos work, te novel metod is more simple and easier to be implemented. It needs only one stabilization parameter bt is still stable. Frtermore, low reglarities for te state and adjoint state variables are needed. Different cases of te admissible control set are discssed and a priori error estimates are proved. Finally, nmerical experiments are addressed to demonstrate te teoretical analysis. Most importantly, we sold point tat tis novel stabilized mixed metod is more competitive. It can easily be extended to solve optimal control problems governed by a bilinear state eqation, te Stokes eqation and so on. Competing interests Te ators declare tat tey ave no competing interests.

20 F et al. Bondary Vale Problems :23 Page 20 of 20 Ators contribtions All ators read and approved te final manscript. Ator details 1 College of Science, Cina University of Petrolem, Qingdao, , Cina. 2 College of Petrolem Engineering, Cina University of Petrolem, Qingdao, , Cina. Acknowledgements Tis work was spported by te National Natral Science Fondation of Cina Nos , , te Promotive Researc Fnd for Excellent Yong and Middle-aged Scientists of Sandong Province No. BS2013NJ001, and te Fndamental Researc Fnds for te Central Universities Nos. 14CX02217A, 15CX08004A, 15CX08011A. Te first ator was partially spported by te Cina Scolarsip Concil No Received: 3 October 2015 Accepted: 12 Janary 2016 References 1. Ciarlet, PG: Te Finite Element Metod for Elliptic Problems. SIAM, Piladelpia Brenner, SC, Scott, LR: Te Matematical Teory of Finite Element Metods. Springer, New York Lions, JL: Optimal Control of Systems Governed by Partial Differential Eqations. Springer, Berlin Li, W, Yan, N: Adaptive Finite Element Metods for Optimal Control Governed by PDEs. Series in Information and Comptational Science, vol. 41. Science Press, Beijing Tiba, D: Lectres on te Optimal Control of Elliptic Eqations. University of Jyväskylä Press, Jyväskylä Hinze, M, Pinna, R, Ulbric, M, Ulbric, S: Optimization wit PDE Constraints. Springer, Berlin Falk, FS: Approximation of a class of optimal control problems wit order of convergence estimates. J. Mat. Anal. Appl. 44, Hinze, M: A variational discretization concept in control constrained optimization: te linear-qadratic case. Compt. Optim. Appl. 30, Meyer, C, Rösc, A: Sperconvergence properties of optimal control problems. SIAM J. Control Optim. 43, Arada, N, Casas, E, Tröltzsc, F: Error estimates for te nmerical approximation of a semilinear elliptic control problem. Compt. Optim. Appl. 23, Cen, Y, Li, W: Error estimates and sperconvergence of mixed finite elements for qadratic optimal control. Int. J. Nmer. Anal. Model. 3, Yan, N, Zo, Z: A RT mixed FEM/DG sceme for optimal control governed by convection diffsion eqations. J. Sci. Compt. 41, Cen, Y, L, Z: Error estimates of flly discrete mixed finite element metods for semilinear qadratic parabolic optimal control problem. Compt. Metods Appl. Mec. Eng. 199, Zo, J, Cen, Y, Dai, Y: Sperconvergence of trianglar mixed finite elements for optimal control problems wit an integral constraint. Appl. Mat. Compt. 217, F, H, Ri, H: A caracteristic-mixed finite element metod for time-dependent convection-diffsion optimal control problem. Appl. Mat. Compt. 218, Gong, W, Yan, N: Mixed finite element metod for Diriclet bondary control problems governed by elliptic PDEs. SIAM J. Control Optim. 49, Raviart, PA, Tomas, JM: A mixed finite element metod for 2nd order elliptic problems. In: Matematical Aspects of Finite Element Metods. Lectre Notes in Matematics, vol. 606, pp Springer, Berlin Hges, TJR, Franca, LP, Hlbert, G: A new finite element formlation for comptational flid dynamics: VIII. Te Galerkin/least-sqares metod for advective-diffsive eqations. Compt. Metods Appl. Mec. Eng. 73, Franca, LP, Stenberg, R: Error analysis of Galerkin least sqares metods for te elasticity eqations. SIAM J. Nmer. Anal. 28, Franca, LP, Frey, SL, Hges, TJR: Stabilized finite element metods: I. Application to te advective-diffsive model. Compt. Metods Appl. Mec. Eng. 95, Franca, LP, Valentin, F: On an improved nsal stabilized finite element metod for te advective-reactive-diffsive eqation. Compt. Metods Appl. Mec. Eng. 190, F,H,Ri,H,Ho,J,Li,H:Astabilizedmixedfiniteelementmetodforellipticoptimalcontrolproblems.J.Sci. Compt doi: /s Brezzi, F, Fortin, M: Mixed and Hybrid Finite Element Metods. Springer, New York 1991