A SPLITTING LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS
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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALSIS AND MODELING Volume 3, Number 4, Pages c 26 Institute for Scientific Computing and Information A SPLITTING LEAST-SQUARES MIED FINITE ELEMENT METHOD FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS HONGFEI FU, HONGING RUI, HUI GUO, JIANSONG ZHANG, AND JIAN HOU Abstract. In tis paper, we propose a splitting least-squares mixed finite element metod for te approximation of elliptic optimal control problem wit te control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for te primal state variable and its flux. Ten, using te first order necessary and also sufficient optimality condition, we deduce anoter two corresponding adjoint state equations, wic are bot independent, symmetric and positive definite. Also, a variational inequality for te control variable is involved. For te discretization of te state and adjoint state equations, eiter RT mixed finite element or standard C finite element can be used, wic is not necessary subject to te Ladyzenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for te control, te states and adjoint states, respectively. Finally, we use some numerical examples to validate te teoretical analysis. Key words. Optimal control, splitting least-squares, mixed finite element metod, positive definite, a priori error estimates.. Introduction Optimal control problems are playing an increasingly important role in modern scientific and engineering numerical simulations. Nowadays, finite element metod seems to be te most widely used numerical metod in practical computation. Te readers are referred to, for example, Refs. [, 2, 3, 4, 5] for systematic introductions of finite element metods and optimal control problems. In tis paper, we are interested in te following convex quadratic optimal control problem wit te control constrained by pointwise inequality: min u U ad J y, σ, u = 2 subject to 2 and 3 y y d 2 + Ω Ω σ σ d 2 + γ divσ + cy = f + Bu, in Ω, σ + A y =, in Ω, y =, on Ω, ξ ux ξ 2, a.e. in Ω U. Ω U u 2 Here γ > is a constant, Ω and Ω U Ω are two bounded domain in R 2, wit Lipscitz boundaries Ω and Ω U. A precise formulation of tis problem including a functional analytic setting is given in te next section. Received by te editors June 22, 25 and, in revised form, December 3, Matematics Subject Classification. 49K2, 49M25, 65N5, 65N3. Corresponding autor: Hongxing Rui. Tis researc was supported by te NSF of Cina 2485, 9336, 57367, 4588, te NSF of Sandong Province BS23NJ, and te Fundamental Researc Funds for te Central Universities 4C227, 5C84A. 6
2 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 6 Classical mixed finite element metods see Refs. [6, 7, 8, 9, ] ave been proved effectively for solving elliptic equations and fluid problems. Tey ave an advantage of approximating te unknown scalar variable and its diffusive flux simultaneously. Besides, tese metods can approximate te unknown variable and its flux to a same order of accuracy. Recently, tere are some researc articles on tese metods for solving optimal control problems, see Refs. [, 2, 3], for example. However, it is well known tat tese metods usually produce a symmetric but indefinite system for elliptic equations. Tus te popular conjugate gradient CG or algebraic multi-grid AMG solvers can not be used for te solution of linear algebraic equation systems. To conquer tese difficulties appeared in using classical mixed finite element metods, least-squares mixed finite element metod, for first-order elliptic mixed system in unknown variable y and unknown velocity flux σ, was introduced by Pelivanov et al. [4]. It is well known tat te least-squares mixed finite element metod as two typical advantages: First, it is not subjected to te Ladyzenkaya- Babuska-Brezzi consistency condition, so te coice of finite element spaces becomes flexible; Second, it results in a symmetric and positive definite system, wic can be solved using tose solvers suc as CG and AMG quickly. Te idea of splitting least-squares was first proposed by Rui et al. in [5] for a reaction-diffusion equation, were by selecting a properly least-squares functional, te autors derived two independent, symmetric and positive definite equations, respectively, for te unknown state variable y and its flux σ. Ten it is applied to solve linear and nonlinear parabolic equations [6], sobolev equations [7], pseudo-parabolic equations [8], and nonlinear convection-diffusion equations [9] and so on. In tis paper, we apply te splitting least-squares mixed finite element metod for te discretization of elliptic optimal control problem. Pointwise inequality constraints on te control variable are considered. We derive optimal a priori error estimates, respectively, for te optimal control u in L 2 Ω U -norm, wic is approximated by piecewise constant or piecewise linear discontinuous elements; for te primal state y and adjoint state z bot in L 2 Ω-norm and H Ω-norm, wic are approximated by standard piecewise linear C finite elements; for te flux state σ and adjoint state ω in Hdiv; Ω-norm, wic are approximated by te lowewt-order RT mixed finite elements or standard piecewise linear C finite elements. Here, te Ladyzenkaya-Babuska-Brezzi consistency condition for te discretization spaces of y and σ is not needed. Tis paper is organized as follows. In Sect. 2, we introduce te optimal control problem and derive te continuous optimality conditions based on te idea of leastsquares. In Sect. 3, a splitting least-squares mixed finite element approximation to te continuous optimal control problem is proposed, and ten we derive te corresponding discrete optimality conditions. In Sect. 4, some a priori error estimates for te states, adjoint states and control are derived under control constrained by pointwise inequality. In Sect. 5, we conduct some numerical experiments to observe te convergence beavior of te numerical sceme. In te last section, some concluding remarks are given. In te following, we employ te standard notations W m,p Ω for Sobolev spaces on Ω wit norm W m,p Ω and seminorm W m,p Ω. We set W m,p Ω = v W m,p Ω : v = on Ω}. For p = 2, we denote H m Ω = W m,2 Ω and H m Ω = W m,2 Ω. In addition, C denotes a general positive constant wic is independent of te spatial mes parameters.
3 62 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU 2. Optimality system based on splitting least-squares In tis section, we first describe te optimal control problem under consideration, and ten derive te continuous optimality system based on te idea of splitting least-squares. Let Hdiv; Ω = τ L 2 Ω 2 : divτ L 2 Ω}, endowed wit te norm given by τ Hdiv;Ω = τ 2 L 2 Ω 2 + divτ 2 L 2 Ω /2. To formulate te optimal control problem, in te rest we sall take te state spaces V = H Ω, W = Hdiv; Ω, te control space and te admissible control set 4 U = L 2 Ω U, U ad = u U : ξ u ξ 2, a.e. in Ω U }, were te bounds ξ, ξ 2 R fulfill ξ < ξ 2. For te optimal control problem -2, we need te following assumptions. H- Te desired states and source function satisfy te following regularity: y d H Ω, σ d H Ω 2, f L 2 Ω. H-2 A = Ax = a i,j x 2 2 is a bounded symmetric and positive definite matrix, i.e., tere are positive constants α and β suc tat α 2 T A β 2, R 2. H-3 c = cx is positive definite and bounded, tat is, tere exist positive constants c and c 2 suc tat < c c c 2. H-4 B is a linear continuous operator from U = L 2 Ω U to L 2 Ω, suc tat Bu, v = B v, u U C u L2 Ω U v L2 Ω, u U, v V, were B is te adjoint operator of B. Te classical mixed weak formula for problems -3 is to find y, σ, u L 2 Ω Hdiv; Ω L 2 Ω U suc tat 5 J y, σ, u = min u U ad J y, σ, u and 6 A σ, τ y, divτ =, τ Hdiv; Ω, divσ, v + cy, v = f + Bu, v, v L 2 Ω. It is clear tat te two state variables σ and y in problem 6 are coupled eac oter, and te discretization of 6 will lead to a saddle-point problem, wic is symmetric but indefinite. To derive a symmetric and positive definite weak formulation for problem 2, we define for a given control u te least-squares functional Fv, τ as follows: 7 Fv, τ = 2 c /2 divτ + cv f Bu 2 L 2 Ω + 2 A /2 τ + A v 2 L 2 Ω 2.
4 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 63 Ten te solution to 2 is corresponding to te least-squares minimization problem: 8 Fy, σ = min Fv, τ. v,τ V W Furtermore, it follows from te first order necessary and sufficient minimization condition, problem 8 becomes: find y, σ V W suc tat A y, v + cy, v = f + Bu, v, v V, 9 c divσ, divτ + A σ, τ = c f + Bu, divτ, τ W. Remark 2.. It is clear tat te two equations in problem 9 are split, and tus can be solved independently. Besides, it is easy to ceck tat tey are also symmetric, coercive and continuous. For te given control set U ad, we now restate te optimal control problem -3 as follows: OCP J y, σ, u = min u U ad J y, σ, u suc tat y, σ, u V W U and A y, v + cy, v = f + Bu, v, v V, c divσ, divτ + A σ, τ = c f + Bu, divτ, τ W. Here te inner product in L 2 Ω or L 2 Ω 2 is indicated by,. Since te objective functional J y, σ, u is convex, it ten follows from Ref. [2] tat te optimal control problem OCP as a unique solution y, σ, u V W U ad. Furtermore, y, σ, u is te solution of OCP if and only if tere is a pair of adjoint state z, ω V W, suc tat y, σ, z, ω, u V W 2 U ad satisfies te following optimality conditions: OCP-OPT A y, v + cy, v = f + Bu, v, v V, c divσ, divτ + A σ, τ = c f + Bu, divτ, τ W. 2 A z, v + cz, v = y y d, v, v V, c divω, divτ + A ω, τ = σ σ d, τ, τ W. 3 γu B z B c divω, u u U, u U ad U, were, U denotes te inner product in U = L 2 Ω U. Remark 2.2. It is clear tat te two variables z and ω in problem 2 are also independent, so tey can be solved separately, too. Besides, te two equations in problem 2 are also symmetric, coercive and continuous. Remark 2.3. [2]. Inequality 3 is equivalent to te following: γu B z B c divω, u = ξ ; 4 γu B z B c divω, u = ξ 2 ; γu B z B c divω =, ξ < u < ξ 2. Define a pointwise projection operator P Uad from U to U ad tat P Uad fx, t = max ξ, minξ 2, fx, t. Ten, te optimality condition 3 can be expressed as 5 u = P Uad γ B z + c divω.
5 64 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU 3. Splitting least-squares mixed finite element approximation In tis section, we consider te approximation of optimal control problem OCP based on te splitting least-squares mixed finite element metod. Let T y and T σ be two families of regular triangulations of Ω, wic could be identical or not. Here y and σ denote te largest diameters of elements in T y and T σ, respectively. Construct two finite element spaces V W V W wit te following approximate properties see Refs. [, 8, 9, ]: Tere exist two integers k k and l suc tat 6 inf v V v v L2 Ω + y v v L2 Ω 2} Cl+ inf τ W τ τ L2 Ω 2 Ck+ σ τ H k+ Ω 2, inf τ W divτ τ L2 Ω C k σ τ H k + Ω 2, y v H l+ Ω, for any v V H l+ Ω and τ W H k+ Ω 2. Here k = k + wen W is cosen as RT mixed finite elements in Ref. [8], and k = k wen W is any one of te oter mixed elements in Refs. [9, ] and standard C elements in Ref. []. Let T u be a family of regular triangulations of Ω U, were u denotes te largest diameters of elements in T u. Associate wit T u is anoter finite element space U U = L 2 Ω U consist of piecewise r-order polynomials on eac element of T u. Here tere is no requirement for te continuity, due to te limited regularity of te optimal control. Let Uad be a closed convex set in U, tat is, 7 U ad = u U : ξ u ξ 2, a.e. in Ω U }. Ten te corresponding splitting least-squares mixed finite element approximation of OCP, wic will be labeled as OCP, is defined as follows: 8 J y, σ, u = min J y, σ, u u Uad were y, σ, u V W U satisfies A y, v + cy, v = f + Bu, v, v V, c divσ, divτ + A σ, τ = c f + Bu, divτ, τ W. It is again well known see, e.g., Ref. [2] tat te optimal control problem OCP as a unique solution y, σ, u V W Uad, and tat a triplet y, σ, u is te solution of OCP if and only if tere is a pair of adjoint state z, ω V W, suc tat y, σ, z, ω, u V W 2 Uad satisfies te following discrete optimality conditions: OCP-OPT 9 A y, v + cy, v = f + Bu, v, v V, c divσ, divτ + A σ, τ = c f + Bu, divτ, τ W. 2 A z, v + cz, v = y y d, v, v V, c divω, divτ + A ω, τ = σ σ d, τ, τ W. 2 γu B z B c divω, u u U, u U ad U. To obtain a priori error estimates for te above numerical scemes 9-2, in te end of tis section, we introduce auxiliary variables y u, σ u V W
6 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 65 and z u, ω u V W, associated wit te optimal control u as follows: A y u, v + cy u, v = f + Bu, v, 22 and 23 c divσ u, divτ + A σ u, τ = c f + Bu, divτ, A z u, v + cz u, v = y u y d, v, c divω u, divτ + A ω u, τ = σ u σ d, τ, for v V and τ W. For simplicity, in te following we denote ỹ := y u, z := z u, σ := σ u, ω := ω u. 4. A priori error estimates In tis section, we are able to derive some a priori error estimates for te optimal control problem OCP-OPT and its splitting least-squares mixed finite element approximation OCP-OPT. Due to te weak regularity of te optimal control problem, we consider piecewise constant elements r = or piecewise linear discontinuous elements r = for te approximation of te optimal control u, and piecewise linear C elements l = are adopted for bot te approximations of te states y and z. Wile for te discretizations of te flux states σ and ω, we use eiter te lowest-order RT mixed finite elements k =, k = or standard piecewise linear C elements k = k =, were in bot cases te Ladyzenkaya- Babuska-Brezzi consistency condition is not satisfied. Let Ω U = τ U : τ U Ω U, ξ < u τu < ξ 2 }, 24 Ω a U = τ U : τ U Ω U, u τu ξ, or u τu ξ 2 }, Ω b U = Ω U \ Ω U Ω a U. It is easy to ceck tat te tree parts do not intersect on eac oter, and Ω U = Ω U Ωa U Ωb U. In tis paper, we assume tat u and T U are regular suc tat measω b U C u see Refs. [2, 22]. Moreover, set Ω U = x Ω U, ξ < ux < ξ 2 }. Ten it is easy to see tat Ω U Ω U. Before deriving te main error estimates for te splitting least-squares mixed finite element approximation of optimal control problem governed by elliptic equations, some interpolation and projection estimates are prepared witout proof. Lemma 4.. []. Let P be te L 2 -projection from U = L 2 Ω U to U suc tat for any u U 25 u P u, ϕ U =, ϕ U. Ten we ave for any u H Ω U. u P u L2 Ω U C u u H Ω U Lemma 4.2. Let I be te standard Lagrange interpolation operator defined in Ref. []. Ten tere is a constant C > suc tat u I u W m,p Ω C 2 m u u W 2,p Ω for u W 2,p Ω, p >, and m = or.
7 66 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU Lemma 4.3. Let R y V be defined as te elliptic projection of y V suc tat 26 A y R y, v + cy R y, v =, w V. It ten follows from Ref. [2] tat y R y L 2 Ω + y y R y H Ω C 2 y y H 2 Ω. Lemma 4.4. [5]. From te approximate property 6 of finite element spaces we know tat for any σ W H 2 Ω 2, tere exists a vector-valued function Q σ W suc tat σ Q σ L2 Ω 2 Ck+ σ σ H k+ Ω 2, divσ Q σ L2 Ω C σ σ H2 Ω 2, were k = for te lowest-order RT mixed finite elements, and k = for piecewise linear C elements. In te following, we wis to demonstrate tree main lemmas wic can be collected to obtain te main conclusion of tis paper. Lemma 4.5. Let y, σ, ω, z and ỹ, σ, ω, z be te solutions of 9-2 and 22-23, respectively. Ten te following estimates old 27 y ỹ H Ω + z z H Ω C u u L 2 Ω U, 28 σ σ Hdiv;Ω + ω ω Hdiv;Ω C u u L 2 Ω U. Proof. It follows from 9 and 22 tat 29 A y ỹ, v + cy ỹ, v = Bu u, v, c divσ σ, divτ + A σ σ, τ = c Bu u, divτ, for v V and τ W. Note tat te two equations in problem 29 are split. By selecting v = y ỹ, τ = σ σ, respectively, and using te assumptions H-2-H-4, we can easily prove tat 3 y ỹ H Ω C u u L 2 Ω U, 3 σ σ Hdiv;Ω C u u L 2 Ω U. Similarly, one can obtain from 2 and 23 tat 32 A z z, v + cz z, v = ỹ y, v, c divω ω, divτ + A ω ω, τ = σ σ, τ, for v V and τ W. Ten wit v = z z and τ = ω ω in 32, and again using te assumptions H-2-H-4, we similarly derive 33 z z H Ω C y ỹ L2 Ω C u u L 2 Ω U, 34 ω ω Hdiv;Ω C σ σ L2 Ω 2 C u u L 2 Ω U, wic ends te proof of Lemma 4.5.
8 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 67 Lemma 4.6. Let y, σ, z, ω, u and y, σ, z, ω, u be te solutions of OCP- OPT and OCP-OPT, respectively. Let U be te piecewise constant element space r =. Assume tat u H Ω U, z H Ω, and divω H Ω. Ten we ave 35 u u L2 Ω U C u + z z L2 Ω + divω ω L2 Ω, were z, ω is defined in 23, and te constant C depends on some norms of te solution z, ω, u, but is independent of te mes parameters u. Let U be te piecewise linear discontinuous element space r =. Furtermore, assume tat u W, Ω U H 2 Ω U and divω W, Ω. Ten we ave 36 u u L 2 Ω U C, were Ω U is defined above, z W, Ω, 3 2 u + z z L 2 Ω + divω ω L 2 Ω were te constant C depends on some norms of te solution z, ω, u, but is independent of te mes parameter u. Proof. Let u I U ad be an approximation of u. Note tat U U and U ad U ad. Recalling te variational inequalities 3 and 2 imply tat 37 Hence, we can deduce tat γu B z B c divω, u u U, γu B z B c divω, u u I U. γ u u 2 L 2 Ω U = γu, u u U γu, u u U = γu B z B c divω, u u U +γu B z B c divω, u u I U, 38 +γu B z B c divω, u I u U +B z z, u u U + B c divω ω, u u γu u, u I u U + γu B z B c divω, u I u U U +B z z, u I u U + B c divω ω, u I u U +B z z, u I u U + B c div ω ω, u I u U +B z z, u u U + B c divω ω, u u U +B z z, u u U + B c div ω ω, u u Te rigt-and side of 38 can be bounded in a standard way. We ere just pay special attentions on te last two terms of te rigt-and side of 38. By selecting v = z z, τ = ω ω in 29 and v = y ỹ, τ = σ σ in 32 we ave 39 and B z z, u u U = z z, Bu u = A y ỹ, z z + cy ỹ, z z = y ỹ, y ỹ = y ỹ 2 L 2 Ω, B c div ω ω, u u U = c div ω ω, Bu u U. 4 = c divσ σ, divω ω + A σ σ, ω ω
9 68 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU = σ σ, σ σ = σ σ 2 L 2 Ω 2. Tus we obtain from 39-4, Lemma 4.5 and Caucy-Scwarz inequality tat γ u u 2 L 2 Ω U γu B z B c divω, u I u U + Cδ u u I 2 L 2 Ω U +Cδ z z 2 L 2 Ω + Cδ divω ω 2 L 2 Ω 2 4 +Cδ z z 2 L 2 Ω + Cδ div ω ω 2 L 2 Ω + Cδ u u 2 L 2 Ω U γu B z B c divω, u I u U + C u u I 2 L 2 Ω U +C z z 2 L 2 Ω + C divω ω 2 L 2 Ω + Cδ u u 2 L 2 Ω U, were δ is an arbitrary small positive number. Tanks to te definition of te control finite element space U, in te following we sall discuss two different cases for estimates of te first two terms on te rigtand side of 4. First let us consider te case tat U is te piecewise constant element space. Let u I be defined as te L2 -projection P u see, Lemma 4.. It is clear tat P u U ad, and 42 u P u L 2 Ω U C u u H Ω U. Moreover, if u H Ω U, z H Ω, and divω H Ω, we conclude from te definition of P in 25 tat γu B z B c divω, P u u U 43 = Θ P Θ P u u Ω U divω 2 H Ω + z 2 H Ω + u 2 H Ω U, C 2 u were Θ := γu B z B c divω. Ten 35 follows from 4-43 wit Cδ = γ/2. Next we consider te case tat U is te piecewise linear element space. Let u I = I u U be defined as te standard Lagrange linear interpolation of u see, Lemma 4.2. It is clear tat I u Uad, and for u W, Ω U H 2 Ω U we ave u I u L 2 Ω U C 2 u u H 2 Ω U, u I u L Ω b U C u u W, Ω b U. Note tat I u = u on Ω a U, ten it follows tat u I u 2 L 2 Ω U = + + u I u 2 Ω U Ω a U Ω b U 44 C 4 u u 2 H 2 Ω U + + C2 u u 2 W, Ω b U measωb U C 3 u u 2 H 2 Ω U + u 2 W, Ω U. Moreover, it follows from 4 tat γu B z B c divω = on Ω U, and we conclude from te definition of Ω b U tat for any element τ U Ω b U, tere is a x suc
10 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 69 tat ξ < ux < ξ 2, and ence γu B z B c divω x =. Terefore, for any τ U Ω b U we ave Tus γu B z B c divω L τ U = γu B z B c divω γu B z B c divω x L τ U C u γu B z B c divω W, τ U. 45 γu B z B c divω, I u u U = + + γu B z B c divω I u u Ω U Ω a U Ω b U = + + γu B z B c divω I u u Ω b U γu B z B c divω L Ω b U I u u L Ω b U measω b U C 3 u u 2 W, Ω U + z 2 W, Ω + divω 2 W, Ω. Ten 36 is proved by inserting into 4 wit Cδ = γ/2. Lemma 4.7. Let y, σ, z, ω and ỹ, σ, z, ω be te solutions of -2 and 22-23, respectively. Assume tat problem 2 is H 2 -regular, and te solutions y, z H Ω H 2 Ω and σ, ω H 2 Ω 2. Ten te following estimates old 46 y ỹ Hs Ω + z z Hs Ω C 2 s y, s =,, 47 σ σ Hdiv;Ω + ω ω Hdiv;Ω C σ, were te constant C depends on some norms of y, σ, z, ω, but is independent of te mes parameters y and σ. Proof. In te following, we divide te proof into two parts. Part I. Note tat σ, ỹ is te least-squares mixed finite element solution of σ, y. Terefore, it follows immediately from Ref. [5] combined wit Lemmas 4.3 and 4.4 tat 48 y ỹ Hs Ω C 2 s y y H2 Ω, s =,, and 49 σ σ Hdiv;Ω C σ σ H2 Ω 2. Part II. To derive corresponding error estimates for z z Hs Ω and ω ω Hdiv;Ω in 46-47, we write z z = z R z + R z z := ρ + π, ω ω = ω Q ω + Q ω ω := ζ + η. Here R z V and Q ω W are defined as in Lemmas , and of course ρ and ζ are bounded in te desired way. In te following, we first estimate π = R z z. By subtracting te first equation in 23 wit tat of 2, we obtain for v V, 5 A z z, v + cz z, v = ỹ y, v,
11 62 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU wic implies 5 A π, v + cπ, v = ỹ y, v, v V. Let v = π = R z z. It ten follows from te standard finite element error estimates and 48 tat 52 R z z H Ω C y ỹ L2 Ω C 2 y y H2 Ω. 53 Similarly, we can obtain anoter error equation c divω ω, divτ + A ω ω, τ = σ σ, τ, τ W. Using te interpolation operator Q, equation 53 can also be written as 54 c divη, divτ + A η, τ = σ σ, τ c divζ, divτ A ζ, τ, τ W. Let τ = η = Q ω ω in 54. It is easy to see tat 55 c 2 divq ω ω 2 L 2 Ω + β Q ω ω 2 L 2 Ω 2 σ σ L 2 Ω 2 Q ω ω L 2 Ω 2 +c divω Q ω L2 Ω divq ω ω L2 Ω +α ω Q ω L 2 Ω 2 Q ω ω L 2 Ω 2 C σ σ L 2 Ω 2 + divω Q ω L 2 Ω + ω Q ω L 2 Ω 2 Q ω ω Hdiv;Ω. Following from Lemma 4.4 and te proved result 49, we deduce 56 Q ω ω Hdiv;Ω C σ σ L2 Ω 2 + divω Q ω L2 Ω + ω Q ω L2 Ω 2 C σ τ H2 Ω 2. τ=σ,ω Incorporating te above results 52, 56 wit te well-known estimates for ρ in Lemma 4.3 and ζ in Lemma 4.4, we derive 57 z z Hs Ω C 2 s y v H2 Ω, s =,, and 58 v=y,z ω ω Hdiv;Ω C σ τ=σ,ω τ H2 Ω 2. Tus Lemma 4.7 is proved. Collecting te bounds given by Lemmas , we ave te following main result. Teorem 4.8. Suppose tat y, σ, z, ω, u and y, σ, z, ω, u are te solutions of OCP-OPT and OCP-OPT, respectively. Assume tat all conditions of Lemmas are valid. Ten for r, s =,, we ave 59 u u L 2 Ω U C + r 2 u + 2 y + σ,
12 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 62 6 y y H s Ω + z z H s Ω C + r 2 u + 2 s y + σ, 6 σ σ Hdiv;Ω + ω ω Hdiv;Ω C + r 2 u + 2 y + σ, were te constant C depends on some norms of y, σ, z, ω, u, but is independent of te mes parameters y, σ and u. Proof. It follows from Lemmas tat 62 u u L 2 Ω U C + r 2 u + z z L2 Ω + divω ω L2 Ω C + r 2 u + 2 y + σ. Moreover, it follows from Lemma 4.5, Lemma 4.7 and 62 tat 63 y y H s Ω + z z H s Ω y ỹ Hs Ω + z z Hs Ω + y ỹ Hs Ω + z z Hs Ω C + r 2 u + 2 s y + σ, 64 σ σ Hdiv;Ω + ω ω Hdiv;Ω σ σ Hdiv;Ω + ω ω Hdiv;Ω + σ σ Hdiv;Ω + ω ω Hdiv;Ω C + r 2 u + 2 y + σ. Tus, Teorem 4.8 follows immediately from Numerical experiments In tis section, we conduct some numerical examples to ceck te efficiency of te spitting least-squares mixed finite element sceme 9-2. Let te spatial domain Ω =, 2 and consider te problem: 65 subject to 66 2 y y d 2 + Ω Ω σ σ d 2 + diva y + cy = f + u, in Ω, y =, Ω u 2 on Ω, wit σ = A y. In fact, te adjoint state z satisfies te following equation: diva z + cz = y yd, in Ω, 67 z =, on Ω. For simplicity we sall adopt te same mes partitions for te states and control in te numerical tests, i.e., Ty = Tσ = Tu. To solve te optimal control problem numerically, we use te C++ software package: AFEpack, it is available at ttp://dsec.pku.edu.cn/ rli/. As for constrained optimal control problems, people pay more attention on te states and te control. Terefore, in te following numerical examples, we mostly center on te primal state variable y, wic is approximated by piecewise linear C elements; te flux state variable σ, wic is
13 622 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU approximated by piecewise linear C elements or te lowest-order RT elements; and te control variable u, wic is discretized using piecewise constant elements or piecewise linear discontinuous elements. Furtermore, in te following tests, we use E u, E y, E y, and E σ to represent te error norms u u L 2 Ω, y y L 2 Ω, y y H Ω, and σ σ Hdiv;Ω, respectively. Example. For te first example, te corresponding analytical solutions for problems wit a constant diffusion coefficient matrix A = I and a constant reaction coefficient c = are given by yx = sinπx sinπx 2, σx = A y, zx = 2 sinπx sinπx 2, ωx = sinπx 2, sinπx, ux = max, min., z + c divω. Te source function f, te desired states y d and σ d can be determined using te above functions. In tis test, we consider te lowest-order RT elements for te approximation of te flux state σ, and piecewise constant elements for tat of te control u. Table displays te errors and convergence orders for te control u, te primal state y, and te flux state σ in corresponding norms. We can see tat te convergence orders for te control u in L 2 Ω-norm, te state y in H Ω-norm, and σ in Hdiv; Ω-norm are almost first-order, wile te convergence order for te state y in L 2 Ω-norm is second-order, wic are consistent wit Teorem 4.8 very well. Figure sows te approximate profiles of te control u and te primal state y. Te two components of te numerical flux state σ are presented in Figure 2. It can be observed tat te numerical figures are agreement wit te analytical solutions very well. Table. σ: te lowest-order RT elements, u: piecewise constant elements. E u Order E y Order E y Order E σ Order E E E E E E E E E E E E E-3..92E E E-. Example 2. For te second example, te corresponding analytical solutions for problems are yx = x x x 2 x 2, σx = A y, zx = 6x x x 2 x 2, ωx = x 2 x 2, x x, ux = max, min.5, z + c divω, were A = I and c =. Te source function f and te desired states y d and σ d can also be determined using te above functions.
14 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP y_ u_ Figure. Te approximate control u left and te approximate state y rigt for Example sigma_2, sigma_, Figure 2. Te approximate flux state σ for Example. In tis example, we consider piecewise linear C elements for te approximation of te flux state σ, and piecewise constant elements for tat of te control u. We list te corresponding numerical errors and convergence orders for te control u, te primal state y and te flux state σ in Table 2. Te convergence orders for te control, te primal state, and te flux state are also well matced wit te teoretical analysis in Teorem 4.8. We also sow te approximate profiles of te control u, te primal state y and te flux state σ in Figures 3-4, respectively. It can be observed tat te numerical results are also agreement wit te analytical solutions very well. Table 2. σ: piecewise linear C elements, u: piecewise constant elements. Eu Order Ey Order Ey Order Eσ E E E-2 6.9E E E E E E E E E E-3..35E E E-3.8 Order Example 3. For te tird example, we consider a reaction-dominated diffusion problem. In tis case, te solution typically as exponential boundary layers on all sides of Ω or internal layers in Ω. We take te analytical solutions for problems
15 624 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU y_ u_ Figure 3. Te approximate control u left and te approximate state y rigt for Example sigma_2, sigma_, Figure 4. Te approximate flux state σ for Example as follows: yx = exp x.52 + x2.52 /ε, σx = ε y, zx = 2 exp x.52 + x2.52 /ε, ωx =,, ux = max, min.5, z + c divω, were ε = 2 and c =. Te source function f and te desired states yd and σd can also be determined using te above functions. In tis example, we consider te lowest-order RT elements for te approximation of te flux state σ, and piecewise linear discontinuous elements for tat of te control u. Some corresponding numerical errors and convergence orders for te control u, te primal state y and te flux state σ are presented in Table 3. It 3 can be observed tat an nearly O 2 -order convergence for te control is derived, wic is consistent wit Teorem 4.8 for te case r =. Te approximate profiles of te control u, te primal state y and te flux state σ are displayed in Figures 5-6, respectively. All tese results also confirms te teoretical analysis. 6. Concluding remarks In tis paper, we derive a priori error estimates for te splitting least-squares mixed finite element discretization of optimal control problem governed by elliptic
16 SPLITTING LEAST-SQUARES MFE FOR ELLIPTIC OCP 625 Table 3. σ: te lowest-order RT elements; u: piecewise linear discontinuous elements. Eu Order Ey Order Ey Order Eσ E E E-.9923E E E E E E E E E E E E E-2. Order y_ u_ Figure 5. Te approximate control u left and te approximate state y rigt for Example sigma_2 sigma_ Figure 6. Te approximate flux state σ for Example 3. equations, were pointwise inequality constraints on te control variable are considered. Tis splitting mixed metod leads to uncoupled positive definite systems for te primal state variable and its flux, also for te adjoint state and flux state variables. Finally, numerical experiments are addressed to verify te teoretical analysis. Acknowledgments Te autors would like to tank te editor and te anonymous referee for teir valuable comments and suggestions on an earlier version of tis paper. Te first autor was partially supported by te Cina Scolarsip Council. References [] P. G. Ciarlet, Te Finite Element Metod for Elliptic Problems, SIAM, Piladelpia, 22. [2] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, SpringerVerlag, Berlin, 97.
17 626 H. FU, H. RUI, H. GUO, J. ZHANG, AND J. HOU [3] W. Liu and N. an, Adaptive Finite Element Metods for Optimal Control Governed by PDEs, Series in Information and Computational Science 4, Science Press, Beijing, 28. [4] M. Hinze, R. Pinnau, M. Ulbric, and S. Ulbric, Optimization wit PDE Constraints, Matematical Modelling: Teory and Applications 23, Springer, Neterlands, 29. [5] D. Tiba, Lectures on te Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland, 995. [6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Metods, Springer-Verlag, 99. [7] J. Douglas Jr. and J. E. Roberts, Global estimates for mixed metods for second order elliptic equations, Mat. Comput., [8] P. A. Raviart and J. M. Tomas, A mixed finite element metod for 2nd order elliptic problems, Lecture Notes in Matematics, vol. 66, Springer, Berlin, 977, [9] F. Brezzi, J. Douglas Jr., M. Fortin and L. D. Marini, Efficient rectangular mixed finite elements in two and tree space variables, RAIRO Model. Mat. Numer. Anal., [] F. Brezzi, J. Douglas Jr. and L. D. Marini, Two family of mixed finite elements for second order elliptic problems, Numer. Mat., []. Cen and W. Liu, Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Int. J. Numer. Anal. Model., [2]. Cen and Z. Lu, Error estimates of fully discrete mixed finite element metods for semilinear quadratic parabolic optimal control problem, Comput. Metods Appl. Mec. Engrg., [3] J. Zou,. Cen and. Dai, Superconvergence of triangular mixed finite elements for optimal control problems wit an integral constraint, Appl. Mat. Comput., [4] A. Pelivanov, G. F. Carey and D. Lazarov, Least-squares mixed finite elements for secondorder elliptic problems, SIAM J. Numer. Anal., [5] H. Rui, S. Kim and S. D. Kim, A remark on least-squares mixed element metods for reactiondiffusion problems, J. Comput. Appl. Mat., [6] H. Rui, S. D. Kim and S. Kim, Split least-squares finite element metods for linear and nonlinear parabolic problems, J. Comput. Appl. Mat., [7] F. Gao and H. Rui, A split least-squares caracteristic mixed finite element metod for Sobolev equations wit convection term, Mat. Comput. Simulation, [8] H. Guo, A remark on split least-squares mixed element procedures for pseudo-parabolic equations, Appl. Mat. Comput., [9] J. Zang and H. Guo, A split least-squares caracteristic mixed element metod for nonlinear nonstationary convection-diffusion problem, Int. J. Comput. Mat., [2] M. F. Weeler, A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., [2] R. Becker and B. Vexler, Optimal control of te convection-diffusion equation using stabilized finite element metods, Numer. Mat., [22] C. Meyer and A. Rösc, Superconvergence properties of optimal control problems, SIAM J. Control Optim., College of Science, Cina University of Petroleum, Qingdao 26658, Cina ongfeifu@upc.edu.cn Scool of Matematics, Sandong University, Jinan 25, Cina xrui@sdu.edu.cn College of Science, Cina University of Petroleum, Qingdao 26658, Cina sdug@63.com College of Science, Cina University of Petroleum, Qingdao 26658, Cina jszang@upc.edu.cn College of Petroleum Engineering, Cina University of Petroleum, Qingdao 26658, Cina oujian@upc.edu.cn
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