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Error analysis of variational discretization solving temperature control problems
Journal of Inequalities and Applications volume 2013, Article number: 450 (2013)
In this paper, we consider variational discretization solving temperature control problems with pointwise control constraints, where the state and the adjoint state are approximated by piecewise linear finite element functions, while the control is not directly discretized. We derive a priori error estimates of second-order for the control, the state and the adjoint state. Moreover, we obtain a posteriori error estimates. Finally, we present some numerical algorithms for the control problem and do some numerical experiments to illustrate our theoretical results.
We are interested in a material plate defined in a two-dimensional convex domain Ω with a Lipschitz boundary ∂ Ω. For the state y of the material, we choose the temperature distribution which is maintained equal to zero along the boundary. We denote thermal radiation or positive temperature feedback due to chemical reactions by the term (see, e.g., ) and assume that there exists a source . This system is governed by the following equation:
The setting above suggests that we may control the temperature distribution y to come close to a given target by acting with an additional distributed source term u, namely the control function. The corresponding optimal control problem is formulated as follows:
where and are strictly convex continuous differentiable functions, as , such that , , , B is a linear continuous operator, and K is defined by
where a and b are two constants.
Optimal control problems have been extensively used in many aspects of the modern life such as social, economic, scientific and engineering numerical simulation . Finite element approximation seems to be the most widely used method in computing optimal control problems. A systematic introduction of finite element method for PDEs and optimal control problems can be found in [1–8]. Concerning elliptic optimal control problems, a priori error estimates were investigated in [9, 10], a posteriori error estimates based on recovery techniques have been obtained in [11–13], a posteriori error estimates of residual type have been derived in [14–22], some error estimates and superconvergence results have been established in [23–27], and some adaptive finite element methods can be found in [28–31]. For parabolic optimal control problems, a priori error estimates are established in [32–34], a posteriori error estimates of residual type are investigated in [35, 36]. Recently, error estimates of spectral method for optimal control problems have been derived in [37, 38], and numerical methods for constrained elliptic control problems with rapidly oscillating coefficients are studied in .
For a constrained optimal control problem, the control has lower regularity than the state and the adjoint state. So most researchers considered using piecewise linear finite element functions to approximate the state and the adjoint state and using piecewise constant functions to approximate the control. They constructed a projection gradient algorithm where the a priori error estimates of the control is first-order in [11, 12]. Recently, Borzì considered a second-order discretization and multigrid solution of elliptic nonlinear constrained control problems in , Hinze introduced a variational discretization concept for optimal control problems and derived a priori error estimates for the control which is second-order in [41, 42]. The purpose of this paper is to consider variational discretization for convex temperature control problems governed by nonlinear elliptic equations with pointwise control constraints.
In this paper, we adopt the standard notation for Sobolev spaces on Ω with the norm and seminorm . We set and denote by . In addition, c or C denotes a generic positive constant.
The paper is organized as follows: In Section 2, we introduce a variational discretization approximation scheme for the model problem. In Section 3, we derive a priori error estimates. In Section 4, we derive sharp a posteriori error estimates of residual type. We present some numerical algorithms and do some numerical experiments to verify our theoretical results in the last section.
2 Variational discretization approximation for the model problem
We now consider a variational discretization approximation for the model problem (1.1). For ease of exposition, we set , , , , and
It follows from the assumptions on that
Then the standard weak formula for the state equation is
where we assume that the function for any , and for any . Thus, the equation above has a unique solution.
Throughout the paper, we impose the following assumptions:
(A1) and are Lipschitz continuous, namely,
(A2) There exists a positive constant m such that
Then the model problem (1.1) can be restated as
It is well known (see, e.g., ) that the control problem (2.3) has a solution , and that if the pair is the solution of (2.3), then there is an adjoint state such that the triplet satisfies the following optimality conditions:
where is the adjoint operator of B.
Lemma 2.1 Suppose that assumptions (A1)-(A2) are satisfied. Let be the solution of (2.4)-(2.6). Then the following equation:
admits a unique solution and .
Proof It follows from that (2.7) has a unique solution. Note that . From the regularity theory of patrial differential equations (see, e.g., ), we have
Because Ω is a two-dimension convex domain, according to embedding theorem, we get
From (A2) and (2.7), we get
Consequently, we complete the proof of equation (2.7). □
We introduce the following pointwise projection operator:
It is clear that is Lipschitz continuous with constant 1. As in , it is easy to prove the following lemma.
Lemma 2.2 Let and be the solutions of (2.4)-(2.6) and (2.7), respectively. Assume that assumptions (A1)-(A2) are satisfied. Then
Remark 2.1 We should point out that (2.6) and (2.9) are equivalent. This theory can be used to more complex situation, for example, K is characterized by a bound on the integral on u over Ω, namely, , we have similar results.
Let be a regular triangulation of Ω, such that . Let , where denotes the diameter of the element τ. Associated with is a finite dimensional subspace of , such that are polynomials of m-order () for all and . Let . It is easy to see that .
Then a possible variational discretization approximation scheme of (2.1) is as follows:
It is well known (see, e.g., ) that control problem (2.10) has a solution , and that if the pair is the solution of (2.10), then there is an adjoint state such that the triplet satisfies the following optimality conditions:
Similar to Lemma 2.2, it is easy to show the following lemma.
Lemma 2.3 Suppose that assumptions (A1)-(A2) are satisfied. Let be the solution of (2.11)-(2.13), and is the solution of the following equation:
Then we have
Remark 2.2 In many applications, the objective functional is uniform convex near the solution u, which is assumed in many studies on numerical methods of the problem, see, for example, [20, 43]. In this paper, we assumed that and are strictly convex continuous differentiable functions, for instance, , which is frequently met, then the exact solution of the variational inequality (2.13) is , and for numerically solving the problem, we can replace by in our program.
3 A priori error estimates
We now derive a priori error estimates of the variational discretization approximation scheme. Just for ease of exposition, let
and is the Fréchet derivative of at u. Similarly to (2.4)-(2.6), we can prove that
where satisfies the following system:
Let be the standard Lagrange interpolation operator such that for any , for all , where P is the vertex set associated with the triangulation , and n is the dimension of the domain Ω, we have the following result:
Lemma 3.1 
Let be the standard Lagrange interpolation operator. For or 1, and , we have
Lemma 3.2 Let and be the solutions of (2.11)-(2.13) and (3.1)-(3.2), respectively. Assume that and is locally Lipschitz continuous. Then there exists a constant C independent of h such that
Proof From , (2.12), (3.2) and embedding theorem , we have
Note that , by using Lemma 3.1, we obtain
Similarly, we can prove that
Then (3.3) follows from (3.5)-(3.6). □
In order to derive sharp a priori estimates, we introduce the following auxiliary problems:
From the regularity estimates (see, e.g., ), we obtain
Lemma 3.3 Let be the solution of (2.11)-(2.13). Suppose that and is locally Lipschitz continuous. Then there exists a constant C independent of h such that
Proof Let and . We have
Similarly, let and , we obtain
From (3.12) and (3.13), we get (3.9). □
Lemma 3.4 Let and be the solutions of (2.4)-(2.6) and (2.11)-(2.13), respectively. Assume that all the conditions in Lemma 3.3 are valid. Then there exists a constant C independent of h such that
Proof It is clear that
By using (2.6) and (2.13), we have
From (3.9) and (3.16), we derive (3.14). □
Now we combine Lemmas 3.2-3.4 to come up with the following main result.
Theorem 3.1 Let and be the solutions of (2.4)-(2.6) and (2.11)-(2.13), respectively. Assume that all the conditions in Lemmas 3.2-3.4 are valid. Then we have
Proof Note that
From (2.4)-(2.5), (3.1)-(3.2) and the regularity estimates, we have
Then, (3.17) follows from (3.9), (3.14) and (3.18)-(3.21). □
4 A posteriori error estimates
We now derive a posteriori error estimates for the variational discretization approximation scheme. The following lemmas are very important in deriving a posteriori error estimates of residual type.
Lemma 4.1 
Lemma 4.2 Let and be the solutions of (2.4)-(2.6) and (2.11)-(2.13), respectively. Then we have
where is defined in (3.2).
Proof It follows from (2.6) and (2.13) that
Let δ be small enough, then (4.2) follows from (4.3). □
Lemma 4.3 Let and be the solutions of (2.11)-(2.13) and (3.1)-(3.2), respectively. Assume that is locally Lipschitz continuous. Then there exists a positive constant C independent of h such that
where is the size of the face , and , are two neighboring elements in , , and are the A-normal and -normal derivative jumps over the interior face l, respectively, defined by
where n is the normal vector on outwards . For later convenience, we defined and when .
Proof Let and , it follows from the Green formula, embedding theorem , Lemma 4.1, (2.12) and (3.2) that
Similarly, we obtain
From (4.5) and (4.6), we derive (4.4). □
Theorem 4.1 Let and be the solutions of (2.4)-(2.6) and (2.11)-(2.13), respectively. Assume that all the conditions in Lemmas 4.2-4.3 are valid. Then there exists a constant C independent of h such that
where and are defined in Lemma 4.3.
Proof Note that
Then (4.7) follows from (4.2), (4.4) and (4.8)-(4.11). □
5 Numerical experiments
For a constrained optimization problem
where is a convex functional on U and K is a convex subset of U, the iterative scheme reads ():
where is a symmetric and positive definite bilinear form, and similarly to , the projection operator is defined: For given find such that
The bilinear form provides a suitable precondition for the projection gradient algorithm. Let . For an acceptable error and a fixed step size , by applying (5.1) to the discretized nonlinear elliptic optimal control problem, we introduce the following projection gradient algorithm (see, e.g., [11, 12]), for ease of exposition, we have omitted the subscript h.
Algorithm 5.1 (Projection gradient algorithm)
Step 1. Initialize ;
Step 2. Solve the following equations:
Step 3. Calculate the iterative error: ;
Step 4. If , stop; else, go to Step 2.
According to the preceding analysis, we construct the following variational discretization algorithm.
Algorithm 5.2 (Variational discretization algorithm)
Step 1. Initialize ;
Step 2. Solve the following equations:
Step 3. Calculate the iterative error: ;
Step 4. If , stop; else, go to Step 2.
It is well known that there are four major types of adaptive finite element methods, namely, the h-methods (mesh refinement), the p-methods (order enrichment), the r-methods (mesh redistribution) and the hp-methods (the combination of h-method and p-method). For an acceptable error , by using a posteriori error estimator and as the mesh refinement indicator and the Algorithm 5.2, we present the following adaptive variational discretization algorithm.
Algorithm 5.3 (Adaptive variational discretization algorithm)
Step 1. Solve the discretized optimization problem with Algorithm 5.2 on the current mesh obtain the numerical solution and calculate the error estimators and ;
Step 2. Adjust the mesh by using estimators and , then update the numerical solution and obtain the new numerical solution on new mesh;
Step 3. If , stop; else, go to Step 1.
All of the following numerical examples were solved numerically with codes developed based on AFEPack which provided a general tool of finite element approximation for PDEs. The package is freely available and the details can be found in .
We consider the following optimal control problems:
the domain Ω is the unit square and .
Example 1 In the first example, we compare the convergence order of in Algorithm 5.1 with that in Algorithm 5.2. The data are as follows:
In Figure 1, we see clearly that in the projection gradient algorithm, , while in the variational discretization algorithm, . In Figure 2, we show the profiles of the exact solution u alongside the solution error.
Example 2 In order to illustrate the reliability and efficiency of the a posteriori error estimates in Theorem 4.1, we use Algorithm 5.3 to solve this example. The data are as follows:
The numerical results based on adaptive mesh and uniform mesh are presented in Table 3. In Figure 3, we show the profiles of the exact solution u alongside the solution error. From Table 3, it is clear that the adaptive mesh generated via the error indicators in Theorem 4.1 are able to save substantial computational work, in comparison with the uniform mesh. Our numerical results confirm our theoretical results.
Pao C: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York; 1992.
Lions J: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.
Ciarlet P: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam; 1978.
He BS: Solving a class of linear projection equations. Numer. Math. 1994, 68: 71–80. 10.1007/s002110050048
Kufner A, John O, Fucik S: Function Spaces. Nordhoff, Leyden; 1977.
Ladyzhenskaya O, Urlatseva H: Linear and Quasilinear Elliptic Equations. Academic Press, New York; 1968.
Lions J, Magenes E: Non Homogeneous Boundary Value Problems and Applications. Springer, Berlin; 1972.
Neittaanmaki P, Tiba D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York; 1994.
Arada N, Casas E, Tröltzsch F: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 2002, 23: 201–229. 10.1023/A:1020576801966
Liu W, Yan N: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing; 2008.
Li R, Liu W, Yan N: A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 2007, 33: 155–182. 10.1007/s10915-007-9147-7
Liu H, Yan N: Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations. J. Comput. Appl. Math. 2007, 209: 187–207. 10.1016/j.cam.2006.10.083
Yan N: A posteriori error estimates of gradient recovery type for FEM of optimal control problem. Adv. Comput. Math. 2003, 19: 323–336. 10.1023/A:1022800401298
Ainsworth M, Oden JT: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York; 2000.
Babuška I, Strouboulis T, Upadhyay CS, Gangaraj SK: A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method. Int. J. Numer. Methods Eng. 1995, 38(24):4207–4235. 10.1002/nme.1620382408
Carstensen C, Verfürth R: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 1999, 36(5):1571–1587. 10.1137/S003614299732334X
Hintermüller M, Hoppe RHW, Iliash Y, Lieweg M: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM Control Optim. Calc. Var. 2008, 14(3):540–560. 10.1051/cocv:2007057
Hintermüller M, Hoppe RHW: Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 2008, 47(3):1721–1743.
Liu W, Yan N: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 2001, 15: 285–309. 10.1023/A:1014239012739
Liu W, Yan N: A posteriori error estimates for control problems governed by nonlinear elliptic equations. Appl. Numer. Math. 2003, 47: 173–187. 10.1016/S0168-9274(03)00054-0
Veeser A: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 2001, 39(1):146–167. 10.1137/S0036142900370812
Verfurth R: A Review of Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley, London; 1996.
Brandts J: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 1994, 68: 311–324. 10.1007/s002110050064
Chen Y, Dai Y: Superconvergence for optimal control problems governed by semi-linear elliptic equations. J. Sci. Comput. 2009, 39: 206–221. 10.1007/s10915-008-9258-9
Chen Y, Liu B: Error estimates and superconvergence of mixed finite element for quadratic optimal control. Int. J. Numer. Anal. Model. 2006, 3: 311–321.
Xing X, Chen Y:-error estimates for general optimal control problem by mixed finite element methods. Int. J. Numer. Anal. Model. 2008, 5(3):441–456.
Zhang Z, Zhu JZ: Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I). Comput. Methods Appl. Math. 1995, 123: 173–187.
Becker R, Kapp H, Rannacher R: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 2000, 39(1):113–132. 10.1137/S0363012999351097
Hoppe RHW, Kieweg M: Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput. Optim. Appl. 2010, 46: 511–533. 10.1007/s10589-008-9195-4
Li R, Liu W, Ma H, Tang T: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 2002, 41(5):1321–1349. 10.1137/S0363012901389342
Vexler B, Wollner W: Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 2008, 47(1):509–534. 10.1137/070683416
Alt W, Mackenroth U: Convergence of finite element approximation to state constrained convex parabolic boundary control problems. SIAM J. Control Optim. 1989, 27: 718–736. 10.1137/0327038
Chen Y, Lu Z: Error estimates for quadratic parabolic optimal control problems by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 2010, 46(11):957–965. 10.1016/j.finel.2010.06.011
Chen Y, Yang J: A posteriori error estimation for a fully discrete discontinuous Galerkin approximation to a kind of singularly perturbed problems. Finite Elem. Anal. Des. 2007, 43(10):757–770. 10.1016/j.finel.2007.03.007
Liu W, Yan N: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 2003, 93: 497–521. 10.1007/s002110100380
Xiong C, Li Y: A posteriori error estimates for optimal distributed control governed by the evolution equations. Appl. Numer. Math. 2011, 61: 181–200. 10.1016/j.apnum.2010.09.004
Chen Y, Huang Y, Yi N: A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations. Sci. China Ser. A 2008, 51(8):1376–1390. 10.1007/s11425-008-0097-9
Chen Y, Yi N, Liu W: A legendre-Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J. Numer. Anal. 2008, 46(5):2254–2275. 10.1137/070679703
Chen Y, Tang Y: Numerical methods for constrained elliptic optimal control problems with rapidly oscillating coefficients. East Asian J. Appl. Math. 2011, 1: 235–247.
Borzì A: High-order discretization and multigrid solution of elliptic nonlinear constrained control problems. J. Comput. Appl. Math. 2005, 200: 67–85.
Hinze M: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 2005, 30: 45–63. 10.1007/s10589-005-4559-5
Hinze M, Yan N, Zhou Z: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 2009, 27: 237–253.
Liu W, Tiba D: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. Numer. Funct. Anal. Optim. 2001, 22: 953–972. 10.1081/NFA-100108317
This work is supported by the Scientific Research Project of Department of Education of Hunan Province (13C338).
The author declares that he has no competing interests.
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Cite this article
Tang, Y. Error analysis of variational discretization solving temperature control problems. J Inequal Appl 2013, 450 (2013). https://doi.org/10.1186/1029-242X-2013-450
- variational discretization
- finite element
- optimal control problems
- a priori error estimates
- a posteriori error estimates