Advanced
SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES OF VARIATIONAL DISCRETIZATION FOR ELLIPTIC CONTROL PROBLEMS†
SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES OF VARIATIONAL DISCRETIZATION FOR ELLIPTIC CONTROL PROBLEMS†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 707-719
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : October 28, 2013
  • Accepted : May 21, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
YUCHUN HUA
YUELONG TANG

Abstract
In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in L 2 -norm and superconvergence between the numerical solution and elliptic projection of the exact solution in H 1 -norm or the gradient of the exact solution and recovery gradient in L 2 -norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results. AMS Mathematics Subject Classification : 49J20, 65M60.
Keywords
1. Introduction
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 [14 , 17 , 18 , 23 , 26] .
For a constrained optimal control problem, the control has lower regularity than the state and the co-state. So most researchers considered using piecewise linear functions to approximate the state and the co-state and using piecewise constant functions to approximate the control. They constructed a projection gradient algorithm in which the control is first-order convergent (see e.g., [16 , 20] ). Recently, Borzì considered a second-order discretization and multigrid solution of constrained nonlinear elliptic control problems in [4] , Hinze introduced a variational discretization concept for optimal control problems and derived a second-order convergence for the control in [11 , 12] .
There has been extensive research on the superconvergence of finite element methods for optimal control problems in the literature, most of which focused upon elliptic control problems. Superconvergence properties of linear and semilinear elliptic control problems are established in [24] and [6] , respectively. Superconvergence of finite element approximation for bilinear elliptic optimal control problems is studied in [32] . Recently, superconvergence of fully discrete finite element and variational discretization approximation for linear and semilinear parabolic control problems are derived in [27 , 28 , 29] .
The literature on a posteriori error estimation of finite element method is huge. Some internationally known works can be found in [1 , 2 , 3 , 5] . Concerning finite element methods of elliptic optimal control problems, a posteriori error estimates of residual type are investigated in [10 , 21 , 33 , 34] , a posteriori error estimates of recovery type are derived in [16 , 20 , 31] . Some error estimates and superconvergence results have been established in [6 , 7 , 36] , and some adaptive finite element methods can be found in [3 , 13 , 15 , 35] . For parabolic optimal control problems, residual type a posteriori error estimates of finite element methods are investigated in [22 , 30] and an adaptive space-time finite element method is investigated in [25] .
The purpose of this paper is to consider the superconvergence and recovery type a posteriori error estimates of variational discretization for elliptic optimal control problems with pointwise control constraints.
We are interested in the following quadratic optimal control problem:
PPT Slide
Lager Image
where Ω is a bounded domain in ℝ 2 with a Lipschitz boundary ∂Ω, the coefficient
PPT Slide
Lager Image
such that ( A ( x ) ξ ) · ξ c | ξ | 2 , ∀ ξ ∈ ℝ 2 , B is a linear continuous operator, yd , ud , f L 2 (Ω), and K is defined by
PPT Slide
Lager Image
where a and b are constants.
In this paper, we adopt the standard notation Wm,q (Ω) for Sobolev spaces on Ω with norm
PPT Slide
Lager Image
and seminorm
PPT Slide
Lager Image
We set
PPT Slide
Lager Image
and denote W m,2 (Ω) by Hm (Ω). In addition, c or C denotes a generic positive constant.
The outline of this paper is as follows. In Section 2, we introduce a variational discretization approximation for the model problem. In Section 3, we derive the convergence. In Section 4, we obtain the superconvergence and a posteriori error estimates. We present some numerical examples to demonstrate 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). For ease of exposition, we set
PPT Slide
Lager Image
and
PPT Slide
Lager Image
It follows from the assumption on A that
PPT Slide
Lager Image
Then the model problem (1) can be restated as:
PPT Slide
Lager Image
It is well known (see e.g., [17] ) that the control problem (2) has a unique solution ( y, u ) ∈ W × K , and a pair ( y, u ) ∈ W × K is the solution of (2) if and only if there is a co-state p W such that the triplet ( y, p, u ) ∈ W × W × K satisfies the following optimality conditions:
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
where B * is the adjoint operator of B .
We introduce the following pointwise projection operator:
PPT Slide
Lager Image
It is clear that Π [a,b] (·) is Lipschitz continuous with constant 1. As in [8] , it is easy to prove the following lemma:
Lemma 2.1. Let ( y, p, u ) be the solution of (3)-(5). Then
PPT Slide
Lager Image
Remark 2.1. We should point out that (5) and (6) are equivalent. This theory can be used to another situation, for example, K is characterized by a bound on the integral on u over Ω, namely, Ω u ( x ) dx ≥ 0, we have similar results.
Let τh be a regular triangulation of Ω, such that
PPT Slide
Lager Image
. Let h = max ττh { hτ }, where hτ denotes the diameter of the element τ . Associated with τh is a finite dimensional subspace Sh of
PPT Slide
Lager Image
, such that χ | τ are polynomials of m-order( m = 1) for all χ Sh and τ τh . Let Wh = { vh Sh : vh | ∂Ω = 0}. It is easy to see that Wh W . Then a possible variational discretization approximation scheme of (2) is as follows:
PPT Slide
Lager Image
It is well known (see e.g., [21] ) that the control problem (7) has a unique solution ( yh, uh ) ∈ Wh × K , and that if the pair ( yh , uh ) ∈ Wh × K is the solution of (7), if and only if there is a co-state ph Wh such that the triplet ( yh, ph, uh ) ∈ Wh × Wh × K satisfies the following optimality conditions:
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Similar to Lemma 2.1, it is easy to show the following lemma:
Lemma 2.2. Suppose ( yh, ph, uh ) be the solution of (8)-(10). Then , we have
PPT Slide
Lager Image
Remark 2.2. According to (11), we can replace uh by
PPT Slide
Lager Image
in our program. Thus the control need not be discretized directly.
3. Convergence analysis
We first introduce the following intermediate variables. For any uh K , let ( y ( uh ), p ( uh )) ∈ W × W satisfies the following system:
PPT Slide
Lager Image
PPT Slide
Lager Image
The following lemmas are very important in deriving the convergence.
Lemma 3.1 ( [9] ). Let πh be the standard Lagrange interpolation operator. For m = 0 or 1,
PPT Slide
Lager Image
and v W 2,q (Ω), we have
PPT Slide
Lager Image
Lemma 3.2. Let ( yh , ph , uh ) and ( y ( uh ), p ( uh )) be the solutions of (8)-(10) and (12)-(13), respectively. Assume that p ( uh ), y ( uh ) ∈ H 2 (Ω). Then there exists a constant C independent of h such that
PPT Slide
Lager Image
Proof . From (9), (13)-(14) and Young’s inequality, we have
PPT Slide
Lager Image
Let δ be small enough, we obtain
PPT Slide
Lager Image
Similarly, we can prove that
PPT Slide
Lager Image
Then (15) follows from (16)-(18).
We introduce the following auxiliary problems:
PPT Slide
Lager Image
PPT Slide
Lager Image
From the regularity estimates (see e.g., [9] ), we obtain
PPT Slide
Lager Image
Lemma 3.3. Let ( yh , ph , uh ) be the solution of (8)-(10). Suppose that y ( uh ), p ( uh ) ∈ H 2 (Ω). Then there exists a constant C independent of h such that
PPT Slide
Lager Image
Proof . Let F 1 = y ( uh ) − yh in (19) and ξh = πhξ . From (8), (12) and (15), we have
PPT Slide
Lager Image
Note that
PPT Slide
Lager Image
Thus,
PPT Slide
Lager Image
Similarly, let F 2 = p ( uh ) − ph in (20) and ζh = πhζ . From (9), (13) and (15), we obtain
PPT Slide
Lager Image
From (24) and (25), we get (21).
Lemma 3.4. Let ( y, p, u ) and ( yh, ph, uh ) be the solutions of (3)-(5) and (8)-(10), respectively. Assume that all the conditions in Lemma 3.3 are valid. Then there exists a constant C independent of h such that
PPT Slide
Lager Image
Proof . By selecting v = uh and v = u in (5) and (10), respectively. From (3)-(4) and (12)-(13), we have
PPT Slide
Lager Image
From (21) and (27), we derive (26).
Now we combine lemmas 3.2-3.4 to come up with the following main result.
Theorem 3.5. Let ( y, p, u ) and ( yh, ph, uh ) be the solutions of (3)-(5) and (8)-(10), respectively. Assume that all the conditions in lemmas 3.2-3.4 are valid. Then we have
PPT Slide
Lager Image
4. Superconvergence and a posteriori error estimates
We now derive the superconvergence and a posteriori error estimates for the variational discretization approximation. To begin with, let us introduce the elliptic projection operator Ph : W Wh , which satisfies: for any ϕ W
PPT Slide
Lager Image
It has the following approximation properties:
PPT Slide
Lager Image
Theorem 4.1. Let ( y, p, u ) and ( yh, ph, uh ) be the solutions of (3)-(5) and (8)-(10), respectively. Assume that all the conditions in lemmas 3.5 are valid. Then we have
PPT Slide
Lager Image
Proof . From (3)-(8), we have the following error equation:
PPT Slide
Lager Image
By using the definition of Ph and choosing wh = Phy yh , we have
PPT Slide
Lager Image
Let us note that B is linear continuous operator. From (26), (29) and Holder inequality, we get
PPT Slide
Lager Image
Similarly, we can derive
PPT Slide
Lager Image
Thus, (30) follows from (33) and (34).
In the second, let us recall recovery operators Rh and Gh , respectively. Let Rhv be a continuous piecewise linear function (without zero boundary constraint). Similar to the Z - Z patch recovery in [37 , 38] , the value of Rhv on the nodes are defined by least-squares argument on an element patches surrounding the nodes. The gradient recovery operator Ghv = ( Rh v x1 , ( Rh v x2 ), where Rh is the recovery operator defined above for the recovery of the control. The details can be found in [16] .
Theorem 4.2. Let ( y, p, u ) and ( yh, ph, uh ) be the solutions of (3)-(5) and (8)-(10), respectively. Suppose that all the conditions in Theorem 4.1 are valid and y, p H 3 (Ω). Then
PPT Slide
Lager Image
Proof . Let yI be the piecewise linear Lagrange interpolation of y . According to Theorem 2.1.1 in [19] , we have
PPT Slide
Lager Image
From the standard interpolation error estimate technique (see, e.g., [9] ) that
PPT Slide
Lager Image
By using (36)-(37), we get
PPT Slide
Lager Image
Therefore,
PPT Slide
Lager Image
From Theorem 4.1 and (39), we derive
PPT Slide
Lager Image
Similarly, we can prove that
PPT Slide
Lager Image
Then (36) follows from (40)-(41).
By using the superconvergence results above, we obtain the following a posteriori error estimates of variational discretization approximation for the elliptic optimal control problems.
Theorem 4.3. Assume that all the conditions in Theorem 4.1 and Theorem 4.2 are valid. Then
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . From Theorem 4.1 and Theorem 4.2, it is easy to obtain the above results.
5. Numerical experiments
In this section, we present some numerical examples which is solved numerically with codes developed based on AFEPack. The package provide a freely available tool of finite element approximation for PDEs and the details can be found in [15] .
We solve the following optimal control problems:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and the domain Ω is the unit square [0, 1] × [0, 1] and B = I is the identity operator and E is the 2 × 2 identity matrix.
Example 1. The data are as follows:
PPT Slide
Lager Image
The numerical results are listed in Table 1 . In Figure 1 , we show the profiles of the exact solution u alongside the solution error.
Numerical results, Example 1.
PPT Slide
Lager Image
Numerical results, Example 1.
PPT Slide
Lager Image
The exact solution u (top) and the error uhu (bottom), Example 1.
The results in Table 1 indicate that ∥ u uh ∥ = O ( h 2 ), ∥ y yh ∥ = O ( h 2 ) and ∥ p ph ∥ = O ( h 2 ). It is consistent with our theoretical result obtained in the Theorem 3.5.
Example 2. The data are as follows:
PPT Slide
Lager Image
The numerical results based on adaptive mesh and uniform mesh are presented in Table 2 . In Figure 2 , we show the profiles of the exact solution u alongside the solution error. From Table 2 , it is clear that the adaptive mesh generated via the error indicators in Theorem 4.3 are able to save substantial computational work, in comparison with the uniform mesh. Our numerical results confirm our theoretical results.
Numerical results, Example 2.
PPT Slide
Lager Image
Numerical results, Example 2.
PPT Slide
Lager Image
The exact solution u (top) and the error uhu (bottom), Example 2.
BIO
Yuchun Hua received M.Sc. form Yunnan University. His research interests include differential equations and dynamical system.
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China.
e-mail:huayuchun306@sina.com
Yuelong Tang received M.Sc. and Ph.D from Xiangtan University. His research interests include numerical optimization and numerical methods for partial differential equations.
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China.
e-mail:tangyuelonga@163.com
References
Ainsworth M. , Oden J.T. 2000 A Posteriori Error Estimation in Finite Element Analysis Wiley Interscience New York
Babuška I. , Strouboulis T. , Upadhyay C.S. , Gangaraj S.K. (1995) A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method Int. J. Numer. Meth. Eng. 38 (24) 4207 - 4235    DOI : 10.1002/nme.1620382408
Becker R. , Kapp H. , Rannacher R. (2000) Adaptive finite element methods for optimal control of partial differential equations: basic concept SIAM J. Control Optim. 39 (1) 113 - 132    DOI : 10.1137/S0363012999351097
Borzì A. (2005) High-order discretization and multigrid solution of elliptic nonlinear constrained control problems J. Comput. Appl. Math. 200 67 - 85
Carstensen C. , Verfürth R. (1999) Edge residuals donminate a posteriori error estimates for low order finite element methods SIAM J. Numer. Anal. 36 (5) 1571 - 1587    DOI : 10.1137/S003614299732334X
Chen Y. , Dai Y. (2009) Superconvergence for optimal control problems gonverned by semilinear elliptic equations J. Sci. Comput. 39 206 - 221    DOI : 10.1007/s10915-008-9258-9
Chen Y. , Liu B. (2006) Error estimates and superconvergence of mixed finite element for quadratic optimal control Inter. J. Numer. Anal. Model. 3 311 - 321
Chen Y. , Yi N. , Liu W. (2008) A legendre galerkin spectral method for optimal control problems governed by elliptic equations SIAM J. Numer. Anal. 46 (5) 2254 - 2275    DOI : 10.1137/070679703
Ciarlet P. 1978 The Finite Element Method for Elliptic Problems North-Holland Amstterdam
Hintermüller M. , Hoppe R.H.W. (2008) Goal-oriented adaptivity in control constatined optimal control of partial differential equations SIAM J. Control Optim. 47 (3) 1721 - 1743    DOI : 10.1137/070683891
Hinze M. (2005) A variational discretization concept in cotrol constrained optimization: the linear-quadratic case Comput. Optim. Appl. 30 45 - 63    DOI : 10.1007/s10589-005-4559-5
Hinze M. , Yan N. , Zhou Z. (2009) Variational discretization for optimal control governed by convection dominated diffusion equations J. Comput. Math. 27 237 - 253
Hoppe R.H.W. , Kieweg M. (2010) Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems Comput. Optim. Appl. 46 511 - 533    DOI : 10.1007/s10589-008-9195-4
Kufner A. , John O. , Fucik S. 1977 Function Spaces, Nordhoff Leiden The Netherlands
Li R. , Liu W. , Ma H. , Tang T. (2002) Adaptive finite element approximation for distributed elliptic optimal control problems SIAM J. Control Optim. 41 (5) 1321 - 1349    DOI : 10.1137/S0363012901389342
Li R. , Liu W. , Yan N. (2007) A posteriori error estimates of recovery type for distributed convex optimal control problems J. Sci. Comput. 33 155 - 182    DOI : 10.1007/s10915-007-9147-7
Lions J. 1971 Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag Berlin
Lions J. , Magenes E. 1972 Non Homogeneous Boundary Value Problems and Applications. Springer-verlag Berlin
Lin Q. , Zhu Q. 1994 The Preprocessing and Postprocessing for the Finite Element Method Shanghai Scientific and Technical Publishers Shanghai
Liu H. , Yan N. (2007) Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations J. Comput. Appl. Math. 209 187 - 207    DOI : 10.1016/j.cam.2006.10.083
Liu W. , Yan N. (2001) A posteriori error estimates for distributed convex optimal control problems Adv. Comput. Math. 15 285 - 309    DOI : 10.1023/A:1014239012739
Liu W. , Yan N. (2003) A posteriori error estimates for optimal control problems governed by parabolic equations Numer. Math. 93 497 - 521    DOI : 10.1007/s002110100380
Liu W. , Yan N. 2008 Adaptive Finite Element Methods For Optimal Control Governed by PDEs Science Press Beijing
Meyer C. , Rösch A. (2004) Superconvergence properties of optimal control problems SIAM J. Control Optim. 43 (3) 970 - 985    DOI : 10.1137/S0363012903431608
Meidner D. , Vexler B. (2007) Adaptive space-time finite element methods for parabolic optimization problems SIAM J. Control Optim. 46 (1) 116 - 142    DOI : 10.1137/060648994
Neittaanmaki P. , Tiba D. 1994 Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications Dekker New Nork
Tang Y. , Chen Y. (2012) Variational discretization for parabolic optimal control problems with control constraints J. Syst. Sci. Complex. 25 880 - 895    DOI : 10.1007/s11424-012-0279-y
Tang Y. , Chen Y. (2013) Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems Front. Math. China 8 (2) 443 - 464    DOI : 10.1007/s11464-013-0239-4
Tang Y. , Hua Y. (2013) A priori error estimates and superconvergence property of variational discretization for nonlinear parabolic optimal control problems J. Appl. Math. Infor. 31 (3-4) 479 - 490    DOI : 10.14317/jami.2013.479
Xiong C. , Li Y. (2011) A posteriori error estimates for optimal distributed control governed by the evolution equations Appl. Numer. Math. 61 181 - 200    DOI : 10.1016/j.apnum.2010.09.004
Yan N. (2003) A posteriori error estimates of gradient recovery type for FEM of optimal control problem Adv. Comput. Math. 19 323 - 336    DOI : 10.1023/A:1022800401298
Yang D. , Chang Y. , Liu W. (2008) A priori error estimate and superconvergence annalysis for an opitmal control problem of bilinear type J. Comput. Math. 26 (4) 471 - 487
Veeser A. (2001) Efficient and reliable a posteriori error estimators for elliptic obstacle problems SIAM J. Numer. Anal. 39 (1) 146 - 167    DOI : 10.1137/S0036142900370812
Verfurth R. 1996 A review of Posteriori Error Esitmation and Adaptive Mesh Refinement Wiley- Teubner London
Vexler B. , Wollner W. (2008) Adaptive finite elements for elliptic optimization problems with control constraints SIAM J. Control Optim. 47 (1) 509 - 534    DOI : 10.1137/070683416
Zhang Z. , Zhu J.Z. (1995) Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I) Comput. Meth. Appl. Math. 123 173 - 187
Zienkiwicz O.C. , Zhu J.Z. (1992) The superconvergence patch recovery and a posteriori error estimates Int. J. Numer. Meth. Eng. 33 1331 - 1382    DOI : 10.1002/nme.1620330702
Zienkiwicz O.C. , Zhu J.Z. (1992) The superconvergence patch recovery (SPR) and adaptive finite element refinement Comput. Meth. Appl. Math. 101 207 - 224
Baiocchi C. , Capelo A. 1984 Variational and Quasi Variational Inequalities J. Wiley and Sons New York
Chan D. , Pang J.S. (1982) The generalized quasi variational inequality problems Math. Oper. Research 7 211 - 222    DOI : 10.1287/moor.7.2.211
Belly C. (1999) Variational and Quasi Variational Inequalities J. Appl.Math. and Computing 6 234 - 266
Pang D. (2002) The generalized quasi variational inequality problems J. Appl.Math. and Computing 8 123 - 245