- Open Access
A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems
Boundary Value Problems volume 2014, Article number: 207 (2014)
In this paper a posteriori error estimate for continuous interior penalty Galerkin approximation of transient convection dominated diffusion optimal control problems with control constraints is presented. The state equation is discretized by the continuous interior penalty Galerkin method with continuous piecewise linear polynomial space and the control variable is approximated by implicit discretization concept. By use of the elliptic reconstruction technique proposed for parabolic equations, a posteriori error estimates for state variable, adjoint state variable and control variable are proved, which can be used to guide the mesh refinement in the adaptive algorithm.
Transient convection diffusion optimal control problems are widely used to model some engineering problems, for example, air pollution problem ,  and waste water treatment . In recent years the numerical approximations of this kind of problems form a hot topic, and many works are contributed to developing effective numerical methods and algorithms. For stabilization methods, we refer to – and for discontinuous Galerkin methods, we refer to , . For more literature, one can refer to the references cited therein.
It is well known that the solutions to convection diffusion problems may have boundary layers with small widths where their gradients change rapidly. Therefore, only using the stable methods to solve convection diffusion optimal control problems is generally not enough. One approach to improve the quality of a numerical solution is to exploit special mesh which is locally refined near the boundary layers, for example, Shishkin-type mesh or adaptive mesh. Note that a priori knowledge of the locations of the boundary layers is necessary to construct Shishkin-type mesh. Using adaptive mesh to resolve the boundary layers seems to be more natural. As we know the key problem of the adaptive finite element method is the a posteriori error estimate. Compared with a posteriori error estimates for stationary convection diffusion optimal control problems (see, , –), the works devoted to a posteriori error estimates for transient convection diffusion optimal control problems are much fewer. In  the authors discuss adaptive characteristic finite element approximation of transient convection diffusion optimal control problems with a general diffusion coefficient, where a posteriori error estimates in norm are derived by dual argument skill for the state and adjoint state variables.
The primary interest of this paper is to derive a posteriori error estimates for the following transient convection diffusion optimal control problem with dominance convection:
The details will be specified in the next section.
In order to improve the quality of the numerical solutions, the continuous interior penalty Galerkin method (CIP Galerkin method) is used to solve the state equation (1.2). This method was firstly proposed in . In ,  the CIP Galerkin method was used to approximate stationary convection diffusion optimal control problems, where a posteriori error estimates in and energy norm were derived. In  the CIP Galerkin method combined with Crank-Nicolson scheme was used to solve transient convection diffusion optimal control problems without constraints and a priori error estimates were deduced.
In the present paper, we apply the CIP Galerkin method combined with the backward Euler method to solve control constrained transient convection diffusion optimal control problems (1.1)-(1.2), where the control is discretized by the implicit discretization method developed in , and the state is approximated by piecewise linear finite element space. Due to the existence of boundary layer or interior layer for the state and adjoint state as well as limited regularity of control variable, we derive a posteriori error estimates for the state and adjoint state, which can be utilized to guide the mesh refinements in the adaptive algorithm. In contrast to , here we use the elliptic reconstruction technique developed in  for parabolic problems instead of dual argument skill to deduce the a posterior error estimates for the state and adjoint state. By use of this technique we can take full advantage of the well-established a posteriori error estimates for stationary convection diffusion optimal control problems in ,  to derive the a posterior error estimate for transient convection diffusion optimal control problems.
The paper is organized as follows. In Section 2 we describe the continuous interior penalty Galerkin scheme for the constrained optimal control problem. In Section 3 a posteriori error estimates are derived. Finally, we briefly summarize the method used, results obtained and possible future extensions and challenges.
Throughout this paper denotes a generic constant independent of mesh parameters and may be different at different occurrence. We use the expression to stand for .
2 The CIP Galerkin approximation scheme
2.1 Problems formulation
Consider the following transient convection diffusion optimal control problems:
Here Ω is a bounded domain in with boundary ∂ Ω. and is the initial value. is a bounded convex set with two constants satisfying . is the reaction coefficient, is a small diffusion coefficient, and is a velocity field. We assume that the following coercivity condition holds:
To consider the CIP Galerkin approximation of the above optimal control problem, we first derive a weak formulation for the state equation. Let be the bilinear form given by
It is easy to check
Then the weak formulation of state equation (2.2) reads as
The existence and uniqueness of solutions to (2.5)-(2.6) can be guaranteed by the theory in . Moreover, by using the Lagrange functional, the first-order necessary (also sufficient here) optimality condition of (2.5)-(2.6) can be characterized by
From the second equation in (2.7), we have that the adjoint state z satisfies transient convection diffusion equations with the strong form
In contrast to the state equation, the velocity field of the adjoint equation is −β .
By the pointwise projection on ,
the optimal condition in (2.7) simplifies to
2.2 Semi-discrete discretization
Let be a regular triangulation of Ω, so that . Let denote the diameter of the element K. Associated with is a finite dimensional subspace of , consisting of piecewise linear polynomials.
To control the convective derivative of the discrete solution sufficiently, a symmetric stabilization form S (see, e.g., ) on was introduced as follows:
where is the stabilization parameter. denotes the collection of interior edges of the elements in . is the size of the edge E. denotes the jump of q across E for defined by
with n being the outward unit normal.
Here the control variable was approximated by variational discrete concept (see ). in general is not a finite element function associated with the space mesh .
By standard argument it can be shown that the following first-order optimality condition holds:
By the pointwise projection operator , we have
2.3 Fully discrete scheme
To define a fully discrete scheme, we introduce a time partition. Let be a time grid with , . Set .
Similar to a semi-discrete scheme, we can derive the discrete first-order optimality condition:
Again by the pointwise projection operator , we obtain
We can see that is a piecewise constant function in time.
For , let
For , , we set , . Note that
for . Then the above optimality conditions can be rewritten as
3 A posteriori error estimates
The objective of this section is to derive a posteriori error estimates for the state, adjoint state and control.
3.1 The estimate for control
To obtain the estimate for control, we introduce an auxiliary problem. For given , let be a solution of the following system:
Here the last inequality was fulfilled due to the implicit discretization of the control variable.
where and was used. Thus we arrive at
Choosing yields the theorem result. □
3.2 The estimate for the state and adjoint state
To this end we first introduce the following elliptic reconstruction definitions for state and adjoint state.
For , we define the elliptic reconstruction and satisfying the following elliptic problems:
Noticing that the CIP Galerkin approximation of can be defined as
Then we have
which implies . We can observe a similar property for the CIP Galerkin approximation of .
Using the above convention, we define and as
for and . We decompose the error as follows:
Nextly we shall derive the estimates of ρ and ξ. For simplicity, we introduce the following notations:
Let and be the discrete operators associated with the state and adjoint state, which are defined by the following for :
The time error estimators are characterized by
Moreover, let , .
In the following we shall deduce the estimates of and . By (3.1) and Definition 3.2 we can derive the following error equations for and .
Given, we deduce
Then we have
Similarly we can deduce the error equation for . □
Letbe a quasi-interpolation operator of Clément type. The following estimates hold for all elements K, all faces E and all functions:
whereanddenote the union of all elements that share at least one point with K and E.
Then we arrive at the following.
The following estimate holds:
Setting in (3.4) leads to
Integrating in time from 0 to T gives
Again integrating in time from 0 to results in
Combining the above two inequalities yields
In the following we shall derive the estimates of . By the definition of the elliptic reconstruction, we can bound as follows:
with an arbitrarily positive constant δ.
For the second term , we can bound it as follows:
Now it remains to estimate . Note that
This term can be estimated by the techniques used in a posterior error estimates for the stationary problem. To this end we introduce an auxiliary problem
For the above auxiliary problem, the following stability estimates (see, e.g., ) hold:
Using the above auxiliary problem, we have
By the definitions of and , we can deduce
where denotes the Clément interpolation of ϕ. Further, we have
Then we derive
It follows from Lemma 3.6 and (3.8) that
Thus we arrive at
Inserting the estimates of , and into (3.6) and setting δ small enough leads to the theorem results. □
Then by Lemmas 3.3 and 3.7 we can deduce the estimates of . □
The second term is the elliptic reconstruction error, which can bounded as follows for :
Combining Lemmas 3.3 and 3.7, we can deduce
Now we turn our attention to estimate . The argument skills are similar to those used in the estimate of . Therefore we just sketch the proof.
Setting in (3.5) leads to
Then integrating the above equation from to T and 0 to T, respectively, leads to
In an analogous way to Lemma 3.7, we can derive the estimate for .
Collecting Lemmas 3.4 and 3.10 and using similar arguments to Theorem 3.8 yields the following.
Similar to Remark 3.9, we can also derive the posteriori error estimates of
3.3 The main results
Using the above estimate and Lemma 3.1, we can derive the posteriori error estimates of .
In this paper a posteriori error estimates were established for time-dependent convection diffusion optimal control problems by the elliptic reconstruction technique. By introducing the elliptic reconstruction, we can take full advantage of the well-established a posteriori error estimates for stationary convection diffusion optimal control problems. There are still many issues needed to be addressed, such as optimal control problems with state constraints and pointwisely imposed control problems. The applications of our approach to these settings will be postponed to our future work.
Parra-Guevara D, Skiba Y: Elements of the mathematical modelling in the control of pollutants emissions. Ecol. Model. 2003, 167: 263-275. 10.1016/S0304-3800(03)00191-1
Zhu J, Zeng QC: A mathematical formulation for optimal control of air pollution. Sci. China, Ser. D 2003, 46: 994-1002. 10.1007/BF02959394
Martínez A, Rodríguez C, Vázquez-Méndez ME: Theoretical and numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim. 2000, 38: 1534-1553. 10.1137/S0363012998345640
Becker R, Vexler B: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 2007, 106: 349-367. 10.1007/s00211-007-0067-0
Hinze M, Yan NN, Zhou ZJ: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 2009, 27: 237-253.
Scott Collis, S, Heinkenschloss, M: Analysis of the Streamline Upwind/Petrov Galerkin Method applied to the solution of optimal control problems. CAAM TR02-01, Rice University, Houston (2002)
Yan NN, Zhou ZJ: A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. J. Comput. Appl. Math. 2009, 223: 198-217. 10.1016/j.cam.2008.01.006
Leykekhman D, Heinkenschloss M: Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 2012, 50(4):2012-2038. 10.1137/110826953
Yan NN, Zhou ZJ: The local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations. Int. J. Numer. Anal. Model. 2010, 7: 681-699.
Dede L, Quarteroni A: Optimal control and numerical adaptivity for advection diffusion-equations. Math. Model. Numer. Anal. 2005, 39: 1019-1040. 10.1051/m2an:2005044
Micheletti S, Perotto S: The effect of anisotropic mesh adaptation on PDE-constrained optimal control problems. SIAM J. Control Optim. 2011, 49: 1793-1828. 10.1137/090758350
Yan NN, Zhou ZJ: A priori and a posteriori error estimates of streamline diffusion finite element method for optimal control problem governed by convection dominated diffusion equation. Numer. Math., Theory Methods Appl. 2008, 1: 297-320.
Fu HF, Rui HX: Adaptive characteristic finite element approximation of convection-diffusion optimal control problem. Numer. Methods Partial Differ. Equ. 2013, 29: 979-998. 10.1002/num.21741
Burman E, Hansbo P: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 2004, 193: 1437-1453. 10.1016/j.cma.2003.12.032
Yan NN, Zhou ZJ: A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations. Front. Math. China 2008, 3: 415-442. 10.1007/s11464-008-0029-6
Burman E: Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations. Commun. Math. Sci. 2011, 9: 319-329. 10.4310/CMS.2011.v9.n1.a16
Hinze M: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 2005, 30: 45-61. 10.1007/s10589-005-4559-5
Makridakis C, Nochetto RH: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 2003, 41: 1585-1594. 10.1137/S0036142902406314
Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.
Lakkis O, Makridakis C: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 2006, 75: 1627-1658. 10.1090/S0025-5718-06-01858-8
Scott LR, Zhang S: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 1990, 54: 483-493. 10.1090/S0025-5718-1990-1011446-7
Verfürth R: A posteriori error estimators for convection-diffusion equations. Numer. Math. 1998, 80: 641-663. 10.1007/s002110050381
Navert, U: A finite element method for convection diffusion problems. PhD thesis, Chalmers Inst. of Tech. (1982)
The authors would like to thank the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant: 11301311, 11201485), the Science and Technology Development Planning Project of Shandong Province (No. 2012GGB01198).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
About this article
Cite this article
Zhou, Z., Fu, H. A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems. Bound Value Probl 2014, 207 (2014). https://doi.org/10.1186/s13661-014-0207-2
- transient convection diffusion optimal control problem
- continuous interior penalty Galerkin method
- elliptic reconstruction
- a posteriori error estimate