A posteriori error estimates for continuous interior penalty Galerkin approximation of transient convection diffusion optimal control problems
© Zhou and Fu; licensee Springer 2014
Received: 15 March 2014
Accepted: 20 August 2014
Published: 25 September 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 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
In contrast to the state equation, the velocity field of the adjoint equation is −β .
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.
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 .
2.3 Fully discrete scheme
To define a fully discrete scheme, we introduce a time partition. Let be a time grid with , . Set .
We can see that is a piecewise constant function in time.
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
Here the last inequality was fulfilled due to the implicit discretization of the control variable.
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.
which implies . We can observe a similar property for the CIP Galerkin approximation of .
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 .
Similarly we can deduce the error equation for . □
whereanddenote the union of all elements that share at least one point with K and E.
Then we arrive at the following.
with an arbitrarily positive constant δ.
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 . □
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.
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.
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.
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).
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