- Open Access
Error analysis of variational discretization solving temperature control problems
© Tang; licensee Springer. 2013
- Received: 23 March 2013
- Accepted: 10 September 2013
- Published: 7 November 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.
- variational discretization
- finite element
- optimal control problems
- a priori error estimates
- a posteriori error estimates
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.
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:
where is the adjoint operator of B.
admits a unique solution and .
Consequently, we complete the proof of equation (2.7). □
It is clear that is Lipschitz continuous with constant 1. As in , it is easy to prove the following lemma.
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 .
Similar to Lemma 2.2, it is easy to show the following lemma.
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.
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 
Then (3.3) follows from (3.5)-(3.6). □
From (3.12) and (3.13), we get (3.9). □
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.
Then, (3.17) follows from (3.9), (3.14) and (3.18)-(3.21). □
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 
where is defined in (3.2).
Let δ be small enough, then (4.2) follows from (4.3). □
where n is the normal vector on outwards . For later convenience, we defined and when .
From (4.5) and (4.6), we derive (4.4). □
where and are defined in Lemma 4.3.
Then (4.7) follows from (4.2), (4.4) and (4.8)-(4.11). □
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 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 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 .
the domain Ω is the unit square and .
Algorithm 5.1, Example 1
16 × 16
32 × 32
64 × 64
128 × 128
256 × 256
Algorithm 5.2, Example 1
16 × 16
32 × 32
64 × 64
128 × 128
256 × 256
Numerical results, Example 2.
This work is supported by the Scientific Research Project of Department of Education of Hunan Province (13C338).
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