Superconvergence of semidiscrete finite element methods for bilinear parabolic optimal control problems

In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence between the semidiscrete finite element solutions and projections of the exact solutions. A numerical example is presented to verify our theoretical results.


Introduction
We consider the following bilinear parabolic optimal control problem: where ∈ R  is a convex polygon with the boundary ∂ , and J = [, T] ( < T < +∞). The coefficient matrix A(x) = (a ij (x)) × ∈ [W ,∞ (¯ )] × is a symmetric and positive definite. Moreover, we assume that f (t, x) ∈ C(J; L  ( )), y  (x) ∈ H   ( ), and the admissible control set K is defined by where  ≤ a < b are real numbers.
There has been a wide range of research on finite element approximation of elliptic optimal control problems. For finite element solving linear and semilinear elliptic control problems, a priori error estimates were investigated in [] and [], and superconvergence were established in [] and [], respectively. Yang et al. [] obtained the superconvergence of finite element approximation of bilinear elliptic control problems. In addition, some similar results of mixed finite element approximation for linear elliptic control problems can be found in [, ].
In recent years, there are a lot of related works on finite element approximation of parabolic optimal control problems, mostly focused on linear or semilinear cases. A priori error estimates of space-time finite element and standard finite element approximation for linear parabolic control problem were derived in [] and []. The superconvergence of variational discretization and standard finite element approximation for semilinear parabolic control problem can be found in [] and [], respectively.
As far as we know, there has been little work done on bilinear parabolic control problems. In this paper, we purpose to obtain the superconvergence properties of semidiscrete finite element method for bilinear parabolic optimal control problems.
We adopt the notation W m,q ( ) for Sobolev spaces on with norm · W m,q ( ) and seminorm | · | W m,q ( ) . We set H   ( ) ≡ {v ∈ H  ( ) : v| ∂ = } and denote W m, ( ) by H m ( ). We denote by L s (J; W m,q ( )) the Banach space of all L s integrable functions from  s for s ∈ [, ∞) and the standard modification for s = ∞. Similarly, we can define the space H l (J; W m,q ( )) and C k (J; W m,q ( )) (see e.g. []). In addition, let c or C be generic positive constants.
The rest of this paper is organized as follows. A semidiscrete finite element approximation of (.)-(.) is presented in Section . Some important intermediate variables and their error estimates are introduced in Section . In Section , a priori error estimates of the approximation scheme are derived. In Section , the superconvergence between projections of the exact solutions and the finite element solutions is obtained. A numerical example is presented to illustrate our theoretical results in the last section.

A semidiscrete finite element approximation
We now consider a standard semidiscrete finite element approximation of (.)-(.). To ease the exposition, we denote L p (J; W m,q ( )) and · L p (J;W m,q ( )) by L p (W m,q ) and · L p (W m,q ) respectively. Let W = H   ( ) and U = L  ( ). Moreover, we denote · H m ( ) and · L  ( ) by · m and · , respectively. Let From the assumptions on A we have The weak formulation of (.)-(.) can be read as follows: It follows from (see e.g. []) that problem (.)-(.) has at least one solution (y, u) and that if the pair (y, u) ∈ (H  (L  ) ∩ L  (H   )) × K is a solution of (.)-(.), then there is a costate p ∈ (H  (L  ) ∩ L  (H   )) such that the triplet (y, p, u) meets the following optimality conditions: As in [], it is easy to get the following lemma.
Lemma . Let (y, p, u) be the solution of (.)-(.). Then Let P  be the space of polynomials not exceeding , and T h be regular triangulations of such that¯ = τ ∈T hτ and h = max τ ∈T h {h τ }, where h τ denotes the diameter of the element τ . Furthermore, we set As in [], we assume that is a closed convex set in U h . We recast a semidiscrete finite element approximation of (.)-(.) as where y h  (x) = R h (y  (x)), and R h is an elliptic projection operator, which will be specified later.
It is well known that (.)-(.) again has a solution (y h , u h ) and that if the pair (y h , u h ) ∈ H  (W h ) × L  (K h ) is a solution of (.)-(.), then there is a costate p h ∈ H  (W h ) such that the triplet (y h , p h , u h ) meets the following conditions: We introduce the averaging operator π c h from U onto U h as where |τ | is the measure of τ . Then we can similarly derive the following lemma.

Error estimates of intermediate variables
In this section, we introduce some important intermediate variables and derive some related error estimates. For Let y h (v), p h (v) meet the following system: If (y, p, u) and (y h , p h , u h ) are the solutions of (.)-(.) and (.)-(.), respectively, then (y, p) = (y(u), p(u)) and (y h , We define an elliptic projection operator R h : W → W h that satisfies and the L  -orthogonal projection operator Q h : U → U h that satisfies They have the following properties (see e.g. []): The following lemmas are very important for a priori error estimates and superconvergence analysis.

A priori error estimates
In this section, we derive a priori error estimates of the approximation scheme (.)-(.). For ease of exposition, we set It is easy to show that As in [], we assume that there exist neighborhoods of the exact solution u or of the approximation solution u h in K and a constant c  >  such that, for any v or v h in this neighborhood, the objective functional satisfies the following convexity conditions:

Superconvergence analysis
In this section, we derive the superconvergence between projections of the exact solutions and approximation solutions. Let u be the solutions of (.)-(.). For a fixed t * ( ≤ t * ≤ T), we divide into the following subsets: It is easy to see that these three subsets do not intersect with each other and =¯ + ∪  ∪¯ -. We assume that u and T h are regular such that meas( -) ≤ Ch (see, e.g., []). Moreover, we suppose that the exact control, state, and costate solutions satisfy Then, we have It follows from the definition of Q h , (.), and (.) that For the first term, at time t * ( ≤ t * ≤ T), we have Hence, By using Hölder's inequalty, the embedding inequality v L  ( ) ≤ C v H  ( ) , and Young's inequality, I  and I  can be estimated as follows: and In addition, noting that y, p ∈ L  (L ∞ ), we have Proof From the definition of R h , (.)-(.), and (.)-(.), for any w h or q h ∈ W h and t ∈ J, we have Hence, letting w h = y h -R h y in (.), (.) follows from (.)-(.), Hölder's inequality, Young's inequality, Gronwall's inequality, (.), and (.). Inequality (.) can be similarly derived.
By using the backward Euler scheme to approximate the time derivative, we introduce the following fully discrete approximation scheme: find (y n h , Example  The data are as follows: f (t, x) = y t (t, x) -div A(x)∇y(t, x) + u(t, x)y(t, x), y d (t, x) = y(t, x) + p t (t, x) + div A * (x)∇p(t, x)p(t, x)y(t, x). where l =  for the control u and the state y, and l =  for the costate p. In Table , the errors |||Q h uu h |||, |||R h yy h |||  , and |||R h pp h |||  on a sequence of uniformly refined meshes are listed. It is consistent with our superconvergence results in Section .
When h = .e-, t =   , and t = ., we plot the profile of u h in Figure .

Conclusions
Although there has been extensive research on a priori error estimates and superconvergence of finite element methods for various optimal control problems, it mostly focused on linear or semilinear elliptic cases (see, e.g., [-, ]). In recent years, there have been considerable related results for finite element approximation of linear or semilinear parabolic 1.25e-2 3.58914e-3 8.83804e-4 1.50701e-3

Figure 1
The numerical solution u h at t = 0.5 in Example 1.
optimal control problems (see, e.g., [-]). Although bilinear optimal control problems are frequently met in applications, they are much more difficult to handle in comparison to linear or semilinear cases. There is little work on bilinear optimal control problems. Recently, Yang et al.
[] investigated a priori error estimates and superconvergence of finite element methods for bilinear elliptic optimal control problems. Hence, our results on bilinear parabolic optimal control problems are new.