New a posteriori error estimates of mixed finite element methods for quadratic optimal control problems governed by semilinear parabolic equations with integral constraint

In this paper, we investigate new L^{\mathrm{\infty }}(L^2) and L^2(L^2)-posteriori error estimates of mixed finite element solutions for quadratic optimal control problems governed by semilinear parabolic equations. The state and the co-state are discretized by the order one Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^{\mathrm{\infty }}(J;L^2(\mathrm{\Omega }))-norm and L^2(J;L^2(\mathrm{\Omega }))-norm for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the optimal control problem.

MSC:49J20, 65N30.

In this paper we consider quadratic optimal control problems governed by the semilinear parabolic equations

where the bounded open set \mathrm{\Omega }\subset {\mathbf{R}}^2 is a convex polygon with the boundary ∂ Ω. J=[0,T]. Let K be a closed convex set in the control space U=L^2(J;L^2(\mathrm{\Omega })), \mathbf{p},{\mathbf{p}}_d\in {(L^2(J;H^1(\mathrm{\Omega })))}^2, y,y_d\in L^2(J;H^1(\mathrm{\Omega })), f,u\in L^2(J;L^2(\mathrm{\Omega })), y_0(x)\in H_0^1(\mathrm{\Omega }). For any R>0, the function \varphi (\cdot )\in W^{1,\mathrm{\infty }}(-R,R), {\varphi }^{\prime }(y)\in L^2(\mathrm{\Omega }) for any y\in H^1(\mathrm{\Omega }), and {\varphi }^{\prime }(y)\ge 0. Assume that the coefficient matrix A(x)={(a_{ij}(x))}_{2\times 2}\in C^{\mathrm{\infty }}(\overline{\mathrm{\Omega }};{\mathbf{R}}^{2\times 2}) is a symmetric 2\times 2-matrix and there are constants c_1,c_2>0 satisfying for any vector \mathbf{X}\in {\mathbf{R}}^2, c_1{\parallel \mathbf{X}\parallel }_{{\mathbf{R}}^2}^2\le {\mathbf{X}}^{t}A\mathbf{X}\le c_2{\parallel \mathbf{X}\parallel }_{{\mathbf{R}}^2}^2. We assume that the constraint on the control is an obstacle such that

Optimal control problems have been successfully utilized in scientific and engineering numerical simulation. Thus they must be solved by using some efficient numerical methods. Among these numerical methods, the finite element method was a good choice for solving partial differential equations. There have been extensive studies in convergence for finite element approximation of optimal control problems. A systematic introduction of the finite element method for optimal control problems can be found in [1–7].

Recently, an adaptive finite element method has been investigated extensively. It has become one of the most popular methods in the scientific computation and numerical modeling. Adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated by a posteriori error estimators. Hence it is an important approach to boost the accuracy and efficiency of finite element discretizations. There are lots of works concentrating on the adaptivity of many optimal control problems, for example, [8–12]. Note that all the above works aimed at the standard finite element method.

In many control problems, the objective functional contains the gradient of state variables. Thus, the accuracy of the gradient is important in numerical discretization of the coupled state equations. Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy by using such methods. When the objective functional contains the gradient of the state variable, mixed finite element methods should be used for discretization of the state equation with which both the scalar variable and its flux variable can be approximated in the same accuracy.

Recently, in [13, 14] we did some primary work on a priori error estimates for nonlinear parabolic optimal control problems by mixed finite element methods. In [15], we considered a posteriori error estimates of triangular mixed finite element methods for semilinear elliptic optimal control problems. The state and the co-state were discretized by the Raviart-Thomas mixed finite element spaces and the control was approximated by piecewise constant functions. In [16], we derived a posteriori error estimates for linear parabolic optimal control problems by the lowest order Raviart-Thomas mixed finite element methods.

This paper is motivated by the idea of the article [17]. We shall use the order one Raviart-Thomas mixed finite element to discretize the state and the co-state. Due to the limited regularity of the optimal control u in general, we therefore only consider a piecewise constant space. Then we derive a posteriori error estimates for the mixed finite element approximation of the optimal control problem. The estimators for the control, the state and the co-state variables are derived in the sense of L^{\mathrm{\infty }}(J;L^2(\mathrm{\Omega }))-norm or L^2(J;L^2(\mathrm{\Omega }))-norm, which are different from the ones in [16].

In this paper, we adopt the standard notation W^{m,p}(\mathrm{\Omega }) for Sobolev spaces on Ω with a norm {\parallel \cdot \parallel }_{m,p} given by {\parallel v\parallel }_{m,p}^p={\sum }_{|\alpha |\le m}{\parallel D^{\alpha }v\parallel }_{L^p(\mathrm{\Omega })}^p, a semi-norm {|\cdot |}_{m,p} given by {|v|}_{m,p}^p={\sum }_{|\alpha |=m}{\parallel D^{\alpha }v\parallel }_{L^p(\mathrm{\Omega })}^p. We set W_0^{m,p}(\mathrm{\Omega })=\{v\in W^{m,p}(\mathrm{\Omega }):v{|}_{\partial \mathrm{\Omega }}=0\}. For p=2, we define H^m(\mathrm{\Omega })=W^{m,2}(\mathrm{\Omega }), H_0^m(\mathrm{\Omega })=W_0^{m,2}(\mathrm{\Omega }), and {\parallel \cdot \parallel }_m={\parallel \cdot \parallel }_{m,2}, \parallel \cdot \parallel ={\parallel \cdot \parallel }_{0,2}. We denote by L^s(0,T;W^{m,p}(\mathrm{\Omega })) the Banach space of all L^s integrable functions from J into W^{m,p}(\mathrm{\Omega }) with the norm {\parallel v\parallel }_{L^s(J;W^{m,p}(\mathrm{\Omega }))}={({\int }_0^T{\parallel v\parallel }_{W^{m,p}(\mathrm{\Omega })}^{s}dt)}^{\frac{1}{s}} for s\in [1,\mathrm{\infty }), and the standard modification for s=\mathrm{\infty }. Similarly, one can define the spaces H^1(J;W^{m,p}(\mathrm{\Omega })) and C^k(J;W^{m,p}(\mathrm{\Omega })). The details can be found in [18].

The plan of this paper is as follows. In the next section, we shall construct the mixed finite element approximation and the backward Euler discretization for quadratic optimal control problems governed by semilinear parabolic equations (1.1)-(1.4). Then, we derive a posteriori error estimates for both the state and the control approximation in Section 3. Finally, we give a conclusion and some future work.

In this section we shall now discuss the mixed finite element approximation and the backward Euler discretization of quadratic semilinear parabolic optimal control problems (1.1)-(1.4). To fix the idea, we shall take the state spaces L^2(\mathbf{V})=L^2(J;\mathbf{V}) and H^1(W)=H^1(J;W), where V and W are defined as follows:

The Hilbert space V is equipped with the following norm:

Let \alpha =A^{-1}, we recast (1.1)-(1.4) as the following weak form: find (\mathbf{p},y,u)\in L^2(\mathbf{V})\times H^1(W)\times K such that

It follows from [17] that optimal control problem (2.1)-(2.4) has a solution (\mathbf{p},y,u), and that if a triplet (\mathbf{p},y,u) is the solution of (2.1)-(2.4), then there is a co-state (\mathbf{q},z)\in L^2(\mathbf{V})\times H^1(W) such that (\mathbf{p},y,\mathbf{q},z,u) satisfies the following optimality conditions:

where (\cdot ,\cdot ) is the inner product of L^2(\mathrm{\Omega }).

In [19], the expression of the control variable was given. Here, we adopt the same method to derive the following operator:

where \overline{z}={\int }_{\mathrm{\Omega }}z/{\int }_{\mathrm{\Omega }}1dx denotes the integral average on Ω of the function z.

Let {\mathcal{T}}_h be regular triangulations of Ω. h_{\tau } is the diameter of τ and h=max{h}_{\tau }. Let {\mathbf{V}}_h\times W_h\subset \mathbf{V}\times W denote the order one Raviart-Thomas space associated with the triangulations {\mathcal{T}}_h of Ω. P_k denotes the space of polynomials of total degree at most k. Let \mathbf{V}(\tau )=\{\mathbf{v}\in P_1^2(\tau )+x\cdot P_1(\tau )\}, W(\tau )=P_1(\tau ). We define

Let L^2({\mathbf{V}}_h)=L^2(J;{\mathbf{V}}_h) and H^1(W_h)=H^1(J;W_h). The mixed finite element discretization of (2.1)-(2.4) is as follows: compute ({\mathbf{p}}_h,y_h,u_h)\in L^2({\mathbf{V}}_h)\times H^1(W_h)\times K_h such that

where y_0^h(x)\in W_h is an approximation of y_0. Optimal control problem (2.13)-(2.16) again has a solution ({\mathbf{p}}_h,y_h,u_h), and that if a triplet ({\mathbf{p}}_h,y_h,u_h) is the solution of (2.13)-(2.16), then there is a co-state ({\mathbf{q}}_h,z_h)\in L^2({\mathbf{V}}_h)\times H^1(W_h) such that ({\mathbf{p}}_h,y_h,{\mathbf{q}}_h,z_h,u_h) satisfies the following optimality conditions:

Next, we define the standard L^2(\mathrm{\Omega })-orthogonal projection {\mathcal{Q}}_h:K\to K_h, which satisfies: for any \tilde{u}\in K,

Similar to (2.12), for variational inequality (2.23), we have the following conclusion [19]. Assume that z_h is known in variational inequality (2.23). The solution of the variational inequality is

Now we consider the fully discrete approximation for the above semidiscrete problem. Let \mathrm{\Delta }t>0, N=\frac{T}{\mathrm{\Delta }t}\in \mathbb{Z}, and t_i=i\mathrm{\Delta }t, i\in \mathbb{Z}. Also, let

The following fully discrete approximation scheme is to find ({\mathbf{p}}_h^i,y_h^i,u_h^i)\in {\mathbf{V}}_h\times W_h\times K_h, i=1,2,\dots ,N, such that

It follows that optimal control problem (2.27)-(2.30) has a solution ({\mathbf{p}}_h^i,y_h^i,u_h^i), i=1,2,\dots ,N, and that if a triplet ({\mathbf{p}}_h^i,y_h^i,u_h^i)\in {\mathbf{V}}_h\times W_h\times K_h, i=1,2,\dots ,N, is the solution of (2.27)-(2.30), then there is a co-state ({\mathbf{q}}_h^{i-1},z_h^{i-1})\in {\mathbf{V}}_h\times W_h such that ({\mathbf{p}}_h^i,y_h^i,{\mathbf{q}}_h^{i-1},z_h^{i-1},u_h^i)\in {({\mathbf{V}}_h\times W_h)}^2\times K_h satisfies the following optimality conditions:

For i=1,2,\dots ,N, let

For any function w\in C(J;L^2(\mathrm{\Omega })), let

Then optimality conditions (2.31)-(2.37) satisfy

Similar to (2.26), the solution of variational inequality (2.44) is

In the rest of the paper, we shall use some intermediate variables. For any control function U_h\in K_h, we first define the state solution (\mathbf{p}(U_h),y(U_h),\mathbf{q}(U_h),z(U_h)) satisfying

For \phi \in W_h, we shall write

where

are bounded functions in \overline{\mathrm{\Omega }}.

Let R_h:W\to W_h be the orthogonal L^2(\mathrm{\Omega })-projection into W_h [20] which satisfies:

Let {\mathrm{\Pi }}_h:\mathbf{V}\to {\mathbf{V}}_h be the Raviart-Thomas projection operator [21] which satisfies: for any \mathbf{v}\in \mathbf{V},

where {\mathcal{E}}_h denote the set of element sides in {\mathcal{T}}_h. We have the commuting diagram property

where and after, I denotes an identity matrix.

Further, the interpolation operator {\mathrm{\Pi }}_h satisfies a local error estimate

The following lemmas are important in deriving a posteriori error estimates of residual type.

Lemma 2.1 Let {\hat{\pi }}_h be the average interpolation operator defined in [22]. For m=0 or 1, 1\le q\le \mathrm{\infty } and \mathrm{\forall }v\in W^{1,q}({\mathrm{\Omega }}^h),

Lemma 2.2 Let {\pi }_h be the standard Lagrange interpolation operator [23]. Then, for m=0 or 1, and \mathrm{\forall }v\in W^{2,q}({\mathrm{\Omega }}^h),

In this section we study new L^{\mathrm{\infty }}(L^2) and L^2(L^2)-posteriori error estimates for the mixed finite element approximation to the semilinear parabolic optimal control problems. Let

It can be shown that (see [13] for some detail discussions)

It is clear that S and S_h are well defined and continuous on K and K_h. Also, the functional S_h can be naturally extended on K. Then (2.1) and (2.27) can be represented as

and

In many applications, S(\cdot ) is uniform convex near the solution u. The convexity of S(\cdot ) is closely related to the second-order sufficient conditions of the optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many applications, there is c>0, independent of h, such that

Firstly, let us derive a posteriori error estimates for the control u.

Theorem 3.1 Let u and U_h be the solutions of (3.6) and (3.7), respectively. Assume that (S_h^{\prime }(U_h)){|}_{\tau }\in H^s(\tau ), \mathrm{\forall }\tau \in {\mathcal{T}}_h (s=0,1), and there is v_h\in K_h such that

Then we have

where

Proof It follows from (3.6) and (3.7) that

Then it follows from assumptions (3.8), (3.9), and the Schwarz inequality that

It is not difficult to show

where z(U_h) is defined in (2.46)-(2.51). Thanks to (3.14), it is easy to derive

Then, by estimates (3.13) and (3.15), we can prove the requested result (3.10). □

To estimate the error {\parallel {\tilde{Z}}_h-z(U_h)\parallel }_{L^2(J;L^2(\mathrm{\Omega }))}^2, we need the following well-known stability results for the following dual equations:

and

where

(3.18)

(3.18′)

Lemma 3.1 Let κ and ϖ be the solutions of (3.16) and (3.17), respectively [24, 25]. Let Ω be a convex domain. Then

and

where |D^{2}v|=max\{|{\partial }^{2}v/\partial x_i\partial x_j|,1\le i,j\le 2\}.

Next, we estimate the errors Y_h-y(U_h) and P_h-\mathbf{p}(U_h).

Theorem 3.2 Let (P_h,Y_h,Q_h,Z_h,U_h) and (\mathbf{p}(U_h),y(U_h),\mathbf{q}(U_h),z(U_h),U_h) be the solutions of (2.38)-(2.44) and (2.46)-(2.51), respectively. Then we have

where

Proof We define {\mathbf{p}}_h^N as follows:

Then from (3.20) we deduce that

Combining (3.20)-(3.21) and the definitions of Y_h and P_h, we can get the following equality:

Let κ be the solution of (3.16) with {\kappa }_0(x)=(Y_h-y(U_h))(x,t^{\ast }), we infer that