A posteriori error estimates for mixed finite element solutions of convex optimal control problems. (English) Zbl 1165.65034

Chen, Yanping; Liu, Wenbin

J. Comput. Appl. Math. 211, No. 1, 76-89 (2008).

A posteriori error estimates for the mixed finite element approximation of a convex optimal control governed by elliptic equations are derived. The authors construct the mixed finite element approximation for the following optimal control problem:
\[ \min_{u\in K\subset L^2(\Omega_U)} [g_1(\mathbf{p})+g_2(y)+h(u)],\tag{1} \]
\[ \operatorname{div} \mathbf{p}=f+Bu \quad \text{in }\Omega \qquad \mathbf{p}=-A \operatorname{grad}y \quad\text{in }\Omega,\qquad y=0 \quad\text{on } \partial \Omega,\tag{2} \]
where the bounded open set \(\Omega \subset\mathbb R^2\), is a convex polygon or has the smooth boundary \(\partial \Omega, \Omega_U\) is a bounded open set in \(\mathbb R^2\) with the Lipschitz boundary \(\partial \Omega_U\), and \(K\) is a closed convex set in \(L^2(\Omega_U)\). Using a weak formula for the state equality (2), the mixed finite element approximation of (1), (2), is introduced as follows: Find \([\mathbf{p}_h, y_h, u_h] \in \mathbf{V}_h \times W_h \times U_h\) such that
\[ \min_{u_h\in K_h \subset U_h} \{g_1(\mathbf{p}_h)+g_2(y_h)+h(u_h)\},\tag{3} \]
\[ \begin{aligned} (A^{-1} \mathbf{p}_h, \mathbf{v}_h)-(y_h, \operatorname{div} \mathbf{v}_h) = 0, &\quad \text{for any } \mathbf{v}_h \in \mathbf{V}_h \subset \mathbf{V}= H(\operatorname{div}, \Omega), \\ \operatorname{div}\mathbf{p}_h, w_h)=(f+Bu_h, w_h), &\quad \text{for any } w_h \in W_h \subset W = L^2 (\Omega), \end{aligned}\tag{4} \]
where \(\mathbf{V}_h \times W_h \subset \mathbf{V} \times W\) denotes the RT, BDM or BDFM space of index \(k\) associated with triangulation or rectangulation \(\operatorname{Im}_h\) of \(\Omega\), where \(k \geq 0, K_h\) is a non-empty closed convex set in \(U_h\). For fixed \(h\) the control problem (3), (4) has a unique solution \([\mathbf{p}_h, y_h, u_h] \in \mathbf{V}_h \times W_h \times U_h\) if and only if there is a co-state \((\mathbf{q}_h, z_h)\in \mathbf{V}_h \times W_h\) such that \((\mathbf{p}_h, y_h, \mathbf{q}_h, z_h, u_h)\) satisfies the following optimality conditions
\[ \begin{aligned} (A^{-1} \mathbf{p}_h, \mathbf{v}_h)-(y_h, \operatorname{div}\mathbf{v}_h) = 0, &\quad \text{for any } \mathbf{v}_h \in \mathbf{V}_h,\\ (\operatorname{div}\mathbf{p}_h, w_h)=(f+Bu_h, w_h), &\quad \text{for any } w_h \in W_h,\\ (A^{-1} \mathbf{q}_h, w_h)-(z_h, \operatorname{div}\mathbf{v}_h) = - (g_{1}' (\mathbf{p}_h), \mathbf{v}_h), &\quad \text{for any } \mathbf{v}_h \in \mathbf{V}_h,\\ (\operatorname{div}\mathbf{q}_h, w_h)= (g'_{2} (y_h), w_h), &\quad \text{for any } w_h \in W_h, \\ (h'(u_h)+B^*z_h, \tilde{u}_h-u_h) \geq 0, &\quad \text{for any }\tilde{u}_h \in K_h. \end{aligned}\tag{5} \]
Main results: A posteriori error estimates for some intermediate errors for the RT, the BDM and the BDFM mixed method, are derived. The authors’ analysis relies on a decomposition of the flux functions in the spirit of a generalized Helmholtz decomposition. Further, a posteriori error estimates for convex optimal control problems in general cases where the convex set \(K\) may not be the whole control space \(U\) and the discretized constraint set \(K_H\) may not be the subset of \(K\), are also presented. Finally equivalent a posteriori error estimates for a class of optimal control problems which are most frequently met in applications are established. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.

Reviewer: Jan Lovíšek (Bratislava)

MSC:

65K10	Numerical optimization and variational techniques
49J20	Existence theories for optimal control problems involving partial differential equations
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

adaptive remeshing schemes; a posteriori error estimator; mixed finite element; interpolation operators; convex optimal control problems; optimality conditions; Raviart-Thomas elements; Brezzi-Douglas-Marini elements

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