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In this paper, we investigate a priori and a posteriori error estimates of H 1 -Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Based on two new elliptic projections, we derive a priori error estimates both for the control variable, the state variable and the co-state variable. The related a priori error estimates for the new projections error are also established. Moreover, a posteriori error estimates for all variables are derived via energy method. Such a posteriori error estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a priori error estimates of H 1 -Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Both semidiscrete and fully discrete schemes are considered. Based on some new elliptic projections, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. The related a priori error estimates for the new projections error are also established. A numerical example is given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces V h and W h is not necessary, thus, we can choose the approximation spaces more flexibly. In order to derive the superconvergence, we will use classical mixed finite element spaces. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. At first, we derive some superconvergence properties for the control variable, the state variables and the adjoint state variables. Then, using a recovery operator, we obtain a superconvergence result for the control variable. Next, combining energy approach with postprocessing method, we derive a posteriori error estimates for optimal control problems. We will show that the method does not involve the jump residuals which make the analysis simpler. Finally, a numerical example is given to demonstrate the theoretical results on superconvergence. [ABSTRACT FROM AUTHOR]