A priori and a posteriori error estimates ofH1-Galerkin mixed finite element methods for elliptic optimal control problems

Abstract

A priori error estimates A posteriori error estimates H 1 -Galerkin mixed finite element methods a b s t r a c tIn this paper, we investigate numerical approximations of H 1 -Galerkin 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. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the standard finite element spaces, the control variable is approximated by piecewise constant functions. We derive a priori and a posteriori error estimates for the control variable, the state variables and the adjoint state variables. Finally, some numerical examples are given to demonstrate the theoretical results about a priori error estimates.

Superconvergence of H¹-Galerkin Mixed Finite Element Methods for Elliptic Optimal Control Problems

“…Therefore, non-standard mixed finite element methods for optimal control problems have been considered. Thus for elliptic optimal control problems, Guo et al [12] established a priori error estimates for a splitting positive definite mixed finite element method and Hou [14] investigated a priori and a posteriori error estimates for H 1 -Galerkin mixed finite element methods from [27,28]. Let us note that the last approach allows to avoid the inf-sup condition while using polynomial approximating spaces of various degree.…”

A priori and a posteriori error estimates of H1‐Galerkin mixed finite element method for parabolic optimal control problems

“…A notable advantage of this method is that it gives flexibility in choosing the approximation spaces. While a priori and a posteriori error estimates for H 1 ‐Galerkin MFEM for elliptic optimal control problem are discussed in another study, the literature seems lack on for parabolic optimal control problems. The present study is motivated by the work of Kröner and Vexler and Pani, whose technical tools are used in the analysis.…”