Error estimates of BDF2 H 1 -Galerkin MFE for parabolic optimal control problems

This paper presents a fully discrete H1-Galerkin mixed finite element approximation for parabolic optimal control problems. For the state and costate variables, the BDF2 scheme and H1-Galerkin mixed finite element method are employed for temporal and spatial discretization, respectively. Since the H1-Galerkin mixed finite element is free from the LBB consistency condition, the unknown scalar and vector functions are therefore approximated by Lagrange elements and Raviart-Thomas mixed elements, respectively. The control is handled via variational discretization. The convergence results of all variables are rigorously derived. Theoretical findings are confirmed by means of some numerical examples.