Mixed Discontinuous Galerkin Time-Stepping Method for Semilinear Parabolic Optimal Control Problems

Abstract

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to semilinear parabolic optimal control problems, where the discontinuous finite element method of the order r (r ≥ 0) is used for the time discretization and the Raviart–Thomas mixed finite element method of the order λ (λ ≥ 0) is used for the space discretization. For λ ≥ 0, r = 0 or 1, we derive a priori error estimates for both the control variable and the state variables. Moveover, we derive a posteriori L²(0,T;L²(Ω)) error estimates for the scalar functions, assuming that only the underlying mesh is static.