TY - JOUR
T1 - Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems
AU - Hou , Tianliang
AU - Chen , Yanping
JO - Journal of Computational Mathematics
VL - 2
SP - 158
EP - 178
PY - 2015
DA - 2015/04
SN - 33
DO - http://doi.org/10.4208/jcm.1211-m4267
UR - https://global-sci.org/intro/article_detail/jcm/9834.html
KW - A priori error estimates, A posteriori error estimates, Mixed finite element,
Discontinuous Galerkin method, Parabolic control problems.
AB - <p style="text-align: justify;">In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.</p>