Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems
Tianliang Hou 1, Yanping Chen 21 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China; School of Mathematics and Statistics, Beihua University, Jilin 132013, China
2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Received 2012-9-23 Accepted 2014-11-17
Available online 2015-3-13
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 L²(0, T ;L²(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.
Key words: A priori error estimates, A posteriori error estimates, Mixed finite element, Discontinuous Galerkin method, Parabolic control problems.
AMS subject classifications: 35K10, 65N30.
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