MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

doi:10.4208/jcm.1211-m4267

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

CLC Number:

35K10
65N30

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MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

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