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A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

  • * Corresponding author: Marita Holtmannspötter

    * Corresponding author: Marita Holtmannspötter 
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  • In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.

    Mathematics Subject Classification: 49M25, 65J10, 65M15, 65M60.


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  • Figure 1.  Refinement of the temporal discretization for $N = 66049$(left) and $N = 263169$(right) nodes

    Figure 2.  Refinement of the spatial discretization for $M = 512$(left) and $M = 4096$(right) time steps

    Figure 3.  Errors $\Vert l-l_\sigma\Vert_{I\times\Omega}$ for different spatial and temporal mesh refinements

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