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

Marita Holtmannspötter; Arnd Rösch; Boris Vexler

doi:10.3934/mcrf.2021014

Article Contents

2021, Volume 11, Issue 3: 601-624. Doi: 10.3934/mcrf.2021014

This issue Previous Article Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations Next Article External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms

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

1.
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
2.
Department of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany

^* Corresponding author: Marita Holtmannspötter
^* Corresponding author: Marita Holtmannspötter

Received: April 2019

Revised: December 2019

Early access: March 2021

Published: September 2021

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Abstract

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.

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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

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

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

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