A priori error estimates for the spacetime finite element discretization of an optimal control problem governed by a coupled linear PDEODE system
Abstract
In this paper we investigate a priori error estimates for the spacetime 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.04448
 arXiv:
 arXiv:2004.04448
 Bibcode:
 2020arXiv200404448H
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control;
 49M25;
 65J10;
 65M15;
 65M60