A priori error estimates for the space-time finite element approximation of a non-smooth optimal control problem governed by a coupled semilinear PDE-ODE system

TL;DR

This paper derives a priori error estimates for a space-time finite element method applied to a coupled PDE-ODE optimal control problem with non-smooth dynamics, demonstrating convergence rates and validating results numerically.

Contribution

It provides the first a priori error estimates for a space-time finite element discretization of a non-smooth coupled PDE-ODE optimal control problem, including control convergence.

Findings

Linear convergence in time for the uncontrolled equation.

Spatial discretization error of order O(h^{3/2 - ε}).

Uniform convergence of discretized controls to the continuous control.

Abstract

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a simplified semilinear 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 a non-smooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the uncontrolled equation, we prove linear convergence in time and an order of $\mathcal{O}(h^{\frac{3}{2}-\varepsilon})$ for the discretization error in space. Our main result regarding the optimal control problem is the uniform convergence of dG(0)cG(1)-discrete controls to $l\in H^1_{\{0\}}(0,T;L^2(\Omega))$. Error estimates for the controls are established via a quadratic growth condition. Numerical experiments are added to illustrate the proven rates of convergence.