# Finite element error estimates for one-dimensional elliptic optimal control by BV functions

## TL;DR

This paper analyzes finite element discretization errors for a one-dimensional elliptic optimal control problem with BV controls, providing error estimates for different control discretization strategies and confirming their optimality through numerical results.

## Contribution

It offers new error estimates for BV control discretization in elliptic optimal control problems, comparing variational and piecewise constant approaches under structural assumptions.

## Key findings

- Variational discretization yields $O(h^2)$ errors for state and control.
- Piecewise constant control discretization achieves $O(h)$ errors.
- Numerical results confirm the optimality of the derived error estimates.

## Abstract

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $L^2$- and $L^\infty$-error for the state and the adjoint state are of order ${\mathcal O}(h^2)$ and that the $L^1$-error of the control behaves like ${\mathcal O}(h^2)$, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $L^2$-error estimates of order $\mathcal{O}(h)$ for the state and $W^{1,\infty}$-error estimates of order $\mathcal{O}(h)$ for the adjoint state. Under the same structural assumption as before we derive an $L^1$-error estimate of order $\mathcal{O}(h)$ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.