Finite element error estimates for elliptic optimal control by BV functions

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler

Published 2019-02-15Version 1

We derive a priori error estimates for two discretizations of a PDE-constrained optimal control problem that involves univariate functions of bounded variation as controls. Using, first, variational discretization of the control problem we prove $L^2$-, respectively, $L^\infty$-error estimates for the state and the adjoint state of order $\mathcal{O}(h^2)$ and show 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 provide numerical results for both discretization schemes indicating that the error estimates are in general optimal.