A priori error estimates for three dimensional parabolic optimal control problems with pointwise control (Q2821804)

The paper deals with a parabolic optimal control problem NEWLINE\[NEWLINE\begin{aligned} &\min_{q,u} J(q,u):=\frac 12 \int_0^T \|u(t)-\hat u\|^2_{L^2(\Omega)}\,dt+\frac \alpha 2\int_0^T|q(t)|^2\,dt,\\ &\text{subject to} \\ & u_t(t,x)-\Delta u(t,x)=q(t)\delta_{x_0}(x),\;(t,x)\in I\times \Omega, \\ &u(t,x)=0,\;,\;(t,x)\in I\times \partial\Omega,\;u(0,x)=0,\;x\in \Omega\subset \mathbb{R}^3, \\ &q_a\leq q(t)\leq q_b\;\text{a.e. in}\;I \end{aligned}NEWLINE\]NEWLINE with the Dirac delta function \(\delta_{x_0},\;x_0\in \Omega\). The problem is approximated using standard continuous linear finite elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, the authors establish a \(\mathcal{O}(\sqrt k+h)\) convergence rate for the control in the \(L^2\)-norm. The result improves the previously known estimate and does not require any relationship between the time step \(k\) and the mesh step \(h\).