Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization

As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder \(Q = \Omega \times (0,T)\) $Q=\mathrm{\Omega }\times (0,T)$ , and that are controlled by the right-hand side \(z_\varrho \) $z_{\varrho }$ from the Bochner space \(L^2(0,T;H^{-1}(\Omega ))\) $L^2(0,T;H^{-1}(\mathrm{\Omega }))$ . So it is natural to replace the usual \(L^2(Q)\) $L^2(Q)$ norm regularization by the energy regularization in the \(L^2(0,T;H^{-1}(\Omega ))\) $L^2(0,T;H^{-1}(\mathrm{\Omega }))$ norm. We derive new a priori estimates for the error \(\Vert \widetilde{u}_{\varrho h} - \overline{u}\Vert _{L^2(Q)}\) $\Vert {\tilde{u}}_{\varrho h}-\overline{u}{\Vert }_{L^2(Q)}$ between the computed state \(\widetilde{u}_{\varrho h}\) ${\tilde{u}}_{\varrho h}$ and the desired state \(\overline{u}\) $\overline{u}$ in terms of the regularization parameter \(\varrho \) $\varrho$ and the space-time finite element mesh size h, and depending on the regularity of the desired state \(\overline{u}\) $\overline{u}$ . These new estimates lead to the optimal choice \(\varrho = h^2\) $\varrho =h^2$ . The approximate state \(\widetilde{u}_{\varrho h}\) ${\tilde{u}}_{\varrho h}$ is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for Q. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. We also provide performance studies for different solvers.