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We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.

We consider standard tracking-type, distributed elliptic optimal control problems with L 2 L^{2} regularization, and their finite element discretization. We are investigating the L 2 L^{2} error between the finite element approximation u ϱ ⁢ h u_{\varrho h} of the state u ϱ u_{\varrho} and the desired state (target) u ¯ \overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ = h 4 \varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.