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 x (0, T), and that are controlled by the right-hand side z((sic)) from the Bochner space L-2(0, T; H-1(Omega)). So it is natural to replace the usual L-2(Q) norm regularization by the energy regularization in the L-2(0, T; H-1(Omega)) norm. We derive newa priori estimates for the error parallel to(u) over tilde (sic)h-(u) over bar parallel to(L2(Q)) between the computed state (u) over tilde ((sic)h) and the desired state (u) over bar in terms of the regularization parameter (sic) and the space-time finite element mesh size h, and depending on the regularity of the desired state (u) over bar. These newestimates lead to the optimal choice (sic) = h(2). The approximate state (u) over tilde ((sic)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.