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=Ω×(0,T), and that are controlled by the right-hand side z_ϱ from the Bochner space L²(0,T;H^-1(Ω)). So it is natural to replace the usual L²(Q) norm regularization by the energy regularization in the L²(0,T;H^-1(Ω)) norm. We derive new a priori estimates for the error ‖u~_ϱh-u¯‖_L2(Q) between the computed state u~_ϱh and the desired state u¯ in terms of the regularization parameter ϱ and the space-time finite element mesh size h, and depending on the regularity of the desired state u¯. These new estimates lead to the optimal choice ϱ=h². The approximate state u~_ϱh 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.