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

Abstract 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 = (0, T) Q=Ω× (0, T), and that are controlled by the right-hand side z_ zϱ from the Bochner space L² (0, T;H^-1 () ) L2 (0, T;H-1 (Ω) ). So it is natural to replace the usual L² (Q) L2 (Q) norm regularization by the energy regularization in the L² (0, T;H^-1 () ) L2 (0, T;H-1 (Ω) ) norm. We derive new a priori estimates for the error u ₇ - u ₋ℂ (ₐ) ‖u~ϱh-u¯‖L2 (Q) between the computed state u ₇ u~ϱh and the desired state u 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 u¯. These new estimates lead to the optimal choice = h² ϱ=h2. The approximate state u ₇ 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.