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) Q=Ω×(0,T) , and that are controlled by the right-hand side z_ zϱ from the Bochner space L²(0,T;H⁻¹(Ω )) 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(Ω )) L2(0,T;H-1(Ω)) norm. We derive new a priori estimates for the error uₕ - ū L²(Q) ‖u~ϱh-u¯‖L2(Q) between the computed state uₕ u~ϱh and the desired state ū 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 ū u¯ . These new estimates lead to the optimal choice = h² ϱ=h2 . The approximate state uₕ 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.