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 $$Q = \Omega \times (0,T)$$ Q = Ω × ( 0 , T ) , and that are controlled by the right-hand side $$z_\varrho $$ z ϱ from the Bochner space $$L^2(0,T;H^{-1}(\Omega ))$$ L 2 ( 0 , T ; H - 1 ( Ω ) ) . So it is natural to replace the usual $$L^2(Q)$$ L 2 ( Q ) norm regularization by the energy regularization in the $$L^2(0,T;H^{-1}(\Omega ))$$ L 2 ( 0 , T ; H - 1 ( Ω ) ) norm. We derive new a priori estimates for the error $$\Vert \widetilde{u}_{\varrho h} - \overline{u}\Vert _{L^2(Q)}$$ ‖ u ~ ϱ h - u ¯ ‖ L 2 ( Q ) between the computed state $$\widetilde{u}_{\varrho h}$$ u ~ ϱ h and the desired state $$\overline{u}$$ u ¯ in terms of the regularization parameter $$\varrho $$ ϱ and the space-time finite element mesh size h, and depending on the regularity of the desired state $$\overline{u}$$ u ¯ . These new estimates lead to the optimal choice $$\varrho = h^2$$ ϱ = h 2 . The approximate state $$\widetilde{u}_{\varrho h}$$ u ~ ϱ 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.