Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization
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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 L^2(0,T;H^-1(Ω)). 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(Ω)) norm. We derive a priori estimates for the error u_ϱ h - u̅_L^2(Q) between the computed state u_ϱ h and the desired state u̅ 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 u̅. These estimates lead to the optimal choice ϱ = h^2. The approximate state 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.
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