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 \(Q = \Omega \times (0,T)\), and that are controlled by the right-hand side \(z_\varrho \) from the Bochner space \(L^2(0,T;H^{-1}(\Omega ))\). So it is natural to replace the usual \(L^2(Q)\) norm regularization by the energy regularization in the \(L^2(0,T;H^{-1}(\Omega ))\) norm. We derive new a priori estimates for the error \(\Vert \widetilde{u}_{\varrho h} - \overline{u}\Vert _{L^2(Q)}\) between the computed state \(\widetilde{u}_{\varrho h}\) and the desired state \(\overline{u}\) in terms of the regularization parameter \(\varrho \) and the space-time finite element mesh size h, and depending on the regularity of the desired state \(\overline{u}\). These new estimates lead to the optimal choice \(\varrho = h^2\). The approximate state \(\widetilde{u}_{\varrho h}\) 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.
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Acknowledgements
The authors would like to acknowledge the computing support of the supercomputer MACH–2 (https://www3.risc.jku.at/projects/mach2/) from the Johannes Kepler Universität Linz and of the high performance computing cluster Radon1 (https://www.oeaw.ac.at/ricam/hpc) from the Johann Radon Institute for Computational and Applied Mathematics (RICAM) on which the numerical experiments were performed. The first and the third authors were partially supported by RICAM. Furthermore, the authors would like to express their thanks to the anonymous referees for their helpful and inspiring comments.
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Communicated by: Stefan Volkwein
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Langer, U., Steinbach, O. & Yang, H. Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization. Adv Comput Math 50, 24 (2024). https://doi.org/10.1007/s10444-024-10123-w
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DOI: https://doi.org/10.1007/s10444-024-10123-w
Keywords
- Parabolic optimal control problems
- Energy regularization
- Space-time finite element methods
- Error estimates
- Solvers