The one-shot method: function space analysis and algorithmic extension by adaptivity

This thesis is concerned with the function space analysis of the one-shot method as well as its algorithmic extension by adaptivity. The convergence of the method is investigated in a general setting and afterwards specified for certain model problems. After analyzing the behavior for the solid fuel ignition model, the results for the viscous Burgers as well as the incompressible Navier-Stokes equations are presented. Starting with distributed optimal control for the first two problems, we additionally consider the boundary control case for the NavierStokes equations. Furthermore, the one-shot algorithm is completed with a grid refinement routine. Numerical results for this method are presented for all three model problems. The method is finally applied to a shape optimization problem, where the optimal shape of an airfoil is requested. Zusammenfassung. Diese Dissertation befasst sich mit der funktionalanalytischen Analysis der One-Shot-Methode wie auch ihrer algorithmischen Erweiterung durch Adaptivität. Die Konvergenz dieser Methode wird sowohl in einem allgemeinen Rahmen als auch für verschiedene Modellprobleme untersucht. Nach der Betrachtung des Verhaltens für das Solid Fuel Ignition Modell folgen die Ergebnisse für die viskose Burgers Gleichung und die inkompressiblen Navier-Stokes Gleichungen. Ausgehend von einem Optimierungsproblem mit verteilter Steuerung für die ersten beiden Probleme, wird das Vorgehen auf ein Randsteuerungsproblem für die Navier-Stokes Gleichungen erweitert. Der One-Shot-Algorithmus wird zusätzlich durch eine Gitterverfeinerung vervollständigt. Es werden numerische Ergebnisse für alle drei Modellprobleme präsentiert. Letztlich wird das Verfahren in der Formoptimierung für ein optimales Tragflügelprofil angewendet. Acknowledgements. First of all, I would like to thank Prof. Dr. Nicolas Gauger for making this research possible. His support, in so many ways, gave me the strength and patience to finish this work. The inspiring and friendly working environment as well as the opportunity to travel have broadened my mind. I wish to thank Prof. Dr. Juan Carlos De Los Reyes, who guided me along the way. Without the many fruitful discussions, in Berlin or Quito, this thesis would never have been possible. In particular, I would like to thank him for hosting me in Quito two times. These stays have been significant for my work and raised my interest in such an amazing culture. This dissertation is the result of three years of work within the research project DGHPOPT funded by the Federal Ministry of Education and Research, to whom I am thankful for the financial support. Many colleagues deserve my gratitude for all the discussions and their support. The great atmosphere at MathCCES made my time in Aachen so enjoyable. Personally, I would like to thank Hanna Schäfer and Stefanie Günther, who have been more than colleagues to me. I also want to thank Martin Frank, Graham Alldredge, Rick Barnard, Kai Krycki, Pascal Richter, Philipp Monreal, Armin Westerkamp, Max Sagebaum and Matthias Sonntag for many discussions, coding help and poker nights. Additionally, I am grateful to my former colleague in Berlin, Thomas Surowiec, who helped me a lot by proofreading the introduction. Lastly, it should not go without mentioning my gratefulness to my family and friends. I thank my parents for their love and support and especially my sister Christine for all the mathematical discussions, happiness and laughter. Thank you for going this way together with me.