Inertial Manifolds and Nonlinear Galerkin Methods

Nonlinear Galerkin methods utilize approximate inertial manifolds to reduce the spatial error of the standard Galerkin method. For certain scenarios, where a rough forcing term is used, a simple postprocessing step yields the same improvements that can be observed with nonlinear Galerkin. We show that this improvement is mainly due to the information about the forcing term that is neglected by standard Galerkin. Moreover, we construct a simple postprocessing scheme that uses only this neglected information but gives the same increase in accuracy as nonlinear or postprocessed Galerkin methods. To the interested reader. iii Acknowledgments I want to express my thanks to my advisors, Prof. Beattie and Prof. Borggaard and my committee for a topic that led me through many fascinating fields of mathematics, including functional analysis, the theory of PDEs and (infinite-dimensional) dynamical systems, numerical analysis (especially spectral methods and efficient time integrators), and reduced order modeling. They always had time for my questions and did a tremendous job in encouraging me whenever nonlinear Galerkin gave disappointing results. I also want to thank them for granting me a Research Assistantship for the Summer and Fall 2005, which gave me the possibility to focus on my thesis work, and for the opportunity to visit the SIAM Annual Meeting 2005 in New Orleans. The student atmosphere at the Math Department is amazing, I have never experienced such a friendly school environment before. It made studying math a pleasure and I made true friends here. I will miss D2 and the great discussions we had over lunch, biking, ATHF [27], the DVD nights until the natural threshold, geek talk etc. TeXmacs (www.texmacs.org) greatly reduced the time I spent typing and debugging the thesis. Last but not least, I want to thank my family for their moral support from a distance and apologize for not keeping in touch as much as I should have.