Westfälische Wilhelms-Universität Münster [chapter]

2017 Geschichte der anorganischen Chemie  
This work is dedicated to my father, Peter Henning, who could not live to see me concluding this thesis, but who always told me that he strongly believes in me and that I could accomplish what ever I want to do. First of all, I sincerely thank my supervisor Prof. Dr. Mario Ohlberger for his offer to join his newly formed working group in Münster, when I finished my Diploma thesis back in 2007. I would like to thank him for the interesting topic, I dealt with and for his numerous advices of any
more » ... ind. I also thank him for giving me the room to find my way with this work, to leave me free to follow my ideas and last but not least I thank him for his strong and kind support during the whole period. I am much obliged to all my former and current colleagues at the Institut für Numerische und Angewandte Mathematik in Münster for the comfortable athmosphere, the pleasant and friendly interaction with one another and for their help, whenever I needed it. I also heartily thank my family, especially my parents, Sabine and Peter Henning, for their encouragement and strong mental support during my whole life. Last but not least, I owe my deepest gratitude to my partner, Kathrin Smetana, for her mathematical help, her patience, her emotional hold and her love. Heterogeneous multiscale finite element methods for advection-diffusion and nonlinear elliptic multiscale problems Summary: In this thesis we introduce a new version of a heterogeneous multiscale finite element method (HMM) for advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. We analyse the method under the restriction of periodicity, stating corresponding a-priori and a-posteriori error estimates. As a reference for the exact solution u , we use the homogenized solution of the original advection-diffusion multiscale problem. We obtained this solution by a technique called 'two-scale homogenization with drift'. This technique was initially introduced by Marušić-Paloka and Piatnitski [73] . Finally, numerical experiments are given to validate the applicability of the method and the achieved error estimates in non-periodic scenarios. Furthermore, we also state a heterogeneous multiscale finite element method for nonlinear elliptic problems. In comparison to preceding works, the nonlinearity affects the gradient of the solution instead of the solution itself. Since this especially results in implementation problems, we present a general combination of the HMM with a Newton scheme. This combination produces new cell problems which must be solved. The implementation is realized with the software toolbox Dune. In order to handle a-posteriori error estimation beyond the periodic setting, we identify an effective macro problem and show that the solution of this problem is equal to the H 1 -limit of a sequence of HMM approximations. Using the localized constituents of the a-posteriori error estimate, we propose algorithms for an adaptive mesh refinement for the coarse macro-grid. Again, this is verified in numerical experiments.
doi:10.1002/9783527693009.ch39 fatcat:a6neoovlrrb53bdsw3nmjg2d4m