Martin Weiser - Böcker

This book deals with the general topic “Numerical solution of partial differential equations (PDEs)” with a focus on adaptivity of discretizations in space and time. By and large, introductory textbooks like “Numerical Analysis in Modern Scientific Computing” by Deuflhard and Hohmann should suffice as a prerequisite. The emphasis lies on elliptic and parabolic systems. Hyperbolic conservation laws are treated only on an elementary level excluding turbulence.Numerical Analysis is clearly understood as part of Scientific Computing. The focus is on the efficiency of algorithms, i.e. speed, reliability, and robustness, which directly leads to the concept of adaptivity in algorithms. The theoretical derivation and analysis is kept as elementary as possible. Nevertheless required somewhat more sophisticated mathematical theory is summarized in comprehensive form in an appendix. Complex relations are explained by numerous figures and illustrating examples. Non-trivial problems from regenerative energy, nanotechnology, surgery, and physiology are inserted.The text will appeal to graduate students and researchers on the job in mathematics, science, and technology. Conceptually, it has been written as a textbook including exercises and a software list, but at the same time it should be well-suited for self-study.

All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered only as far as it gives insight into the construction of algorithms. In the exercises, a complete FE-solver for stationary 2D problems is implemented in Matlab/Octave. Contents:Finite Element FundamentalsGrids and Finite ElementsAssemblySolversError EstimationMesh RefinementMultigridElastomechanicsFluid MechanicsGrid Data StructureFunction Reference

Das Lehrbuch behandelt adaptive Algorithmen für die numerische Lösung von elliptischen und parabolischen Gleichungen, wobei der Schwerpunkt auf der algorithmischen Effizienz liegt und die notwendige Theorie so elementar wie möglich gehalten wird. Die zweite Ausgabe wurde korrigiert und um eine constraints-basierte Formulierung der Dirichlet-Randbedingungen und einen Abschnitt über spektrale verzögerte Korrekturmethoden erweitert.

Mathematical optimization and machine learning are closely related. This proceedings volume of the Thematic Einstein Semester 2023 of the Berlin Mathematics Research Center MATH+ collects recent progress on their interplay in topics such as discrete optimization, nonlinear programming, optimal control, first-order methods, multilevel optimization, machine learning in optimization, physics-informed learning, and fairness in machine learning.