### Einführung in die Numerik

Die Vorlesung führt in die Theorie und Praxis des numerischen Lösens von Grundaufgaben der Analysis und Linearen Algebra auf Computern ein. Behandelt werden neben den fundamentalen Themen "Rundungsfehler", "Konditionierung" und "Approximation" vor allem numerische Algorithmen zur Lösung von linearen Gleichungssystemen und Eigenwertaufgaben sowie zur Berechnung von Integralen und zur Bestimmung von Nullstellen. Diese Algorithmen werden hinsichtlich ihrer Komplexität, Lösungsgenauigkeit und Stabilität untersucht. Dabei werden nur elementare Methoden der Analysis und Linearen Algebra benötigt.

### Finite Elements

Many practical applications involve the description of the state of a solid body, a fluid or just any region of space. As examples consider the gravitational field within and outside of an inhomogeneous body, the temperature of a solid body, the flow of water in the subsurface, the flow of gases in a complicated duct, the propagation of sound or water waves or the mechanical stress in a bridge. In the initial part of this course, we will give some background on how to derive the equations of mathematical physics that describe all these phenomena, as well as the functional analytical tools to analyse them. However, the key aim of this course is to introduce the most versatile numerical method for solving such problems, the Finite Element Method.  The course will involve detailed tutorials on how to implement finite element methods, as well as a rigorous numerical analysis of the resulting approximation errors.

### Multiscale Methods & Homogenization Theory

Composite materials containing two or more finely mixed constituents are widely used nowadays in industry, due to their properties. PDEs (partial differential equations) with highly oscillating coefficients naturally arise in the modeling of composite materials. The major difficulty of direct numerical solution of such problems is the tremendous size of the computation. This course aims to provide an introduction of homogenization theory and multiscale methods for solving second-order elliptic boundary value problems with highly oscillating coefficients at an acceptable computational cost. Basic knowledge of PDEs, functional analysis, and finite element methods is required.