Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

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The lecture is suitable for Master and PhD students of mathematics.
The students of economathematics can use it as component AKap.NAaA-a "Selected Topics in Numerical Analysis and Algorithms" in the module of the same name.

Topics of the Lecture:

Numerical methods, nowadays also termed methods from the area of scientific computing, are usually taught in universities in a traditional way. All methods discussed are based on Taylor's series expansions, using no knowledge whatsoever about the problem to be approximated. The advantage is that such methods are generally applicable, the downside is that they are not always as efficient and accurate as desired or even required. Mimetic methods, i.e. methods that mimic properties of the underlying problem and its solutions, are more specific to the problem, and hence bear the promise to produce more accurate solutions, in a more efficient way, often with less computational effort.

From a mathematical point of view, one may say (but it is a vague argument) that mimetic methods restrict solutions to a subspace of possible solutions, with the subspace being much more amenable to the original problem. A common misunderstanding about mimetic methods is that this class of methods is restricted to discretisation processes only. This is not correct: mimetic methods have also been developed for the solution of linear systems, for nonlinear systems, for model order reduction, and other areas.

In this lecture series, the concept of mimetic methods will be explained, and the advantage as compared to traditional methods. We will discuss the construction of mimetic methods for several areas in scientific computing, including discretisation, linear and nonlinear solution techniques and model order reduction, also using port-Hamiltonian systems. We will discuss various industrial applications, most importantly the simulation of semiconductor devices and for drilling applications.

Contents

• We will cover a variety of mimetic methods, from all areas of scientific computing
• We will start with mimetic methods for spatial discretization, mainly covering finite difference, finite volume, finite element and mixed finite element methods
• We will discuss so-called symplectic methods for time integration
• Mimetic modifications to iterative solution methods for large linear systems will be discussed
• We will give an introduction to mimetic decomposition methods for indefinite matrices
• Nonlinear variable transformations/correction transformation will be explained in detail, leading to an independent choice of variables for the linear and nonlinear systems
• We will discuss structure-preserving model order reduction
• Guest speakers will cover port-Hamiltonian systems and their application to practical problems

Literature:

Previous knowledge: Analysis I - III, basic knowledge of ordinary differential equations.

Exercises:
For the exercises we recommend ...
Sheets.

Criteria: Regular participation and participation in the exercise groups, as well as reaching 50% of the possible points on the first seven or the remaining exercise sheets and at least 2/3 of the possible points for the practical tasks.