Aufgrund der fundamentalen Bedeutung partieller Differentialgleichungen zur Beschreibung von Phänomenen in angewandten Wissenschaften ist deren Analyse ein Kerngebiet der Mathematik. Durch Computer lassen sich die Lösungen für eine Vielzahl dieser Gleichungen näherungsweise bestimmen. Die dabei verwendeten numerischen Verfahren sollen auf möglichst exakte Approximationen führen und deren Genauigkeit verifizieren. Die Least-Squares Finite-Elemente-Methode (LSFEM) und die unstetige Petrov-Galerkin (DPG) Methode sind solche Verfahren. Sie werden in dieser Dissertation untersucht. Der erste Teil der Arbeit untersucht die Genauigkeit der mittels LSFEM berechneten Näherungen. Dazu werden Eigenschaften der zugrundeliegenden Differentialgleichungen mit den Eigenschaften der LSFEM kombiniert. Dies zeigt, dass die Abweichung der berechneten Näherung von der exakten Lösung einem berechenbaren Residuum asymptotisch entspricht. Ferner wird ein Verfahren zu Berechnung einer garantierten oberen Fehlerschranke eingeführt. Während etablierte Fehlerschätzer den Fehler signifikant überschätzt, zeigen numerische Experimente eine äußerst geringe Überschätzung des Fehlers mittels der neuen Fehlerschranke. Die Analyse der Fehlerschranken für das Stokes-Problem offenbart ein Beziehung der LSFEM und der LBB Konstanten. Diese Konstante ist entscheidend für die Existenz und Stabilität von Lösungen in der Strömungslehre. Der zweite Teil der Arbeit nutzt diese Beziehung und entwickelt ein auf der LSFEM basierendes Verfahren zur numerischen Berechnung der LBB Konstanten. Der dritte Teil der Arbeit untersucht die DPG Methode. Dabei werden existierende Anwendungen der DPG Methode zusammengefasst und analysiert. Diese Analyse zeigt, dass sich die DPG Methode als eine leicht gestörte LSFEM interpretieren lässt. Diese Interpretation erlaubt die Anwendung der Resultate aus dem ersten Teil der Arbeit und ermöglicht dadurch eine genauere Untersuchung existierender und die Entwicklung neuer DPG Methoden. / The analysis of partial differential equations is a core area in mathematics due to the fundamental role of partial differential equations in the description of phenomena in applied sciences. Computers can approximate the solutions to these equations for many problems. They use numerical schemes which should provide good approximations and verify the accuracy. The least-squares finite element method (LSFEM) and the discontinuous Petrov-Galerkin (DPG) method satisfy these requirements. This thesis investigates these two schemes. The first part of this thesis explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this thesis introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound. The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuška-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this thesis utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant. The third part of this thesis investigates the DPG method. It analyses an abstract framework which compiles existing applications of the DPG method. The analysis relates the DPG method with a slightly perturbed LSFEM. Hence, the results from the first part of this thesis extend to the DPG method. This enables a precise investigation of existing and the design of novel DPG schemes.

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Hochwasserschadensprozesse werden von den drei Komponenten des Hochwasserrisikos bestimmt – der Gefahr, der Exposition und der Vulnerabilität. Dabei bleiben wichtige Einflussgrößen auf die Vulnerabilität, wie die private Hochwasservorsorge aufgrund fehlender quantitativer Informationen unberücksichtigt. Diese Arbeit entwickelt daher eine robuste statistische Methode zur Quantifizierung des Einflusses von privater Hochwasservorsorge auf die Reduzierung der Vulnerabilität von Haushalten bei Hochwasser. Es konnte gezeigt werden, dass in Deutschland private Hochwasservorsorgemaßnahmen den durchschnittlichen Hochwasserschaden pro Wohngebäude um 11.000 bis 15.000 Euro reduzieren. Hochwasserschadensmodelle mit Expertenwissen und datengestützten Methoden sind dabei am besten in der Lage Unterschiede in der Vulnerabilität durch private Hochwasservorsorge zu erkennen. Die über Hochwasserschadenprozesse erhobenen Daten und Modellannahmen sind von Unsicherheit geprägt und so sind auch Schätzungen mit. Die Bayesschen Modelle, die in dieser Arbeit entwickelt und angewandt werden, nutzen Annahmen über Schadensprozesse als Prior und empirische Daten zur Aktualisierung der Wahrscheinlischkeitsverteilungen. Die Modelle bieten Hochwasserschadensschätzungen als Verteilung, welche die Bandbreite der Variabilität der Schadensprozesse und die Unsicherheit der Modellannahmen abbilden. Hochwasserschadensmodelle, hinsichtlich der Prognoseerstellung und Anwendbarkeit. Ins Besondere verbessert die Verwendung einer Beta–Verteilung die Zuverlässigkeit der Modellergebnisse im Vergleich zu den häufig genutzten Gaußschen oder nicht parametrischen Verteilungen. Der hierarchische Bayessche Ansatz schafft eine verbesserte Parametrisierung von Wasserstand-Schadens-Funktionen und ersetzt so die Notwendigkeit empirischer Daten durch regional- und Ereignis-spezifisches Expertenwissen. Auf diese Weise kann die Vorhersage bei einer zeitlich und räumlichen Übertragung des Models verbessert werden. / Flood damage processes are influenced by the three components of flood risk - hazard, exposure and vulnerability. In comparison to hazard and exposure, the vulnerability component, though equally important is often generalized in many flood risk assessments by a simple depth-damage curve. Hence, this thesis developed a robust statistical method to quantify the role of private precaution in reducing flood vulnerability of households. In Germany, the role of private precaution was found to be very significant in reducing flood damage (11 - 15 thousand euros, per household). Also, flood loss models with structure, parameterization and choice of explanatory variables based on expert knowledge and data-driven methods were successful in capturing changes in vulnerability, which makes them suitable for future risk assessments. Due to significant uncertainty in the underlying data and model assumptions, flood loss models always carry uncertainty around their predictions. This thesis develops Bayesian approaches for flood loss modelling using assumptions regarding damage processes as priors and available empirical data as evidence for updating. Thus, these models provide flood loss predictions as a distribution, that potentially accounts for variability in damage processes and uncertainty in model assumptions. The models presented in this thesis are an improvement over the state-of-the-art flood loss models in terms of prediction capability and model applicability. In particular, the choice of the response (Beta) distribution improved the reliability of loss predictions compared to the popular Gaussian or non-parametric distributions; the Hierarchical Bayesian approach resulted in an improved parameterization of the common stage damage functions that replaces empirical data requirements with region and event-specific expert knowledge, thereby, enhancing its predictive capabilities during spatiotemporal transfer.

Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.

This paper presents results of a two year study, at Master’s level, which was undertaken to investigate how two Grade 9 Arts and Culture teachers incorporated mathematics in their Arts and Culture lessons in their classrooms in South Africa. Data from concept mapping activities and subsequent interviews with both teachers were collected and analysed using typological methods of analysis. Data collected from the study revealed that teachers still continue to grapple with the notion of integration. Lack of proper training and insufficient teacher knowledge seem to be the challenging factors for teachers to navigate successfully through the notion of integrated teaching and learning. Drawing from the theory of situated learning, this paper argues that although integration between mathematics and Arts and Culture is desirable in teaching and learning, it is problematic in practice. The analysis from this study raises important pedagogical issues about the link between ‘integrated teaching’ and ‘teacher training-and-content knowledge’.

We present several possible forms of mathematical competitions for University students. One of them is Blitz Mathematical Olympiad. It is a team competition, when all teams receive the same problem and are allotted 10-15 minutes to come up with a solution. This cycle is repeated 6-8 times with different problems. Modern Internet technologies allow us to organize Blitz Mathematical Olympiads for the teams which are in different cities and even countries.

Didactic proposals on modelling in mathematics education mostly give priority to models which describe, explain as well as partially forecast and provide mathematical solutions to real situations. A view of the modelling concept of informatics, which also initiates rapidly generalised deliberations of models, can also make a contribution to the spectrum of models, which are treated in a meaningful sense in mathematics lessons so as to expand some interesting aspects. In this paper, this is illustrated by means of conceptual design models – and, here, especially of process models – using the example of elevator organisation in a multi-storey construction.

This research studies the interaction among the following contexts: natural language, geometrical language and musical language and it can provide new instruments to accord didactical situations and for a deeper understanding of communication processes. It springs from the consideration that the geometrical transformations are usually used in the compositional processes and the “role of the music to learn geometrical transformations” is actually a new study. In the field of the theory of situations by G. Brousseau (1986) we can assume to be in front of a learning teaching-situation including non-teaching situation as the teacher of musical instruments, while transmitting the knowledge of musical language (theoretical-practical) didn’t have the intention to transmit the geometrical transformation.

In this workshop, we present a model of problem that we call Research Situation for the Classroom (RSC). The aim of a RSC is to put students in a role close to a mathematical researcher in order to make them work on mathematical thinking/skills. A RSC has some characteristics : the problem is close to a research one, the statement is an easy understandable question, school knowledge are elementary, there is no end, a solved question postponed to new questions... The most important characteristic of a RSC is that students can manage their research by fixing themselves some variable of the problem. So, a RSC is completely different from a problem that students usually do in France. For short : there is no final answer, students can try to resolve their own questions : a RSC is a large open field where many sub-problems exist; the goal for the students is not to apply a technique: the goal is, as for a researcher, to search. These type of situations are particularly interesting to develop problem solving skills and mathematical thinking. They can also let students discover that mathematics are “alive” and “realistic”. This workshop will be split into two parts. First, we propose to put people in the situation of solving a RSC to make them discover practically what is it. After, we present the model of a RSC and some results of our experimentations.

The transition from school to tertiary study of mathematics is rightly coming under increasing scrutiny in research. This paper employs Tall’s model of the three worlds of mathematical thinking to examine key variables in teaching and learning as they relate to this transition. One key variable in the transition is clearly the teacher/lecturer and we consider the perspectives of both teachers and lecturers on teaching related matters relevant to upper secondary and first year tertiary calculus students. While this paper deals with a small part of the data from the project, which aims to model the transition, the results provide evidence of similarities and differences in the thinking of teachers and lecturers about the transition process. They also show that each group lacks a clear understanding of the issues involved in the transition from the other’s perspective, and there is a great need for improved communication between the two sectors.