The basic concepts and algorithms for the numerical solution of problems from linear algebra and analysis are covered:

• error analysis: condition of a problem, stability of an algorithm
• approximation theory: interpolation by polynomial and spline functions
• numerical methods for linear systems of equations
• linear curve fitting
• eigenvalue problems
• numerical integration: interpolation and Gaussian quadrature
• nonlinear and parameter-dependent systems of equations

Most problems in science, technology, and engineering can be modeled by a set of differential equations. In general, these equations are too complex to be solved analytically. This course provides the necessary tools and methods to treat initial value problems numerically.

The following topics will be covered:

• explicit and implicit one-step methods (Runge-Kutta methods)
• error estimation and step size control
• multistep methods (Adams and BDF methods)
• consistency, stability, and convergence
• methods for stiff problems

Mathematical modelling, numerical methods, and software for the following topics:

• elastic bodies
• special cases of beams and plane strain/stress state
• finite element space discretisation
• specific time integration schemes

• first-order differential equations: autonomous first-order differential equations, variation of constants, explicitly solvable cases, initial value problems
• existence and uniqueness: functional-analytical foundations, Banach fixed-point theorem, Picard-Lindelöf theorem, the continuability of solutions, Peano's existence theorem
• qualitative behaviour: Grönwall's lemma, continuous dependency on data, upper and lower functions
• linear differential equations: homogeneous linear systems, matrix exponential function, variation of constants, nth-order differential equations
• stability: dynamical systems, phase space, Hamiltonian systems, asymptotic behaviour, stability theory according to Lyapunov

• vector analysis: vectors, subspaces, linear independence, basis, dimension, scalar product, orthogonality, projections, vector product
• matrix calculus: definition, calculation rules, base change, linear mappings, description of linear mappings via matrices, linear systems of equations (description via matrices, structure of solutions, Gaussian algorithm), invertibility, calculation of inverse, normal equations and linear least squares, determinants, eigenvalues and eigenvectors (diagonalizability, principal axis theorem)
• differentiation (multidimensional): scalar and vector fields, curves, contour lines, total and partial differentiability, directional derivation, implicit differentiation, inverse function theorem, differentiation rules (in particular: inverse function and chain rule), Taylor expansion, extremes under constraints (scalar functions of several variables), gradient fields, potentials, divergence and rotation, applications
• integration (multidimensional): normal domains, multiple integrals over normal domains

The theory and numerical analysis of differential-algebraic equations are discussed, in particular:

• application fields (electrical circuits and multibody mechanical systems)
• relation with singularly perturbed problems
• solution theory and index concepts
• normal form for linear DAEs
• numerical aspects

Mathematical modelling, numerical methods, and software for the following topics:

• elastic bodies
• special cases of beams and plane strain/stress state
• finite element space discretisation
• specific time integration schemes

The following topics will be covered:

• spline functions and spline spaces
• B-splines
• Bézier splines (Bézier polynomials, de Casteljau's algorithm, Bézier curves, Bézier polynomials over triangles, tensor product Bézier surfaces)
• B-spline smoothing (de Boor's algorithm)

Most problems in science, technology, and engineering can be modeled by a set of differential equations. In general, these equations are too complex to be solved analytically. This course provides the necessary tools and methods to treat initial value problems numerically.

The following topics will be covered:

• explicit and implicit one-step methods (Runge-Kutta methods)
• error estimation and step size control
• multistep methods (Adams and BDF methods)
• consistency, stability, and convergence
• methods for stiff problems

The theory and numerical analysis of differential-algebraic equations are discussed, in particular:

• application fields (electrical circuits and multibody mechanical systems)
• relation with singularly perturbed problems
• solution theory and index concepts
• normal form for linear DAEs
• numerical aspects

A presentation of the proseminar can be found at the following link.

To describe real-world processes, one often makes use of partial differential equations, which, in general, cannot be solved analytically. In this course, we will discuss and study the mathematical techniques required for solving such equations numerically. The focus lies on the discretization of boundary value problems for elliptic differential equations with finite difference or finite element methods. At the end of the course, these ideas will be applied to parabolic differential equations.

The following topics will be covered:

• approximation methods for elliptic problems
• theory of weak solutions
• consistency, stability, and convergence
• approximation methods for parabolic problems

Mathematical modelling, numerical methods, and software for the following topics:

• elastic bodies
• special cases of beams and plane strain/stress state
• finite element space discretisation
• specific time integration schemes

The basic concepts and algorithms for the numerical solution of problems from linear algebra and analysis are covered:

• error analysis: condition of a problem, stability of an algorithm
• approximation theory: interpolation by polynomial and spline functions
• numerical methods for linear systems of equations
• linear curve fitting
• eigenvalue problems
• numerical integration: interpolation and Gaussian quadrature
• nonlinear and parameter-dependent systems of equations