Sem3 LUH Nonlinear phenomena

leibniz hannover logoApplications  @  LUH  30 ECTS credits

Mathematical models of nonlinear phenomena

The curriculum at the Institute of Applied Mathematics (IfAM) at Leibniz University Hannover is motivated by practical questions from natural sciences, engineering, economics and other fields. Therein, mathematical modelling, theory and numerical methods are addressed. Reliable models for complicated situations are based on natural scientific laws, which can often be described with the help of nonlinear partial differential equations. Consequently, during their semester at LUH, students will be offered rigorous analysis classes, modern numerical analysis, optimization, and courses on application-driven scientific computing. Below you can find a course catalogue listing the subjects taught within MathMods in Hannover. Note: Not all courses will be offered in each semester. The courses offered in winter 2025/26 will be: (to be announced soon)

 

Below you can find information about the subjects for this semester.

 

  • Numerical methods of partial differential equations with the finite element method [10 credits]

    Numerical methods of partial differential equations with the finite element method

    • ECTS credits 10
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to the numerical solution of partial differential equations (PDEs).
      The students will learn how to design numerical algorithms, establish their theoretical properties, and how to realize them in
      implementations.

    • Topics

       

      Recapitulation of characteristic concepts of numerical mathematics
      Brief review of mathematical modeling of differential equations
      Discretization with Finite differences (FD) for elliptic boundary value problems
      Discretization with Finite elements (FEM) for elliptic boundary value problems
      Numerical solution of the discretized problems
      Time-dependent problems: Methods for parabolic and hyperbolic problems
      Numerical methods for nonlinear and coupled problems


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A choice between (5 ECTS)

  • Modelling and numerical methods for phase-field fracture in continuum mechanics [5 credits]

    Modelling and numerical methods for phase-field fracture in continuum mechanics

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to numerical modeling of phase-field fracture problems and more
      general numerical modeling of coupled variational inequality systems. Besides a bit recapitulation of continuum mechanics,
      modern nonlinear and linear algorithms are introduced and analyzed.

    • Topics

       

      VI (variational inequality) and propagating interfaces
      Relation to obstacle problems
      Basic from continuum mechanics
      Phase-field fracture systems
      Treating crack irreversiblity: Penalization and primal-dual active set methods
      Nonlinear solution schemes
      Numerical analysis for a simplified problem


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  • Space-time modeling, goal-oriented error control, and adaptivity for continuum mechanics applications [5 credits]

    Space-time modeling, goal-oriented error control, and adaptivity for continuum mechanics applications

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to space-time formulations and space-time
      Galerkin finite element modeling of nonstationary partial differential
      equations. As one application, space-time goal-oriented error
      control with adjoints will be discussed in greater detail.

    • Topics

       

      Space-time modeling
      Space-time Galerkin finite element discretization
      Goal-oriented error control
      Adaptivity in space and time
      Applications to Navier-Stokes (fluid flow) and fluid-structure interaction


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A choice between (total 10 ECTS)

  • Nonlinear functional analysis [10 credits]

    Nonlinear functional analysis

    • ECTS credits 10
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      Nonlinear Functional Analysis offers a framework to formulate various
      problems as nonlinear operator equations in infinite-dimensional Banach spaces.
      In this lecture some of the solving methods are presented and applied.

    • Topics

       

      Implicit function theorem
      The degrees of Brouwer and Leray-Schauder
      Local and global bifurcation methods
      Theory of maximal monotone operators
      Applications, e.g. to structured population dynamics


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  • Nonlinear Optimization 1 [10 credits]

    Nonlinear Optimization 1

    • ECTS credits 10
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to the fundamental theory and algorithms of
      constrained nonlinear optimization problems (NLPs).
      The students will learn how to design, analyze, and implement
      nonlinear optimization algorithms.

    • Topics

       

      Necessary and sufficient optimality conditions, regularity, stability;
      line search and trust region methods, quadratic optimization problems
      and active set methods, penalty, augmented Lagrangian and SQP methods;
      various application examples


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  • Semigroups & Evolution Equations [10 credits]

    Semigroups & Evolution Equations

    • ECTS credits 10
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      Semigroup theory provides a functional analytic approach to
      numerous evolution problems where the initial state of a system uniquely
      determines the future of the system. In this lecture strongly continuous
      and analytic semigroups will be investigated and used to solve linear and semilinear equations.

    • Topics

       

      Unbounded operators
      Strongly continuous and analytic semigroups
      Characterization theorems
      Interpolation theory
      Linear and semilinear hyperbolic and parabolic Cauchy problems
      Applications e.g. to transport, heat, wave, and Schrödinger equations


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  • Numerical methods for algorithmic systems and neural networks [5 credits]

    Numerical methods for algorithmic systems and neural networks

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to mathematical tools needed for modern techniques
      in algorithmic systems and neural networks

    • Topics

       

      Basic concepts from numerical mathematics
      Feedforward neural networks
      Modern extensions of neural networks


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  • Linear optimization [5 credits]

    Linear optimization

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to linear optimization theory and the Simplex method

    • Topics

       

      Polyhedral theory, simplex method, duality theory, various applications


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  • Optimal control with ODE models [5 credits]

    Optimal control with ODE models

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to nonlinear optimization theory in Banach spaces
      and its application to optimal control with ODE models

    • Topics

       

      Regularity (Robinson), necessary optimality conditions,
      approximation property, sufficient optimality conditions;
      optimal control for multipoint ODE boundary value problems,
      maximum principle, several application examples


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  • Dynamic optimization [5 credits]

    Dynamic optimization

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course is devoted to algorithms for optimal control with ODE models.
      The students will implement the multiple shooting method, then formulate
      and solve a concrete problem (e.g., moon landing).

    • Topics

       

      Problem formulation and multiple shooting discretization
      for optimal control and parameter estimation with ODE models,
      algorithms for deterministic and stochastic multistage NLPs,


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  • PDE constrained optimization in engineering and physics [5 credits]

    PDE constrained optimization in engineering and physics

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      The course is devoted to infinite dimensional
      optimization problems which are gouverned by partial
      differential equations.

    • Topics

       

      Linear quadratic optimal control problems
      modelling examples in applications
      existence and uniqueness results
      adjoined state
      discretization and optimization: FEM


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A choice between (5 ECTS)

  • Nonlinear PDEs: Elliptic equations [5 credits]

    Nonlinear PDEs: Elliptic equations

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      Various natural phenomena can be described by partial differential equations.
      For realistic models the equations are mostly nonlinear. In this lecture
      existence results for nonlinear elliptic boundary value problems
      are presented based on fixed point methods and variational methods.

    • Topics

       

      Fixed point theorems of Schauder and Leray-Schauder
      The direct method of calculus of variations
      Mountain-Pass Lemma
      Sub-supersolution method


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  • Nonlinear PDEs: Parabolic equations [5 credits]

    Nonlinear PDEs: Parabolic equations

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      Parabolic equations are an important tool in modeling time-dependent phenomena.
      This course presents techniques the mathematical
      treatment of nonlinear parabolic equations.

    • Topics

       

      Existence of weak solutions for semilinear and quasilinear equations
      Compactness arguments
      Monotone operators & Galerkin method
      Porous medium equation


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  • Partial differential equations of mathematical biology [5 credits]

    Partial differential equations of mathematical biology

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      This course gives an introduction to partial differential equations
      arising in mathematical biology and their treatment by mathematical
      analysis.

    • Topics

       

      Diffusion equation, travelling wave solutions, comparison principle for
      parabolic equations, long-term behaviour and blow-up in reaction-diffusion equations,
      energy methods, Keller-Segel system;
      spread of species, reaction-diffusion systems, pattern formation via
      Turing mechanism, chemotaxis


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  • Implementing finite element methods for advanced applications [5 credits]

    Implementing finite element methods for advanced applications

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      The students will learn to implement Galerkin finite elements
      in C++. Moreover, they will learn how to analyze computational results
      with the help of numerical analysis tools.

    • Topics

       

      Creating a mesh
      Creating a sparsity pattern
      Implementing finite elements
      Solving Poisson's problem


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  • Seminar: Models and applications in sciences [5 credits]

    Seminar: Models and applications in sciences

    • ECTS credits 5
    • University Leibniz University Hannover
    • Semester 3
    • Objectives

       

      In this seminar, each student will present a research paper or book chapter
      on current research in mathematical models of nonlinear phenomena.

    • Topics

       

      Various topics in solid mechanics, fluid flow, biology, electromagnetism and other areas
      according to students' own interests and preferences.


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