MM834: Partial differential equations: theory, modelling and simulation

A Bachelor’s degree in mathematics, physics or computer science.
The course cannot be chosen by students, who passed MM546.

• have knowledge of calculus, linear algebra, real analysis, integration theory and Banach spaces.
• be able to use numerical methods to solve algebraic and ordinary differential equations.
• be familiar with the fundamentals of python programming for scientific computing applications.

as e.g. achieved in the courses in the courses MM536 (Calculus for mathematics), MM538 (Algebra and linear algebra), MM533 (Mathematical and Numerical Analysis), MM547 (Ordinary differential equations: theory, modelling and simulation), and MM548 (measure- and integration theory, Banach spaces).

The learning objectives of the course are that the student demonstrates the ability to:

• Be able to deal with partial differential equation models for complex processes in science.
• Classify 2nd order PDEs and describe their characteristic properties.
• Analyze and simulate partial differential equations using appropriate, advanced methods and modern software.
• Construct, implement and analyze numerical methods to compute (approximate) solutions to partial differential equations.
• Understand the mathematical theory for numerical methods for PDEs.
• Design and perform reliable simulations of PDE models for complex processes in science.
• Give a seminar presentation of the individual project and answer supplementary questions and further questions on the course content.

The course gives knowledge and understanding of:

• Mathematical modelling and numerical analysis in science and engineering
• Theories, methods and practices in the field of applied mathematics.
• Advanced models and methods in mathematics

The course consists of a selection of the following main topics. While certain fundamentals are always included, the course focus may shift according to the teacher's preferences.

• Classification of 2nd order PDEs: Elliptic, parabolic and hyperbolic problems.
• Elliptic boundary value problems and Galerkin Finite Elements.
• Variational formulation, ellipticity, and the Lax-Milgram theorem.
• Sobolev spaces, Cauchy-Schwarz and Poincare inequalities.
• The poisson equation: Variational form, ellipticity, and FEniCS implementation.
• Galerkin's method, Galerkin orthogonality, best approximation, and error analysis.
• Finite elements for the Poisson equation, error bounds by duality.
• Neumann, Dirichlet, and Robin boundary conditions.
• div-grad operators and FEniCS.
• Parabolic PDEs: The heat equation.
• Runge-Kutta time stepping in variational form.
• SDIRK methods and L-stability.
• Simulation of the heat transfer.
• Parabolic-elliptic systems: Navier-Stokes equations
• Chorin’s projection method.
• Incremental pressure correction – IPC method
• Simulation of incompressible flow with heat transfer
• Adaptive calibration of PDE models
• Model reduction techniques for large-scale dynamical systems

Planned lessons

Total number of planned lessons: 84
Hereof:

Common lessons in classroom/auditorium: 84

For mediating and discussing the learning material and on putting it in perspective, a combination of prepared lecture slides and dynamic derivations on the blackboard is used. Whenever suitable, the presented material is illustrated by examples, sketches and figures as well as computer demonstrations. The latter tool is also used to demonstrate how theoretically proven facts show in practical implementations.

The course content is supplemented by practical exercises to solve theoretical and scientific computing problems, the latter including the implementation of numerical algorithms.

The students will be given ample opportunity to discuss the teaching material with the teacher and their peers.

Other planned teaching activities:

• preparation of exercises in study groups
• preparation of projects
• contributing to online learning activities related to the course