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.

Students taking the course are expected to:

• Have knowledge of calculus, linear algebra, real analysis, integration theory and Banachspaces.
• Be able to use python scripting and numerical methods to solve algebraic and ordinary differential equations.

The course builds on the knowledge acquired 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, Banachspaces).

In relation to the competence profile of the degree it is the explicit focus of the course to:

• Give the competence to :
1. handle complex and development-oriented situations in study and work contexts.
2. Identify needs and plan individual learning
• Give skills to:
1. analyze practical and theoretical problems using numerical simulations based on suitable mathematical models.
2. Analyze qualitative and quantitative properties of mathematical models.
3. Explain and evaluate errors in modeling and simulation
4. Explain and select relevant analytic and solution models
5. Formulate and present problems and solutions to fellow students and partners
• Give knowledge and understanding of:
1. mathematical modelling and numerical analysis of problems in natural science and engineering.
2. theory, methods and praxis within applied mathematics.

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.

The following main topics are contained in the course:

• 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

• preparation of exercises in study groups
• preparation of projects