# Finite Elements in Fluids

Copyright: © Michel Make

Lecture “Finite Elements in Fluids” is an elective course suitable for students in the Master programs “Computational Engineering Science”, “Simulation Sciences”, “General Mechanical Engineering”, and others.

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## Calendar WS 2021/22

Lecture: Wednesday 10:30-12:00 E1 and hybrid

Exercise: Wednesday 12:00-12:45 E1 and hybrid

Office hour Prof. Behr: to be announced

Instructors

Lecture: Prof. Marek Behr, Ph.D.

Exercise: Anna M. Ranno, M.Sc., Blanca Ferrer Fabón, M.Sc.

Content

This lecture gives an insight into mathematical basics and essential approaches of the finite elements method in the context of fluid mechanics. Covered are the advection-diffusion equation, the Stokes equations as well as the Navier-Stokes equations. Multiphase systems will also be mentioned.

Topics of numerical instability, which can result from certain discretizations, will be discussed. This especially concerns effects of high Peclet numbers and incompatible interpolation functions. For this, relevant stabilization methods will be presented.

The module consists of a lecture with 2 SWS and an exercise with 1 SWS, and carries 4 ECTS points.

Topics

• Introduction and preliminaries
• kinematic descriptions of the flow field
• basic conservation equations
• basic ingredients of the finite element method
• Steady transport problems
• problem statement
• Galerkin approximation
• early Petrov-Galerkin methods
• stabilization techniques
• Unsteady convective transport
• method of characteristics
• classical time and space discretization techniques
• stability and accuracy analysis
• Taylor-Galerkin methods
• discontinuous Galerkin method
• space-time formulations
• Compressible flow problems
• Unsteady convection-diffusion problems
• time discretization procedures
• space discretization procedures
• stabilized space-time formulations
• Viscous incompressible flows
• steady Stokes problem
• stead Navier-Stokes problem
• unsteady Navier-Stokes problem

Literature

1. A. Donea, A. Huerta, Finite Elements for Flow Problems, Wiley, 2003, ISBN 0-471-49666-9.

Examination

Points are awarded during the semester based on Matlab assignments in the framework of the exercise. A 30-minute oral exam will be offered on several alternate dates in February and March.