# Stabilized and Discontinuous Galerkin Finite Element Methods for Computational Fluid Dynamics

- englischsprachige Lehrveranstaltung -

While the finite element method is widely used for simulating structural mechanics problems, its industrial use for computational fluid mechanics applications is not yet widely established. In this course we investigate the difficulties that arrise when dealing with flow problems while using a finite element framework, in contrast to finite volume methods which are currently the standard in industry. At the same time, we discuss the potential benefits of using the finite element method, and elaborate on the different approaches to mitigate the aforementioned issues. Examples are stabilized methods, discontinuous Galerkin formulations and suitable velocity/pressure interpolation pairs. Towards the end of the course, we shift focus to turbulence modeling and the variational multiscale method. In general, this course will discuss various advanced topics in finite element theory, and familiarize the student with some open research questions concerning the finite element method for computational fluid dynamics.

## Course content

• Prototypical fluid mechanics equations: the advection(-diffusion), Burgers, Stokes and Navier-Stokes equations.
• The basic required functional analysis theory.
• Analysis of the model equations with an emphasis on the challenges with regards to the finite element method.
• Stabilized methods; Galerkin Least-Squares (GLS), Artificial diffusion, Streamline-upwind Petrov-Galerkin (SUPG).
• Suitable interpolation pairs in mixed methods (e.g. Taylor-Hood).
• Discontinuous Galerkin methods.
• A short introduction to the physics of turbulence.
• Classical turbulence models: Reynolds-averaged Navier-Stokes (RANS) and Large eddy formulation (LES).
• The variational multiscale method.

## Prerequisites

This course requires a solid knowledge on the finite element method and continuum mechanics