Finite Element Methods for Flow Simulations
Lecturer:  D. Arndt 
Class data: 
Tue/Thu 1113h, Mathematikon (INF 205), SR11

 Start: October 17, 2017 
Module  MM35

Summary
Due to the ubiquity of fluids, the simulations of flow problem is of great importance in
many applications: Examples include the design process of new aircraft or cars, simulations
of air flow in the interior of buildings, simulation of the natural convection in the earthâ€™s
mantle, simulation of fusion reactors or understanding the dynamo effect in astrophysical
bodies. The simulation of all these phenomena in experiments is often very complicated
and expensive. Hence, there is an increasing desire to perform numerical simulations be it
to complement the experiments or to replace them.
This lecture will focus on mathematical tools and techniques
for suitable discretizations of flow problems. In particular, we will consider existence and uniqueness theory for
the Stokes, Oseen and NavierStokes equations both on the continuous and on the discrete level.
The course will mainly follow Verfürth's lecture notes on Computational Fluid Dynamics
but we will also consider some extensions like pressurerobust error estimates, graddiv stabilization and stabilizations restoring infsup stability.
Prerequisites
The course requires a basic understanding of finite element discretizations for partial differential equations such as taught in the lecture "Numerik partieller Differentialgleichungen".
Knowledge about mixed finite elements methods is favourable but not strictly required. We will do some recapitualtions in the first few lectures.
Announcements
Oral exams will take place on February 6 and February 8. Further information on relevenat topics can be found here.
Schedule
 Lecture 01: Notation, Modelling of flows, Transport theorem, Cauchy Theorem
(3.3 The Cauchy Stress tensor)
 Lecture 02: Conservation Laws, Basic equations of fluid dynamics
 Lecture 03: Initial and Boundary Conditions, Theory and FEM for scalar problems(Hilbert spaces, Sobolov spaces)
 Lecture 04: LaxMilgram, RitzGalerkin discretization, FEM discretization, BrambleHilbert
 Lecture 05: ClementInterpolation, Error estimates for elliptic problems, Theory of the Stokes problem, infsup stability
 Lecture 06: Conforming FEM for the Stokes problem, not infsup stable elements
 Lecture 07: Repetition of the infsup condition and the closed range theorem.
 Lecture 08: Wellposedness and error estimates for the discretized Stokesn problem, infsup stable FEpairs
 Lecture 09: More stable FEpairs, PetrovGalerkin stabilization
 Lecture 10: PetrovGalerkin stabilization, existence and error estimates
 Lecture 11: The timedependent Stokes problem, Vectorvalued functions, Gelfand triple
 Lecture 12: Existence for linear elliptic instationary problems, existence and uniqueness for liner mixed problems
 Lecture 13: Existence and uniqueness for the semidiscretized problem, error estimates
 Lecture 14: The fully discretized problem, coupled and uncoupled approaches, Helmholtz decomposition
 Lecture 15: The continuous and the discretized stationary Oseen problem, existence and uniqueness
 Lecture 16: Error estimates for the stationary Oseen problem, graddiv stabilization
 Lecture 17: PetrovGalerkin stabilization for the stationary Oseen problem, the instationary Oseen problem existence, uniqueness and error estimates
 Lecture 18: The stationary incompressible NavierStokes problem, existence, uniqueness for small data, typical behavior for increasing Reynolds number
 Lecture 19: Structure and regularity of weak solutions, Existence and convergence of solutions to the discretized NSE
 Lecture 20: The instationary incompressible NavierStokes problem, existence of solutions
 Lecture 21: Regularity and uniqueness of (continuous) solutions, Existence of solutions for the semidiscretiozed equations
 Lecture 22: Convergence results for the semidiscretized instationary incompressible NavierStokes equations
 Lecture 23: Turbulence Modeling, Kolomogorov's 5/3 law
 Lecture 24: Turbulence Modeling, LES
 Lecture 25: H(div)conforming FEM for Stokes, Introduction
 Lecture 26: H(div)conforming FEM for Stokes, SIPG, existence, uniqueness, error estimates
Literature
Homework problems
 Exercise Sheet 1
 Exercise Sheet 2
 Exercise Sheet 3
 Exercise Sheet 4
 Exercise Sheet 5
 Exercise Sheet 6
 Exercise Sheet 7
 Exercise Sheet 8
 Exercise Sheet 9