Finite Element Methods for Flow Simulations
Lecturer: | D. Arndt |
Class data: |
Tue/Thu 11-13h, 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 earth’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 Navier-Stokes 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 pressure-robust error estimates, grad-div stabilization and stabilizations restoring inf-sup 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: Lax-Milgram, Ritz-Galerkin discretization, FEM discretization, Bramble-Hilbert
- Lecture 05: Clement-Interpolation, Error estimates for elliptic problems, Theory of the Stokes problem, inf-sup stability
- Lecture 06: Conforming FEM for the Stokes problem, not inf-sup stable elements
- Lecture 07: Repetition of the inf-sup condition and the closed range theorem.
- Lecture 08: Well-posedness and error estimates for the discretized Stokesn problem, inf-sup stable FE-pairs
- Lecture 09: More stable FE-pairs, Petrov-Galerkin stabilization
- Lecture 10: Petrov-Galerkin stabilization, existence and error estimates
- Lecture 11: The time-dependent Stokes problem, Vector-valued 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, grad-div stabilization
- Lecture 17: Petrov-Galerkin stabilization for the stationary Oseen problem, the instationary Oseen problem existence, uniqueness and error estimates
- Lecture 18: The stationary incompressible Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes 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