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

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

H.-G. Roos, M. Stynes, L. Tobiska: Robust Numerical Methods for Singularly Perturbed Differential Equations - Convection-Diffusion-Reaction and Flow Problems
G. Kanschat: Lecture notes Mixed Finite Elements
R. Verfürth: Lecture notes Numerische Strömungsmechanik
R. Verfürth: Lecture notes Computational Fluid Dynamics