Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term.
In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square.
The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.:Acknowledgement III
Notation IV
1 Introduction 1
1.1 Existence of solutions 2
1.2 Transformation into a fourth-order problem 4
2 Asymptotic analysis 6
2.1 A fourth-order problem in 1D 6
2.2 A fourth-order problem in 2D 14
2.2.1 Asymptotic expansion 19
2.2.2 Estimation of the residual 26
2.2.3 Asymptotic expansion without compatibility conditions 30
3 Solution decomposition and layer-adapted meshes 32
3.1 Solution decomposition 32
3.2 Layer-adapted meshes 33
3.3 Interpolation errors on layer-adapted meshes 36
4 Galerkin method and stabilisation 41
4.1 Discrete problem and stabilised formulation 41
4.2 A priori error estimates 44
5 Numerical results 48
5.1 Numerical evaluation of inf-sup constants 48
5.1.1 Theoretical aspects 48
5.1.2 Numerical results for β0 and B0 50
5.2 Convergence studies 53
5.2.1 Uniformity in ε 54
5.2.2 Convergence order 55
5.2.3 Necessity of stabilisation 56
5.2.4 Further experiments without known exact solution 56
6 Conclusions and outlook 60
A Numerical study of the stability estimate (2.35) 62
Bibliography 67
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:29042 |
| Date | 05 November 2015 |
| Creators | Höhne, Katharina |
| Contributors | Roos, Hans-Görg, Tobiska, Lutz, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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