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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:14-qucosa-188322 |
| Date | 16 November 2015 |
| Creators | Höhne, Katharina |
| Contributors | Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, Prof. Dr. rer. nat. habil. Hans-Görg Roos, Prof. Dr. rer. nat. habil. Hans-Görg Roos, Prof. Dr. rer. nat. habil. Lutz Tobiska |
| Publisher | Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
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