Title: A Mass conserving mixed stress formulation for incompressible flows
Language: English
Authors: Lederer, Philip Lukas 
Qualification level: Doctoral
Advisor: Schöberl, Joachim 
Issue Date: 2019
Number of Pages: 142
Qualification level: Doctoral
Abstract: 
This work deals with the introduction and the analysis of a new finite element method for the discretization of incompressible flows. The main focus essentially lies on the discussion of the linear incompressible Stokes equations. These equations describe the physical behaviour and the relation -- derived from the fundamental Newtonian laws -- between the fluid velocity and the pressure (-gradient). Where the standard variational formulation of the Stokes equations demand a Sobolev regularity of order one for the velocity, we give an answer to the question if it is possible to define a variational formulation demanding a weaker regularity property of the velocity. With respect to a formally equivalent representation of the Stokes equations, we answer this question by the introduction of a new function space used for the definition of the gradient of the velocity. The resulting variational formulation is well-posed if we assume that the divergence of the velocity is square integrable. Thereby, with respect to the standard formulation, where all partial derivatives have to be square integrable, this is a reduced regularity property. We present certain properties of the new defined function space and discuss a proper continuous trace operator and the density of smooth functions. Motivated by this new variational formulation, we present and analyze a new finite element method in the rest of this work. For the approximation of the velocity we can now choose a conforming discrete space. This results in a (physically correct) incompressibility of the velocity field, thus exact mass conservation is provided. For the approximation of the gradient of the velocity we define new matrix-valued finite element shape functions, which are normal-tangential continuous across element interfaces. We present a detailed stability analysis and prove optimal convergence order of the discretization error.
Keywords: Navier Stokes; inf-sup condition; divergence-free; mixed finite elements; weakly symmetric
URI: https://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-122519
http://hdl.handle.net/20.500.12708/5335
Library ID: AC15327919
Organisation: E101 - Institut für Analysis und Scientific Computing 
Publication Type: Thesis
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