Stabilized Finite Element Methods for Coupled Incompressible Flow Problems

by Daniel Arndt
Doctoral thesis
Date of Examination:2016-01-19
Date of issue:2016-01-28
Advisor:Prof. Dr. Gert Lube
Referee:Prof. Dr. Gert Lube
Referee:Prof. Dr. Dr. Andreas Dillmann
Referee:Prof. Dr. Guido Kanschat
Persistent Address: http://dx.doi.org/10.53846/goediss-5485

 

 

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Abstract

English

In this thesis, a finite element discretization of the incompressible Navier-Stokes equations for a non-isothermal and electrically conducting fluid in a possibly rotating frame of reference is considered. In particular, the Oberbeck-Boussinesq model is combined with resistive incompressible magnetohydrodynamics. In order to account for instabilities and to diminish unphysical oscillations a stabilization for the incompressibility constraint as well as a local projection approach for various terms is considered. For the spatial discretization inf-sub stable ansatz spaces for velocity and pressure are used. With respect to the semi-discretization in space stability analysis is performed and quasi-optimal and semi-robust error estimates are derived for each coupling separately. Additionally, it is investigated how the results for the Navier-Stokes equations extend from Dirichlet to outflow boundary conditions. With respect to the full discretization a segregation scheme based on a pressure-correction projection method is considered. Also for this model stability is investigated and quasi-optimal and semi-robust error estimates are derived. Numerical results are used to validate the analytical findings and to investigate the influence of the stabilization terms apart from convergence on the numerical solutions. Some analytical results confirm the proposed rates of convergence in space and time and show the importance of grad-div stabilization. Afterwards, more realistic test cases such as the expulsion of the magnetic field by a fast rotating electrically conducting fluid or Rayleigh–Bénard convection in an inertial or rotating frame of reference are considered to obtain a suitable design for the stabilization parameters. Finally, the effectiveness of the implementation is demonstrated.
Keywords: Finite Element Methods; Navier-Stokes; Oberbeck-Boussinesq; Magnetohydrodynamics; Rotating Frame of Reference; Local Projection Stabilization; Segregation Scheme; Pressure-Correction Projection Method
 
Statistik