The work at hand is addressed to methods from the fictitious domain context, combined with the finite element method.
Fictitious domain techniques are suitable in case partial differential equations have to be solved on a geometrically complex or time-dependent domain.
Own suggestions based on penalization and regularization, utilizing variants of Nitsches method in order to impose boundary conditions in a weak sense, are introduced and analyzed. The techniques presented are generalizations of methods due to Glowinski et al.
The underlying model equations of order two are non-linear reaction-diffusion-convection equations of rather general type, able to describe a lot of real-world situations. In addition, incompressible Stokes and Navier-Stokes systems are introduced, being special cases of the original model equations in some sense.
The numerical analysis is carried out with respect to linearized versions of the original model equations. Within this process, symmetrical reaction-diffusion, diffusion-convection-reaction and a version of the Oseen equations are treated separately, in order to deal with different typical problems appearing in each case adequately.
Following that, besides symmetrical problems, tasks like dominant convection, as well as the circumvention of a discrete inf-sup condition in case the Oseen problem are discussed. Streamline diffusion/Galerkin least squares techniques playing a key role in that matter.
For implicit description of the embedded boundary the well established level set method is used, describing the boundary as a zero level set of suitable functions. This allows for an easy and flexible handling of the geometrical aspects.
Algorithmic tasks are addressed, followed by tests regarding numerical accuracy. In the end, several applications are presented in order to show the potential of the new methods: examples regarding plain flow problems, including one with moving boundary, and the Boussinesq equations on complex multi-connected domains.