The BoSSS Discontinuous Galerkin solver for incompressible fluid dynamics and an extension to singular equations.

This PhD thesis contains three major aspects: (1) the BoSSS software framework (or simply BoSSS code) itself, (2) an incompressible Navier-Stokes solver that is based on the BoSSS framework and finally (3) the development of the Extended Discontinuous Galerkin (XDG) method.

One major result is the BoSSS software framework (or simply BoSSS code) itself. Its core aspects are discussed form both, software engineering and mathematical point of view. The software design itself features some novel aspects. Up to our knowledge, it is the first time someone implemented a large-scale, MPI-parallel CFD-application in the C# -language.

The implemented BoSSS software library is a general tool for for the discretization of a systems of balance equations by means of the Discontinuous Galerkin (DG) method.

On the foundation of this software library, a solver for incompressible single-phase problems, based on the projection method, was developed.

Since the solution of the Poisson equation proofed to be the dominating operation in the incompressible Navier-Stokes solver, the Conjugate Gradient solver was ported to GPU (Graphics processing Unit), yielding an acceleration factor in the range of 5 to 20 in comparison to CPU.

By the XDG method, it becomes possible to treat equations with singularities, i.e. jumps and kinks in the the solution, without regularizing these singularities (i.e. without ``smearing them'' out). The final aim of the XDG method is the treatment of immiscible multiphase flows. Since in single-phase settings it is commonly accepted that fractional-step - approaches like the Projection method offer better performance than `overall'-schemes, which assemble a large nonlinear, differential-algebraic system from the Navier-Stokes equations, it seems beneficial to extend these ideas to multiphase flows. Therefor, solvers for singular scalar equations were developed: for the Poisson equation with jump, as a proptotype for elliptic steady-state problems and for the instationary Heat equation as an example for a time-dependent equation with moving interface.