Although the field of computational fluid dynamics has gone through a remarkable series of successes, the accurate prediction of rarefied gas flows remains a challenging task. This is not only an issue from the perspective of scientific computing, but already the modeling turns out to be very cumbersome. It is especially owned to the increased importance of non-equilibrium thermodynamics, in which already the lack of intuition states a major obstacle. A well established framework to deal with those effects is given by kinetic theory. Within this theory, the Boltzmann equation is the main tool to describe any kind of flow based on a statistical per- spective of molecular dynamics. But solving the Boltzmann equation is very expensive in the sense of computational time. This is due its high dimensionality, since the equation is not only posed in space and time, but also in phase space. The phase space thereby spans the space of all possible velocities of the particles. In order to make a transition in scales, moment methods yield a flexible framework to derive macroscopic equations. But when deriving those, a closure problem occurs, since there is always one unknown more than equations. Finding such a closure is the major challenge, where both, mathematical considerations and physical intuition, meet. In the case of rarefied gas dynamics, this task is still an open research topic. Recently, the regularized 13-moment equations (R13) have been introduced (see [1]), which overcome some of the drawbacks, that are inherent to common moment systems.

The talk begins with an overview over the capabilities and some details of the R13 equations in multidimensional settings. Analytical solutions for a linearized subsystem will be presented, which give first insight about the predictive qualities of the model, e.g. flow around a sphere or a driven cavity. This is then followed by a presentation of numerical approaches that have been undertaken thus far. Special emphasis thereby will be put on the finite element method and the special challenges that appear within. Those challenges range from the handling of compli- cated non-standard boundary conditions to Stokes-like saddle-point structures that demand for a special treatment. The focus is always put on general geometries, which means that especially curved boundaries are of interest. Problems arising due to curvature effects will be discussed in the context of boundary condition treatment.

[1] H. STRUCHTRUP and M. TORRILHON, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, (2003).