



Abstract:  
For the mathematical modeling of twophase flow phenomena we use the incompressible (Navier)Stokes equations in each phase. The coupling of the phases is achieved by a surface tension force at the interface (coupling of the stresses normal to the interface) and a noslip condition on the velocity tangential to the interface. For the evolution of the interface a level set technique is utilised. For the finite element discretization of onephase flow problems the TaylorHood P2P1 pair is a popular choice due to the quadratic convergence and LBBstability. For twophase flow problems however, the P1 P2 discretization with unfitted meshes leads to a rather poor approximation quality of O(h) as P1 elements are not able to represent discontinuities in the solution. Enriching the P1 space with Heaviside jump functions one can recover the optimal approximation property, but numerical experiments indicate that the P2P1X velocitypressure pair is not LBB stable. In the enriched pressure space has been reduced by omitting the extended basis functions with small supports, which cause the instability. Introducing the socalled ghost penalty stabilization for the pressure space results in a discrete infsup stability for a modified bilinear form. As opposed to the reduced XFEM space, the ghost penalty method does not need to reduce the approximation space and thus may lead to smaller errors in the solution. The added stability terms lead to a modified Schur complement and therefore the preconditioners have to be adapted in order to solve the system matrix efficiently. New preconditioning strategies developed in [3] are presented here. For a constructed Stokes model problem with an analytical solution both stabilization methods are compared with respect to the discretization errors and convergence rates. For a realistic, fully coupled NavierStokes rising droplet problem the stabilization methods are compared with respect to the resulting droplet position and velocity. 



Biography:  
Since September 2015 Thomas Ludescher is working as a PhD student at the Chair for Numerical Mathematics at RWTH Aachen University under the supervision of Prof. Arnold Reusken and Dr. Sven Groß. In 2015, he has received his Master in Computational Engineering Science at RWTH Aachen University and 2013 his Bachelor in the same study program. His research interests are stabilized discretizations for incompressible twophase flows and the development of solvers and preconditioners for that particular problem class. 



