Stabilized CutFEM for the Discretization of Two-phase Incompressible Flows in 3D

For the mathematical modeling of two-phase 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 no-slip 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 one-phase flow problems the Taylor-Hood P2-P1 pair is a popular choice due to the quadratic convergence and LBB-stability. For two-phase 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 P2-P1X velocity-pressure 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 so-called ghost penalty stabilization for the pressure space results in a discrete inf-sup 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 Navier-Stokes rising droplet problem the stabilization methods are compared with respect to the resulting droplet position and velocity.

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 two-phase flows and the development of solvers and preconditioners for that particular problem class.