Cut finite element methods for incompressibleflows with unfitted interfaces

Abstract: Problems with time-evolving domains are frequently occurring in computationalfluid dynamics and many other fields of science and engineering.Unfitted methods, where the computational mesh does not conform to thegeometry, are of great interest for handling such problems, since they removethe burden of mesh generation. We work towards the goal of developingan unfitted solver for Navier-Stokes equations on time-evolving domainsby developing and presenting cut finite element (CutFEM) splitting methodsfor solving Navier-Stokes equations. These CutFEM splitting methodsuse Nitsche’s method for incorporating boundary conditions and employpatch-based ghost penalty stabilization of the cut elements to achieve stabilityand optimal order error estimates. Numerical benchmarks are used toverify the methods and implementations. The methods are tested against aproblem with known analytical solution, the Taylor-Green vortex, and alsocompared to the classical Deutsche Forschungsgemeinschaft (DFG) benchmarkproblem with channel flow around a cylinder. For both benchmarks,the methods was shown to be stable when satisfying the parabolic Courant–Friedrichs–Lewy (CFL) condition, and to produce optimal convergencerates.