The stable XFEM for two-phase flows

The extended finite element method (XFEM) is frequently used in order to incorporate discontinuous solution properties into the approximation space. Discontinuities inside elements, as they e.g. occur in two-phase/free-surface flows with implicit interface descriptions, can thereby be accounted for appropriately. Yet, the XFEM is often prone to ill-conditioning of the global system matrix which reduces the performance of iterative solution techniques significantly. This paper introduces and studies new approaches to circumvent the problem of ill-conditioning with the XFEM. The stable XFEM, a recently proposed approach, is employed and studied in this work. Approximation properties and iterative solver performance are systematically compared for the different approaches which should improve the conditioning. Two recommended settings are finally used to solve a 3D free-surface flow benchmark problem and a classical two-phase flow problem. To our best knowledge, this is the first time that the stable XFEM is applied to industrially relevant flow problems.