Cut finite element methods for interface problems

Interface problems modeled by Partial Differential Equations (PDEs) appear in a wide range of fields such as biology, fluid dynamics, micro and nano-technologies. In computer simulations of such problems a fundamental task is to numerically solve PDEs in/on domains defined by deformable interfaces/geometries. Those interfaces may for example model fractures, cell membranes, ice sheets or an airplane wing. This thesis is devoted to the development of Cut Finite Element Methods (CutFEM) that can accurately discretize PDEs with weak or strong discontinuities in parameters and solutions across stationary or evolving interfaces and conserve physical quantities. On evolving geometries, spatial discretization based on the cut finite element method is combined with a finite element method for the time discretization using discontinuous piecewise polynomials. Different strategies have been investigated throughout this thesis in order to achieve optimal approximation properties, well-posed resulting linear systems, and conservation of physical quantities. In Paper I and III, we consider two-phase flow problems where the interface separates immiscible incompressible fluids with different densities and viscosities. In Paper I we develop a numerical algorithm for obtaining high order approximations of the mean curvature vector and hence the surface tension force. Coupling this strategy with a space-time cut finite element discretization of the Navier-Stokes equations gives us a method that can accurately capture discontinuities across evolving interfaces and be used to accurately simulate the dynamics of two-phase flow problems, see Paper I. In Paper II, we develop a cut finite element discretization for linear hyperbolic conservation laws with an interface and show that stability and conservation can be obtained when using appropriate penalty parameters. In Paper III, discrete conservation of the total mass of surfactant is obtained by introducing a new weak formulation of the convectiondiffusion interface problem modeling the evolution of insoluble surfactants, with the help of the Reynold’s transport theorem. In Paper IV we propose a new stabilization for the discretization of the Darcy interface problem using cut finite element methods. The new stabilized cut finite element method preserves the divergencefree property of Hdiv conforming elements also in an unfitted setting. Thus, with the new scheme optimal approximation can be obtained for the velocity, pressure, and the divergence with control on the condition number of the resulting system matrix