459 IGPM459.pdf        November 2016
TITLE High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces
AUTHORS Christoph Lehrenfeld, Arnold Reusken
ABSTRACT In this contribution we treat a special class of recently developed unfitted finite element methods for the discretization of mass and surfactant transport equations in incompressible two-phase flow problems. For the two-phase flow problem we restrict to a sharp interface model for the fluid dynamics, which consists of the Navier-Stokes equations for the bulk fluids with an interfacial surface tension force term in the momentum equation. In case of solute transport this Navier-Stokes equation is coupled with a convection-diffusion equation. If surfactants are present, a convection-diffusion equation on the (evolving) interface is used for modeling the surfactant transport. In this contribution we restrict to the transport equations for solute and surfactants. A key difficulty in the numerical simulation of two-phase flow problems is an accurate numerical approximation of the interface. We consider a setting, as in e.g. level set methods, in which the triangulations are not fitted to the interface. Recently, significant progress has been made in the construction, analysis and application of so-called unfitted FEM for the discretization of such transport equations. We present an overview of these recently developed methods with emphasis on the construction of higher order discretizations.
KEYWORDS unfitted finite element method, isoparametric finite element method, interface problems, surface PDEs