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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 twophase flow problems. For the twophase flow problem we restrict to a sharp interface model for the fluid dynamics, which consists of the NavierStokes equations for the bulk fluids with an interfacial surface tension force term in the momentum equation. In case of solute transport this NavierStokes equation is coupled with a convectiondiffusion equation. If surfactants are present, a convectiondiffusion 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 twophase 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 socalled 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
