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Preprint-No.: <   459   >   Published in: November 2016   PDF-File: IGPM459.pdf
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