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Parametric finite element methods for two-phase flow and dynamic biomembranes

Presented by: 
Robert Nürnberg Imperial College London
Date: 
Friday 17th July 2015 - 09:00 to 10:15
Venue: 
INI Seminar Room 1
Abstract: 
In this talk I will discuss recent advances in the numerical analysis of front-tracking methods for two moving interface problems. In the first part of the talk I will concentrate on a parametric finite element approximation of two-phase flow. Here two fluids evolve in a domain, separated by an interface. Stress balance conditions on the interface lead to surface tension terms involving curvature. We employ a variational approximation of curvature that originates from the numerical
approximation of geometric evolution equations, such as mean curvature flow. The arsising finite element approximation of two-phase flow can
be shown to be unconditionally stable.
In the second part of the talk, building on the concepts introduced in the first part, I will present a numerical method for the
approximation of dynamic biomembranes. Once again two phases of fluid are separated by an interface, but here the interface is endowed with
an elastic curvature energy. This models the properties of the lipid bilayer structure of the biomembrane's cell walls. Combining ideas on
the numerical approximation of Willmore flow, two-phase flow and  surface PDEs on an evolving manifold we are able to introduce a stable
parametric finite element method for the evolution of biomembranes.

Suggested review articles:
Deckelnick, K., Dziuk, G., and Elliott, C. M. (2005). Computation of geometric partial differential equations and mean curvature flow.
Acta Numer., 14, 139--232.
Seifert, U. (1997). Configurations of fluid membranes and vesicles.Adv. Phys., 46, 13--137.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons