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My research focuses on the numerical analysis of, and scientific computation
for, finite element approximations
for nonlinear and possibly degenerate partial differential equations,
as well as for geometric evolution equations. In recent years
I have been working on nonlinear reaction diffusion systems arising in
mathematical biology, on thin film flows under the influence of surfactants
and van der Waals forces and on Cahn-Hilliard type system. In addition
I have been concerned with a finite element approximation
of a phase field model that describes void electromigration. The model can be
extended to include the effect of both stressmigration and grain boundaries.
Below you can find a few simulations for the problems mentioned above.
More recently my research has focussed on the parametric finite element
approximation of
geometric evolution equations, as well as the coupling of such equations
to a partial differential equation in the bulk. My work on purely geometric
evolution equations includes approximations for mean curvature flow,
surface diffusion and Willmore flow. The approximations are characterized by
an implicit tangential motion of vertices, which ensures that the meshes stay
nice during the evolution. In addition, the developed method is able to
consider triple junctions, which in 3D leads to surface clusters, as well
as anisotropic surface energies, which is relevant for certain applications in
Materials Science.
In applications the evolution of an interface is often coupled to
quantities that solve a PDE in the bulk. Extending the above work on geometric
evolution equations to such situations often yields robust numerical methods
that can be shown to satisfy an energy law. In particular, in my work I often
consider a so-called unfitted front-tracking approach, which means that the
parametric approximation of the interface is totally independent from the bulk
mesh. Examples for the coupling of an interface evolution to a bulk PDE are the
Stefan problem with Gibbs--Thomson law, the Mullins--Sekerka problem,
void electromigration, possibly including stressmigration,
two phase flow, possibly including surfactants, as well as
dynamic models for biomembranes.
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