Dr Robert Nürnberg, Department of Mathematics, Imperial College London


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. . .

Simulations last modified: 12/02/2015