  In this paper, we propose an approach for solving PDEs on evolving surfaces
using a combination of the trace finite element method and a fast marching
method. The numerical approach is based on the Eulerian description of the
surface problem and employs a time-independent background mesh that is not
fitted to the surface. The surface and its evolution may be given implicitly,
for example, by the level set method. Extension of the PDE off the surface is
not required. The method introduced in this paper naturally allows a surface to
undergo topological changes and experience local geometric singularities. In
the simplest setting, the numerical method is second order accurate in space
and time. Higher order variants are feasible, but not studied in this paper. We
show results of several numerical experiments, which demonstrate the
convergence properties of the method and its ability to handle the case of the
surface with topological changes.
