**Simulation of crystal growth**

The blue shape resembling a christmas tree above is a so called dendrite, a crystal that has grown from a small nucleus into an undercooled melt (red in the figure). The reason that the pattern becomes this complicated is that a simple form, such as plane or circular, is unstable and will develop into the 'arms' or 'fingers' that are visible in the picture. This is a fundamental phenomenon in solidification and phase change in general, which is of great importance both in practice and in principle. In the particular case studied above the phenomenon is complicated further by the fact that the melt is flowing from left to right, giving rise to the asymmetry in the shape.

This picture was produced by solving equations describing the phase change numerically in two dimensions. In these simulations adaptive finite element methods have been used together with so called 'phase -field' models for the phase change. The computations become quite challenging, since very high accuracy is needed in order to capture the time dependent evolution correctly.