Simulation and numerical analysis of dendritic growth

Dendritic growth is a nonlinear process, which falls into the category of self-organizing pattern formation phenomena. It is of great practical importance, since it appears frequently and, in the case of alloys, affects the engineering properties of the resulting solid. In the first part of this article we report on an analysis of spatially semi-discrete approximations to the Stefan problem for two-dimensional, pure dendrites. A priori error estimates for the temperature field, the parametrization of the free boundary, relevant geometric and measuring quantities are presented and discussed. The second part describes a new algorithm for the two--dimensional Stefan problem. Here the free boundary is represented as a level set. This allows to handle topological changes of the free boundary. The accuracy of the method is verified and various numerical simulations, including topological changes of the free boundary, are presented.