  We consider a non-isothermal multi-phase field model. We subsequently
discretize implicitly in time and with linear finite elements. The arising
algebraic problem is formulated in two variables where one is the multi-phase
field, and the other contains the inverse temperature field. We solve this
saddle point problem numerically by a non-smooth Schur-Newton approach using
truncated non-smooth Newton multigrid methods. An application in grain growth
as occurring in liquid phase crystallization of silicon is considered.
