We present discretization and solver methods for a model of the solid-gas phase in a crystal growth apparatus. The model equations are coupled Eulerian and heat-transfer equations with flux boundary conditions. For a more detailed discussion we consider simpler equations and present time- and space-decomposition methods as solver methods to decouple the multi-physics processes. We present the error analysis for the discretization and solver methods. Numerical experiments are performed for the Eulerian and heat-transfer equation using decomposition methods. We present a real-life application of a crystal growth apparatus, based on underlying stationary heat conduction. Finally we discuss further error analysis and application to a more complex model of crystal growth.