  We implement an adaptive mesh algorithm for calculating the space and time
dependence of the atomic density field during materials processing. Our
numerical approach uses the systematic renormalization-group formulation of the
phase field crystal model to provide the underlying equations for the complex
amplitude of the atomic density field--a quantity that is spatially uniform
except near topological defects, grain boundaries and other lattice
imperfections. Our algorithm is a hybrid formulation of the amplitude
equations, combining Cartesian and polar decompositions of the complex
amplitude. We show that this approach leads to an acceleration by three orders
of magnitude in model calculations of polycrystalline domain formation in two
dimensions.
