Efficient Computation of Dendritic Microstructures Using Adaptive Mesh Refinement

We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, allowing simulations on very large lattices. We present computations on a $2^{17}\times 2^{17}$ lattice, but note that this is not an upper limit. Time-dependent calculations in two dimensions are in good agreement with the predictions of solvability theory for high undercoolings, but predict higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.