We study the evolution of solidification microstructures using a phase-fieldmodel computed on an adaptive, finite element grid. We discuss the details ofour algorithm and show that it greatly reduces the computational cost ofsolving the phase-field model at low undercooling. In particular we show thatthe computational complexity of solving any phase-boundary problem scales withthe interface arclength when using an adapting mesh. Moreover, the use ofdynamic data structures allows us to simulate system sizes corresponding toexperimental conditions, which would otherwise require lattices greater that$2^{17}\times 2^{17}$ elements. We examine the convergence properties of ouralgorithm. We also present two dimensional, time-dependent calculations ofdendritic evolution, with and without surface tension anisotropy. We benchmarkour results for dendritic growth with microscopic solvability theory, findingthem to be in good agreement with theory for high undercoolings. At lowundercooling, however, we obtain higher values of velocity than solvabilitytheory at low undercooling, where transients dominate, in accord with aheuristic criterion which we derive.
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