The efficient spectral Galerkin method to the phase-field models in polar geometry

We construct in this paper an efficient spectral Galerkin approximation in combination with scalar auxiliary variable (SAV) method to the Allen–Cahn model and Cahn–Hilliard model in polar geometry. Since the spectral methods cannot be directly applied to the non-rectangular regions, the disk region is firstly mapped to the rectangular region by the polar transformation that will lead to singularity at the pole, then providing appropriate pole conditions and basis functions are fundamental for constructing efficient algorithms. Moreover, the accuracy and stability of the proposed approximation are verified by numerical experiments.