Employ a multigrid algorithm to solve the shape-transformation phase-field model

This paper presents an original three-dimensional shape-transformation phase-field model solved by a highly efficient, optimally complex multigrid algorithm. The model extends the classical Allen–Cahn equation through the novel introduction of a static coupling term, $\alpha {|{\varphi }^2-1|}^2(\psi -\varphi )$, which forces alignment to a target phase while preserving interface sharpness via a localization factor that vanishes at equilibrium ($\varphi =\pm 1$). We develop a customized V-cycle multigrid solver with four key components: (1) Gauss–Seidel relaxation for high-frequency error damping, (2) residual computation, (3) restriction operators, and (4) prolongation operators. For stability and efficiency, an operator-splitting technique decouples nonlinear diffusion and coupling terms, solved sequentially. Rigorous theoretical analysis proves $L^2$-stability under a time-step constraint and establishes Q-linear convergence of Newton-multigrid iterations. Numerical experiments demonstrate $>99\text{\%}$ similarity in topology-changing transformations, confirming the method’s robustness and computational superiority.