Abstract Unsteady phase-field problems such as the Allen-Cahn equation play a central role in the simulation of interface evolution in materials science and multi-phase flows, and at the same time present unique challenges in least-squares discontinuous Galerkin (LS-DG) formulations: time derivatives and numerical traces destroy the standard coercivity of the bilinear form, making rigorous energy stability and error analysis highly nontrivial. To overcome this difficulty, we propose an LS-DG scheme combined with a Crank-Nicolson time discretization for the Allen-Cahn equation. By introducing a stabilization term into the weighted least-squares functional and a trace perturbation term, we recover coercivity in the discontinuous setting and obtain a fully discrete LS-DG scheme for the Allen-Cahn equation with rigorous proofs of unique solvability, discrete energy stability, and optimal error estimates. The proposed method integrates the strengths of discontinuous Galerkin and least-squares finite element methods, and naturally provides a posteriori error estimators for adaptive refinement. We rigorously prove unique solvability, discrete energy stability, and optimal error estimates. Numerical experiments not only confirm the theoretical convergence but also demonstrate the effectiveness of the adaptive refinement strategy and its advantages over classical DG methods in terms of accuracy and efficiency.
Zou et al. (Tue,) studied this question.
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