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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.<\/jats:p>","DOI":"10.1007\/s00366-026-02328-y","type":"journal-article","created":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T04:16:20Z","timestamp":1777954580000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Least-squares discontinuous Galerkin Crank-Nicolson method for the Allen-Cahn equation: stability, error estimates, and adaptive simulation"],"prefix":"10.1007","volume":"42","author":[{"given":"Guang-an","family":"Zou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saisai","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yutong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaofeng","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,5]]},"reference":[{"key":"2328_CR1","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.camwa.2023.10.028","volume":"152","author":"D Acosta-Soba","year":"2023","unstructured":"Acosta-Soba D, Guill\u00e9n-Gonz\u00e1lez F, Rodr\u00edguez-Galvan JR (2023) A structure-preserving upwind dg scheme for a degenerate phase-field tumor model. 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