In this paper, a reduced-order finite element (FE) method preserving the unconditional energy-stability is proposed to simulate the Allen-Cahn phase field model, based on the proper orthogonal decomposition (POD) method with the snapshot technique. We first derive the full order FE formulation of the Allen-Cahn model and compute its FE full solutions, from which we choose a few spatio-temporal solutions as snapshots. Based on the POD technique, we then build a set of optimal POD bases maximizing the energy content in the original ensemble data, and in the new low-dimensional space spanned by the POD bases, we establish a low-order numerical model of stable reduced-order FE (SROFE) formulation for the Allen-Cahn phase field model. We also prove error estimates of the SROFE solutions of the Allen-Cahn phase field model. Finally, some numerical results are provided to test the validity of the SROFE formulation.

本文基于快照技术，在适当的正交分解（POD）方法的基础上，提出了一种保留无条件能量稳定性的降阶有限元（FE）方法来模拟Allen-Cahn相场模型。我们首先导出Allen-Cahn模型的全阶有限元公式，并计算其有限元完全解，然后从中选择一些时空解作为快照。然后，基于POD技术，构建一组最佳POD基，以最大化原始集合数据中的能量含量，并在POD基跨越的新低维空间中，建立稳定降阶的低阶数值模型。 Allen-Cahn相场模型的二阶有限元（SROFE）公式。我们还证明了Allen-Cahn相场模型的SROFE解决方案的误差估计。最后，