In this paper, we propose the unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element method and the flexibility of the unfitted Nitsche's method. We also use tailored ghost penalty terms to enhance its robustness. We establish optimal $hp$ convergence rates for both elliptic interface problems and interface eigenvalue problems. Additionally, we demonstrate spectral accuracy for model problems in terms of polynomial degree.

翻译：本文提出了一种非适配谱元法，用于求解椭圆界面问题及其对应的特征值问题。该方法的新颖之处在于，它结合了谱元法的谱精度和非适配Nitsche方法的灵活性。我们还使用了定制化的虚拟惩罚项以增强其鲁棒性。针对椭圆界面问题和界面特征值问题，我们建立了最优的$hp$收敛速率。此外，通过多项式阶数的验证，我们展示了模型问题中的谱精度。