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：本文提出非拟合谱元法求解椭圆界面问题及相应的特征值问题。该方法的新颖性在于将谱元法的谱精度与非拟合Nitsche方法的灵活性相结合。同时，我们采用定制化虚拟罚项增强其鲁棒性。针对椭圆界面问题与界面特征值问题，我们建立了最优$hp$收敛速率。此外，通过多项式阶数对模型问题验证了其谱精度。