We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. Numerical experiments suggest that the presence of an interface seems to be of minor importance for the continuation of the solution beyond the data domain. On the other hand, certain convexity properties of the geometry are crucial as has already been observed for many other problems without interfaces.

翻译：我们研究了一种面向界面条件稳定性的稳定化非拟合规有限元方法在界面上的唯一延拓问题。界面通过背景网格的等参变换进行逼近，并将相应的几何误差纳入误差分析中。为应对离散非一致性可能导致的失稳效应并处理界面条件，我们引入了适应性正则化项。这使我们能够基于条件稳定性推导误差估计。数值实验表明，界面的存在对解在数据域外的延拓似乎影响较小。另一方面，几何的某些凸性性质至关重要——正如许多无界面问题中已观察到的结论。