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：我们研究了一种针对问题条件稳定性设计的稳定化非拟合有限元方法在界面上实现唯一延拓。该方法采用背景网格的等参变换逼近界面，并将相应的几何误差纳入误差分析。为应对离散非协调性可能引发的失稳效应并处理界面条件，我们引入了自适应正则化项，从而基于条件稳定性导出误差估计。数值实验表明，界面的存在对解在数据域外的延拓影响较小。另一方面，几何的凸性性质具有关键作用——这一结论已在许多无界面问题中有所观察。