We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An $hp$ a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

翻译：我们考虑在笛卡尔网格上可靠实现一种自适应高阶无拟合有限元方法，以求解具有几何弯曲奇异性的椭圆界面问题。我们将先前针对光滑界面的可靠单元合并算法扩展，使其能够自动生成分段光滑界面的诱导网格。针对一种新的无拟合有限元方法（其有限元函数在每个子区域内是共形的），推导了$hp$后验误差估计。数值算例验证了该方法的竞争性性能。