  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.
