An Unfitted Interface Penalty DG--FE Method for Elliptic Interface Problems

We propose an unfitted interface penalty Discontinuous Galerkin-Finite Element Method (UIPDG-FEM) for elliptic interface problems. This hybrid method combines the interior penalty discontinuous Galerkin (IPDG) terms near the interface-enforcing jump conditions via Nitsche method-with standard finite elements away from the interface. The UIPDG-FEM retains the flexibilities of IPDG, particularly simplifying mesh generation around complex interfaces, while avoiding its drawback of excessive number of global degrees of freedom. We derive optimal convergence rates independent of interface location and establish uniform flux error estimates robust to discontinuous coefficients. To deal with conditioning issues caused by small cut elements, we develop a robust two-dimensional merging algorithm that eliminates such elements entirely, ensuring the condition number of the discretized system remains independent of interface position. A key feature of the algorithm is a novel quantification criterion linking the threshold for small cuts to the product of the maximum interface curvature and the local mesh size. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed method.