We design an unfitted interface penalty DG-FE method (UIPDG-FEM) for an elliptic interface problem, which uses the interior penalty discontinuous Galerkin methods locally along the interface together with additional penalty terms on the interface (or the Nitsche's trick) to deal with the jump conditions, and uses the finite element methods away from the interface. Moreover, the trick of merging elements is used to keep the condition number of the algebraic system not affected by the interface position. The proposed UIPDG-FEM not only possesses flexibilities of the IPDG method, in particular, simplifying the process of merging elements near a complex interface, but also avoids its drawback of larger number of global degrees of freedom. The convergence rates of the UIPDG-FEM solution are optimal and independent of the interface position. Furthermore, a uniform estimate of the flux value is established in terms of the discontinuous physical coefficients. A two dimensional merging algorithm is also presented, which is guaranteed to succeed under appropriate assumptions on the interface. Numerical examples are given to verify the theoretical results.

翻译：针对椭圆界面问题，我们设计了一种非拟合界面惩罚DG-FE方法（UIPDG-FEM）。该方法沿界面局部使用内部惩罚间断伽辽金方法，并结合界面上的附加惩罚项（或尼采技巧）处理跳跃条件，同时在远离界面的区域采用有限元方法。此外，通过单元合并技巧确保代数系统的条件数不受界面位置影响。所提出的UIPDG-FEM不仅保留了IPDG方法的灵活性（特别是简化了复杂界面附近的单元合并过程），还避免了其全局自由度数量过多的缺点。该方法解的收敛阶达到最优且与界面位置无关，进一步建立了基于间断物理系数的通量一致估计。本文还提出了二维合并算法，该算法在界面满足适当假设时保证成功。数值算例验证了理论结果。