We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions exactly and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet-Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Numerical examples are provided to showcase the convergence properties of the method under $h$ and $p$ refinements.

翻译：我们提出了一种高阶浸入式有限元（IFE）方法，用于求解与网格无关的椭圆界面问题。本文发展的IFE函数精确满足界面条件，并具有最优逼近能力。该新型IFE空间的构建依赖于基于界面Frenet-Serret标架的非线性变换，可将界面局部映射为直线段，这一特性使得不同阶次IFE函数的构建过程兼具成本效益与鲁棒性。这类新型浸入式有限元函数在局部上与界面问题的常规弱形式相容，因此可直接应用于标准内罚间断伽辽金格式中，无需在界面上增加额外罚项。数值算例展示了该方法在$h$和$p$细化下的收敛性。