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：我们提出一种高阶浸入有限元方法，用于解决采用界面无关网格的椭圆界面问题。本文发展的浸入有限元函数精确满足界面条件并具有最优逼近能力。该新型浸入有限元空间的构建依赖于基于界面Frenet-Serret标架的非线性变换，将界面局部映射为线段，此特性使得任意阶数的浸入有限元函数构建过程兼具成本效益与鲁棒性。这类新型浸入有限元函数局部适配界面问题的标准弱形式，因而可直接应用于标准内罚间断伽辽金格式，无需对界面施加额外惩罚项。数值算例展示了该方法在$h$和$p$细化下的收敛特性。