This paper is devoted to the construction and analysis of immersed finite element (IFE) methods in three dimensions. Different from the 2D case, the points of intersection of the interface and the edges of a tetrahedron are usually not coplanar, which makes the extension of the original 2D IFE methods based on a piecewise linear approximation of the interface to the 3D case not straightforward. We address this coplanarity issue by an approach where the interface is approximated via discrete level set functions. This approach is very convenient from a computational point of view since in many practical applications the exact interface is often unknown, and only a discrete level set function is available. As this approach has also not be considered in the 2D IFE methods, in this paper we present a unified framework for both 2D and 3D cases. We consider an IFE method based on the traditional Crouzeix-Raviart element using integral values on faces as degrees of freedom. The novelty of the proposed IFE is the unisolvence of basis functions on arbitrary triangles/tetrahedrons without any angle restrictions even for anisotropic interface problems, which is advantageous over the IFE using nodal values as degrees of freedom. The optimal bounds for the IFE interpolation errors are proved on shape-regular triangulations. For the IFE method, optimal a priori error and condition number estimates are derived with constants independent of the location of the interface with respect to the unfitted mesh. The extension to anisotropic interface problems with tensor coefficients is also discussed. Numerical examples supporting the theoretical results are provided.

翻译：本文致力于三维浸入式有限元方法的构建与分析。与二维情况不同，界面与四面体棱边的交点通常不共面，这使得基于界面分段线性逼近的原始二维浸入式有限元方法无法直接推广至三维情况。我们通过离散水平集函数逼近界面的方法解决了该共面性问题。从计算角度来看，该方法非常便捷，因为实际应用中精确界面通常未知，而仅有离散水平集函数可用。由于二维浸入式有限元方法也尚未采用此思路，本文提出了适用于二维和三维情况的统一框架。我们考虑基于传统Crouzeix-Raviart元的浸入式有限元方法，以面积分值作为自由度。该浸入式有限元的创新性在于：基函数在任意三角形/四面体上具有唯一可解性，且无需任何角度限制（即使对于各向异性界面问题），这优于以节点值作为自由度的浸入式有限元。在形状正则三角剖分下，我们证明了浸入式有限元插值误差的最优阶估计。针对该浸入式有限元方法，我们推导了最优先验误差和条件数估计，其常数独立于界面相对于非拟合网格的位置。此外，讨论了该方法向具有张量系数的各向异性界面问题的推广。文中提供了支持理论结果的数值算例。