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.

本文致力于构建和分析三维浸入式有限元 (IFE) 方法。与 2D 情况不同，界面与四面体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 2D IFE 方法扩展到 3D 情况并不简单。我们通过一种通过离散水平集函数来近似界面的方法来解决这个共面性问题。从计算的角度来看，这种方法非常方便，因为在许多实际应用中，确切的接口通常是未知的，并且只有离散水平集函数可用。由于 2D IFE 方法中也没有考虑这种方法，因此在本文中，我们为 2D 和 3D 情况提供了一个统一的框架。我们考虑一种基于传统 Crouzeix-Raviart 元素的 IFE 方法，使用面的整数值作为自由度。所提出的 IFE 的新颖之处在于，即使对于各向异性界面问题，任意三角形/四面体上的基函数也没有任何角度限制的不解，这优于使用节点值作为自由度的 IFE。IFE 插值误差的最佳边界在形状正则三角剖分上得到证明。对于 IFE 方法，最佳先验误差和条件数估计是使用与未拟合网格的界面位置无关的常数导出的。还讨论了张量系数对各向异性界面问题的扩展。提供了支持理论结果的数值例子。