Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements with arbitrary interface-cutting configuration. Optimal a priori error estimates are rigorously proved in both energy and $L^2$ norms, with the constant in the error bound independent of the interface location and its dependence on coefficient contrast explicitly specified. Numerical examples are provided not only to verify our theoretical results but also to demonstrate the applicability of this IFE method in tackling some real-world 3D interface models.

浸入式有限元方法旨在解决界面未拟合网格上的界面问题。然而，大部分研究，尤其是分析，主要限于二维情况。在本文中，我们对在由长方体组成的笛卡尔网格上求解三维椭圆界面问题的三线性浸入式有限元方法进行了先验分析。我们为具有任意界面切割配置的界面元素建立三线性 IFE 函数的迹不等式和逆不等式。最优先验误差估计在能量和${升}^2$范数，误差界中的常数独立于界面位置及其对系数对比度的依赖，明确指定。提供的数值示例不仅是为了验证我们的理论结果，而且是为了证明这种 IFE 方法在处理一些真实世界的 3D 界面模型中的适用性。