  In this paper, we develop and analyze a trilinear immersed finite element
method for solving three-dimensional elliptic interface problems. The proposed
method can be utilized on interface-unfitted meshes such as 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 L2 norms. 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.
