  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.
