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This paper is devoted to the construction and analysis of immersed finiteelement (IFE) methods in three dimensions. Different from the 2D case, thepoints of intersection of the interface and the edges of a tetrahedron areusually not coplanar, which makes the extension of the original 2D IFE methodsbased on a piecewise linear approximation of the interface to the 3D case notstraightforward. We address this coplanarity issue by an approach where theinterface is approximated via discrete level set functions. This approach isvery convenient from a computational point of view since in many practicalapplications the exact interface is often unknown, and only a discrete levelset function is available. As this approach has also not be considered in the2D IFE methods, in this paper we present a unified framework for both 2D and 3Dcases. We consider an IFE method based on the traditional Crouzeix-Raviartelement using integral values on faces as degrees of freedom. The novelty ofthe proposed IFE is the unisolvence of basis functions on arbitrarytriangles/tetrahedrons without any angle restrictions even for anisotropicinterface problems, which is advantageous over the IFE using nodal values asdegrees of freedom. The optimal bounds for the IFE interpolation errors areproved on shape-regular triangulations. For the IFE method, optimal a priorierror and condition number estimates are derived with constants independent ofthe location of the interface with respect to the unfitted mesh. The extensionto anisotropic interface problems with tensor coefficients is also discussed.Numerical examples supporting the theoretical results are provided.

An immersed Crouzeix-Raviart finite element method in 2D and 3D based on discrete level set functions