  This article presents an immersed finite element (IFE) method for solving the
typical three-dimensional second order elliptic interface problem with an
interface-independent Cartesian mesh. The local IFE space on each interface
element consists of piecewise trilinear polynomials which are constructed by
extending polynomials from one subelement to the whole element according to the
jump conditions of the interface problem. In this space, the IFE shape
functions with the Lagrange degrees of freedom can always be constructed
regardless of interface location and discontinuous coefficients. The proposed
IFE space is proven to have the optimal approximation capabilities to the
functions satisfying the jump conditions. A group of numerical examples with
representative interface geometries are presented to demonstrate features of
the proposed IFE method.
