  In this article, we study superconvergence properties of immersed finite
element methods for the one dimensional elliptic interface problem. Due to low
global regularity of the solution, classical superconvergence phenomenon for
finite element methods disappears unless the discontinuity of the coefficient
is resolved by partition. We show that immersed finite element solutions
inherit all desired superconvergence properties from standard finite element
methods without requiring the mesh to be aligned with the interface. In
particular, on interface elements, superconvergence occurs at roots of
generalized orthogonal polynomials that satisfy both orthogonality and
interface jump conditions.
