  This article presents and analyzes a $p^{th}$-degree immersed finite element
(IFE) method for elliptic interface problems with nonhomogeneous jump
conditions. In this method, jump conditions are approximated optimally by basic
IFE and enrichment IFE piecewise polynomial functions constructed by solving
local Cauchy problems on interface elements. The proposed IFE method is based
on a discontinuous Galerkin formulation on interface elements and a continuous
Galerkin formulation on non-interface elements. This $p^{th}$-degree IFE method
is proved to converge optimally under mesh refinement. In addition, this
article addresses the stability of this IFE method and has established upper
bounds for its condition numbers which are optimal with respect to the mesh
size but suboptimal with respect to the contrast of the discontinuous
coefficient.

Numerical Analysis;Numerical Analysis