  This paper presents a new parameter free partially penalized immersed finite
element method and convergence analysis for solving second order elliptic
interface problems. The optimal approximation capabilities of the immersed
finite element space is proved via a novel new approach that is much simpler
than that in the literature. A new trace inequality which is necessary to prove
the optimal convergence of immersed finite element methods is established on
interface elements. Optimal error estimates are derived rigorously even though
the curved interface is approximated by line segments. The new method and
analysis have also been extended to problems with variable coefficients.
Numerical examples are also provided to confirm the theoretical analysis and
efficiency of the new method.
