  This paper presents a new parameter free partially penalized immersed finite
element method and convergence analysis for solving second order elliptic
interface problems. A lifting operator is introduced on interface edges to
ensure the coercivity of the method without requiring an ad-hoc stabilization
parameter. 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 with the
constant independent of the interface location relative to the mesh. The new
method and analysis have also been extended to variable coefficients and
three-dimensional problems. Numerical examples are also provided to confirm the
theoretical analysis and efficiency of the new method.
