  This article presents new immersed finite element (IFE) methods for solving
the popular second order elliptic interface problems on structured Cartesian
meshes even if the involved interfaces have nontrivial geometries. These IFE
methods contain extra stabilization terms introduced only at interface edges
for penalizing the discontinuity in IFE functions. With the enhanced stability
due to the added penalty, not only these IFE methods can be proven to have the
optimal convergence rate in the H1-norm provided that the exact solution has
sufficient regularity, but also numerical results indicate that their
convergence rates in both the H1-norm and the L2-norm do not deteriorate when
the mesh becomes finer which is a shortcoming of the classic IFE methods in
some situations. Trace inequalities are established for both linear and
bilinear IFE functions that are not only critical for the error analysis of
these new IFE methods, but also are of a great potential to be useful in error
analysis for other IFE methods.
