  In this article, we introduce a new partially penalized immersed finite
element method (IFEM) for solving elliptic interface problems with
multi-domains and triple-junction points. We construct new IFE functions on
elements intersected with multiple interfaces or with triple-junction points to
accommodate interface jump conditions. For non-homogeneous flux jump, we enrich
the local approximating spaces by adding up to three local flux basis
functions. Numerical experiments are carried out to show that both the Lagrange
interpolations and the partial penalized IFEM solutions converge optimally in
L2 and H1 norms.
