  We introduce an enriched unfitted finite element method to solve 1D elliptic
interface problems with discontinuous solutions, including those having
implicit or Robin-type interface jump conditions. We present a novel approach
to construct a one-parameter family of discontinuous enrichment functions by
finding an optimal order interpolating function to the discontinuous solutions.
In the literature, an enrichment function is usually given beforehand, not
related to the construction step of an interpolation operator. Furthermore, we
recover the well-known continuous enrichment function when the parameter is set
to zero. To prove its efficiency, the enriched linear and quadratic elements
are applied to a multi-layer wall model for drug-eluting stents in which
zero-flux jump conditions and implicit concentration interface conditions are
both present.
