  This paper is concerned with a 1D Schr\"odinger scattering problem involving
both oscillatory and evanescent regimes, separated by jump discontinuities in
the potential function, to avoid "turning points". We derive a non-overlapping
domain decomposition method to split the original problem into sub-problems on
these regions, both for the continuous and afterwards for the discrete problem.
Further, a hybrid WKB-based numerical method is designed for its efficient and
accurate solution in the semi-classical limit: a WKB-marching method for the
oscillatory regions and a FEM with WKB-basis functions in the evanescent
regions. We provide a complete error analysis of this hybrid method and
illustrate our convergence results by numerical tests.
