  Motivated by the paraxial narrow-angle approximation of the Helmholtz
equation in domains of variable topography that appears as an important
application in Underwater Acoustics, we analyze a general Schr\"odinger-type
equation posed on two-dimensional variable domains with mixed boundary
conditions. The resulting initial- and boundary-value problem is transformed
into an equivalent one posed on a rectangular domain and is approximated by
fully discrete, $L^2$-stable, finite element, Crank--Nicolson type schemes. We
prove a global elliptic regularity theorem for complex elliptic boundary value
problems with mixed conditions and derive $L^2$-error estimates of optimal
order. Numerical experiments are presented which verify the optimal rate of
convergence.
