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Motivated by the paraxial narrow-angle approximation of the Helmholtzequation in domains of variable topography that appears as an importantapplication in Underwater Acoustics, we analyze a general Schr\"odinger-typeequation posed on two-dimensional variable domains with mixed boundaryconditions. The resulting initial- and boundary-value problem is transformedinto an equivalent one posed on a rectangular domain and is approximated byfully discrete, $L^2$-stable, finite element, Crank--Nicolson type schemes. Weprove a global elliptic regularity theorem for complex elliptic boundary valueproblems with mixed conditions and derive $L^2$-error estimates of optimalorder. Numerical experiments are presented which verify the optimal rate ofconvergence.

Crank-Nicolson Finite Element Discretizations for a 2D Linear Schr\"odinger-Type Equation Posed in a Noncylindrical Domain