Crank-Nicolson Finite Element Discretizations for a 2D Linear   Schr\"odinger-Type Equation Posed in a Noncylindrical Domain
  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. 
D. C. Antonopoulou, G. D. Karali, M. Plexousakis, G. E. Zouraris
math.NA