Discontinuous Galerkin methods for the linear Schrodinger equation in non-cylindrical domains

The convergence of a discontinuous Galerkin method for the linear Schrodinger equation in non-cylindrical domains of R-m, m >= 1, is analyzed in this paper. We show the existence of the resulting approximations and prove stability and error estimates in finite element spaces of general type. When m = 1 the resulting problem is related to the standard narrow angle 'parabolic' approximation of the Helmholtz equation, as it appears in underwater acoustics. In this case we investigate theoretically and numerically the order of convergence using finite element spaces of piecewise polynomial functions.