Galerkin methods for a Schroedinger-type equation with a dynamical boundary condition in two dimensions

This is the author's PDF version of an article published in ESAIM: Mathematical modelling and numerical analysis© 2015. The definitive version is available at http://www.esaim-m2an.org/In this paper, we consider a two-dimensional Schodinger-type equation with a dynamical boundary condition. This model describes the long-range sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a Crank-Nicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution