Galerkin methods for a Schrödinger-type equation with a dynamical boundary condition in two dimensions∗
1 Department of Mathematics, University of Chester, Thornton Science Park, CH2 4NU, UK.
Revised: 31 July 2014
In this paper, we consider a two-dimensional Schrödinger-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. Due to the complexity of the analyzed problem, we use very refined technics in order to derive estimates of the numerical error in the H1-norm.
Mathematics Subject Classification: 65M60 / 65M12 / 65M15 / 76Q05
Key words: 2-D Schrödinger equation / finite element methods / error estimates / noncylindrical domain / Neumann boundary condition / cubic splines / Crank–Nicolson time stepping / dynamical boundary condition / underwater acoustics
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