Galerkin methods for the 'Parabolic Equation' Dirichlet problem in a variable 2-D and 3-D topography

The problem analyzed in this paper is a model for the Narrow Angle parabolic approximation of Helmholtz equation in environments in R-n, n = 2,3, of variable topography used in underwater acoustics. By applying a horizontal bottom transformation combined with an exponential one, we present this Schrodinger-type Dirichlet initial and boundary-value problem in a weak formulation and prove the uniqueness of weak solution. Further, we construct Galerkin semidiscrete and Crank-Nicolson fully discrete schemes. We prove stability of numerical solution, analyze the error and prove estimates of optimal order in the L-2-norm. For the 2-D case, we numerically verify the optimal order of accuracy and present numerical results for some standard Benchmark acoustical problems. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.