Error Estimates for Finite Element Methods for a Wide–angle Parabolic Equation

We consider a model initial– and boundary–value problem for the third– order wide–angle parabolic approximation of underwater acoustics with depth– and range–dependent coefficients. We discretize the problem in the depth variable by the standard Galerkin finite element method and prove optimal–order L–error estimates for the ensuing continuous–in–range semidiscrete approximation. The associated o.d.e. systems are then discretized in range, first by a second–order accurate Crank– Nicolson type method, and then by the fourth–order, two–stage Gauss–Legendre, implicit Runge–Kutta scheme. We show that both these fully discrete methods are unconditionally stable and possess L–error estimates of optimal rates. Dedicated to Professor Robert Vichnevetsky on the occasion of his 65 birthday.