Error estimates for finite element methods for a wide-angle parabolic equation

Abstract

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 discritize the problem in the depth variable by the standard Galerkin finite element method and prove optimal-order L 2-error estimates for the ensuing continuous-in-range semidiscrete approximation. The associated ODE 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 2-error estimates of optimal rates.