Galerkin Methods for Parabolic and SCHR{\"O}DINGER Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics

TL;DR

This paper develops and analyzes Galerkin-finite element methods for solving parabolic and Schrödinger equations with dynamical boundary conditions, with applications to underwater acoustics, providing optimal error estimates and comparing different modeling approaches.

Contribution

It introduces error estimates for Galerkin methods applied to these equations with dynamical boundaries and compares models in underwater acoustics.

Findings

Optimal convergence rates in $L^2$ and $H^1$ norms.

Dynamical boundary conditions can be effectively approximated.

Alternative models may provide better approximations in certain scenarios.

Abstract

In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schr\"odinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in $L^2$ and $H^1$ are proved for the accociated semidiscrete and fully discrete Crank-Nicolson-Galerkin approximations. The problem involving the Schr\"odinger equation is motivated by considering the standard `parabolic' (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long-range sound propagation in the sea, in the specific case of a domain with a rigid bottom of variable topography. This model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations. In the (real) parabolic case, numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type.