  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.
