  Modern laser systems are able to generate short and intense laser pulses
ionizing matter in the poorly explored barrier-suppression regime. Field
ionization in this regime is studied analytically and numerically. For
analytical studies, both the classical and the quantum approaches are used. Two
approximations to solve the time-dependent Schr\"odinger equation are proposed:
the free electron approximation, in which the atomic potential is neglected,
and the motionless approximation, in which only the external field term is
considered. In the motionless approximation, the ionization rate in extremely
strong fields is derived. The approximations are applied to several model
potentials and are verified using numeric simulations of the Schr\"odinger
equation. A simple formula of the ionization rate both for the tunnel and the
barrier-suppression regimes is proposed. The formula can be used, for example,
in particle-in-cell codes for simulations of the interaction of extremely
intense laser fields with matter.
