  We apply a fundamental definition of time delay, as the difference between
the time a particle spends within a finite region of a potential and the time a
free particle spends in the same region, to determine results for
photoionization of an electron by an extreme ultraviolet (XUV) laser field
using numerical simulations on a grid. Our numerical results are in good
agreement with those of the Wigner-Smith time delay, obtained as the derivative
of the phase shift of the scattering wave packet with respect to its energy,
for the short-range Yukawa potential. In case of the Coulomb potential we
obtain time delays for any finite region, while - as expected - the results do
not converge as the size of the region increases towards infinity. The impact
of an ultrashort near-infrared probe pulse on the time delay is analyzed for
both the Yukawa as well as the Coulomb potential and is found to be small for
intensities below $10^{13}$ W/cm$^2$.
