  In chiral $p$-wave superconductors, magnetic flux patterns may appear
spontaneously when translational symmetry is broken such as at surfaces, domain
walls, or impurities. However, in the candidate material Sr$_2$RuO$_4$ no
direct signs of such magnetic fields have been detected experimentally. In this
paper, the flux pattern at the edge of a disk-shaped sample is examined using
the phenomenological Ginzburg Landau approach. The detailed shape of the flux
pattern, including self-screening, is computed numerically for different
surface types by systematically scanning a range of boundary conditions.
Moreover, specific features of the electronic structure are included
qualitatively through the coefficients in the Ginzburg Landau functional. Both
the shape and the magnitude of the flux pattern are found to be highly
sensitive to all considered parameters. In conclusion, such spontaneous
magnetic flux patterns are not a universal feature of chiral $p$-wave
superconductors.
