  We investigate the systematics of fission barriers in superheavy elements in
the range Z = 108-120 and N = 166-182. Results from two self-consistent models
for nuclear structure, the relativistic mean-field (RMF) model as well as the
non-relativistic Skyrme-Hartree-Fock approach are compared and discussed. We
restrict ourselves to axially symmetric shapes, which provides an upper bound
on static fission barriers. We benchmark the predictive power of the models
examining the barriers and fission isomers of selected heavy actinide nuclei
for which data are available. For both actinides and superheavy nuclei, the RMF
model systematically predicts lower barriers than most Skyrme interactions. In
particular the fission isomers are predicted too low by the RMF, which casts
some doubt on recent predictions about superdeformed ground states of some
superheavy nuclei. For the superheavy nuclei under investigation, fission
barriers drop to small values around Z = 110, N = 180 and increase again for
heavier systems. For most of the forces, there is no fission isomer for
superheavy nuclei, as superdeformed states are in most cases found to be
unstable with respect to octupole distortions.
