  The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear
time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to
discontinuities of the material parameters, the solution of the Maxwell
equations is less than $H^2$-regular in space. For the ADI scheme, we prove a
rigorous time-discrete error bound with a convergence rate that is half an
order lower than the classical one. Our statement imposes only assumptions on
the initial data and the material parameters, but not on the solution. To
establish this result, we analyze the regularity of the Maxwell equations in
detail in an appropriate functional analytical framework. The theoretical
findings are complemented by a numerical experiment indicating that the proven
convergence rate is indeed observable and optimal.
