  An implicit Euler--Maruyama method with non-uniform step-size applied to a
class of stochastic partial differential equations is studied. A spectral
method is used for the spatial discretization and the truncation of the Wiener
process. A discrete analogue of maximal $L^2$-regularity of the scheme and the
discretised stochastic convolution is established, which has the same form as
their continuous counterpart.
