  The Douglas--Rachford and Peaceman--Rachford splitting methods are common
choices for temporal discretizations of evolution equations. In this paper we
combine these methods with spatial discretizations fulfilling some easily
verifiable criteria. In the setting of linear dissipative evolution equations
we prove optimal convergence orders, simultaneously in time and space. We apply
our abstract results to dimension splitting of a 2D diffusion problem, where a
finite element method is used for spatial discretization. To conclude, the
convergence results are illustrated with numerical experiments.
