  We prove first-order convergence of the semi-explicit Euler scheme combined
with a finite element discretization in space for elliptic-parabolic problems
which are weakly coupled. This setting includes poroelasticity,
thermoelasticity, as well as multiple-network models used in medical
applications. The semi-explicit approach decouples the system such that each
time step requires the solution of two small and well-structured linear systems
rather than the solution of one large system. The decoupling improves the
computational efficiency without decreasing the convergence rates. The
presented convergence proof is based on an interpretation of the scheme as an
implicit method applied to a constrained partial differential equation with
delay term. Here, the delay time equals the used step size. This connection
also allows a deeper understanding of the weak coupling condition, which we
accomplish to quantify explicitly.
