  The Darwin field model addresses an approximation to Maxwell's equations
where radiation effects are neglected. It allows to describe general
quasistatic electromagnetic field phenomena including inductive, resistive and
capacitive effects. A Darwin formulation based on the Darwin-Amp\`ere equation
and the implicitly included Darwin-continuity equation yields a non-symmetric
and ill-conditioned algebraic systems of equations received from applying a
geometric spatial discretization scheme and the implicit backward
differentiation time integration method. A two-step solution scheme is
presented where the underlying block-Gauss-Seidel method is shown to change the
initially chosen gauge condition and the resulting scheme only requires to
solve a weakly coupled electro-quasistatic and a magneto-quasistatic discrete
field formulation consecutively in each time step. Results of numerical test
problems validate the chosen approach.
