  A formulation is presented to efficiently model ionic conduction inside, i.e.
across and along, grain boundaries. Efficiency and accuracy is achieved by
reducing it to a two-dimensional manifold while guaranteeing the conservation
of mass and charge at the intersection of multiple grain boundaries. The
formulation treats the electric field and the electric current as independent
solution variables. We elaborate on the numerical challenges this formulation
implies and compare the computed solution with results from an analytical
solution by quantifying the convergence towards the exact solution. Towards the
end of this work, the model is firstly applied to setups with extreme values of
crucial parameters of grain boundaries to study the influence of the ionic
conduction in the grain boundary on the overall battery cell voltage and,
secondly, to a realistic microstructure to show the capabilities of the
formulation.
