  Mathematical models for excitable tissue with explicit representation of
individual cells are highly detailed and can, unlike classical homogenized
models, represent complex cellular geometries and local membrane variations.
However, these cell-based models are challenging to approximate numerically,
partly due to their mixed-dimensional nature with unknowns both in the bulk and
at the lower-dimensional cellular membranes. We here develop and evaluate a
novel solution strategy for the cell-based KNP-EMI model describing ionic
electrodiffusion in and between intra- and extracellular compartments with
explicit representation of individual cells. The strategy is based on operator
splitting, a multiplier-free formulation of the coupled dynamics across
sub-regions, and a discontinuous Galerkin discretization. In addition to
desirable theoretical properties, such as local mass conservation, the scheme
is practical as it requires no specialized functionality in the finite element
assembly and order optimal solvers for the resulting linear systems can be
realized with black-box algebraic multigrid preconditioners. Numerical
investigations show that the proposed solution strategy is accurate, robust
with respect to discretization parameters, and that the parallel scalability of
the solver is close to optimal - both for idealized and realistic two and three
dimensional geometries.
