  Mathematical modelling of ionic electrodiffusion and water movement is
emerging as a powerful avenue of investigation to provide new physiological
insight into brain homeostasis. However, in order to provide solid answers and
resolve controversies, the accuracy of the predictions is essential. Ionic
electrodiffusion models typically comprise non-trivial systems of non-linear
and highly coupled partial and ordinary differential equations that govern
phenomena on disparate time scales. Here, we study numerical challenges related
to approximating these systems. We consider a homogenized model for
electrodiffusion and osmosis in brain tissue and present and evaluate different
associated finite element-based splitting schemes in terms of their numerical
properties, including accuracy, convergence, and computational efficiency for
both idealized scenarios and for the physiologically relevant setting of
cortical spreading depression (CSD). We find that the schemes display optimal
convergence rates in space for problems with smooth manufactured solutions.
However, the physiological CSD setting is challenging: we find that the
accurate computation of CSD wave characteristics (wave speed and wave width)
requires a very fine spatial and fine temporal resolution.
