In this talk, we discuss parallel simulation approaches and methods for the massively parallel simulation of electrophysiology which are parallel not only in space but also in time. Given that modern parallel machines only provide "more and more" cores, it is natural to assume that largher machines will not provide significant speedup any more, as parallelization is only done in space. We therefore consider space time discretization and time-parallel solution methods, which allow for looking at the whole trajetory at once.

Wheres these methods are by now well established for linear parabolic equatiosn, the non-linearities arising from the ion-channels makes the space time solution more difficult.

We start our talk with adaptive and parallel space-time discretizations. Our adaptive€ approach is based on locally structured mesh hierarchies which are glued along their interfaces by a non-conforming mortar element discretization. To further increase the overall efficiency, we keep the spatial meshes constant over suitable time windows in which error indicators are accumulated. This approach facilitates strongly varying mesh sizes in neighboring patches as well as in consecutive time steps. For stability reasons, for the transfer of the dynamic variables between different spatial approximation spaces, a discrete $L^2$-projection is used. We derive a spatial preconditioner for the arising non-linear elliptic problems.

We also discuss space-time multigrid methods for electrophysiology and explain the difficulties in the construction of efficient space-time solution methods. We compare our method to a standard adaptive refinement strategy using unstructured meshes. As it turns out, our novel adaptive scheme provides a very good balance between reduction in degrees of freedom and overall parallel efficiency.

We will close our talk with some recent results for a GPU based solution method for the eikonal equation. Here, we go the opposite way and simplify the model in order to get a short time-to-solution . We shortly comment on how high-fidelty models such as mono- or bi-domain can be combined with low-fidelty models such as the eikonal equation in the context of uncertainty quantification and illustrate this using our numerical results.