Summary: The authors introduce and study numerical methods that are fully adaptive in both three-dimensional (3D) space and time to challenging multiscale cardiac reaction-diffusion models. In these methods, temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration, while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators. In contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time (so-called method of lines), our method discretizes first in time and then in space (so-called Rothe method) – an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology (KARDOS code library).
With this method, the evolution of a complete heartbeat, from the excitation to the recovery phase, is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models, coupled with either a variant of the simple FitzHugh-Nagumo model or the more complex phase-I Luo-Rudy ionic model. The numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction-diffusion models on 3D domains up to moderate sizes. In particular, the method accurately resolves the evolution of the intra- and extracellular potentials, gating variables, and ion concentrations during the excitation, plateau, and recovery phases.