Scalable approximation and solvers for ionic electrodiffusion in cellular geometries
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The activity and dynamics of excitable cells are fundamentally regulated and moderated by extracellular and intracellular ion concentrations and their electric potentials. The increasing availability of dense reconstructions of excitable tissue at extreme geometric detail pose a new and clear scientific computing challenge for computational modelling of ion dynamics and transport. We introduce a scalable numerical algorithm for solving the time-dependent and nonlinear KNP-EMI (Kirchhoff-Nernst-Planck Extracellular-Membrane-Intracellular) equations describing ionic electrodiffusion for excitable cells with an explicit geometric representation of intracellular and extracellular compartments and interior interfaces. Our solution strategy is based on an a mixed finite element discretization of ion concentrations and electric potentials in intracellular and extracellular domains, and an algebraic multigrid-based, inexact block-diagonal preconditioner for GMRES. Such a solution strategy is motivated and studied via spectral analysis of the corresponding discrete operators. Numerical experiments using realistic geometries, with up to 100M unknowns per time step and up to 256 cores, demonstrate that this solution strategy is robust and scalable with respect to the problem size, time discretization and number of cores.