Dense cell-by-cell systems of PDEs: approximation, spectral analysis, and preconditioning

Abstract In the present study, we consider the Extra-Membrane-Intra (EMI) model for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc boundary conditions, relevant for modelling portions of excitable tissues, composed of several cells. In particular, we study two geometrical settings in the context of computational cardiology and neuroscience. The Galerkin approximations of the considered system of PDEs lead to large linear systems of algebraic equations, where the coefficient matrices depend on the number N of cells and the fineness parameters. We present a structural and spectral analysis of the related matrix-sequences as the fineness parameters tends to zero. Based on the theoretical results, we propose preconditioners and specific multilevel solvers. Numerical experiments are presented and critically discussed, showing that a monolithic multilevel solver is efficient and robust with respect to all the problem and discretization parameters. In particular, we include numerical results for an increasing number of cells N , both for idealized geometries (with N exceeding $$10^5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>5</mml:mn> </mml:msup> </mml:math> ) and for realistic, densely populated 3D tissue reconstructions.