Convergence analysis of BDDC preconditioners for composite DG discretizations of the cardiac cell-by-cell model

A balancing domain decomposition by constraints (BDDC) preconditioner is constructed and analyzed for the solution of composite discontinuous Galerkin discretizations of reaction-diffusion systems of ordinary and partial differential equations arising in cardiac cell-by-cell models. The latter are different from the classical bidomain and monodomain cardiac models based on homogenized descriptions of the cardiac tissue at the macroscopic level, and therefore they allow the representation of individual cardiac cells, cell aggregates, damaged tissues, and nonuniform distributions of ion channels on the cell membrane. The resulting discrete cell-by-cell models have discontinuous global solutions across the cell boundaries, and hence the proposed BDDC preconditioner is based on appropriate dual and primal spaces with additional constraints which transfer information between cells (subdomains) without influencing the overall discontinuity of the global solution. A scalable convergence rate bound is proved for the resulting BDDC cell-by-cell preconditioned operator, while numerical tests validate this bound and investigate its dependence on the discretization parameters.