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

Abstract: 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. Unlike classical Bidomain and Monodomain cardiac models, which rely on homogenized descriptions of cardiac tissue at the macroscopic level, the cell-by-cell models enable 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 exhibit discontinuous global solutions across the cell boundaries. Therefore, the proposed BDDC preconditioner employs appropriate dual and primal spaces with additional constraints to transfer information between cells (subdomains) without affecting 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.