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.
Application of the parallel BDDC preconditioner to the Stokes flow
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematical and Statistical Sciences, University of Colorado Denver（科罗拉多大学丹佛分校数学与统计科学系）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague（布拉格捷克理工大学机械工程学院数学系）；Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague（布拉格捷克理工大学土木工程学院数学系）
A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem
