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

TL;DR

This paper develops and analyzes a BDDC preconditioner tailored for composite DG discretizations of cardiac cell-by-cell models, enabling efficient solution of complex, discontinuous reaction-diffusion systems in cardiac simulations.

Contribution

It introduces a novel BDDC preconditioner specifically designed for composite DG discretizations in cardiac cell-by-cell models, with proven scalability and validated numerical performance.

Findings

Proven scalable convergence rate bound for the preconditioner.

Numerical validation confirms the theoretical convergence bound.

Preconditioner effectively handles discontinuities across cell boundaries.

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.