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

TL;DR

This work addresses solving EMI-based cell-by-cell cardiac models with discontinuous potentials using a DG discretization. It develops a DG-compatible BDDC preconditioner based on enriched dual and primal spaces to transfer information across cell boundaries without eliminating discontinuities, and proves a scalable convergence bound for the preconditioned Schur system. Theoretical results include energy-norm bounds and a projection operator bound that underpin convergence, while extensive 2D numerical tests demonstrate scalability, quasi-optimality, and robustness to time-step size. The approach enables efficient large-scale simulations of cellular-level cardiac tissue and sets the stage for 3D extensions and integration into HPC libraries for parallel cardiac electrophysiology computations.

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.

Figures (7)

• Figure 1: Exemplification of homogenization: the intracellular space $\Omega_i$ of a cell (depicted in orange, on the left side) is assumed to co-exist at every point of the myocardium $\Omega$ (representation on the right side, in green) with the extracellular space $\Omega_e$ (depicted in light blue) and the cell membrane $\Gamma$.
• Figure 2: Simplified EMI model. In this example, we depict a simple situation with only two cells, whose intracellular spaces are denoted by $\Omega_1$ and $\Omega_2$, while $\Omega_0$ denotes the extracellular surrounding; $F_{1,2}$ is assumed to be the intersection of the cell membrane between the two cells.
• Figure 3: Schematic partition of degrees of freedom of the substructure $\Omega_i$ (left) and zoom over the common geometric boundary $\partial \Omega_i \cap \partial \Omega_j$ (right). The index sets below the Figures are related to the substructure $\Omega_i$.
• Figure 4: Example of global enumeration of degrees of freedom for DG-type discretizations, in a non-overlapping domain decomposition setting. Simple case of three subdomains: extracellular $\Omega_0$ in blue, intracellular $\Omega_1$ and $\Omega_2$ in green and red.
• Figure 5: Time evolution at different time steps.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Remark 1
• Lemma 2
• Lemma 3
• Theorem 1: Trace theorem
• Lemma 4
• proof